Ultrasonics: data, equations, and their practical uses

ULTRASONICS Data, Equations, and Their Practical Uses

CRC_DK8307_FM.indd i

11/14/2008 11:58:03 AM

CRC_DK8307_FM.indd ii

11/14/2008 11:58:03 AM

ULTRASONICS Data, Equations, and Their Practical Uses Edited by

Dale Ensminger Foster B. Stulen

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC_DK8307_FM.indd iii

11/14/2008 11:58:04 AM

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-0-8247-5830-1 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Ultrasonics : Data, Equations, and Their Practical Uses / editors, Dale Ensminger and Foster B. Stulen. p. cm. “A CRC title.” Includes bibliographical references and index. ISBN 978-0-8247-5830-1 (alk. paper) 1. Ultrasonic waves--Industrial applications. 2. Ultrasonics. I. Ensminger, Dale. II. Stulen, Foster B. III. Title. TA367.U35 2009 620.2’8--dc22

2008034102

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

CRC_DK8307_FM.indd iv

11/14/2008 11:58:04 AM

Contents Preface .............................................................................................................................. vii Editors ............................................................................................................................... ix Contributors ...................................................................................................................... xi 1

Oscillatory Motion and Wave Equations ..............................................................1 Dale Ensminger

2

Ultrasonic Horns, Couplers, and Tools ................................................................27 Dale Ensminger

3

Advanced Designs of Ultrasonic Transducers and Devices Using Finite Element Analysis ...........................................................................129 Foster B. Stulen and Robert B. Francini

4

Piezoelectric Materials: Properties and Design Data ...................................... 185 Dale Ensminger

5

Magnetostriction: Materials and Transducers .................................................235 Dale Ensminger

6

Pneumatic Transducer Design Data ................................................................... 273 Dale Ensminger

7

Properties of Materials ........................................................................................285 Dale Ensminger

8

Ultrasonics-Assisted Physical and Chemical Processes .................................323 Dale Mangaraj, B. Vijayendran, and Dale Ensminger

9

Advances in Generation and Detection of Ultrasound in the Field of Nondestructive Testing/Evaluation ..........................................365 Allan F. Pardini, Gerald J. Posakony, and Theodore T. Taylor

10

Medical Ultrasound: Therapeutic and Diagnostic Imaging ...........................407 Foster B. Stulen

11

Mechanical Effects of Ultrasonic Energy ..........................................................447 Dale Ensminger

12

Criteria for Choosing Ultrasonics ...................................................................... 471 Dale Ensminger

Index ................................................................................................................................481

v

CRC_DK8307_FM.indd v

11/14/2008 11:58:04 AM

CRC_DK8307_FM.indd vi

11/14/2008 11:58:04 AM

Preface Ultrasonics is a form of acoustical energy, generally pitched above the audible range of frequencies. The interesting phenomena attributable to ultrasonics, particularly in its early history, elicited many suggestions for its use. Some of these suggestions seem to imply that ultrasonics has a mysterious and magical power. Many suggestions, however, have proved within the past century to be practical in various areas of science and industry. All these practical applications have benefited greatly from recent technological advances in electronics and computer science. However, the ultimate effectiveness of any process depends on the knowledge available to the developer and user about the principles on which the process is based. In ultrasonic applications, the basic principles to consider are those related to both ultrasonics and its application. Ultrasonics processes involve considerations of phenomena such as wave propagation, chemical reactions, thermal effects resulting from absorption of energy, effects attributable to cavitation, and other stress-, friction-, and momentum-related factors. Any successful application of ultrasonics must be related in a practical way to some acoustic property or properties of the media being irradiated ultrasonically. This book includes data and functions gathered from several areas of science and technology for the purpose of facilitating the use of ultrasonic energy. The objectives of this work are to evaluate the practicality of new ideas and to make important data available so that the design of ultrasonic systems meets the needs of present and new developments. These developments include applications in both standard and harsh environments. Chapters 1 and 2 discuss basic ultrasonic equations that relate to factors affecting wave motion and the application of these factors to the design of ultrasonic systems such as vibrating bars, horns, plates, and rings. Chapter 3 goes more deeply into designs that apply finite elements to ultrasonic drivers (transducers), large horns, and blades. Chapter 4 includes properties and design data applicable to piezoelectric materials and the uses of piezoelectric transducers. Chapter 5 provides information relative to the design of magnetostrictive transducers. Chapter 6 covers pneumatic and liquid transducers and their design and uses. Chapter 7 examines mechanical and physical properties of materials, including those properties necessary for the welding and forming of materials. Chapter 8 reviews the chemical properties and compatibilities of materials, and discusses the chemical effects of ultrasound. Several practical applications are presented. Chapter 9 is an update of nondestructive testing applications. This chapter provides examples of modern equipment that can be readily procured and configured for use in many demanding ultrasonic nondestructive testing applications. Electromagnetic acoustic transducers (EMATs) and lasers and their uses are described. Chapter 10 presents inclusive coverage of medical uses of ultrasound today. Medical ultrasound, therapies, and diagnostic imaging as well as low-intensity and high-intensity applications are discussed. Low-intensity areas include imaging devices and blood flow measurements such as Doppler methods. Surgery has benefited considerably from ultrasonic applications in such areas as removal of cataracts, lithotripsy, the treatment of prostate cancer, atrial fibrillation treatment, wrinkle reduction, fat removal, wound cleaning and healing, sonophoresis, ultrasonic welding in vivo, and the dissolving of clots.

vii

CRC_DK8307_FM.indd vii

11/14/2008 11:58:04 AM

viii

Preface

Chapter 11 discusses the mechanical effects of ultrasonic energy. This chapter reviews some of the mechanical factors previously discussed, while emphasizing some new processes as well. Chapter 12 summarizes the history of ultrasonics. It discusses how prior problems that faced the advancement of the industry have been overcome and projects further advances. We appreciate the cooperation that we have received from all the experts who have contributed their chapters to this book. We also appreciate the help and cooperation of the publisher who has waited patiently for the manuscript to be completed. We hope that the reader will find this book to be a time-saving and dependable source of valuable information.

CRC_DK8307_FM.indd viii

11/14/2008 11:58:04 AM

Editors Dale Ensminger joined the Battelle staff in 1948 while a student at the Ohio State University. He began active research in ultrasonics and acoustics in February 1950 and since then has been responsible for or participated in more than 980 projects in acoustics, ultrasonics, and related areas. He received a bachelor’s degree in mechanical engineering in March 1950, and a bachelor’s degree in electrical engineering in June 1950. Between 1950 and 1953, he took additional courses in mathematics and physics at the Ohio State University on a part-time basis in support of the needs presented by ultrasonics research. Ensminger is the author of approximately 150 articles and holds several patents on applications of ultrasonics. He has written a book Ultrasonics—The Low- and High-Intensity Applications, published in February 1973. A second edition of this book, Ultrasonics: Fundamentals, Technology, Applications, was published in 1988. This book is presently being updated. Ensminger is the author of the chapter “Acoustic Dewatering” in the book Advances in Solid–Liquid Separation, edited by H. S. Muralidhara and published by Battelle Press in 1986. He was an editorial reviewer for Nondestructive Testing Handbook, 2nd Edition, Volume Seven, Ultrasonic Testing, published by the American Society for Nondestructive Testing in 1991. He is the author of A Handbook on Ultrasonic Methods of Nondestructive Testing written for the U.S. Army through the Watertown Arsenal in 1973. He is also the author of a major section on nondestructive testing by ultrasonics appearing in a handbook used by the U.S. Air Force and published in 1971. Ensminger is a member of the Acoustical Society of America, Ultrasonics Industry Association, the Society for Nondestructive Testing, and ASM International. Since his retirement, Ensminger has continued as a staff member (senior research scientist) of the Battelle Memorial Institute, where he has been employed since his college days (June, 1948 until May, 2008). Dr. Foster B. Stulen received his PhD from the Massachusetts Institute of Technology in 1980. His dissertation was on the use of frequency analysis of the myoelectric signal to quantify muscle fatigue. In 1979, he joined Battelle in Columbus, Ohio, one of the world’s largest contract R&D firms. There, he was mentored by Dale Ensminger and Dr. Naga Senapati in the field of ultrasonics. Dr. Stulen spent 18 years at Battelle developing and testing concepts ranging from large ultrasonic degassers for oil field applications to ultrasonic mosquito chasers. He even worked on the proverbial kitchen sink, acoustic emission monitoring from the spalling of porcelain. At Battelle, Dr. Stulen developed considerable expertise in modeling and analyzing power ultrasonic systems. One of his last projects there was to model an ultrasonic surgical system for Ethicon Endo-Surgery (EES), a Johnson & Johnson company. As a result, he accepted a position with that company in 1997. Since then, Dr. Stulen has made significant contributions to the Harmonic™ product line as it is known worldwide today. Dr. Stulen left EES for 1 year to join a medical device start-up company as the chief technology officer. The business strategy was to identify innovations at universities and academic health centers and develop them through NIH Small Business Innovation Research grants. During two grant cycles, the team of engineers working with him submitted nearly 40 grants at a success rate approaching 50%. ix

CRC_DK8307_FM.indd ix

11/14/2008 11:58:04 AM

x

Editors

Dr. Stulen is a guest lecturer at Miami University in Oxford, Ohio, where he created a senior technical elective course—Medical Device Design. He is also an associate editor for the Journal of Medical Devices, a publication of the ASME. He has numerous publications and has been awarded more than 21 patents with several additional patents pending.

CRC_DK8307_FM.indd x

11/14/2008 11:58:04 AM

Contributors Dale Ensminger Battelle Memorial Institute Columbus, Ohio Robert B. Francini Kiefner and Associates, Inc. Columbus, Ohio Dale Mangaraj Innovative Polymer Solutions Battelle Memorial Institute Columbus, Ohio Allan F. Pardini Applied Physics and Materials Characterization Sciences Pacific Northwest National Laboratory Richland, Washington

Gerald J. Posakony Applied Physics and Materials Characterization Sciences Pacific Northwest National Laboratory Richland, Washington Foster B. Stulen Ethicon Endo-Surgery Cincinnati, Ohio Theodore T. Taylor Applied Physics and Materials Characterization Sciences Richland, Washington B. Vijayendran Battelle Memorial Institute Columbus, Ohio

xi

CRC_DK8307_FM.indd xi

11/14/2008 11:58:04 AM

CRC_DK8307_FM.indd xii

11/14/2008 11:58:04 AM

1 Oscillatory Motion and Wave Equations Dale Ensminger

CONTENTS 1.1 Introduction ............................................................................................................................1 1.2 Elementary Vibratory Systems .............................................................................................2 1.2.1 The Pendulum............................................................................................................. 3 1.2.1.1 Simple and Compound Pendulums ..........................................................4 1.2.1.2 Torsional Pendulums ..................................................................................7 1.2.2 The Simple Spring/Mass Oscillator ........................................................................8 1.2.2.1 Effect of the Mass of the Spring on the Spring/Mass Oscillator ..............................................................................9 1.2.2.2 Effect of Losses on the Spring/Mass Oscillator .....................................9 1.3 Impedance, Resonance, and Q ........................................................................................... 10 1.3.1 Electrical and Mechanical Q ................................................................................... 11 1.4 Acoustic Wave Equations .................................................................................................... 12 1.4.1 Plane Wave Equation ............................................................................................... 12 1.4.2 The General Wave Equation ................................................................................... 14 1.4.3 Transverse Wave Equation for Flexible String ..................................................... 14 1.4.4 Transverse Wave Equation for a Membrane ........................................................ 15 1.4.5 Transverse Wave Equation for Bars ....................................................................... 16 1.4.6 The Plate Wave Equation......................................................................................... 17 1.4.6.1 Lamb Waves ............................................................................................... 17 1.4.7 Love Waves (5) .......................................................................................................... 19 1.4.8 Laplace Operators .................................................................................................... 19 1.5 Horns...................................................................................................................................... 20 1.6 Effects Due to Material Geometry and Elastic Properties .............................................22 1.7 Summary of Physical Factors in Ultrasonic Technology ................................................ 24 Further Readings .......................................................................................................................... 25

1.1

Introduction

The two properties of a medium that enable it to conduct acoustic waves are (1) its mass and (2) its elastic properties. The simple presentation in this chapter of the manner in which each of these two properties influence acoustic wave propagation is intended to assist the reader in understanding how to apply the principles and data presented in later chapters. The temptation to offer a more extensive treatise has been suppressed in the interest of providing a useful reference book, rather than a textbook. 1

CRC_DK8307_CH001.indd 1

10/24/2008 6:28:52 AM

Ultrasonics: Data, Equations, and Their Practical Uses

2

In this chapter, the functions of mass and elasticity are demonstrated in the presentation of fundamental wave equations germane to the field of ultrasonics. In general, specific solutions to these equations are presented in later chapters, where they are most appropriately correlated with principles and data. The concept of equivalent circuits is introduced for its value in the design of acoustic systems, including transducers and active elements of the systems.

1.2

Elementary Vibratory Systems

Ultrasonic energy is acoustic energy, and acoustic principles apply to any aspect of its study or use. There are many types of acoustic waves. They propagate through a medium as a succession of mechanical actions and reactions. The particle motion at a fixed position x in a wave is oscillatory. Although the general wave equation does not imply linear conditions and harmonic particle motion, common concepts and usage of ultrasonic energy involve harmonic motion. An acoustic wave is generated by any force that produces oscillatory vibrations within a continuous medium. The vibrations are passed from element to element, due to its inertial and elastic properties, at a rate corresponding to the velocity of sound in the medium. The mass of the element is determined by its volume and the density of the medium. The amplitude of the element motion is determined by the forces exerted upon it, its mass, and the elastic conditions surrounding it. These principles are fundamental to the development and understanding of wave motion equations. Force is defined as follows: an influence that, if applied to a free body, results chiefly in an acceleration of the body and sometimes in elastic deformation and other effects. The force, F, required to produce an acceleration, a, in a mass, m, is given by F = ma

(1.1)

A force, F, applied to any mass (or object) equals exactly the sum of all opposing forces, that is, the sum of all components of force acting in a direction opposite that of the applied force. (According to Newton’s 3rd law of motion for every action, there is an equal and opposite reaction.) The energy in an acoustic system also obeys the law of conservation of energy. The law of conservation of energy states that energy cannot be created or destroyed. A wave passing through a medium loses energy to the medium by various absorption mechanisms. The dissipated energy is not destroyed, but it is changed; that is, kinetic energy is changed to thermal energy. Equation 1.1 may be written in the form of a differential equation, that is F
 ma  m

d 2x dt 2

The opposing forces in a linear system may be represented by a proportionality constant, k, multiplied by the displacement, x, that is F− = kx

(1.2)

as F− is equal in magnitude and opposite F+.

CRC_DK8307_CH001.indd 2

10/24/2008 6:28:54 AM

Oscillatory Motion and Wave Equations

3

Therefore, we can write m

d 2x  kx dt 2

or d 2x k
x0 2 dt m

(1.3)

Equation 1.3 is the differential equation for harmonic vibration. The acceleration is proportional to the displacement and always directed toward the origin. Its general solution is well-known, that is x  A cos
t
B sin
t (1.4)

 C e j
t
D ej
t

where A, B, C, and D are constants to be determined from the initial conditions of motion, j2 is –1, ω is angular frequency (=2πf ) , ω2 = k/m, and f is frequency. The constants A, B, C, and D are related as follows: A=C+D B = j(C − D) The vibratory motion of simple mechanisms, such as pendulums and mass-loaded springs, operating within a linear range are described in a relationship similar to Equation 1.3. The frequency equation is ω2 equated to the coefficient of the term (k/m) in Equation 1.3, that is
2 

1.2.1

k m

(1.5)

The Pendulum

The pendulum is a simple example of oscillatory motion. Basic configurations of pendulums are (1) the simple pendulum (Figure 1.1a), (2) the compound pendulum (Figure 1.1b), and (3) the torsional pendulum (Figure 1.1c). In accordance with the law of conservation of energy, the motion of the pendulum is sustained by the alternating exchange of energy from one form to another (potential and kinetic) until the energy of the system is fully dissipated by various loss mechanisms.

ᐉ

ᐉ

M (a)

θ

ᐉ

d

θ

θ (b)

(c)

FIGURE 1.1 Common pendulums: (a) simple pendulum, (b) compound pendulum, and (c) torsional pendulum.

CRC_DK8307_CH001.indd 3

10/24/2008 6:28:54 AM

Ultrasonics: Data, Equations, and Their Practical Uses

4 1.2.1.1

Simple and Compound Pendulums

The simple pendulum is illustrated (Figure 1.1a) by a mass, m, on the free end of a string oscillating about a fixed support (or axis). The string is assumed to be inextensible and of negligible weight. A body that vibrates like a pendulum but with its mass distributed in any geometrical manner from or about its axis and not concentrated is called a physical or compound pendulum (Figure 1.1b). The simplest example of a compound pendulum is a uniform rod suspended at one end and free to vibrate like a pendulum if displaced and released. Another familiar form of the compound pendulum is a mass located at any position between the axis of rotation and the free end of a uniform bar. The simple torsional pendulum (Figure 1.1c) is represented as a rigid disk attached to one end of a light shaft, the other end of which is fixed. The rigid disk of Figure 1.1c may be replaced by any other configuration. The motions of each of these systems is described by the general relationship d 2 T()  0 dt 2 Ι

(1.6)

Therefore, the frequency relationships for any type of pendulum are dependent upon the ratio T(θ)/I, where T is the torque applied about the fixed point (or axis) of the pendulum, I the moment of inertia of the mass of the system about the fixed axis, and θ the angle in a vertical plane between the equilibrium position and the line through the support axis and the center of gravity of the pendulum (Figure 1.1a). The pull of gravity on the mass, m, and the distance between the point of support and the center of gravity determines the torque for the simple and the compound pendulums. Torque in the torsional pendulum is related to the elastic properties of the shaft plus gravitational factors attributable to the geometry of the oscillating body. For the simple and compound pendulums, the torque is the force of gravity resolved normal to a straight line through the support axis and the center of gravity, mg sin θ, multiplied by the distance,
, between the support axis and the center of gravity (Figure 1.2). This torque is always in a direction that tends to reduce θ, and is T()  mg
sin 

(1.7)

I  m
2

(1.8)

For the simple pendulum,

where m is the mass attached to the inextensible string and
the length of the string plus the distance to the center of the mass. Substituting these quantities into Equation 1.6 gives d 2 g sin 
0 dt 2


(1.6a)

for the simple and the compound pendulums. Equation 1.6a is nonlinear because of the term sin θ. Exact solutions are obtainable for only a few nonlinear differential equations. A common approach is to substitute θ for sin θ,

CRC_DK8307_CH001.indd 4

10/24/2008 6:28:55 AM

Oscillatory Motion and Wave Equations

5

ᐉ

θ mg sin θ

m

θ (T = mgᐉ sin θ)

mg FIGURE 1.2 Torque on a simple pendulum due to the pull of gravity.

ignoring the higher order terms in the sin θ series. This changes Equation 1.6a into the form of the linear Equation 1.3, for which the solution is applicable only to a very small θ. Substituting higher order terms from the sin θ series sin    

 3  5 7  9

 3! 5! 7 ! 9!

into Equation 1.6a would increase its accuracy for larger θ, but the resulting equation is nonlinear again, calling for an approximate method of solution. When a pendulum is released from a large angle θ and allowed to vibrate freely, the amplitude of swing will decrease gradually toward zero as energy is lost to the atmosphere and sin θ approaches θ. Substituting θ for sin θ (for very small θ) into Equation 1.6a gives d 2 g
0 dt 2


(1.6b)

which is similar in form to Equation 1.3. Therefore


g


(1.9)

which is the frequency equation commonly used for the pendulum in elementary physics. For the purposes of this book, the conventional method leading to Equation 1.6b is adequate. Finding the frequency of vibration of a pendulum consisting of any combination of physical features involves summing all components of torque acting on the system and dividing this sum by the total of the moments of inertia of all elements in the system.

CRC_DK8307_CH001.indd 5

10/24/2008 6:28:55 AM

Ultrasonics: Data, Equations, and Their Practical Uses

6

O ᐉ

L

θ ms r

FIGURE 1.3 Uniform bar with end mass load.

For the uniform bar mounted at one end and freely vibrating as a compound pendulum (Figure 1.1b), these quantities are 1 I  m
2 3

(1.10)


T  mg sin  2

(1.11)

for long slender rod about one end and

where the center of gravity is located at a distance
/2 from one end of the bar. For very small θ


3g 2


(1.12)

or f

1 2


3g 2


(1.12a)

Adding a mass to the end of a uniform bar (Figure 1.3) increases the torque and the moment of inertia by amounts attributable to the weight of the added mass and its distance from the axis O. If
is the length of the bar (extending from the axis O to the surface of the mass) and r the distance from the same surface point to the center of mass, the values of total torque and moment of inertia for very small θ are T = −(msLgθ + mrg
θ/2) I = (mr
2/3) + msL2 L=
+r


CRC_DK8307_CH001.indd 6

(m sL
m r (L  r)/2)g 1 m sL2
m r (L  r)2 3

(1.13)

10/24/2008 6:28:55 AM

Oscillatory Motion and Wave Equations

7

For the special case in which ms is negligible, r = 0 and
 3

gh L2

(1.13a)




g L

(1.13b)

If mr is negligible,

or the angular frequency of a simple pendulum. 1.2.1.2

Torsional Pendulums

The simple torsional vibrator is represented in Figure 1.1c by a rigid round plate of uniform thickness attached to the end of a light, cylindrical, elastic rod, or bar of length L. The opposite end of the bar is mounted rigidly to the support. The axes of the plate and the bar coincide. Within the elastic range, rotating the plate through an angle θ applies a torque, kθ, to the bar or T = kθ

(1.14)

where k is a torsional spring constant related to the dimensions of the bar, its geometry, and the shear modulus, G, of the material of the bar. For a round bar of length L and diameter d, assuming that the applied torque produces strictly shear stress in the bar without axial deformation, the spring constant, k, is k

d 4 G 32L

(1.15)

For a body rotating about a fixed axis, the moment of inertia of the body with respect to the axis multiplied by the angular acceleration is equal to the moment of the external forces acting on the body with respect to the axis of rotation, or I

d 2  k dt 2

(1.16)

For a round plate of uniform thickness and diameter D, I

m pD 2 8

(1.17)

where mp is the mass of the plate. Therefore, rearranging Equation 1.16, substituting for I and k, and simplifying gives d 2  d 4 G 
   0 dt 2  4m pD 2L 

(1.16a)

for the round plate on the end of a cylindrical rod.

CRC_DK8307_CH001.indd 7

10/24/2008 6:28:56 AM

Ultrasonics: Data, Equations, and Their Practical Uses

8

Equation 1.16a is similar in form to Equation 1.3. Therefore, the angular frequency of a torsional pendulum consisting of a round plate of uniform thickness mounted coaxially on a cylindrical rod and oscillating within the linear elastic range of the rod is


d 4 G 4m pD 2L

(1.18)

The torsional pendulum may consist of any of numerous geometrical structures other than the coaxial uniform bar and disk. The resonance frequency for any combination of such elements is determined according to the general rule derived from Equations 1.6 and 1.13b: (1) total all torques acting upon the pendulum element and divide this sum (gravitational and elastic quantities) by the sum of all moments of inertia of the components of the oscillatory system and (2) equate ω2 to the coefficient corresponding to k/m of Equation 1.3 if linear assumptions are applicable. Nonlinear equations are usually solved by approximate methods. 1.2.2

The Simple Spring/Mass Oscillator

The torsional pendulum is a type of spring/mass oscillator, because the rod is like a spring. The principles leading to Equation 1.3 are illustrated by the simple spring/mass oscillator of Figure l.4. A force, F, moves the mass toward the right to a distance x from the equilibrium position where the spring exerts an opposing force equal to F on the mass. For a linear spring, this reactive force is Fs = −kx

(1.2b)

where k is the spring constant in newtons per meter. If losses in the spring and between the mass and the slide are negligible, the force exerted on the mass by the compressed spring will accelerate the mass toward the neutral position when the force, F, is removed. These are the exact conditions assumed in the derivation of Equation 1.3. Thus the frequency of the undamped, or lossless, spring/mass oscillator is f

1 k 2 m

(1.19)

FS = −kx 1

2 m

x Frictionless surface FIGURE 1.4 Simple spring/mass oscillator.

CRC_DK8307_CH001.indd 8

10/24/2008 6:28:56 AM

Oscillatory Motion and Wave Equations 1.2.2.1

9

Effect of the Mass of the Spring on the Spring/Mass Oscillator

In practical cases, the mass of the spring affects its natural resonance frequency. If the mass of the spring is uniformly distributed along its length, the natural resonance frequency of the mass-loaded spring is f

1 k  m 2 m
s 3

(1.19a)

where ms is the mass of the spring. 1.2.2.2

Effect of Losses on the Spring/Mass Oscillator

A spring/mass oscillator loses energy by mechanisms such as friction, internal absorption, and air resistance. The equilibrium equation for such a system is m

d 2x dx
Rm
kx  Ft dt 2 dt

(1.20)

where Rm(dx/dt) represents the sum of all losses in the system and Ft is a driving force. When Ft suddenly drops to zero, the amplitude of vibration decays at a rate determined primarily by the loss factor, Rm. The resonant frequencies of vibration and the rate of decay are determined by solutions to Equation 1.20 for which Ft = 0, or m

d 2x dx
Rm
kx  0 dt 2 dt

(1.20a)

for which the solution is  k k  x  et K1 cos t
 2 sin t M′ M′  

(1.21)

The damped angular frequency is
d 

k t 
02   2 M′

(1.22)

where ω 0 is the undamped angular frequency, α = Rm/2m is the damping factor, K1 and K2 are the amplitudes of displacement, and M′ 

4m 2k 2 4km  R m

Here, m also includes the corrections for the mass of the spring. A second method of writing the solution to Equation 1.20a is x = K e−t cos(
dt + φ)

CRC_DK8307_CH001.indd 9

(1.21a)

10/24/2008 6:28:57 AM

Ultrasonics: Data, Equations, and Their Practical Uses

10

where K and φ are amplitude and phase constants, respectively. Equation 1.21a provides a means of determining α and, therefore, Rm, by measuring the rate of decay of the freely vibrating system.

1.3

Impedance, Resonance, and Q

The relationships between forces, motions, and losses within a mechanical vibratory system, and transmission of acoustic waves in general, are similar to those that exist between voltages, current, and impedances within electrical circuitry. Equivalent circuits and acoustic impedances are very useful and important concepts in analyzing and designing acoustic systems. In more general Equation 1.20, the vibrational amplitude of the system caused by the force, Ft, is determined by the amplitude and frequency of Ft and by the reactions of m, R m, and k. When Ft = 0, as in Equation 1.20a, the energy stored within the system is dissipated and the amplitude of vibration decays according to Equations 1.21 and 1.21a. Note the comparisons between the mathematical relationships describing the motions in the mechanical system and the electrical functions of Figure 1.5, a closed-loop series RLC circuit. The integro-differential equation that relates the loop current, i, with the source voltage, Vt, in Figure 1.5, is Vt  V1
V2
V3 L

(1.23)

di 1
R ei
∫ idt dt C

where V1 is the voltage across the inductor, L, due to current, i; V2 the voltage across the resistor, Re, due to the current, i; and V3 the voltage across the capacitor, C, due to the current, i. Substituting v = dx/dt converts Equation 1.20 to Ft  m

V1

i Vt

R

dv
R m v
k ∫ vdt dt

(1.24)

V2

L C

V3

FIGURE 1.5 Simple RLC electrical circuit.

CRC_DK8307_CH001.indd 10

10/24/2008 6:28:57 AM

Oscillatory Motion and Wave Equations

11

which is identical in form with Equation 1.23. Comparing Equations 1.23 and 1.24 shows equivalences between the two systems, which are: Mass, m, is equivalent to inductance, L. Force across a mass precedes the motion by 90°. Voltage across a coil precedes the current by 90°. Mechanical resistance, Rm, is equivalent to electrical resistance, Re. Force across Rm is in phase with the motion. Voltage across Re is in phase with current. Particle velocity, v, is equivalent to current, i. Elastic constant, k, is equivalent to the inverse of capacitance (1/C). Force across a lossless spring lags the motion of the spring by 90°. Voltage across a capacitor lags current by 90°. These comparisons show the logic behind equivalent circuits for use in designing and analyzing acoustic systems. Electrical impedance, Ze, is the vectorial sum of the electrical resistance and reactances, or Ze = Re + j(XL − Xc)

(1.25)

The mechanical impedance, Zm, is the vectorial sum of the mechanical resistance and reactances, or Zm = Rm + j(Xm − Xs)

(1.26)

where XL = ωLe is the inductive reactance of the electrical circuit, Xc = 1/ωCe the capacitive reactance of the electrical circuit, Xm = ωLm the reactance due to the density or mass of the material of the medium, and Xs = k/ω the reactance corresponding to the elastic properties of the material of the medium. Resonance occurs when |XL|=|Xc| and when |Xm|=|Xs|. 1.3.1

Electrical and Mechanical Q

The quality factor, Q, is a measure of the sharpness of resonance of an oscillatory system. In a series resonance circuit, it is defined as Q = Xm/Rm =
rm/Rm =
r/(
2 −
1) =
r/2

(1.27)

where ωr is the angular frequency at resonance, ω1 the angular frequency below resonance at which the amplitude of displacement in a driven system is 0.707 times its amplitude at resonance, ω2 the angular frequency above resonance at which the amplitude of displacement is 0.707 times its amplitude at resonance, and ω2 − ω1 the bandwidth of the system. In electrical series resonant circuitry, the voltage that appears across either the inductor or capacitor is Q times the voltage inserted in series with the circuit. In a mechanically vibrating system, it is convenient to relate the ratio of mechanical displacement amplitude at a position of maximum displacement at resonance (such as a free end of a longitudinally resonant uniform bar) to the amplitude at that same position under the same driving force, but at a frequency well removed from resonance. This ratio is the Q of the mechanical system. The stresses are affected similarly. Their distribution within the system depends upon the geometry and physical properties of the material of the system.

CRC_DK8307_CH001.indd 11

10/24/2008 6:28:57 AM

Ultrasonics: Data, Equations, and Their Practical Uses

12

Xm

E

Xs

Rm

FIGURE 1.6 Equivalent parallel circuit for a simple mechanically resonant system.

Most common ultrasonic systems are represented by equivalent parallel circuits or combinations of series and parallel circuits. Figure 1.6 is an equivalent parallel circuit for a simple, mechanically resonant system. For levels of Q in a parallel circuit higher than 10, Zr = QX

(1.28)

where Zr is resistive impedance at resonance. X is the reactance in ohms of either the total equivalent inductor or the total equivalent capacitor at resonance. Therefore, the relative impedance of a high-Q circuit is maximum at resonance. The shapes of the curves are identical to those of the series RLC circuit in which current peaks at resonance. When the Q of a parallel resonant circuit falls below 10, two different conditions may be called resonance. In maximum impedance parallel resonance, the impedance is maximum, but is not resistive. In resistive impedance parallel resonance, the parallel impedance is a pure resistance, but the impedance is not maximum. The characteristic acoustic impedance is a resistive component of the acoustic impedance of a material, the product of density and velocity of sound. Characteristic acoustic impedances are listed with other acoustic properties of materials in Chapter 6. In an active ultrasonic system, an electrical source supplies the energy to drive a mechanical system. In a passive device, electrical energy is produced in a system by reaction to mechanical forces. Electromechanical coefficients relate these parameters, so that both mechanical and electrical quantities are included within one equivalent circuit for analytical and design purposes. Use of equivalent circuits in the design of transducers is demonstrated in Chapters 4 and 5. Other practical uses of equivalent circuits are demonstrated in other chapters.

1.4 1.4.1

Acoustic Wave Equations Plane Wave Equation

The general method of deriving the plane wave equation is to consider the motion of plane waves in a medium in which the velocity of sound is c. Attenuation of the wave is assumed to be zero. The waves are free to move in both the positive and negative direction of x described by the equation   f1(x  ct )
f2 (x
ct )

CRC_DK8307_CH001.indd 12

(1.29)

10/24/2008 6:28:57 AM

Oscillatory Motion and Wave Equations

13

where f1(x − ct) refers to waves moving in the positive direction of x and f2(x + ct) to waves moving in the negative direction of x. If f1(x − ct) and f2(x + ct) are continuous, Equation 1.29 may be differentiated twice with respect to x (keeping t constant), giving  2  f1′′(x  ct )
f2′′(x
ct ) x 2

(1.30)

Similarly, differentiating Equation 1.29 twice with respect to t (keeping x constant) gives  2  c 2f1′′(x  ct )
c 2f2′′(x
ct ) t 2

(1.31)

or, by comparing Equations 1.30 and 1.31, 2  2 2    c t2 x 2

(1.32)

As a premise for its derivation, Equation 1.29 is an obvious general solution to the plane wave equation in Equation 1.32. The characteristic of a true plane wave is that pressures and motions at every position in a plane normal to the direction of propagation are equal in amplitude and phase. These conditions are seldom, if ever, met in actual practice. The type of wave is not defined by the velocity, c, in Equation 1.32. The equation describing a longitudinal wave in a thin bar (d << λ) is a specific example of Equation 1.32. The bar is assumed to be so slender that the effects of Poisson’s ratio can be neglected. Consider a uniform, narrow, homogeneous, elastic bar as being composed of a series of incremental elements of density ρ, longitudinal thickness dx, and cross-sectional area S (Figure 1.7). Losses are assumed to be negligible. Balancing the forces of momentum of an element Fm  ma  S a

 2 dx t2

(1.33)

 2 t2

m = ρsdx = Mass of element Propagation direction

Fx

x

Fx+dx

dx

FIGURE 1.7 Forces on an increment normal to the direction of propagation of a longitudinal plane wave in a uniform slender bar.

CRC_DK8307_CH001.indd 13

10/24/2008 6:28:58 AM

Ultrasonics: Data, Equations, and Their Practical Uses

14

where m = ρSd,  is the displacement of the element dx, and t the time against the elastic reaction of a neighboring element located between x and (x + dx), in the same manner used in deriving the spring equation, Fe  YS Fx   YS

d dx

 2 dx x 2

(1.34)

dF   Fx
dx   dFx
x dx   dx

leads to S

 2  2 dx  YS 2 dx 2 t x

2  2 Y  2 2     c o t2  x 2 x 2

(1.35)

(1.35a)

where c o  (Y/)0.5

(l.36)

is the velocity of sound and Y is Young’s modulus of elasticity in the medium of the bar. 1.4.2

The General Wave Equation

In the general wave equation, the Laplace operator, ∇ 2, replaces the one-dimensional operator to account for wave motion along all coordinates of space. The general wave equation is, therefore, more realistic and far more useful to the study of acoustic energy transmission and systems. The general wave equation is  2  c 2∇ 2 t2

(1.37)

The Laplace operator, ∇ 2, is expressed in any system of coordinates that best suits the boundaries; that is, Cartesian coordinates for a rectangular configuration, cylindrical coordinates for a cylindrical configuration, and spherical coordinates for a spherical system. In the Cartesian coordinate system, Equation 1.37 reduces to Equation 1.35, when the y and z components of wave propagation are zero. Laplace operators common to various coordinate systems used in acoustics are presented in Section 1.4.8. 1.4.3

Transverse Wave Equation for Flexible String

The wave equation for transverse vibrations on a flexible string is in the form of Equation 1.35a. Balancing the distributed mass of the string and forces tending to return the displaced string to its equilibrium position (Figure 1.8) leads to 2  2 T  2 2     c F t2  x 2 x 2

CRC_DK8307_CH001.indd 14

(1.38)

10/24/2008 6:28:58 AM

Oscillatory Motion and Wave Equations

15

Ty′(x+dx)

−T

T y(x+dx) y (x)

− Ty′ (x)

y x FIGURE 1.8 Forces tending to return a displaced string to its equilibrium position.

m=0 n=1

FIGURE 1.9 Lowest mode of vibration of a circularly clamped membrane.

where  is the amplitude of vibration of the transverse wave in the string; cF the velocity of a transverse wave along the string; T the equilibrium, uniform tension in the string; and ε the mass per unit length of string. 1.4.4

Transverse Wave Equation for a Membrane

The analysis of flexural waves in a membrane (Figure 1.9) is more complicated than that of a vibrating string. The wave equation for the membrane is  2  c 2∇ 2 t2

(1.39)

where  is the amplitude of vibration of the transverse wave in the membrane; cm the velocity of transverse waves in the membrane; T the uniform surface tension, or force per unit length with which the part of the membrane on one side of any line drawn in the membrane pulls on the part on the other side; σ the mass per unit area of the membrane; and ∇2 the Laplace operator.

CRC_DK8307_CH001.indd 15

10/24/2008 6:28:59 AM

Ultrasonics: Data, Equations, and Their Practical Uses

16

A type of membrane of common interest in acoustical applications is “circular,” that is, clamped along a circular periphery. The following additional conditions are assumed for solving the wave equation for such circular membranes: The membrane is vibrating with circular symmetry. The membrane is thin and uniform, having negligible stiffness. The membrane is perfectly elastic. The membrane has no damping. The membrane vibrates with very small displacement amplitudes. Tension at equilibrium is distributed uniformly throughout the membrane in all directions. Mass per unit area is constant throughout the membrane. 1.4.5

Transverse Wave Equation for Bars

The transverse wave equation for bars is derived by considering bending moments, M, and shear forces, Fy, Figure 1.10, leading to a fourth-order differential equation 4  2 2 2      c t2 x 4

(1.40)

where  is the amplitude of lateral displacement from the rest position at x, c the bar velocity of sound in the medium, κ the radius of gyration of the cross-sectional area S or κ 2 = −RM/YS with R as the radius of curvature of the neutral axis or plane at position x, M the bending moment, and Y Young’s modulus. For a round bar of radius a,  = a/2

(1.41)

For a rectangular bar of thickness t (measured in the y direction), 

t 12

(1.42)

−(Fy)(x+dx)

Mx

−M(x+dx) x

x+dx

(Fy)x FIGURE 1.10 Bending moments and shear forces in an elastic bar in flexure.

CRC_DK8307_CH001.indd 16

10/24/2008 6:28:59 AM

Oscillatory Motion and Wave Equations 1.4.6

17

The Plate Wave Equation

The plate wave equation is ∇ 4


3(1  2 )  2 0 Yh 2 t2

(1.43)

where Y is Young’s modulus of elasticity of the material of the plate, h the half thickness of the plate, and σ the Poisson’s ratio of the material. The equation for a bar in flexure can be derived from Equation 1.43 by applying boundary conditions applicable to a thin bar in flexure. 1.4.6.1

Lamb Waves

Lamb waves are plate waves. There are two types of Lamb waves: symmetrical and asymmetrical. There exist an infinite number of each of these types of waves. Each travels at a velocity determined by its frequency and the thickness, density, and elasticity of the medium. A symmetrical wave has many of the characteristics of a longitudinal wave. The particle motion along the central plane of the plate is parallel to the plane. Particle motion in every other plane of the plate is a lateral response to the successive compressions and extensions in the direction of propagation, characterized by a vector with one component parallel to the midplane and the other normal to the midplane. The motion on one side of the midplane is a mirror image of that on the other side (Figure 1.11a). Asymmetrical Lamb waves are flexural. The particle motion at the center of the plate is transverse (see Figure 1.11b). The particle motion of both the symmetrical and the asymmetrical Lamb wave modes is elliptical at the surface of the plate. The rate at which Lamb waves propagate along a plate is properly termed phase velocity, cp. Separate wave equations may be written for each of the two mode components of Lamb waves, longitudinal and shear; for longitudinal waves, ∇ 2 

1  2

c L2  t 2

(1.44)

(a)

(b) FIGURE 1.11 Displacement characteristics of Lamb waves: (a) symmetrical Lamb wave and (b) asymmetrical Lamb wave.

CRC_DK8307_CH001.indd 17

10/24/2008 6:29:00 AM

Ultrasonics: Data, Equations, and Their Practical Uses

18 and for shear waves,

∇ 2 

1  2 c s2  t 2

(1.45)

where cL is the velocity of longitudinal waves in the plate and cs the velocity of shear waves in the plate. Solutions to Equations 1.44 and 1.45 for infinite plates bounded by two planes parallel to and equidistant from the yz axis are [4,14] for symmetrical waves,  c 2p  2 1  2 x e j(
t( 2z/ )) 1   A cosh

cL  

(1.46a)

 c 2p  2π 1 − 2 x e j(ωt −( 2z/ )) 1   B sinh cs 



(1.46b)

and for asymmetrical waves,  c p2  2 1  2 x e j(
t(2  ))

2   A sinh

cL  

(1.47a)

 c 2p  2 1  2 x e j(
t( 2z/ )) 2   B cosh

cs  

(1.47b)

Under the same conditions, for infinite plates bounded by two planes parallel to and equidistant from the yz axis, the frequency equations are as follows. For symmetrical Lamb waves: tanh
h tanh
h

tanh
h tanh
h

tanh
h tanh
h

CRC_DK8307_CH001.indd 18

(c (c

2 s

2 L

(c (c

2 p

2 L

(c (c

2 p

2 p

)  c )/c c

4

)  c )/c c

4

 c 2p / c s2c 2p 2 p

2 2 L p

 c s2 / c s2c p2 2 p

2 2 L p

)  c )/c c

 c s2 / c s2c 2p 2 L

2 2 L p

 4

(c (c

2 L

( 2 L

(c

)(

)

 c p2 c s2  c 2p c s6

2c s2

 c 2p

)(

)

4

)

 c 2p c 2p  c s2 c s6

(

2c s2  c p2 c L2

2 p

 c L2 c p2  c s2 c 6s

(

)(

)

4

)

cp  cs  cL

(1.48a)

cs  cp  cL

(1.48b)

c L2

)

4

cs  cL  cp

(1.48c)

2c s2  c 2p c L2

10/24/2008 6:29:00 AM

Oscillatory Motion and Wave Equations

19

and for asymmetrical Lamb waves: tanh
h tanh
h

tanh
h tanh
h

tanh
h tanh
h

1.4.7

(c (c

2 s

2 L

(c (c

2 p

2 L

(c (c

2 p

2 p

)  c )/c c

 c 2p / c s2c 2p 2 p

2 2 L p

1  4

)  c )/c c

1  4

)  c )/c c

1  4

 c s2 / c s2c 2p 2 p

2 2 L p

 c s2 / c s2c p2 2 L

2 2 L p

(c

(2c 2 s

(c (c

2 s

 c 2p

)(

)

4

c L2

)

 c 2p c L2  c 2p c 6s

(2c 2 p

 c p2

)(

)

4

c L2

2 s

 c p2

)(

)

4

c L2

(1.49a)

)

cs  cp  cL

(1.49b)

)

cs  cL  cp

(1.49c)

 c s2 c L2  c p2 c 6s

(2c 2 p

2 s

cp  cs  cL

 c s2 c 2p  c L2 c 6s

Love Waves (5)

Love waves are typified by waves in the upper layer of the Earth during an earthquake. They are confined to a layer, in a layered medium in which material properties differ significantly between these layers. Particle motion within a Love wave is in shear and parallel with the bounding surfaces of the layer. They are dispersive (i.e., velocity changes with frequency). Complete analysis of the propagation characteristics of these waves is quite complicated. The most common use of Love waves in ultrasonics is in acoustic delay lines. 1.4.8

Laplace Operators

Laplace operators for use with the various two-dimensional and three-dimensional wave equations are: l. Two-dimensional (thin membranes, and so on): In rectangular coordinates, 2 2
2 2 x y

(1.50)

1    1  2  r
r  r   r  r 2   2 

(1.51)

∇2 

In polar coordinates, ∇2 

where r and φ are the polar coordinates of distance and direction for the wave traveling in a radial direction. The polar coordinates are related to the xy coordinates by x = r cos φ and y = r sin φ

CRC_DK8307_CH001.indd 19

10/24/2008 6:29:00 AM

Ultrasonics: Data, Equations, and Their Practical Uses

20 2. Three-dimensional: In rectangular coordinates,

2 2 2
2
2 2 x y z

(1.52)

2 1    1  2  r 
2  2
2  r  r   r  r     z

(1.53)

∇2  In cylindrical coordinates, ∇2 

In spherical coordinates, ∇2 

    2 1   2  1 1 r
sin 
r 2  r   r  r 2 sin      r 2 sin 2   2

(1.54)

where θ is the angle between the r and the z axis of the Cartesian system and φ the angle between the rz plane and the xz plane of the Cartesian system. Relative to the Cartesian system of coordinates, x = r sin θ cos φ y = r sin θ sin φ z = r cos θ

1.5

Horns

High-intensity applications of ultrasonics have brought about the development of solid horns of a variety of designs. In most high-intensity applications, the main purpose of the horn is to provide an amplitude of vibration at its load (or output) end that is greater than that at the driven (or input) end. This is the reverse of the usual objective of the traditional horn—for example, those used in musical acoustics—which converts higher amplitude vibrations at the input end to lower amplitude vibrations, distributed over a larger area, at the output end (gaseous medium). A horn, whether the medium is gas or solid, acts as: (1) a particle velocity transformer, (2) an impedance transformer, (3) a filter, or (4) a guide for concentrating acoustic waves into a more directional radiation pattern. These effects are produced by guiding the acoustic waves through tapered structures that conform to certain conditions. The analysis of wave motion in a horn is very complicated, and certain practical approximations are made in deriving design equations. Common simplifying assumptions applied to the design of horns, whether they are to be used in a fluid or solid medium, include: 1. The acoustic pressure amplitudes remain within the linear range of the compressibility or elasticity curve for the fluid or solid. Thus, second-order terms can be neglected. This assumption appears to be more appropriate in the design of solid horns used in ultrasonics than for fluid types, when high intensities are involved.

CRC_DK8307_CH001.indd 20

10/24/2008 6:29:01 AM

Oscillatory Motion and Wave Equations

21

2. The acoustic wave is propagated through the horn in the form of plane waves moving parallel to the axis; all particle motion is parallel to this axis; the energy distributes itself uniformly across each plane normal to the axis at distance x from the source; and the pressure, particle velocity, particle amplitude, or intensity are functions only of time t and distance x measured along the axis. For the solid ultrasonic horn, the limits of this assumption are determined by Poisson’s ratio. (Another method is to consider the wave front to be conforming to a curvature dictated by the geometry of the horn: spherical, cylindrical, or other.) 3. The walls of the horn are sufficiently rigid to prevent them from radiating in a direction normal to the surface. Energy radiated from these walls would reduce the effectiveness of the horn. There is sufficient mismatch between solids and a surrounding gaseous atmosphere to justify this assumption with the solid horns used in power ultrasonics. If the horn is submerged within a liquid, the accuracy of this assumption depends upon the geometry of the horn and the differences between acoustic impedances of the solid and the liquid. 4. The taper of the horn is sufficiently gradual that the assumed plane waves can maintain contact with the walls; otherwise, the purpose of the horn is defeated. Contour is the important factor that establishes an acoustical transmission line as a horn, and if the wave front cannot maintain contact with this contour, it loses its effectiveness in guiding the spread (or condensation) of the wave in the intended manner. When the taper is too great, the assumed plane waves moving from the small to the large end lose contact with the walls and spread out as spherical waves in free space. 5. The taper of the solid horn is such that acoustic rays initiated at a driving source in the plane of the large end do not reflect from the contoured surface(s) in a manner that causes large differences in phase between reflected rays. The solid horn equation, which is the one of most interest in the high-intensity ultrasonics field, is derived by the same procedure that led to Equation 1.35. The derivation takes into account the taper-related changes in the momentum between successive elements (as illustrated for the horn in Figure 1.12) leading to 1  2ξ 1  S   2   0 c o2  t 2 S x x x 2

(1.55)

Sx SL

So Fx

x

−Fx+dx

dx L

FIGURE 1.12 Forces on an increment normal to the direction of propagation of a longitudinal wave in a tapered, slender bar, or horn.

CRC_DK8307_CH001.indd 21

10/24/2008 6:29:01 AM

Ultrasonics: Data, Equations, and Their Practical Uses

22 and

 2v 1 S  v
2

2 v0 c x 2 S x x

(1.55a)

where co is the longitudinal bar velocity of sound in the horn material, ξ the displacement amplitude at x, t the time, S the cross-sectional area at x, v the velocity amplitude at x, and ω = 2πf, where f is the frequency. When S(x) is constant, Equation 1.55a reduces to Equation 1.35a, the plane wave equation; that is, the taper function 1  S  0 S x x The assumptions leading to Equation 1.55 apply to slender horns, which include uniform bars.

1.6

Effects Due to Material Geometry and Elastic Properties

The term velocity of sound deserves some discussion. Tabulations of velocities of sound in solid media usually include bar velocity, co (Equation 1.36); bulk velocity, cB; shear wave velocity, cs; and surface (or Rayleigh) wave velocity, cR. These terms are interrelated by material properties of density and elasticity. Velocity of sound also depends upon the direction of particle motion relative to the direction of wave propagation and upon various geometrical and structural factors. The path followed by the wave (e.g., Rayleigh and Love waves) is another determining factor. The important elastic constants of a material are: Bulk modulus of elasticity, K, which is the ratio of a tensile or compressive stress, triaxial and equal in all directions (e.g., hydrostatic pressure) to the relative change that it produces in volume [10]. Compressibility, B, is the reciprocal of bulk modulus. Bulk modulus is generally used in relationships applicable to solids; compressibility is used in relationships for fluids. Young’s modulus of elasticity, Y, which is the rate of change of unit tensile or compressive stress with respect to unit tensile or compressive strain for the condition of uniaxial stress within the proportional limit [10]. Shear modulus of elasticity (modulus of rigidity), which is the rate of change of unit shear stress with respect to unit shear strain for the condition of pure shear within the proportional limit [10]. Poisson’s ratio, σ, which is the ratio of lateral unit strain to longitudinal unit strain under the condition of uniform uniaxial longitudinal stress within the proportional limit [10]. These quantities are interrelated as follows: Y = 2(1 + )  = Y/2(1 + )

CRC_DK8307_CH001.indd 22

10/24/2008 6:29:01 AM

Oscillatory Motion and Wave Equations

23

= (Y − 2)/2 K = Y/3(1 − 2σ) = (3K − Y)/6K The longitudinal bulk velocity or velocity of sound in an infinite, homogeneous medium is cB 

Y(1  ) (1  )  co (1
)(1  2 ) (1
) (1  2 )

(1.56)

and the velocity of shear waves is cs 

Y  1   co 2(1
) ρ 2(1
)

(1.57)

where µ is the shear modulus of elasticity. The longitudinal velocity of sound in a thin plate of homogeneous material with large lateral and longitudinal dimensions is cp 

Y 1  co (1  2 ) 1  2

(1.56a)

The velocity of Rayleigh surface waves is given approximately by c R = K Rc s

(1.58)

where KR is a constant determined by the relationship between the longitudinal velocity and the shear velocity. The relationship between cB, cs, and cR is given by   c 6R 8c R4 16  c s2  2 24 
c   16 1    R 0  2 c 6s c s4 c 2B  c R2    cs

(1.59)

where cR is the particular solution for which its value is slightly less than cs. Even the slenderest of rods undergoing dynamic tension and compression experiences the effects of Poisson’s ratio. Its effect on the velocity of sound in a very slender bar is insignificant, but as the ratio of cross-sectional dimensions to the wavelength in the medium increases, Poisson’s ratio becomes an important factor. The effect of Poisson’s ratio is illustrated in Figure 1.13. The following equations may be used to estimate the effects of Poisson’s ratio on the velocity of sound in bars of uniform cross-section: 2

1    c′ 2 2  1   [B
R ] 6  co

(1.60)

for rectangular cross-section and 1  d  c′  1   4  co

CRC_DK8307_CH001.indd 23

2

(1.61)

10/24/2008 6:29:02 AM

Ultrasonics: Data, Equations, and Their Practical Uses

24 (1−σs) B

T

(1−σs) R

s = ∆L/L σ = ∆d /sd = ∆b/sb

(1+s) L

L

B

R

−T FIGURE 1.13 Effect of Poisson’s ratio on a volume under uniaxially applied stress.

for uniform cylinders, where (d/λ) ≤ 0.4, B and R are the dimensions of the rectangular cross-section, d is the diameter of the cylindrical section, σ Poisson’s ratio, f the design frequency, co the bar velocity of sound in the material of the bar, and c′ the velocity of sound in the bar corrected for Poisson’s ratio. The calculated half-wavelength of a resonant uniform bar based upon bar velocity of 200,000 in./s and a frequency of 20 kHz is 5.0 in. Letting σ = 0.3 for the same material and design frequency, a more nearly accurate length of a rectangular bar with cross-section of 0.5 × 1.0 sq. in. or a uniform cylinder 1.0 in. in diameter is ~4.99 in., according to Equations 1.60 and 1.61. The actual difference in lengths is 0.2%. Large-area horns are specific examples in which geometry requires including Poisson’s ratio in their design formulae. These designs will be included with those of other geometrical types in Chapter 2.

1.7

Summary of Physical Factors in Ultrasonic Technology

Among the physical characteristics to consider for effective ultrasonic technology are: Elastic properties. Moduli and Poisson’s ratio, a factor in acoustic impedance. Density. Influences the velocity of sound, affects the reaction of a structure to a mechanical stimulus, is an important factor in acoustic impedance of a substance.

CRC_DK8307_CH001.indd 24

10/24/2008 6:29:02 AM

Oscillatory Motion and Wave Equations

25

Velocity of sound. A quantity determined by the elastic properties of a medium, its density, the direction of particle displacement, its boundary conditions and structure. Boundary conditions. Determine allowable modes of vibration, determine allowable frequencies. Geometrical factors. Determine sonic velocities, determine modes, determine coupling methods and impedances. Acoustic impedance and matching. Determine effectiveness of energy transfer between elements in an ultrasonic system, important factors in transducer design and horn and load designs. Isotropy and anisotropy. These factors affect transmission of ultrasonic energy and the performance of an ultrasonic vibrator. An isotropic medium exhibits the same elastic properties along any axis. An anisotropic material exhibits different values when measured along axes in different directions. Anisotropy causes scattering of energy and thus contributes to wave attenuation.

Further Readings 1. A. H. Church, Mechanical Vibrations, John Wiley and Sons, Inc., New York and London, 1963. 2. E. B. Cole, The Theory of Vibrations for Engineers, The MacMillan Company, New York, 1979. 3. P. L. L. M. Derks, The Design of Ultrasonic Resonators with Wide Output Cross-Sections, Philips, Eindhoven, The Netherlands, 1984. 4. D. Ensminger, Ultrasonics, Second edition, Marcel Dekker, New York, 1988. 5. K. F. Graff, Wave Motion in Elastic Solids, Ohio State University Press, Columbus, OH, 1975. 6. T. F. Hueter and R. H. Bolt, Sonics, John Wiley and Sons, Inc., New York; Chapman and Hall, Limited, London, 1955. 7. E. Kinsler and A. R. Frey, Fundamentals of Acoustics, John Wiley and Sons, Inc., New York; Chapman and Hall, Limited, London, 1950. 8. T. Baumeister and L. S. Marks, Eds, Standard Handbook for Mechanical Engineers, Seventh Edition, McGraw-Hill Book Company, New York, 1967. 9. S. S. Rao, Mechanical Vibrations, Second Edition, Addison-Wesley Publishing Company, Reading, MA, 1990. 10. R. J. Roark and W. C. Young, Formulae for Stress and Strain, Fifth Edition, McGraw-Hill Book Company, New York, NY, 1975. 11. R. F. Steidel, Jr., An Introduction to Mechanical Vibrations, Second Edition, John Wiley and Sons, Inc., New York, NY, 1979. 12. S. Timoshenko, Vibration Problems in Engineering, Third Edition, D. Van Nostrand Company, Inc., New York, NY, 1955. 13. V. Neufeldt, Editor-in-Chief, Third College Edition Webster’s New World Dictionary of American English, Webster’s New World, Cleveland and New York, 1991. 14. D. C. Worlton, Lamb waves at ultrasonic frequencies, Hanford Atomic Products Operation, Report HW-60662, Richland, Washington, published by the U.S. Atomic Energy Commission, June 9, 1969.

CRC_DK8307_CH001.indd 25

10/24/2008 6:29:02 AM

CRC_DK8307_CH001.indd 26

10/24/2008 6:29:02 AM

2 Ultrasonic Horns, Couplers, and Tools Dale Ensminger

CONTENTS 2.1 Introduction .......................................................................................................................... 29 2.2 Definitions............................................................................................................................. 29 2.3 Specific Solutions to the Horn Equation ...........................................................................30 2.3.1 Free–Free Uniform Bar............................................................................................30 2.3.2 Double Cylinder or Stepped Horn ........................................................................ 32 2.3.2.1 Stress Concentration at a Step in a Horn ............................................... 33 2.3.2.2 Step at a Position Other than a Node ..................................................... 35 2.3.3 Cross-Sections Other than Circular ...................................................................... 37 2.3.4 Exponentially Tapered Horn .................................................................................. 37 2.3.4.1 Velocity Distribution, v ............................................................................ 38 2.3.4.2 Particle Displacement Distribution,  .................................................... 38 2.3.4.3 Particle Acceleration Distribution, a ...................................................... 39 2.3.4.4 Stress Distribution, s ................................................................................. 39 2.3.4.5 Length of the Half-Wave Horn, ℓ............................................................ 39 2.3.4.6 Mechanical Impedance at x ..................................................................... 39 2.3.4.7 Exponentially Tapered Horn of Rectangular Cross Section and Constant Width (Width Small Compared with Wavelength) ...................................................................................... 39 2.3.5 Slender Wedge-Shaped Horn .................................................................................40 2.3.5.1 Velocity and Displacement Nodes.......................................................... 41 2.3.5.2 Stress Distribution, s .................................................................................42 2.3.5.3 Length of the Half-Wave Wedge-Shaped Horn, ℓ ................................42 2.3.5.4 Impedance of the Wedge-Shaped Horn.................................................43 2.3.5.5 Amplification Factor for Wedge-Shaped Horns ...................................44 2.3.6 Conically Tapered Horns ........................................................................................44 2.3.6.1 Area Increasing with x ............................................................................. 45 2.3.6.2 Area Decreasing with Increasing x ........................................................ 48 2.3.7 Catenoidal Horns ..................................................................................................... 50 2.3.8 Hyperbolic Horns .................................................................................................... 52 2.4 Horn Design and Performance Factors ............................................................................ 55 2.4.1 Effects of Poisson’s Ratio on Performance............................................................ 56 2.4.1.1 Corrections to Velocity of Sound in Tapered Horns ............................ 58 2.4.2 Effects of Losses on Horn Performance................................................................ 61 2.4.2.1 Losses Associated with the Load into which the Horn Works .......... 61 2.4.2.2 Internal Damping Factors ........................................................................ 61

27

CRC_DK8307_CH002.indd 27

10/29/2008 3:48:07 PM

Ultrasonics: Data, Equations, and Their Practical Uses

28 2.4.3

2.5

2.6

Effect of Temperature on Horn Performance ...................................................... 68 2.4.3.1 Phonons ...................................................................................................... 68 2.4.3.2 Thermoelastic Effects on Horn Performance ........................................ 69 Combinations of Geometrical Forms ................................................................................ 69 2.5.1 Uniform Bars of Equal Diameter but Different Materials ................................. 70 2.5.2 Horn and Lumped Mass ......................................................................................... 71 2.5.2.1 Lumped Mass Attached to a Uniform Bar ............................................ 71 2.5.2.2 Lumped Mass Attached to Exponential Horn...................................... 72 2.5.3 Horns with Distributed Loads ............................................................................... 72 2.5.3.1 Matching a Uniform Bar to a Horn ........................................................ 73 2.5.4 Multiple Mode Systems ........................................................................................... 74 2.5.4.1 Longitudinal Mode Horn Driving a Flexural Bar ................................ 75 2.5.4.2 Moments of Inertia.................................................................................... 76 2.5.4.3 General Solution to the Transverse Wave Equation for Thin Bars .............................................................................................. 78 2.5.4.4 Specific Solutions to the Flexural Bar..................................................... 81 2.5.4.5 Stresses in Bars in Flexure ....................................................................... 93 2.5.4.6 Coupling between Driver and Flexural Bar .......................................... 94 Plates ...................................................................................................................................... 94 2.6.1 General Solution to the Plate Equation in Polar Coordinates ........................... 96 2.6.2 General Solution to the Plate Equation in Rectangular Coordinates ............... 97 2.6.3 Specific Solutions to the Plate Wave Equations ................................................... 97 2.6.3.1 Circular Plate of Uniform Thickness and Radius a Clamped at the Outer Circumference ..................................................................... 97 2.6.3.2 Free Circular Plate of Uniform Thickness and Radius a..................... 98 2.6.3.3 Circular Plate with Fixed Center ............................................................ 98 2.6.3.4 Circular Plates Simply Supported All Around..................................... 99 2.6.4 Annular Plates of Uniform Thickness ................................................................ 100 2.6.4.1 Annular Plates Clamped on Outside and Inside ............................... 100 2.6.4.2 Annular Plate Clamped on Outside and Simply Supported on the Inside............................................................................................. 101 2.6.4.3 Annular Plates Clamped on Inside and Simply Supported on the Outside ......................................................................................... 101 2.6.4.4 Annular Plate Simply Supported on Inside and Outside Circumferences ....................................................................................... 102 2.6.4.5 Annular Plate Clamped on Inside Circumference and Free on Outside Circumference ..................................................................... 103 2.6.4.6 Annular Plate Simply-Supported at the Inner Circumference and Free at the Outer Circumference................................................... 104 2.6.4.7 Free–Free Annular Plate ........................................................................ 104 2.6.4.8 Annular Plate Clamped at the Outer Circumference and Free at the Inner Circumference .................................................................... 105 2.6.4.9 Annular Plate Simply Supported at the Outer Circumference and Free at the Center ............................................................................ 105 2.6.5 Rectangular Plates ................................................................................................. 106 2.6.5.1 Rectangular Plate Simply Supported on All Sides ............................. 109 2.6.5.2 Rectangular Plate Simply Supported on Two Opposite Sides and Clamped on the Remaining Two Sides ........................................ 110 2.6.5.3 Other Rectangular Plate Conditions .................................................... 113

CRC_DK8307_CH002.indd 28

10/29/2008 3:48:08 PM

Ultrasonic Horns, Couplers, and Tools

29

2.7

Rings and Hollow Cylinders ............................................................................................ 114 2.7.1 Pure Radial Vibration ............................................................................................ 114 2.7.2 Flexural Modes of Rings ....................................................................................... 116 2.7.3 Hollow Cylinders ................................................................................................... 117 2.8 Wide and Large-Area Horns ............................................................................................ 118 2.8.1 Wide Blade-Type Horns ........................................................................................ 119 2.8.2 Large-Area Block-Type Horns .............................................................................. 121 2.8.3 Other Designs—Large, Cup-Shaped Horns ...................................................... 122 2.9 Additional Design Factors ................................................................................................ 126 2.9.1 Stress Concentration Factors ................................................................................ 127 2.9.2 Treating Materials at High Temperature ............................................................ 127 2.9.3 Treating Materials in a Harsh Chemical Environment .................................... 127 References .................................................................................................................................... 128

2.1

Introduction

The equations presented in this chapter are for the purpose of simplifying the design procedure of the vibratory structures in an ultrasonic system. In general, these systems are for power applications, that is, applications for promoting a physical or chemical change in a product.

2.2

Definitions

Ultrasonic horns. In the context of this chapter, an ultrasonic horn is an element operating in a longitudinal mode used for the efficient transfer of ultrasonic energy from a source element (transducer or another horn) to a second horn, coupler, tool, or load. In this sense, it is a transmission line, generally (but not always) providing a change of amplitude of vibration between the input and the output ends of the horn. Other appropriate terms are transmission line, velocity, displacement, or impedance matching transformer. Horns fall into two major categories: (a) slender horns, in which the effects of Poisson’s ratio are considered to be negligible and (b) wide area horns, in which the effects of Poisson’s ratio cannot be neglected. Effects of internal losses on horn performance also are considered. Couplers. In the context of this chapter, couplers have a broad meaning, generally covering resonant members of an ultrasonic assembly or portions of resonant members designed and impedance-matched into the remaining assembly for efficient transfer of energy from one element of the system to another or into a load. Couplers can include horns, sections of horns, bolts, diaphragms, flexural plates, flexural bars, and so on. Tools. Primarily, tools include elements attached to an ultrasonic device for the purpose of imparting specific actions or effects to a load. Typical examples include ultrasonic drilling, machining, or forming tools; special tips for soldering, metal welding, metallurgical applications, and activities in harsh chemical environments; and special transmission-line elements for transferring ultrasonic energy along guided paths into otherwise inaccessible regions.

CRC_DK8307_CH002.indd 29

10/29/2008 3:48:08 PM

Ultrasonics: Data, Equations, and Their Practical Uses

30

2.3

Specific Solutions to the Horn Equation

The basic horn equation is presented in Chapter 1, Equations 1.55 and 1.55a, in terms of particle displacement, , 1 2 1 S  2   0 c 2 t 2 S x x x 2

(1.55)

and in terms of particle velocity, v, and implying harmonic motion, 2 v 1 S v
2 v0

x 2 S x x c 2

(1.55a)

where x is the distance from one end of the horn to the reference position, in meters; S the cross-sectional area of the horn at distance x, in square meters; ω the angular frequency (=2πf); f the frequency of vibration, in hertz; and c the velocity of sound in the medium of the horn, in meters per second. Slender horns, including uniform bars and tapered structures, usually are designed using bar velocity values for the materials of the horns, ignoring Poisson’s ratio when the diameter or lateral dimensions are <20% of the wavelength calculated. Ignoring Poisson’s ratio leads to calculated values for horn lengths that are greater than the actual resonant lengths. The extent of the error is illustrated as follows, using a Poisson’s ratio of 0.3 as a norm: 1. 2. 3. 4. 5.

When d is 0.10λ, the error is 0.0247σ2 (0.2% for σ = 0.3). When d is 0.15λ, the error is 0.055σ 2 (0.5% for σ = 0.3). When d is 0.20λ, the error is 0.0987σ 2 (0.9% for σ = 0.3). When d is 0.25λ, the error is 0.1542σ 2 (1.4% for σ = 0.3). When d is 0.30λ, the error is 0.222σ 2 (2.0% for σ = 0.3).

where λ is the wavelength based upon bar velocity, co, and frequency f. Temperature and load variations may cause shifts in frequency that exceed 1.5%. Most driving systems for high-power applications of ultrasonic energy operate over a range of frequencies that compensate for variations in load and horn performance. If accuracies greater than those obtained by neglecting Poisson’s ratio in the design of slender horns are necessary, the horns can be “fine-tuned” by removing the excess length by carefully machining to the length that produces resonance at the desired frequency. Wide horns must take Poisson’s ratio into consideration. Effects of Poisson’s ratio are discussed more extensively in Section 2.4.1. 2.3.1

Free–Free Uniform Bar

Referring to Equations 1.55 and 1.55a, dS 0 dx

CRC_DK8307_CH002.indd 30

10/29/2008 3:48:08 PM

Ultrasonic Horns, Couplers, and Tools

31

D1

D2

x λ/4 λ/2 FIGURE 2.1 Double-cylinder half-wave resonator.

This relationship reduces Equations 1.55 and 1.55a to the plane wave equation for which the general solution, in terms of the displacement, , is
x
x      A cos
B sin cos(
t ) c c  

(2.1)

or in terms of particle velocity  
x  
x   v  
 A cos  
B sin   sin(
t )  c c   

(2.2)

For a longitudinally half-wave resonant uniform bar (Figure 2.1) 
x  cos(
t)   m cos   c 

(2.3)


x  v 
m cos  sin(
t )  c 

(2.4)

and

where m is a maximum displacement and is located at x = 0. The acceleration at any point, x, along the bar is 
x  a  
2m cos  cos(
t)  
2  c 

(2.5)

Stress, s, at x in a half-wave resonant bar is s s  jY

jY dv
dx


 m 
x  sin  sin(
t)  c  c

(2.6)

(2.6a)

where ω_m__is the maximum velocity and occurs at x = 0, Y is Young’s modulus of elasticity, and j = √−1 .

CRC_DK8307_CH002.indd 31

10/29/2008 3:48:08 PM

Ultrasonics: Data, Equations, and Their Practical Uses

32

For a uniform bar in half-wave longitudinal resonance, the maximum stress occurs at the midpoint between the free ends (x = ℓ/2, where ℓ is the length of the bar). The mechanical impedance at a point x along a member vibrating in a longitudinal mode is defined by the force due to the strain across a plane at and normal to x divided by the particle velocity (or velocity potential) at x, that is Zx = Fx/vx For the half-wave-resonant uniform bar Zx  

sxS SY 
x  
x  j tan   jSρc tan   c   c  vx c

(2.7) ____

where the term ρc is called the characteristic acoustic impedance of the c = √Y/ρ medium. Standing alone, it is a pure resistance with dimensions FTL−3 (force × time over length).

2.3.2

Double Cylinder or Stepped Horn

For the horn of Figure 2.1, with the step occurring at the midplain along the length, momentum of the elements on either side of the step leads to the identity 1 v1 S 2    2 v 2 S1

(2.8)

where 1 is the particle displacement at x = 0, 2 the particle displacement at x = ℓ, v1 the particle velocity at x = 0, v2 the particle velocity at x = ℓ, S1 the cross-sectional area of the horn at x = 0, and S2 the cross-sectional area of the horn at x = ℓ. Equations 2.3 through 2.8 are applicable to the stepped horn of Figure 2.1 if the values of m and x are correlated with the cross section. For values of x ≤ λ/4, the displacement, x, is 
x  x  1 cos  cos(
t)  c 

(2.9)


x  vx 
1 cos  sin(
t )  c 

(2.10)


x  ax 
21 cos  cos(
t)  c 

(2.11)

Velocity, vx, is

and acceleration, ax, is

For values of λ/4 < x ≤ λ/2, the displacement, x, is 
 x   2 cos  (
 x) cos(
t ) c 

CRC_DK8307_CH002.indd 32

(2.12)

10/29/2008 3:48:08 PM

Ultrasonic Horns, Couplers, and Tools

33

Velocity, vx, is 
 vx 
 2 cos  (
 x) sin(
t) c 

(2.13)

The amplification factor for the half-wave, double-cylinder horn is therefore determined by comparing vx2 when x = ℓ (Equation 2.13), with vx1 when x = 0. By comparing impedances (see Equation 2.7) vx2 Zx1 S1  2    vx1 Zx2 S 2 1

(2.14)

or 2 

2.3.2.1

S1 1 S2

(2.15)

Stress Concentration at a Step in a Horn

When double cylinders are used as half-wave horns with step at the node (or λ/4 position), the maximum stress occurs at the step. The maximum stress is calculated approximately by using the stress in the smaller diameter sector. From Equations 2.6 and 2.6a, this maximum stress is sm  Y


 2
S1 Y 1 c c S2

(2.16)

A horn designed according to Figure 2.1 will fail in fatigue at the junction between the two elements. This junction is a position of maximum stress, and the abrupt change in diameter increases the stress near the surface of the smaller diameter segment and the nodal face of the larger diameter. In practice, these horns are designed with a highly polished fillet located between the larger diameter and the smaller diameter segments to reduce the stress concentration at the junction. The stress concentration factor, k, for various fillet radii can be determined by using the relationship [1] (Figure 2.2) 2

 2h   2h   2h  k  K1
K 2  
K 3  
K 4    D  D  D

3

(2.17)

where For 0.25 ≤ h/r ≤ 2.0 K1 K2 K3 K4

0.927 + 1.149 √(h/r) − 0.086h/r 0.011 − 3.029 √(h/r) + 0.948h/r 0.304 + 3.979 √(h/r) − 1.737h/r 0.366 − 2.098 √(h/r) + 0.875h/r

For 2.0 ≤ h/r ≤ 20.0 1.225 + 0.83l √(h/r) − 0.010h/r −1.831 − 0.318 √(h/r) − 0.049h/r 2.236 − 0.522 √(h/r) + 0.176h/r −0.630 + 0.009 √(h/r) − 0.117h/r

Calculated concentration factors for typical horn designs are given in Table 2.1. The probable endurance of a double cylinder can be estimated using Equation 2.16, values for k calculated from Table 2.1, and the design maximum output displacement, 2. The concentrated stress, sc, is sc = ksm (2.18) where sm is the value of stress calculated using Equation 2.16.

CRC_DK8307_CH002.indd 33

10/29/2008 3:48:09 PM

Ultrasonics: Data, Equations, and Their Practical Uses

34

h

+

D r

FIGURE 2.2 Double cylinder with square shoulder and fillet.

TABLE 2.1 Stress Concentration Factors for Longitudinal Stress in the Fillet between Sections of a Double Cylinder-Type Ultrasonic Horn (Figure 2.2) h

h/r

k

h/r

k

h/r

k

h/r

k

h/r

k

1.4807 1.2799 1.2026 1.1538 1.0892 1.0516 1.0165 2.0024 1.9453 1.8456 1.7829 1.6991 1.6490 1.5984

0.5

1.5526 1.5132 1.4496 1.4122 1.3689 1.3501 1.3453 2.0872 2.0277 1.9232 1.8249 1.7658 1.7100 1.6502

0.75

1.6911 1.6457 1.5689 1.5228 1.4663 1.4378 1.4194 2.1660 2.1047 1.9961 1.9260 1.8292 1.7679 1.6995

1.0

1.8076 1.7569 1.6710 1.6182 1.5511 1.5145 1.484

1.25

1.9100 1.8558 1.7622 1.7040 1.6280 1.5842 1.5435

2.3974 2.3262 2.1936 2.1016 1.9606 1.8569 1.7125 3.4329 3.3333 3.1406 3.0004 2.7709 2.5875 2.3053

4.0

2.5986 2.5271 2.3876 2.2904 2.1400 2.0280 1.8695 3.6550 3.5547 3.3651 3.2310 3.0182 2.8536 2.6080

5.0

2.7728 2.6934 2.5441 2.4395 2.2764 2.1537 1.9775 3.8489 3.7436 3.5449 3.4044 3.1812 3.0081 2.7483

6.0

2.9290 2.8460 2.6827 2.5799 2.4077 2.2773 2.0882 4.0275 3.9176 3.7104 3.5640 3.3313 3.1506 2.8783

8.0

3.2033 3.1140 2.9453 2.8262 2.6384 2.4946 2.2835 4.1936 4.0793 3.8642 3.7123 3.4710 3.2834 2.9847

(0.25 ≤ h/r ≤ 2.0) D1/20 D1/15 D1/10 D1/8 D1/6 D1/5 D1/4 D1/20 D1/15 D1/10 D1/8 D1/6 D1/5 D1/4

0.25

1.5

1.75

2.0

(2 ≤ h/r ≤ 20.0) D1/20 D1/15 D1/10 D1/8 D1/6 D1/5 D1/4 D1/20 D1/15 D1/10 D1/8 D1/6 D1/5 D1/4

3.0

10

CRC_DK8307_CH002.indd 34

12

14

16

18

10/29/2008 3:48:09 PM

Ultrasonic Horns, Couplers, and Tools

35

The most practical diameter ratio for the double-cylinder horn falls within D1 ≤ 2D2, where D1 is the diameter of the larger segment of the horn and D2 the diameter of the smaller segment of the horn, each segment being λ/4 in length, neglecting the fillet. 2.3.2.2

Step at a Position Other than a Node

Stepped horns may be of any configuration or combination of configurations either in cross section or in longitudinal directions. Common practice in designing stepped horns is to locate the step at a velocity node. This practice provides a location for mounting the horn with minimum loss to supporting structures. In addition, the amplification of displacement and velocity is maximum when the step is located at a nodal position. In the half-wave horn, the sectors before and after the nodal position are quarter-wavelengths. There may be instances that require the step to occur away from the nodal position or, with a similar effect, a tool may be attached to the end of a horn. In the latter case, the tool may consist of a material entirely different from that of the horn. Under these circumstances, the design of the unit and the determination of the amplification factor involve matching the mechanical impedances of the adjoining sectors at the junction. Matching of elements is covered later in this chapter. However, at this point it seems appropriate to illustrate the effects of locating the step of a double cylinder at a position away from the node. The condition is illustrated in Figure 2.3. In Figure 2.3 1. xn is the distance from the larger free end to the nodal position. 2. 3. 4. 5. 6.

xh is the distance from the larger free end of the horn. −xt is the distance from the smaller free end of the horn. A represents the position at the step. Distance xh to A (designated xhA) may be greater than or less than λ/4. Distance −x to A (designated xtA) may be less than or greater than λ/4.

At A, ZhA = ZtA, that is, in order for the horn to perform properly at the design frequency, the impedances of the two segments at A must be matched. 
x  jYS1 sin  hA   c  ZhA  
x hA  c cos   c 

(2.19)

A

D1

D2

xn xhA

xtA

FIGURE 2.3 Double cylinder with step located away from velocity node.

CRC_DK8307_CH002.indd 35

10/29/2008 3:48:09 PM

Ultrasonics: Data, Equations, and Their Practical Uses

36


x  jYS 2 sin  tA   c  ZtA  
x  c cos  tA   c 

(2.19a)

or equating ZhA and ZtA and simplifying, S 
x  
x  tan  hA   2 tan  tA   c   c  S1

(2.20)

Therefore x tA 

 S1 c 
x   tan1  tan  hA    c 
 S2

(2.21)

In most cases xhA is defined and the horn is matched by a length determined by using Equation 2.21. xtA ≄ λ/4 ≄ xhA xtA + λ hA > λ/2 where λ = c/f. The displacement or velocity amplification factor is determined as follows: ωxt t = 2 cos ( −___ c ) cos ωt ωxh h = 1 cos ( ____ c ) cos ωt The momentum of an element, dx located between (x hA − dx) and xhA is ωhρS1dx and between (xhA + dx) it is ωtρS2dx Equating momentums at A and simplifying, hS1 = tS2 −ωxtA ωxhA _____ S11cos ( c ) = S22 cos (______ c ) from which the amplification factor is 2  1


x  S1 cos  hA   c   
x tA  S 2 cos   c 

(2.22)

When the step is located away from the node, the stresses in elements adjacent to the step are in phase. The stress in each of the two elements is found by applying Equation 2.6a to the appropriate section.

CRC_DK8307_CH002.indd 36

10/29/2008 3:48:09 PM

Ultrasonic Horns, Couplers, and Tools 2.3.3

37

Cross-Sections Other than Circular

The cross section of a slender longitudinally resonant member may include any of a wide selection of geometries such as triangles, rectangles, trapezoids, pentagons, hexagons, heptagons, octagons, or any other polygon or design shape. They may contain linear grooves or cavities (e.g., hollow cylinders). A single horn may combine two or more crosssectional features. The guidelines include: 1. Maintaining a stiffness that inhibits flexural modes at the operating frequency 2. Keeping lateral dimensions sufficiently small to qualify as slender (large-area horns are presented in Section 2.8) 3. Having the cross-sectional area for any configuration be continuous as a function of x for its applicable distance The equations governing displacement, velocity, acceleration, stress, and impedance are the same for these structures as for corresponding horn and bar types of circular cross section. The center of gravity of the cross sections of all coupled horns or elements and the axes of coupling bolts should coincide. 2.3.4

Exponentially Tapered Horn [2]

The function describing the longitudinal geometry of the exponential horn (Figure 2.4) is S = Soe−γx where γ is a taper factor [= (1/ℓ)ln(So/Sℓ)] and So the cross-sectional area at the large end of the horn. The horn Equation 1.55a converts to d2v dv
2 v0 
2 dx dx c 2

(2.23)

based upon the simple harmonic motion of a linearly active horn, that is v

d  j
t dt

D1 D2

x λ/2 FIGURE 2.4 Exponentially tapered horn.

CRC_DK8307_CH002.indd 37

10/29/2008 3:48:09 PM

Ultrasonics: Data, Equations, and Their Practical Uses

38 or

 = −j(v/ω) 2.3.4.1

Velocity Distribution, v

The solution to Equation 2.23 for a half-wave horn is    2  2 2 2
 
 v  Vo cos 2  x  sin 2  x  ex/2 4  c 4 c 
2 2 2 2   4 c  

(2.24)


x c′
x  x/2   v  Vo cos sin e c′ 2
c′  

(2.24a)

or

where

2 2     c′  c 2 4

1/ 2

from which c′ 

c  2c 2 1 4
2

 ln(S o/S
)   c 1
   2

(2.25) 2

Vℓ = −Vo(So/Sℓ)1/2 = jωℓ For a round horn Vℓ = −Vo(Do/Dℓ) where Do is the diameter of the cross section at the large end and Dℓ the diameter at the smaller end. 2.3.4.2

Particle Displacement Distribution,


Distribution of particle displacement for the exponentially tapered horn is
x c′
x  x/2  sin   o  cos  e  c′ 2
c′ 

(2.26)


o (S o/S
)1/2

(2.27)

ℓ = −o(Do/Dℓ)

(2.27a)

For a round horn

CRC_DK8307_CH002.indd 38

10/29/2008 3:48:09 PM

Ultrasonic Horns, Couplers, and Tools

39

The displacement and velocity node occurs where tan

2.3.4.3


x 2
 c′ c′

(2.28)

Particle Acceleration Distribution, a

Particle acceleration is given by a

dv 
2 dt

or
x c′
x  x/2  a 
2ο  cos  sin e c′ 2
c′   2.3.4.4

(2.29)

Stress Distribution, s

The stress distribution, s, is s

jY dv Y 
 2c′  x/2
x  j V0 
e sin
dx
 c′ 4
 c′

(2.30)

Maximum stress occurs when ds/dx = 0, that is, when tan

2.3.4.5

2

x  c′ c ′

(2.31)

Length of the Half-Wave Horn, ℓ

The length of the half-wave horn, ℓ, is
 2.3.4.6

c′ c′ 
2f

(2.32)

Mechanical Impedance at x

Mechanical impedance at x is Zm = sS/v or 
 2c ′  
x  jSY 
tan    c′   c′ 4
 Zm    c ′  
x  
1     tan   2
c′   

(2.33)

where S = So e−γx. 2.3.4.7

Exponentially Tapered Horn of Rectangular Cross Section and Constant Width (Width Small Compared with Wavelength)

The cross-sectional area of an exponentially tapered horn with rectangular cross-section and constant width is S = 2Wyoe−γx

CRC_DK8307_CH002.indd 39

10/29/2008 3:48:09 PM

Ultrasonics: Data, Equations, and Their Practical Uses

40

yo is half the height of the horn at x = 0 So/Sℓ = yo/yℓ

(2.34)

ℓ = −o(yo/yℓ)1/2

All equations for particle displacement, velocity and acceleration, stress, length and mechanical impedance applicable to round horns are also valid for the slender rectangular exponential horn. The same is true for any geometry in which the area changes exponentially with increasing x. To determine these quantities for any other geometry in terms of a dimension of a leg or element of the cross section, it is necessary to determine how that particular element varies with x by its relationship to the area. For example, Area of a circle is πr2 = πD2/4. Area of a square is W2, where W is the length of the side of the square. Area of a rectangle of sides A and B is A × B.

2.3.5

Slender Wedge-Shaped Horn [3]

Methods of analyzing the wedge-shaped horn are illustrated in Figure 2.5. The case in which the area decreases as x increases is presented here to satisfy the purposes of ultrasonic design. Here Sx = S1 − kx where k = (S1 − S2)/ℓ Substituting S  k x

1 S k  S x S1  kx

into the general horn equation gives d2v k dv
2 v0 
dS 2 S1  kx dx c 2

(2.35)

for the wedge-shaped horn at resonance. y2

y1 y2

y1

x

x x′

ᐉ

ᐉ

x′

FIGURE 2.5 Wedge-shaped horns.

CRC_DK8307_CH002.indd 40

10/29/2008 3:48:09 PM

Ultrasonic Horns, Couplers, and Tools

41

Equation 2.35 is a Bessel equation. It may be written in terms of v = f(S) rather than in terms of v = f(x), as d 2 v 1 dv

a2v  0 dS 2 S dS

(2.36)

v = Ajo(aS)

(2.37)


 S1
 v  AJ o  
x    c  S1  S 2  

(2.37a)

where a2 = ω2/k2c2. The solution to Equation 2.36 is

or

where Jo is a Bessel function of the first kind of order zero and A is a constant of integration. At x = 0 in Equation 2.37a 
 S1
  v  V1  AJ o     c  S1  S 2   Therefore A

V1


 S1
  Jo     c  S1  S 2  

and the velocity distribution, v, is 
 S1
 V1J o    x     c  S1  S 2 v 
 S1
  Jo     c  S1  S 2  

0x


(2.38)

The value of Jo[x] is symmetric about x = 0, as J o [x]  1 

2.3.5.1

x2 x4 x6 x8

 22 [1!]2 2 4 [2 !]2 26 [3 !]2 28 [4 !]2

(2.39)

Velocity and Displacement Nodes

The velocity and displacement nodes occur where 
 S1
  x   0 Jo      c  S1  S 2

CRC_DK8307_CH002.indd 41

(2.40)

10/29/2008 3:48:10 PM

Ultrasonics: Data, Equations, and Their Practical Uses

42 2.3.5.2

Stress Distribution, s

The stress distribution, s, is given by s j

Y v c 2 v j
x
x



S1   x  J1     dv
V1  c  S1  S 2  dx c 

S1   Jo     c  S1  S 2  

(2.41)

where J1 is a Bessel function of the first kind of order 1. Therefore, for the half-wave resonant horn with area decreasing with increasing x, the stress at any position x may be calculated using the following equation: 

S1   x  J1      c  S1  S 2 s  jcV1 

S1   Jo     c  S1  S 2  

(2.42)

The maximum stress occurs when 

S 1   x  J1      c  S1  S 2

(2.43)

is a maximum or a minimum. 2.3.5.3

Length of the Half-Wave Wedge-Shaped Horn, ℓ

The stress at the free ends of a half-wave, longitudinally resonant horn is zero. The displacement velocity at each of these ends is maximum. Therefore, the stress and velocity equations provide starting points for determining the length of the horn. The defined boundary conditions are S1, S2, ω, and c. The stress can be zero only at positions where J1[f(x)] = 0. The zeros of J1[f(x)] coincide with f(x) = 0, 3.8317, 7.0156, 10.1735, 13.3237, 16.4706, 19.6159, 22.7601, 25.9037, 29.0468, 32.1897, … At x = 0, the function, f(x), can only be zero when S2 is zero, that is, the horn is a full wedge with the end at ℓ being like a sharp knife-edge. The length of a slender wedge-shaped horn in longitudinal half-wave resonance is the difference in values of x between successive zeros of J1[f(x)] in Equation 2.42. For the full wedge, using Equation 2.42, S2 is zero at x = ℓ. Therefore, the length, ℓ, occurs between x = 0 and x = ℓ = 3.8317c/ω, that is, between the first two zeros of J1[f(x)].

CRC_DK8307_CH002.indd 42

10/29/2008 3:48:10 PM

Ultrasonic Horns, Couplers, and Tools

43

The half-wavelength, ℓ, of a slender horn with equal end areas (S2 = S1) is that of a uniform bar given by ℓ = πc/ω. These quantities identify the limits for half-wavelength of wedge-shaped horns. That is 1. For the full wedge, ℓ = 3.8317c/ω. 2. For the wedge in which S2 = S1, ℓ = πc/ω. All other half-wavelengths lie between these two limits. Obviously only specific ratios of S1/S2 are easily fitted into the relationships stated previously. These specific ratios of S1/S2 may be determined as follows: 1. 2. 3. 4.

At x = 0, J1[f(xn)] = 0. At x = ℓ, J1[f(xn+1)] = 0. Dividing 1 by 2, S1/S2 = f(xn)/f(xn+1) = C. Substituting C into f(xn).

Using this method to determine the ratios that match the second and the third zeroes, the third and the fourth zeroes, the fourth and the fifth zeroes, the fifth and the sixth zeroes, and the sixth and the seventh zeroes, the following corresponding sequence of lengths are obtained. There is generally no need to evaluate area ratios smaller than those given in Table 2.2, because (a) they obviously approach the values of a uniform bar and (b) little increase in amplitude of vibration can be gained by smaller ratios. Wedge-shaped horns with area ratios falling between 1.5 and infinity are the most likely to find use in power applications of ultrasonic energy. As a shortcut to design, the curve of Figure 2.6 can be used to determine reasonably accurate lengths of slender wedge-shaped horns.  C  1 c
 f(x n
1 )  C 
2.3.5.4

(2.44)

Impedance of the Wedge-Shaped Horn

The impedance at x of the half-wave resonant wedge-shaped horn is given by Zx = Fx/Vx: 

S1    S
 S1x
S 2x  J1    x   1 
    c  S1  S 2 Zx  jc 
 S
 S1x
S 2x   Jo   1   S1  S 2  c  

(2.45)

TABLE 2.2 Lengths of Wedge-Shaped Half-Wave Resonant Horns Corresponding to Specific End Area Ratios f(xn) f(xn+1) S1/S2 = f(xn)/f(xn+1) S2/S1 ℓ

CRC_DK8307_CH002.indd 43

3.8317 0 ∞ 0 3.8317c/ω

7.0156 3.8317 1.831 0.54617 3.1840c/ω

10.1735 7.0156 1.450 0.6896 3.1573c/ω

13.3237 10.1735 1.310 0.7636 3.1529c/ω

16.4706 13.3237 1.236 0.8091 3.1449c/ω

10/29/2008 3:48:10 PM

Ultrasonics: Data, Equations, and Their Practical Uses

44 3.80 3.70

3.50

3.30 π 3.10

0

0.2

0.4

0.6

0.8

1.0

S1/S2 FIGURE 2.6 Wedge length as a function of taper.

2.3.5.5

Amplification Factor for Wedge-Shaped Horns

The amplification factor for wedge-shaped horns is the ratio of velocity potentials at the ends (x = 0, ℓ). From Equation 2.38, at x = 0, v = V1 = j
o At x = ℓ, 
 S 2
  V1J o     c  S1  S 2   v  V2  
S1
 Jo    c S1  S 2 

(2.46)


S 2
 Jo   V2 
 c (S1  S 2 )    V1 o 
S1
 Jo    c (S 1  S 2 ) 

(2.47)

The amplification factor is

2.3.6

Conically Tapered Horns [4]

Design equations are given for conically tapered horns in which (a) the area increases with increasing x (Figure 2.7a) and (b) the area decreases with increasing x (Figure 2.7b). There are certain advantages and also disadvantages to considering the analysis of conical horns for power applications by either approach.

CRC_DK8307_CH002.indd 44

10/29/2008 3:48:10 PM

Ultrasonic Horns, Couplers, and Tools

45 y2

y1

y1

x

x

λ/2

λ/2

x′

y2

(a)

x′

(b)

FIGURE 2.7 Conically tapered horns.

2.3.6.1

Area Increasing with x

The area as a function of x is Sx 

   D 2  D1    x
D1  4 


2

The derivative of Sx with respect to x is dS x   2 [(D 2  D1 )x

D1 ][D 2  D1 ] dx 2
Assuming simple harmonic motion for displacement at resonance and substituting for S, the horn equation for the cone with positive taper is d 2 v 2(D 2  D1 ) dv
2 v0

dx 2 [f(
, D, x)] dx c 2

(2.48)

The general solution to Equation 2.48 is v

D 2  D1 
 [f(
, D, x)] 
 [f(
, D, x)]  A cos  
B sin    c  ( D 2  D1 )  c  (D 2  D1 )  [f(
, D, x)] 

(2.49)

where f(ℓ, D, x) = D1ℓ + (D2 − D1)x = ℓD1 − (D1 − D2)x The constants A and B are determined by applying the following boundary conditions: at x  0, v  V1

at x 
, v  V2

at x  0,


dv  0, 0 dx

from which A

CRC_DK8307_CH002.indd 45





D 2   c
D1V1 
D1  D 2  D1  sin   cos   
D 1
(D 2  D1 )  c  c D 2  D1   c D 2  D1  

10/29/2008 3:48:10 PM

Ultrasonics: Data, Equations, and Their Practical Uses

46

Substituting these values in Equation 2.49 and simplifying leads to v


D1V1 
x  c(D 2  D1 ) 
x 
 sin   j
 cos  f(
, D, x)  c c  

D 1 

(2.50)

The equation for the full cone, where D1 = 0 is determined by writing v as a function of (ℓ − x) which leads to v

V2 x


c
  
cos c (
 x) 
sin c (
 x)

(2.50a)

The amplification factor (V1/V2 = 1/2) is determined by noting that v = V2 at x = ℓ, so that B





c
D1V1  (D 2  D1 ) D1 
D1  
sin  cos    
(D 2  D1 ) 
D1  c D 2  D1  c  c D 2  D1  

giving V2 

D1V1 D2

 V1   D1  V    D  2 2



c ( D 2  D 1 )


sin   j
 2 cos c

D c  1  

c ( D 2  D 1 )

  sin  cos c

c 
D 1 

1



(2.51)

1 2

Note: Equation 2.51 gives an infinite (or unrealistic) value for the amplification factor of a full cone, that is, where D1 = 0. A practical approach is to use very small real dimensions for the pointed end of the horn, which will provide satisfactory working values for amplification factor. The velocity node occurs where tan


x

D 1    c c  D 2  D1 

(2.52)

For the full cone (D1 = 0), the velocity node occurs where tan




(
 x )  c c

(2.52a)

The length of the horn is determined by the fact that, at x=ℓ

dv/dx = 0

from which tan

CRC_DK8307_CH002.indd 46


c
(D 2  D1 )2

 2 2 c

D1D 2
c 2 (D 2  D1 )2

(2.53)

10/29/2008 3:48:10 PM

Ultrasonic Horns, Couplers, and Tools

47

and for the full cone (D1 = 0) tan





 c c

(2.53a)

The wavelength of any tapered bar at resonance is greater than that for a uniform bar. For this reason, ℓ ≠ c/2f, where c is the velocity of sound in a thin, uniform bar unless D1 = D2 (a uniform bar). If ℓ = λ/2 = c/2f, ωℓ/c would equal π, sin(ωℓ/c) = 0, and cos(ωℓ/x) would equal −1. Under these circumstances, dv/dx could equal zero when x = ℓ only if V1 = 0 or D1 = D2. The validity of determining the length of the cone by using Equations 2.53 and 2.53a might be better shown by writing v in terms of f(ℓ − x) as follows:  (D  D1 )V2  
D 2  c

v 2 cos (
 x)  sin (
 x)   c c
 [f(
, D, x)]   D 2  D1 

(2.50b)

and applying the boundary conditions that at x = 0, dv/dx = 0. The results are the same, that is, the solutions lead to Equations 2.53 and 2.53a. Stress distribution, s, is given by s j

Y v
x

s j

Y V1(D 2  D1 )2
[f(
, D, x)]2

 
x 
2 (
D1 )[f(
, D, x)]
c 2 (D 2  D1 )2 
 sin (  x )
x cos   c
c(D 2  D1 )2 c    

(2.54)

For the full cone, D1 = 0, and s j

YV1
x 2


x c
x   x cos c 
sin c 

(2.54a)

ds/dx = 0  j


2 (
D1 )[f(
, D, x)]2
2c 2 (D 2  D1 )3 x
x YV1 cos ( D 2  D 1 )2  2 2 3 c D  D f (
, D, x )] c
( ) [ 2 1 

2{
2
D1[f(
, D, x)]
c 2 (D 2  D)2 }

2 [f(
, D, x)]2
x 
sin  3
c(D 2  D1 )[f(
, D, x)] c 

(2.55)

Therefore, at the position of maximum stress tan


2
D1[f(
, D, x)]2
2c 2 (D 2  D1 )3 x
x
 c c 2(D 2  D1 ){(
2
D1[f(
, D, x)]
c 2 (D 2  D1 )2 )

2 [f(
, D, x)]2 }

(2.56)

For the full cone, D1 = 0, and the point of maximum stress is where tan

CRC_DK8307_CH002.indd 47


x 2
cx  2 c 2c 
2x 2

(2.56a)

10/29/2008 3:48:11 PM

Ultrasonics: Data, Equations, and Their Practical Uses

48

The mechanical impedance, Zm, of the conical horn, area increasing with x, is Zm  sS/v 

jY[f(
, D, x)] c
x   4
2c 
D1
(D 2  D1 )tan c 
 c2
x  c   (D 2  D1 )2 x  [(
D1 )2
2 (D 2  D1 )2
(
D1 )(D 2  D1 )x] tan
c  


(2.57)

For the full cone, D1 = 0, and Zm j

2.3.6.2

D 22Yx  c
x   x cot 2  4c

c 

(2.57a)

Area Decreasing with Increasing x

The area as a function of x is Sx 

 2  D x  2 [
D1  (D1  D 2 )x]2 4 4


(2.58)

The horn equation for this case is 2(D1  D 2 ) d2v dv
2 v0 
2 dx
D1  (D1  D 2 )x dx c 2

(2.59)

assuming that the solution is simple harmonic. See Figure (2.7b). The solution to Equation 2.59 for 0 ≤ x ≤ λ/2 is v

(D1  D 2 )V1 
D1
x c
x  cos  sin    j
 c c 
D1  (D1  D 2 )x  (D1  D 2 )


(2.60)

For the full cone, D2 = 0, and v


V1  c
x
x   j
 cos  sin  c c  (
 x ) 



(2.60a)

The amplification factor is V2  2 D1

c(D1  D 2 )

  cos  sin V1 1 D 2 c c

D 2

(2.61)

The velocity node occurs where tan

CRC_DK8307_CH002.indd 48


x

D 1  c c(D1  D 2 )

(2.62)

10/29/2008 3:48:11 PM

Ultrasonic Horns, Couplers, and Tools

49

For the full cone, the velocity node occurs where tan


x

 c c

(2.62a)

The length, ℓ, is determined by the fact that, at x = ℓ, dv/dx = 0. (D1  D 2 )2 V1 dv  dx [
D1  (D1  D 2 )x]2 
x  c 2 (D1  D 2 )2

2
D1[
D1  (D1  D 2 )x] 
x   x cos  sin   2 c
c(D1  D 2 ) c    

(2.63)

Letting x = ℓ (in Equation 2.63) leads to tan



c(D1  D 2 )2

 2 c c (D1  D 2 )2

2
2D1D 2

(2.64)





 c c

(2.64a)

Y dv
dx

(2.65)

For the full cone (D2 = 0), tan The stress-distribution, s, is given by s j s j

Y(D1  D 2 )2 V1
[
D1  (D1  D 2 )x]2


x
2
D1[
D1  (D1  D 2 )x]
c 2 (D1  D 2 )2 
x  x cos   sin 2 c
c(D  D ) c 1 2   For the full cone, s j

YV1
(
 x)2


x c 2

2
(
 x)
x   x cos  sin  c
c c  

(2.65a)

At the position of maximum stress, ds/dx = 0, from which tan

 2c 2 (D  D 2 )3 x 
2
D1[
D1  (D1  D 2 )x]2
x
   2 2 21  2 2 2 2 c c  {
[
D1  (D1  D 2 ) x ]
c (D1  D 2 ) }(D1  D 2 ) 

(2.66)

For the full cone, D2 = 0, the point of maximum stress occurs where tan

CRC_DK8307_CH002.indd 49


x
2c 2x 
2
(
 x)2  c c
2 (
2  x 2 )
c 2

(2.66a)

10/29/2008 3:48:11 PM

Ultrasonics: Data, Equations, and Their Practical Uses

50

The mechanical impedance, Zm, is Zm = sS/v or Zm  j

Y(D1  D 2 )2 [f(
, D, x)] 4
2




D1[
D1  (D1  D 2 )x] c 2 
x 
tan x    c ( D 1  D 2 )2 c 
2     c
x  
D1  (D1  D 2 )tan c
   

(2.67)

For the full cone, D2 = 0, and the mechanical impedance is 
 c2 
x  x x

(
 )   tan  2 2 YD1 (
 x)  c c 
 Zm  j   c
x 4

2  
 tan
c   2.3.7

(2.67a)

Catenoidal Horns

The equation of the catenary (Figure 2.8) is a x y  (ex/a
ex/a )  a cosh a 2

(2.68)

a kx y  (ekx/a
ekx/a )  a cosh 2 a

(2.68a)

If a = r1 = D1/2 corresponds to the radius at x = 0, the cross-sectional area at x is S  y 2 

r12 2kx/r1 kx (e
2
e2kx/r1 )  S1 cosh r1 4

(2.69)

Differentiating Equation 2.69 with respect to x gives dS 2k kx kx S1 cosh  sinh dx r1 r1 r1

S2

S1

x ᐉ FIGURE 2.8 Catenoidal horn.

CRC_DK8307_CH002.indd 50

10/29/2008 3:48:11 PM

Ultrasonic Horns, Couplers, and Tools

51

Therefore, the equation for the catenoidal horn (Figure 2.8) is d 2 v  2k kx  dv
2
 tanh 
v0 2 dx r1  dx c 2  r1

(2.70)

assuming simple harmonic motion at resonance. The general solution to Equation 2.70 is v

1 (A cos k ′x
B sin k ′x) r1 cosh(kx/r1 )

(2.71)

where k′ 


2 k 2  2 c2 r1

and a k  cosh1(D 2/D1 )
At x = 0, dv/dx = 0, and v = V1. Therefore A = r1V1 and B=0 Then v

V1 cos(k ′x)  j
 kx cosh r1

(2.71a)

The displacement velocity, V2, at x = ℓ is V2 

V1 cos(k ′
) k
cosh r1

(2.72)

The amplification factor is V1 r2 D  cos k ′
 2 cos k ′
V2 r1 D1

(2.73)

The length, ℓ, of a half-wave catenoidal horn is 
  k′ 1
2  2 2 c


CRC_DK8307_CH002.indd 51

  r  c   2
cosh1  2   2 ω  r1      1 r2 cosh r  1  

2

(2.74)

10/29/2008 3:48:11 PM

Ultrasonics: Data, Equations, and Their Practical Uses

52

The velocity and displacement node occurs where cos k′x = 0, or x

  2k ′



(2.75)


4k 2 2 2  2 c D1 2

The stress along the catenoidal horn is given by s j

YV1 cos k ′x  Y V kx   j
ak ′ tan k ′x  tanh 
x
a cosh(kx/a) a

(2.76)

At the point of maximum stress, sm, ds/dx = 0, from which tan

 1
r12k 2
2r12 x kx tanh
r1k ′ tan k ′x   2  r1  r1 2c 2 

(2.77)

1 dS 2x  2 S dx x
b2 The mechanical impedance, Zm, at x is Zm 

2.3.8

 sS YS  kx kx  j 1  tanh
r1k ′ tan k ′x cosh 2 v r1 r1
r1  

(2.78)

Hyperbolic Horns

The hyperbola is defined as a set of points so located in a plane that the difference of its distances from two fixed points (foci) is a given positive constant denoted 2a (Figure 2.9). When the vertices and the focal points lie on the y-axis and its center is located at (0, 0), the contour of the hyperbolic horn of circular cross section, with its axis coinciding with the x-axis (Figure 2.10), may be described by Equation 2.79. y2 x2  2 1 a2 b

(2.79)

y

D2 = 2r2 0

x

−y

D1 = 2r1

FIGURE 2.9 The hyperbola.

FIGURE 2.10 Hyperbolic horn.

CRC_DK8307_CH002.indd 52

ᐉ

10/29/2008 3:48:12 PM

Ultrasonic Horns, Couplers, and Tools

53

The ratios ±a/b are the slopes of the asymptotes to the hyperbola. In Equation 2.79, y is the radius of the horn at any position from x = 0 to x = ℓ. Thus, at x = 0, y = ±a, and a = r1, the radius at the small end of the horn. At x = ℓ and y = r2. The area of the horn at position x is  x2  S  y 2  r 2  a 2  2
1 b 

(2.80)

dS 2a 2x  dx b2

(2.81)

Applying Equation 2.81 to the general equation for horns vibrating in simple harmonic motion at half-wave resonance, Equation 1.55a, leads to d2v 2x dv
2 v0
2
2 dx x
b2 dx c 2

(2.82)

as the equation describing the motion of a resonant, half-wave hyperbolic horn. The solution to Equation 2.82 is of the form v  eKx [C1 cos Ax
C 2 sin Ax]

(2.83)

where K, C1, C2, and A are constants, the values of which are determined by applying the boundary conditions: 1. At x = 0, v = C1 = V1 (the velocity amplitude at x = 0) dv/dx = 0 2. At x = ℓ, v = V2 (the velocity amplitude at x = ℓ) dv/dx = 0 where A = ω/c′ and c′ is the apparent velocity of sound in the horn. c′ differs from the bar velocity, c, due to the taper of the horn. The negative exponent applies because principles of momentum determine that the velocity amplitude at the small end of the horn is greater than that at the large end. After applying these boundary conditions, Equation 2.83 may be written K   v  V1eKx cos Ax
sin Ax  A  

(2.83a)

The first derivative of Equation 2.83a is dv K    V1eKx [ A sin Ax
K cos Ax]  V1KeKx cos Ax
sin Ax  dx A   K2   sin Ax  V1eKx  A
A  

(2.84)

and the second derivative is d2v K 2  Kx  V A e [A cos Ax  K sin Ax]  
1   dx 2 A 

CRC_DK8307_CH002.indd 53

(2.85)

10/29/2008 3:48:12 PM

Ultrasonics: Data, Equations, and Their Practical Uses

54 From these equations,

A2 


2
2 2  K  c2 c′ 2

(2.86)

2
(
2
b 2 )

(2.87)

and K

The equation describing the velocity distribution is determined by using Equations 2.84 through 2.87, to obtain

) V1 cos
x
2 (r22  r12 ) sin
x 

(

 (
2/c 2 )(ω 2/c ′ 2 ) x

ve

c′



r22

c′ 

(2.88)

where c′ 

c


 2r24
4(r22  r12 )2 r24

and the length, ℓ, is given by c′ c′ = ___ ℓ = __ ω 2f

(2.89)

and b2 

r12
2  r12 )

(r22

(2.90)

The constant, b, must be known to plot the contour of the horn. The velocity as a function of displacement, , is v

d  j
 dt

(2.91)

The velocity and displacement node occurs where tan

ωx πr22  c′ 2 (r22 − r12 )

(2.92)

At the node, x > ℓ/2 from the small end of the horn. The stress distribution is s j

YV1 ( e


)

(
2/c 2 )  (
2/c ′ 2 ) x

  2
(r22  r12 )
2
2 
x   2  cos  2 2 c c′  c′   r2 c′ 
2 (r22  r12 )
2
2 
 2 r22 c2 c′  c′

CRC_DK8307_CH002.indd 54


x   sin  c′  

(2.93)

10/29/2008 3:48:12 PM

Ultrasonic Horns, Couplers, and Tools

55

where Y is Young’s modulus of elasticity for the material of the horn. Maximum stress occurs where tan

r22c 2  4 (r22  r12 ) c c′ 2  c 2
x  c′ 2r22c c′ 2  c 2
2(c′ 2  2c 2 ) (r22  r12 )

(2.94)

At the position of maximum stress, x < ℓ/2 from the small end of the horn. The mechanical impedance, which is necessary in matching unlike segments for a resonant system, is   x2 ja 2Y  2
1 sS  b Zm   2 2  v r r  2 ( )
x 
x
2 2 1 sin  cos c′ r2 c′  

(2.95)

 2 (r 2  r 2 ) 1 
x  1 1 
x  1 2 (r22  r12 ) 1   2 2 1  2  2  cos  2  sin 
 2 2 c c′  c′  c′ r2 c c′  c′   r2 c′  where s is the stress as a function of x, S the cross-sectional area as a function of x, and v the velocity as a function of x. The ratio of displacements at the ends of the horn is 1 v1   e( 2 v2

2.4

)

(
2/c 2 ) (
2/c ′ 2 )


(2.96)

Horn Design and Performance Factors

Several factors enter into the choice of materials and the design of ultrasonic horns. The basic considerations fall under two major headings: 1. Application objectives and requirements 2. Power source flexibility and capability Application objectives and requirements determine the specifications for the ultrasonic system and the environment in which it will operate. Power source flexibility and capability determine what the designer has at his disposal (whether commercially available, designable, or readily on hand) for driving the horn to perform the requirements of the project. The primary factors affecting horn design and performance include: 1. 2. 3. 4.

Effects related to Poisson’s ratio Effects related to internal losses and thermal conductivity Effects related to loading variations Effects related to design anomalies

CRC_DK8307_CH002.indd 55

10/29/2008 3:48:12 PM

Ultrasonics: Data, Equations, and Their Practical Uses

56

The primary factor to consider in designing ultrasonic horns for power applications is velocity of sound in the material of the horn. Velocity of sound in solid media is variously defined elsewhere in this book as follows: 1. Bar velocity of sound. The rate at which an acoustic wave in which the motion is strictly rarefaction and compression in the axial direction travels along a thin homogeneous bar in which the lateral dimensions are very small compared to the wavelength of the disturbance. This velocity is given by co 

Y 

(1.36)

where Y is Young’s modulus of elasticity and ρ is the density of the material of the bar. This is the basic quantity, velocity of sound, c, appearing in the solutions to the horn equation of Section 2.3.1 through 2.3.7. Its derivation does not take into account the effects of Poisson’s ratio on the velocity of sound. 2. Bulk velocity of sound. The rate at which an acoustic wave, compressional in the direction of propagation, travels through an infinite, solid medium having lateral dimensions that are much larger than the length of the wave. This velocity is cB  co

1 (1
)(1  2 )

(1.56)

where σ is Poisson’s ratio of the medium. The bulk velocity, cB, is always >co due to the effects of Poisson’s ratio. 3. Shear velocity of sound. The rate at which an acoustic wave consisting of shear, or transverse, motion only travels through a medium. This velocity is cs  co

1   2(1
) 

(1.57)

where µ is shear modulus of elasticity in the medium. 2.4.1

Effects of Poisson’s Ratio on Performance

The equations of Section 2.3 are useful for designing horns if the lateral dimensions of the horns and tools are small compared with the wavelength. The slenderness of the horns is considered to be sufficient reason to ignore the effects of Poisson’s ratio. A stress applied along the axis of a rod produces a strain parallel to the axis and a corresponding strain of opposite sign in a direction normal to the axis. Poisson’s ratio is defined as the ratio of the normal strain to the axial strain and it is a characteristic of all solid metals. The effect is to divide the total kinetic energy of the vibrating system between the desired longitudinal mode and a lateral component of motion. The effects may range from (a) that attributable to the inertia of the mass associated with the lateral dimensions of the horn to (b) actual spurious structural resonances that may coincide, or nearly coincide, with the resonance frequency of the intended mode of vibration and thus disrupt the performance of the horn. The latter condition is a factor in the design of wide and large area horns considered in Section 2.8. Lord Rayleigh [5] analyzed theoretically the effect of Poisson’s ratio on the period of vibration of a longitudinally resonant solid cylinder or wire. The analysis is based upon

CRC_DK8307_CH002.indd 56

10/29/2008 3:48:12 PM

Ultrasonic Horns, Couplers, and Tools

57

comparing the total kinetic energy of the system due to the longitudinal displacement plus that due to the lateral displacement caused by Poisson’s ratio with the kinetic energy of the longitudinal motion alone. The effect of the lateral inertia on the period of vibration, T, is equivalent to adding a mass to the end of a slender bar if Poisson’s ratio can be ignored: it increases the period of vibration. Rayleigh assumes a simplified lateral function of displacement but the correction is valid for cylindrical rods to ratios of diameter to wavelength of ∼0.4, according to Derks [6]. Rayleigh defined the kinetic energy associated with the longitudinal component of displacement as 2

kinetic energy  U 

S
 d  dx 2 ∫o  dt 

(2.97)

where ρ is the density of the medium, S the cross-sectional area of the transmission line,  the displacement at x, d/dt the particle velocity at x, and x the direction of propagation. For a round wire or bar, the corresponding kinetic energy associated with radial expansions and contractions due to Poisson’s ratio, σ, is 2

dη  S 2r 2 U  ρ∫ ∫   dxr dr  0 0  dt  4


r




2

d 2  ∫0  dt dx  dx

(2.98)

where η is the lateral displacement of the particle at distance r from the axis. Particle motion in the round, uniform, homogeneous bar, resonating in a free–free longitudinal mode is described by   o cos


x sin
t c

(2.99)

d
x 
o cos cos
t dt c and d 2 
2o
x  sin cos
t dt dx c c

(2.100)

Using these quantities in Equations 2.99 and 2.100 gives
2S
o2 sin 2
t 4

(2.101)


4S 2r 2
o2 cos 2
t 8c 2

(2.102)

U and U 

The total absolute value of kinetic energy is U
U 

CRC_DK8307_CH002.indd 57


2S
o2 4


2 2r 2   1
2c 2   

(2.103)

10/29/2008 3:48:13 PM

Ultrasonics: Data, Equations, and Their Practical Uses

58

For the cylindrical half-wave resonator ω 2 2d 2 
2 2r 2    U
U  U 1
 U 1
2  2c  8c 2   

(2.104)

In terms of Y and c, the total kinetic energy is U
U 

S
2Y
ο2 
2 2r 2 
1  4c 2  2c 2 

(2.105)

For the uniform bar in longitudinal half-wave resonance c 2  ( f )2  Y/ρ 

S
2Y
o2 4
2  2 T 4U

(2.106)

Therefore, T2 ∝ U T2 c* 2 U   c2 U
U (T
T)2

(2.107)

where T is the period of vibration when the effect of Poisson’s ratio, σ, is ignored and T + δT is the period of vibration corresponding to taking σ into account. The corrected value of velocity of sound in the uniform bar due to Poisson’s ratio is, therefore c* 2 

S
2Y
o2 4[U
U]

(2.108)

The ratio of corrected value due to Poisson’s ratio to bar value of velocity of sound is c* 2 1 1  
2 2r 2
2 2d 2 c2 1
1
2c 2 8c 2

(2.109)

As long as the wavelength of the propagating wave is large compared with the lateral dimensions of the rod, the effects of Poisson’s ratio on bar velocity are insignificant. By assuming that σ 2k2d2/8 is very small, Derks [6] linearized the Rayleigh correction to c* 1  1  2k 2d 2 c 16

(2.110)

where k = ω/c. 2.4.1.1

Corrections to Velocity of Sound in Tapered Horns

2.4.1.1.1 Exponential Horns The procedures of the previous section for correcting velocity of sound for effects due to Poisson’s ratio in uniform bars can be extended to other geometrical configurations. The procedure will be illustrated for exponential horns only. For slender horns, the corrections due to σ are very slight and are usually ignored, because other factors have greater

CRC_DK8307_CH002.indd 58

10/29/2008 3:48:13 PM

Ultrasonic Horns, Couplers, and Tools

59

influence on the performance of these horns. However, Poisson’s ratio cannot be ignored in the design of horns having at least one significantly large lateral dimension compared with a wavelength. Not only does it influence the velocity of sound in horns of large cross section, wide blade-types, and special designs such as flared or cup shaped horns, but it also is an important cause of spurious, deleterious modes of vibration in these designs. For the round, exponentially tapered horn
x c ′
x  x/2   sin e   o cos 2 c c′ 
′ 

(2.26)

The kinetic energy associated with longitudinal motion alone is 2

U ex 

S
 d    dx 2 ∫0  dt 

(2.111)

d
x c′
x  x/2   e 
o cos sin cos(
t ) dt c′ 2
c′   S = Soe−γx U ex 

S
2o2 2

r = roe−γx 2


x c ′
x  ∫ᒌ cos c′  2
sin c′  ex dx


c ′
x
x  2 c ′ 2
x  2
x
cos  cos sin sin 2 dx ∫ᒌ  2 c′
c′ c′ 4
c′  S
2o2
  2c ′ 2   o 1
 4 4
2  

U ex 

S o
2o2 2

(2.112)

(2.113)




(2.113a)

Comparing the exponential horn with the uniform bar of equal length (λ/2), diameter (do), and material T2 U c′ 2   2 2 Tex U ex c

(2.114)

where c′ is the velocity corresponding to an exponentially tapered horn, ignoring σ, which leads to U c ′ 2  2c ′ 2  2 
1 U ex c 4
2

(2.115)

When the lengths are identical and the maximum diameter of the exponential horn equals the diameter of the uniform bar, the lateral inertia of the uniform bar is greater than that of the exponentially tapered horn. The apparent velocity of sound (ignoring σ) in the tapered horn is greater than that of the uniform bar according to Equation 2.25, that is c′ 2 

CRC_DK8307_CH002.indd 59

c2  2c 2   1 4
2   

(2.25)

10/29/2008 3:48:13 PM

Ultrasonics: Data, Equations, and Their Practical Uses

60

The effect of Poisson’s ratio is determined from the relationship U ex c* 2  ex U ex
U ex c′ 2

(2.116)

where Uex is given by Equation 2.113a, and c′, the apparent velocity of sound in the exponential horn ignoring Poisson’s ratio, is given by Equation 2.25. S 2r 2 U ex  4

2




d 2  ∫o  dt dx  dx

S 2r 2 2   2c′
2  U ex  o o o 
 c′  4  4

2




∫o ex sin 2


x dx c′

d 2
x   2c′
2o  x/2 sin  
e dt dx c c′ 2 ′   2 c*ex  c′ 2

2
c 2   2c 2   2
c 2
 2ro 2
2 [1  e
] 1  4
2   

or U ex 

S oo2 2ro2
2 4c ′ 2

e
   2c′ 2
4
2   1      8  

(2.117)

The total kinetic energy is U ex
U ex 

S o
2o2
  2c′ 2  S oo2 2ro2
2
1
 4 4
2  4c ′ 2 

  2c′ 2
4
2   1  e
     8   

(2.118)

from which 2 c*ex U ex 2c′ 2
  c′ 2 U ex
U ex 2γc′ 2

2ro2
2 [1  e
]

(2.119)

where c*ex is the velocity of sound in an exponential horn corrected for Poisson’s ratio and c′ the velocity of sound in an exponential horn taking only the taper into consideration. In relation to the bar velocity of sound, c (which does not take σ into consideration), 2 c ex  c′ 2

1   2c 2 ]  
 [1  e ] 8
c 2

 2ro2 [4
2

1
 

(2.120)

and 2 c ex  c2  2c 2   2ro2  1  4
2   1
8
c 2  

CRC_DK8307_CH002.indd 60

1   4
2   2c 2  [1  e
]   2
  

(2.121)

10/29/2008 3:48:13 PM

Ultrasonic Horns, Couplers, and Tools 2.4.2

61

Effects of Losses on Horn Performance

Losses that affect horn performance fall into two general categories: 1. Those that are associated with the load against or into which the horn works 2. Internal damping 2.4.2.1

Losses Associated with the Load into which the Horn Works

This condition is illustrated in a very simplified manner by the equivalent circuit of Figure 2.11. The transformer, Tr, represents the multiple degrees of coupling between the horn and the variety of loads to which the horn might be subjected. In general, the impedance reflected by the load into the horn is ignored in designing a horn for commercial use. The load impedance may vary between (a) that of another member resonant at the resonance frequency of the driving horn, (b) that of a nonresonant member that can be coupled to the horn with modification in such a manner that the two resonate as a single unit (half-wave), and (c) that of a highly viscous load (tightly coupled horn with tip immersed in an extremely viscous liquid). Conditions described by case (c) may cause significant effects on the performance of the horn, such as increasing the dynamic period of the horn and causing excessive heating in high-power applications of ultrasonics. High internal losses produce similar effects, as discussed later. In most commercial applications of power ultrasonics involving treating high-loss types of loads the horn is either energized under no load condition and the energy stored is dissipated in the load as the horn is suddenly pressed against or into it or the damping aspect of the load impedance is insignificant as far as the performance of the horn is concerned. In either case, the horn is designed as a “no-load” item. Impedance matching of horns and elements is discussed in Section 2.5. 2.4.2.2

Internal Damping Factors

The design of horns for power applications of ultrasonic energy is based upon tabulated properties (elastic constants, densities, velocity of sound, fatigue limits) of selected metals. Chemical compatibility with materials to be treated and elastic properties as functions of temperature are often critical considerations in choosing materials. Thermal conductivity and Q affect the performance of the horn. High thermal conductivity is especially important to the performance of horns of large cross sections. When velocity of sound, c, in a bar material is determined dynamically, the result includes the effect of the internal loss factor, R m. It is this value of c that is used to design

Lh

Tr RL Ch

CL

LL

FIGURE 2.11 Equivalent circuit of load with losses.

CRC_DK8307_CH002.indd 61

10/29/2008 3:48:13 PM

Ultrasonics: Data, Equations, and Their Practical Uses

62

Rh

Lh

Ch

FIGURE 2.12 Equivalent circuit of a horn with internal losses.

a horn. The primary interests in Rm are (a) quality control: identification of materials of quality inferior to that expected of an alloy and (b) monitoring performance: discerning between incipient damage (fatigue) in operating horns and downward shifts in frequency due only to thermoelastic effects. Figure 2.12 is an equivalent circuit of a horn with internal losses (damping). The horn equation may be modified to account for internal damping by adding the term Rm/ρc2 as follows: 1 2 R m 2 1 S ξ 2
  0 c 2 t 2 c 2 xt S x x x 2

(2.122)

2 v 1 S v
R m v
2 j v0

x 2 S x x c 2 x c 2

(2.123)

or

assuming harmonic motion and homogeneous material (i.e., the intrinsic loss mechanisms are distributed uniformly throughout the volume of the horn). Equations 2.122 and 2.123 may be used wherever the narrow horn equations are applicable, such as (a) uniform bars or (b) tapered horns. The first step in solving either Equation 2.122 or 2.123 is to identify the type of horn and thereby determine the quantity (1/S)∂S/∂x. 2.4.2.2.1 Uniform Bars with Losses In a uniform bar, the taper is zero; therefore, dS/dx = 0, and Equation 2.123 may be written d2v
R m dv
2 v0 j
2 dx c 2 dx c 2

(2.123a)

for which the solution is  
R2  2 v  e j(
R mx/2c ) V1 cos  1
2m 2  x 4 c  c  j

CRC_DK8307_CH002.indd 62


Rm R2 R2   1
2m 2 sin  1
2m 2  x  2c 4 c 4 c   c

(2.124)

10/29/2008 3:48:14 PM

Ultrasonic Horns, Couplers, and Tools

63

Therefore, the effect of damping is to change the apparent velocity of sound to 1 c 2 2  2 R  2c 2
R m 1
2 m2 c

c′d  c

(2.125)

The stress in a half-wave uniform bar in which Rm ≠ 0 is s j j

Y dv
dx   Y j(
R m x/2c2 )  R3 R2  R2 e V1 j 3m 3 cos
1
2m 2  2m 2  1 sin  c 8  c  c 4  4 c    

(2.126)

R2
x 1
2m 2 c 4 c

(2.127)

where

 When Rm = 0 s  j

Y
x
x
x V1 sin  jcV1 sin  jc
m sin jy c c c c


S m c

Xsin


x sin(
t) c

(2.6a)

The complex impedance, Zmd, due to considering Rm ≠ 0 is Zmd = sS/v or   S Y   1 Zmd   o 2   4 4 2 2 2  2 2  2c   16 c  R m ( 4 c
R m ) tan 

(2.128)

3 2 2 4 2 2  c
j[R m
4 2c 2 (R m  4 2c 2 )] 4 2c 2
R m tan

 {4R m 4  R m [R m  16 4c 4 ]tan 2 }

which reduces to Zm j

SoY 
x  tan  jspc tan   c  c

(2.7)

when Rm = 0. 2.4.2.2.2 Exponentially Tapered Horns with Losses For the exponentially tapered horns, for which S = Soe−γx, Equation 2.123 becomes d2v 
R m  dv
2 j v0  

 dx 2  c 2  dx c 2

CRC_DK8307_CH002.indd 63

(2.123b)

10/29/2008 3:48:14 PM

Ultrasonics: Data, Equations, and Their Practical Uses

64 for which the solution is v  e(
j(
R m/c



2 ))( x/2 )

1/ 2  (c 2
j
R m )2 
 Vo cos 1   x c 4
2 2c 2  

2 1/ 2

c 2
j
R m 2 1/ 2

  c 2
j
R m   c 2  1     2
c   

sin

 c 2
j
R m  
 1    c 2
c  

   x   

(2.129)

The apparent phase velocity, cd′, of an exponentially tapered horn with losses is therefore 1/2

2   (c 2
j
R m )   c′d  c 1     2
ρc   

(2.130)

which converts to c′ 

c  2c 2 1 4
2

(2.25)

when Rm = 0. 2.4.2.2.3 Conically Tapered Horns with Losses For the conically tapered horn in which the area decreases with increasing x, Sx 

D x2   2 [
D1  (D1  D 2 )x]2 4 4


Taking the derivatives of S and inserting the function (1/S)∂S/∂x into Equation 2.123 gives  2v  2(D1  D 2 ) j
R m  v
2 

v0  2 c 2  x c 2 x  [
D1  (D1  D 2 )x]

(2.123c)

The solution to Equation 2.123c is of the form (D1  D 2 )V1e j(
R m/c )K1x [
D1  (D1  D 2 )x] 2

v


D1K 2
x c
x  cos  (1
K 3 )sin  
c  c  ( D1  D 2 )

(2.131)

The constants K1, K2, and K3 are determined by applying the boundary conditions when x = 0 where v = V1

CRC_DK8307_CH002.indd 64

10/29/2008 3:48:14 PM

Ultrasonic Horns, Couplers, and Tools

65 v 0 x  2v
2   2 V1 2 x c

and the condition described by the horn equation, Equation 2.123c. The final solution is  (D  D 2 )V1e( j
R m /c2 )x  
D1 j
R m
D 1 
x c 
x  v  1 cos  1
sin    2
 c 2c (D1  D 2 )  c   [
D1  (D1  D 2 )x]   (D1  D 2 )

(2.132)

Equation 2.132 reduces to the undamped solution, Equation 2.60, when Rm = 0. 2.4.2.2.4 The Meaning of Rm The quantity Rm represents a summation of mechanically related loss mechanisms within the horn material. The significant mechanisms include thermoelastic effects, interaction with thermal phonons, and dislocation damping. Plastic deformation and fatigue damage are irreversible effects associated with dislocation movement. Therefore, dislocations are probably the most significant of the internal loss mechanisms in horns used in power applications of ultrasonic energy. Fatigue strength (endurance limit) is of major importance in choosing materials for horns to be used in power applications of ultrasonic energy. In most cases, Rm is considered only intuitively in designing ultrasonic horns. For the materials most commonly used (titanium, aluminum alloys, steel alloys, monel), its effect is usually insignificant and has already entered into the measurement of bar velocity of sound. Heat generated by losses in the horn and the performance of the horn in manners related to Q (such as damping, resonance amplitudes, and bandwidth) are usually anticipated in the design by experience and available performance data. Occasionally, one may wish to measure Rm, and for this purpose, modern instrumentation provides relatively simple means. Such measurements are particularly useful for quality control of horn materials. Increased Rm and decreased velocity of sound (determined by accurate measurements of dimensions and half-wave resonance frequency) are indicators of a decrease in the quality of a horn material. The fact that ultrasonic horns typically vibrate in longitudinal half-wave resonance suggests logical methods of making the measurement. Because Rm represents the energy absorption losses within the material of a vibrating member, it must have a direct relationship to the rate at which the amplitude of displacement of the member in resonance decays. It also must affect the quality factor, Q, of the system. Thus, R m may be determined either (a) by measuring the damping factor of a freely suspended bar in longitudinal half-wave resonance or (b) by measuring the Q of this suspended bar. Q is also determined by the damping factor. The damping capacity is affected by the magnitude of strain, frequency of vibration, temperature, composition, grain size, heat treatment, aging, cold work, and state of magnetization (for ferroelectric materials) [7]. The bar is suspended in a nonconstrained manner, usually by means of extremely light wires. The bar is energized by means of a suitable noncontacting source (such as electromagnetic means). The amplitude of vibration is measured at one end by means of a noncontacting device (such as capacitance gage). Fiber optic techniques provide very accurate measurements of very low-amplitude displacements. Electronic instrumentation is provided for displaying and recording amplitude, time, and frequency (Figure 2.13).

CRC_DK8307_CH002.indd 65

10/29/2008 3:48:14 PM

Ultrasonics: Data, Equations, and Their Practical Uses

66

Amplitude, frequency, time recorder

Light wire suspensions Bar

Energizer

L/4

L/4 L

Noncontacting displacement gage (capacitance, fiber optic)

FIGURE 2.13 Electronic instrumentation for determining Q, R m, and damping factor of a resonating bar.

The relationship between Rm, Q, and damping factor may be explained by using the electrical equivalent tuned circuit consisting of an inductance, resistance, and capacitance in series. In the electrical circuit, the total voltage across the inductors is given by the equation Erms = Irms Ze = Irms(Re + jXL ) = Irms(Re + j
Le)

(2.133)

where Erms is the RMS voltage across the coil; Irms the RMS current through the coil; Re the electrical resistance across the coil, ohms; XL the inductive reactance of the coil at frequency, f, ohms; and Le the inductance of the coil. The Q of the coil in an electrical circuit is Q = XL/Re = tan θ

(2.134)

where θ is the phase angle between current and voltage. In the mechanical system, letting Force, F, correspond to voltage Ems Velocity, v, correspond to current, Irms Resistance, Rm, correspond to electrical resistance, Re Mechanical reactance, Xm, correspond to electrical reactance gives the correlation Q = Xm/Rm = tan θ = f/∆f

(Q ≥ 10)

(2.135)

The Q of the resonant bar is determined by measuring frequency at resonance, fo, the frequency above resonance, f2, and the frequency below resonance, f1, at which the power is one-half the power at the resonance peak. Because power is proportional to the square of the velocity, hence of the square of amplitude of displacement, the half power points occur at the frequencies at which the amplitude is 1/√ 2 (=0.707) times the amplitude at resonance. The Q is determined using Equation 2.135 when ∆f << fo. The damping factor is determined by measuring the decay rate of amplitude of vibration of a freely resonant bar (i.e., the logarithmic decrement, δ). The logarithmic decrement relates the peaks in the amplitude of vibration of successive cycles as the vibrations of the freely resonant bar die down. The relationship is as follows: oe(−t/T)δ = t

CRC_DK8307_CH002.indd 66

(2.136)

10/29/2008 3:48:14 PM

Ultrasonic Horns, Couplers, and Tools

67

where o is the amplitude peak from which the measurement is started, t the time measured from t = 0 to the peak amplitude at which the measurement is stopped, and T the period of a cycle of vibration. Thus, when t = T, oe−δ = 1 the amplitude relationship between successive peaks, o and 1. The Q is given by Q = /

(2.137)

Therefore, combining Equations 2.135 and 2.137, Q

Xm f   o   tan  R m ∇f 

(2.138)

Comparing δ of Equation 2.137 with α of Equation 1.21 and Equation 1.27 shows that δ = αT. The actual value of R m may be determined by comparing Equations 1.21 and 2.135. Equation 1.21 relates undamped angular frequency of a spring, ωo, and the damping factor, α, to the damped frequency, ωd. Equation 2.135 relates Rm, undamped velocity of sound, c, density, ρ, and cross-sectional area, S, to the damped velocity of sound, cd. Combining Equations 1.21 and 2.135
o


′d 
o2   2  1


2 2 Rm  2 2 2 2  S
o


(2.139)

from which, by squaring,  2 
o2 

R 2 2
o2  2 m2 2 2 2 R  S
1
2 m  S 2
o2
2

(2.140)

or    T 

R m S
 R m  fd S
fd′

(2.140a)

(2.141)

where fd′ is the frequency of the damped bar at resonance. Therefore Rm 

S
fd′ 

(2.141a)

where Sℓ is the volume of the bar. The dimensions of Rm are FTL−1.

CRC_DK8307_CH002.indd 67

10/29/2008 3:48:14 PM

Ultrasonics: Data, Equations, and Their Practical Uses

68

Because Q = π/δ, the angular frequency of the damped vibrator is
d 
o2   2 
o2 

2  2R m
d2 2 
 o  2S 2
2 4Q 2

(2.142)

or  1  ω o2  ω′d2 1
 4Q 2 

(2.143)

where ωo is the angular frequency that would occur if R m = 0, that is, based upon c = (Y/ρ)0.5 ω′d is the measured angular frequency of the damped bar in longitudinal half-wave resonance. A typical value for the Q of aluminum alloys is ≈104. Therefore, for an aluminum bar, ω′d is lower than the hypothetical ωo only by a factor of ∼5 × 10−4. Damping in good-grade materials used for ultrasonic horns is insignificant. One value of understanding these principles is the quality control or evaluation potential they afford. Large differences from expected performance, especially in frequencies that are lower than those anticipated, are reasons to suspect the quality of the materials. Abnormal frequency shifts may be due to 1. A poor alloy grade. 2. Fatigue failure, especially if there is a continuous, irreversible frequency shift during performance. 3. Heat—this can occur due to continuous high-amplitude performance or from high load absorption. If no failure occurs, the shift will reach an equilibrium when heat energy removed equals the rate of heat produced by losses. Measurements of Rm or c are used as a means of quality control or for laboratory analysis of materials. 2.4.3

Effect of Temperature on Horn Performance

In Section 2.3.2, thermoelastic effects and interaction with thermal phonons were also mentioned as significant mechanically related loss mechanisms. The latter will be discussed first. 2.4.3.1

Phonons

By analogy with photons (the elementary quantity, or quantum, of radiant energy according to quantum theory of radiation, regarded as a discrete quantity and having momentum equal to hν/c, where h is Planck’s constant, ν is the frequency of the radiation, and c is the speed of light in a vacuum), the phonon is a quantized elastic wave. Phonons are standing elastic waves of extremely high frequency (on the order of 1012 Hz). The energy associated with these waves is the internal energy of the material (in the present case, a solid). The energy associated with a phonon is hν where h is Planck’s constant [8] (h = 0.626176(36) × 10−34 J Hz−1) [9]. The Handbook of Chemistry and Physics, 66th edition [9], defines internal energy as a “mathematically defined thermodynamic function of state, interpretable through statistical

CRC_DK8307_CH002.indd 68

10/29/2008 3:48:14 PM

Ultrasonic Horns, Couplers, and Tools

69

mechanics as a measure of the molecular activity of a system.” It appears in the first law of thermodynamics as U = du − dq − dw

(2.144)

where du is the increment of specific internal energy, dq the increment of heat, and dw the increment of work done by the system per unit mass. High-power ultrasonic applications are performed at frequencies that are very low compared with the frequencies of phonon modes. Therefore, the acoustic wavelength in an ultrasonic horn is typically very large compared with that of the thermal phonons, and the phonons experience a nearly spatially uniform strain. There exists an equilibrium distribution of thermal phonons corresponding to any given distribution of stress within the horn. The ultrasonic vibrations cause a continual change in the equilibrium conditions. The redistribution of phonons due to the changing equilibrium conditions lags the changing equilibrium conditions, and as a result, energy is dissipated. Attenuation attributable to phonons at very high ultrasonic frequencies, where the length of the propagating wave approaches that of the mean-free-path of the phonon, differs in mechanism from that at low frequencies. Here the ultrasound is of low energy, and its interaction with thermal phonons will cause scattering or local absorption of some of the ultrasonic energy and thus contribute to attenuation of ultrasound. This high-frequency type of energy loss is not a concern in high-intensity ultrasonic devices. 2.4.3.2

Thermoelastic Effects on Horn Performance

Horns tend to heat up during continuous high-intensity operation due to internal and load losses. The temperature usually reaches an equilibrium point that depends upon the environment of the horn, the intensity at which it is being driven, and the material properties of the horn. In correlation with the decrease in elastic moduli, density, and possibly of changes in values of energy loss factors, the resonance frequency of a horn decreases with increasing temperature. The rate at which change occurs depends upon the metal of the horn. Some alloys, such as certain molybdenum alloys, retain good elastic properties to very high temperatures. (Molybdenum oxidizes very rapidly at high temperatures and therefore must be used in a protective nonoxidizing atmosphere for processing at high temperatures.) If the horn suffers no damage during operation, its performance characteristics will return to normal upon cooling to room temperature. Acoustic properties of materials are presented in Chapter 6. Thermal properties of certain important materials in acoustic design are included, such as effects of temperature on moduli and density, environmental compatibilities (heat, chemical), and temperature/ stress limitations.

2.5

Combinations of Geometrical Forms

Applications of high-intensity ultrasonic energy to any process involves a series of structures resonant at a common frequency. As the resonant frequency of each structure in the series must match that of every other structure in the series, the mechanical impedances of these elements must also match those of the elements to which they are joined. In some cases, adjoining elements may be similar in structure, but this is not the general situation.

CRC_DK8307_CH002.indd 69

10/29/2008 3:48:15 PM

Ultrasonics: Data, Equations, and Their Practical Uses

70

A basic system includes (a) a transducer, (b) usually a booster horn, and (c) a horn to deliver the ultrasound to the load object. The transducer is a complex structure consisting in its simplest form of the energy converter (piezoelectric plates, magnetostrictive stack) and an element to transfer the generated ultrasound to the next element in the series. The booster is a horn that resonates at the frequency of the driving transducer. Its configuration is usually designed to meet the manufacturer’s needs for mounting his system. The load horn may be of any of the simple geometrical structures, the slotted wide or large-area designs, or a combination of simple geometrical form coupled to another geometrical form, such as a lumped mass or long and slender uniform bar to be used as a tool. Usually the attachment is much shorter than a wavelength in the material of the attachment at the driven frequency. However, the attachment may be of any desired geometry and size that can be impedance-matched to the driving horn. Structures in an ultrasonic power line may be complicated by variations in geometry and material. Each configuration or material is a separate element of the structure and these elements must be combined in such a way as to allow the structure to resonate at the specified frequency of the system with minimum loss. Therefore, not only must each structure of the series be matched in mechanical impedance with its neighboring structure in the series, but also each element of each structure must be similarly matched to its neighbors to maintain proper resonance conditions in the overall system. The final dimensions of the structures are determined by a procedure of matching mechanical impedances of successive elements. 2.5.1

Uniform Bars of Equal Diameter but Different Materials

The simplest forms for matching impedances are uniform bars of equal diameters or identical cross sections of different materials. Ignoring the effects of means used for connecting the two segments (silver brazing, pressure, bolting, etc.), the two segments are matched according to the mechanical impedance, Zm, of a uniform bar (Equation 2.7). In Figure 2.14, the length of the bar segment on the left is ℓ1 (λ1/2 > ℓ1 > λ1/4), and consists of a metal with density, ρ1, and bar velocity, c1. The length of the segment on the right is ℓ2 (ℓ2 ≤ λ 2/4), and the bar consists of material of density, ρ2, and bar velocity of sound, c2. In practice, ℓ2 is generally specified. The cross-sectional area of both elements is identical. The matching condition is that Zx1 = Zx2 or jSρ1c1 tan



1

 jS 2c 2 tan 2 c1 c2 Material B

Material A

xb

xa λ/2 FIGURE 2.14 Resonant bars of uniform cross section but different materials.

CRC_DK8307_CH002.indd 70

10/29/2008 3:48:15 PM

Ultrasonic Horns, Couplers, and Tools

71

or tan



1  2c 2

 tan 2  1c1 c1 c2

(2.145)

All terms of Equation 2.145 are specified with the exception of ℓ1. By definition, ℓ1 ≥ λ1/4, which places it in the second quadrant (ωℓ1/c1 ≥ 90°). 2.5.2

Horn and Lumped Mass

A lumped mass is an element having dimensions that are very small compared with a wavelength of sound. The stiffness of any material overlapping the lateral dimensions of the driving horn is assumed to be infinite and the motion of the entire mass is assumed to be everywhere in phase. The mass may be a change in dimensions of a horn near the load end or it may be an entirely different material attached for specific types of applications. According to these specifications, the length of the horn, ℓ, is less than λ1/2 and greater than λ1/4. The mechanical impedance of the mass is ZM = −jωM = −jωρ × Volume 2.5.2.1

(2.146)

Lumped Mass Attached to a Uniform Bar

In this case, the impedance of the uniform bar, Equation 2.7 is set equal to ZM of the mass (Figure 2.15), that is 
x  Zm  Zm  jSc tan   c  or tan




Μ
ρ (Volume of mass)   2 c1 S 1c1 S 1c1

(2.147)

S

M

L FIGURE 2.15 Mass-loaded resonant bar.

CRC_DK8307_CH002.indd 71

10/29/2008 3:48:15 PM

Ultrasonics: Data, Equations, and Their Practical Uses

72

When the mass is a change in dimensions but not of material at the end of a bar (either reduction or increase in cross-sectional area, S) tan




( Volume of mass)  c1 Sc1

(2.147a)

Similar procedures are used in determining dimensions of stepped horns driving a mass load. 2.5.2.2

Lumped Mass Attached to Exponential Horn

According to Equation 2.33, the mechanical impedance, Zm, of the exponential horn is 
 2c ′  
x  x jS o Y 
e tan    c′   c′ 4
 Zm  c ′
x  
1  tan
2 c′  

(2.33)

Letting Zm = ZM leads to tan



 c′


2Me
 ω  2c′  Μ
c′e

SoY 
 c′ 4
 2
2 2e
( Volume of mass) 
 2c′  Μ
c′e
SoY 

 c′ 4
 2

(2.148)


2 2(Volume of mass) 
 c′  
 2c′
( Volume of mass) SoY 
e
 c′ 2 4


(2.148a)



or tan



 c′

2

The length, ℓ of the exponentially tapered portion is determined by iteration using Equations 2.148 (ℓ > λ/4). The lengths of mass-loaded horns of any taper for which the mechanical impedance relationship is known are determined by following the procedures of Sections 2.5.2.1 and 2.5.2.2: equate the mechanical impedance, Zm, of the horn with that of the mass at the junction between the two and solve for the length of horn that satisfies the equation with ℓ > λ/4. To be treated as a lumped mass, all dimensions of M must be very small compared with λ in the material of the mass at the resonance frequency, fo. Otherwise, it must be considered as a distributed mass. 2.5.3

Horns with Distributed Loads

Seldom will two adjacent sections of a resonant structure be properly described by any term other than as a distributed mass. Its impedance is always complex, as illustrated in the previous sections dealing with bars and horns of various geometries. Several variations in

CRC_DK8307_CH002.indd 72

10/29/2008 3:48:15 PM

Ultrasonic Horns, Couplers, and Tools

73

impedance functions may be present within a half-wave structure, such as a piezoelectric transducer for power applications. (Piezoelectric transducer design data are presented in Chapter 4.) 2.5.3.1

Matching a Uniform Bar to a Horn

The matching principle is demonstrated in Section 2.5.1 for two bars of equal cross section but different materials. The same procedure may be followed for matching any distributed mass to any other distributed mass vibrator (horn). For example, to match a uniform bar to an exponentially tapered horn, note that the impedance of the horn at x = ℓ1 is 
 2c ′  

 jS o Y1 
tan  1  e
1   c′   c′ 4
 Zh  c ′

 
1  tan 1  c′ 
2 

(2.149)

and the impedance of the bar is Zh = −jSbρ2c2 tan(
ℓ2/c2)

(2.150)

Impedance matching occurs when Zh = Zb. Usually the application requirements determine the dimensions of the bar. A horn is chosen that seems best for use with the specified bar. If the application calls for a long bar (λ/2 or greater), the most desirable condition is to make ℓ2 = nλ/2 where n = 1, 2, 3, 4, …. In this way, a standard horn may be used without altering its length. The stresses at the junction are minimum so that the two elements may be joined safely by threaded elements, silver brazing, or other acceptable means. It is not always possible to make the attachment a resonant length. Very often the application requires a short bar attachment. The length of a standard horn has to be adjusted so that its mechanical impedance matches that of a bar that is short compared with a half-wavelength, or of a bar for which the length exceeds n half-wavelengths by a fraction of a half-wavelength. When the length of the bar ℓ2 < λ/4, ℓ1 > λ/4, the impedance relationship for determining ℓ1 is

2 c2 2 
1


 c′  e S b2c 2c′ tan 2
S o Y1 
 c′ 4
 2 c2

(2.151)


 2c ′  

 S o Y1 
tan  1  e
1  c′   c′

4
  S b2c 2 tan 2 c′

1  c2 
1  tan  c′  2


(2.152)



tan 1  c′

e
1
S b2c 2 tan

or

The bar length, ℓ2, is specified and the horn length, ℓ1, is determined using Equation 2.152.

CRC_DK8307_CH002.indd 73

10/29/2008 3:48:15 PM

Ultrasonics: Data, Equations, and Their Practical Uses

74

Stress at the junction between the horn and the bar may be determined using the equation for stress distribution in a uniform bar, letting x = ℓ2, that is S b  jYb


m



sin 2  j2c 2
m sin 2 c2 c2 c2

(2.153)

where m is the maximum displacement at the free end of the bar. Often short bars of tool steel are silver brazed to a horn to perform certain functions such as materials forming. Silver brazed joints have fillets. The stress concentrations due to the fillets can be estimated using Equations 2.17, 2.18, and 2.153. If this concentrated stress exceeds the endurance limit of the soldered joint, the tool is too long and the joint will fail in fatigue very rapidly. Impedance matching of other geometrical combinations is accomplished in a manner similar to the examples given in this section. 2.5.4

Multiple Mode Systems

A system consisting of one or more elements of distributed elastic mass is capable of vibrating in more than one mode (such as longitudinal, flexural, or torsional motion). The design problem is to control the modes of vibration throughout the system to deliver the proper type of activity to the application zone. Design objectives might include any of the following: 1. Having strictly one specific mode of vibration, with all others suppressed within a vibratory member—preferably by designing the component so that no outside structural constraints are needed for this mode suppression. In the ultrasonics industry, horns are commonly designed to operate in the longitudinal mode. If a structure vibrates in more than one mode at the same frequency, the total vibrational energy of the structure is shared by all modes, thus reducing the energy available to the desired mode. 2. Sharing the available energy between modes. This condition is usually undesirable. However, there are certain exceptions where applications are best satisfied by the motion that combined modal action can provide at the load end of an ultrasonic transmission line. In the case of slender rods (or wires) that are several longitudinal half-wavelengths long, it is virtually impossible to prevent flexural modes from occurring simultaneously with longitudinal modes. In certain types of applications, the elliptical motion afforded by the combination of modes is beneficial, for example, in disintegrating friable substances obstructing small passageways. 3. Transferring energy from one element vibrating in one pure mode to a second element in the vibratory system vibrating in a different pure mode. A horn vibrating in a longitudinal mode may be used to drive a uniform bar in flexure to bond thin strips or to spot-weld thin plastic sheets. Other combinations might include a longitudinally vibrating horn driving a thin diaphragm, plate, or cylinder in flexure (e.g., transducers mounted on the bottom of an ultrasonic cleaning tank). In the present sense, a multiple-mode system is defined as one in which a source operating at resonance in one mode excites resonance in a second member vibrating in a different mode from that of the first. Coupled elements in such a system must be matched in mechanical impedance, as in any other coupled system. In addition, some thought should be given to the coupling conditions themselves, that is, as much as is possible, design so that the stress patterns in each of the connected elements conform at the junction to enhance rather

CRC_DK8307_CH002.indd 74

10/29/2008 3:48:15 PM

Ultrasonic Horns, Couplers, and Tools

75

than inhibit the intended modal patterns. Poor coupling conditions, including method and position, will produce poor output activity. In some cases, tabulations of theoretical data for bars and plates considered in the following sections may extend beyond practical limits from an applications standpoint. These are included for completeness, particularly with respect to frequency limits for conditions intermediate to those presented. For example, mass loading conditions, such as (a) a mass on the end of cantilever, (b) a mass loading a plate with any boundary conditions, or (c) a water load on any plate, are not included. In each case, the load would reduce the fundamental frequency, and, depending upon the loading conditions, would generally alter the modal patterns. Other partial or flexible constraints could lead to an infi nite number of conditions, complicating frequency and modal problems. One might apply fi nite element methods more practically to analyze these complicated systems. 2.5.4.1

Longitudinal Mode Horn Driving a Flexural Bar

The transverse wave equation for bars is given in Equation 1.40 as 4 2 YI 4 2 2   c      o t 2 x 4 S x 4

(1.40)

where  is the amplitude of lateral displacement from the rest position x; co the bar velocity of sound in the medium of the bar; κ the radius of gyration of the cross-sectional area S, or κ 2 = −RM/YS with R as the radius of curvature of the neutral axis or plane at position x, M the bending moment, Y Young’s modulus of the bar material, S the cross-sectional area at x; and I the moment of inertia of the area at x (=Sκ 2). In using Equation 1.40 for flexural bars, it is assumed that the displacement and slope at any position along the bar are small enough that variations in angular momentum may be neglected. Its derivation is based upon a balance between the bending moments (M) and shear forces (Fx) across a lateral element dx. These moments and shear forces across an element, dx, are represented by Mx − M(x+dx) − (Fy)(x+dx)dx = 0

(2.154)

where (M)(x
dx)  (M)x


M dx x

and (Fy )(x
dx)  (Fy )x


Fy x

dx

However, these terms for M(x+dx) and (Fy)(x+dx) are defined here by the first two terms of Taylor’s series, ignoring second-order and higher terms involving (dx)2, and so on, for small dx. Therefore, Equation 1.40 is still an approximation, even for small values of displacement and slope, but it is very nearly correct if the amplitude of vibration is small compared with the length of the bar. Because the amplitudes of flexural vibrations of bars as used in power applications of ultrasonics generally are very small compared with the lengths of the bars, use of Equation 1.40 is justified for the derivations that follow.

CRC_DK8307_CH002.indd 75

10/29/2008 3:48:15 PM

Ultrasonics: Data, Equations, and Their Practical Uses

76 2.5.4.2

Moments of Inertia

The moment of inertia of an area with respect to a given axis is the limit of the sum of the products of the elementary areas into which the area may be conceived to be divided and the square of their distance, y, from the given axis; that is I  ∫ y 2 dA  A 2

(2.155)

Referring to the rectangle of Figure 2.16, the area is A = 2by and dA = 2b dy. Therefore, the moment of inertia of the rectangle about the axis 0 − 0 is I∫

h 2

0

2 by 2 dy 

2 by 3 3

h/2

 0

bh 3 12

(2.156)

Equations for determining moment of inertia, section modulus (I/c), and radius of gyration of cross sections that might be used in flexural elements in ultrasonic applications are given in Table 2.3. Here c is the distance from the neutral axis to the outer fibers of the member. For elements other than those listed in Table 2.3 but for which the cross-sectional area can be defined in terms of their dimensions, use Equation 2.155. The performance of all beams in flexural vibration, whether they are uniform in cross section or complex in structure, is dependent upon the moment of inertia of a solid. The moment of inertia of a solid body with respect to a given axis is the limit of the sum of the products of the masses, dm, of each of the elementary particles into which the body may be conceived to be divided and the square of their distance from the given axis. Thus I m  ∫ y 2 dm  ∫ y 2

dm g

(2.157)

where dm = dw/g represents the mass of an elementary particle, dw the weight of the elementary particle, g gravity, and y the distance of the particle from the given axis. The moment of inertia of a solid of elementary thickness about an axis is equal to the moment of inertia of the area of one face of the solid about the same axis multiplied by the mass per unit volume of the solid times the elementary thickness of the solid. Polar moment of inertia is another important term in the design of vibrating structures. The polar moment of inertia is equal to the sum of the moments of inertia about any two axes at right angles to each other in the plane of the area and intersecting at the pole (Figure 2.17), that is Ip = Ix + Iy where Ix is the moment of inertia of area A about axis XX and IY the moment of inertia of area A about axis YY. y O

O

h

I = bh3/12

b FIGURE 2.16 Moment of inertia for rectangle.

CRC_DK8307_CH002.indd 76

10/29/2008 3:48:15 PM

Ultrasonic Horns, Couplers, and Tools

77

TABLE 2.3 Moment of Inertia (I), Section Modulus (I/c), and Radius of Gyration (κ = Î/(I/A)) of Various Cross Sections Relative to Flexural Elements Used in Ultrasonic Applications Cross Section

h

Moment of Inertia, I

Section Modulus, I/c

Radius of Gyration, κ

bh 3 12

bh 2 6

bh 3 3

bh 3 3

bh 3 3

b2
4 bb1
b12 3 h 36( b
b1 )

b2
4 bb1
b12 2 h 12(2 b
b1 )

h 2( b2
4 bb1
b12 )

d 4 r 4  64 4

d 3 r 3  32 4

r d  2 4

 4 (D  d 4 ) 64

 D4  d4 D 32

  (R 4  r 4 ) 4

 R 4  r4  R 4

c b

c

h

h  0.577 h 3

b b1 c h

c

6( b
b1 )

1 2 b
b1 h 3 b
b1

b

d r R D

d r c1

c2

8   r4     8 9 

r

R 2
r2 2 

I  0.1908r 3 c2

D2
d2 4 9 2  64 6

I  0.2587 r 3 c1 c1 = 0.4244r a

a 3 b 4

a 2 b 4

5 3 4 R 16

5 3 R 8

a 2

b c

5 R 24

R

(continued)

CRC_DK8307_CH002.indd 77

10/29/2008 3:48:15 PM

Ultrasonics: Data, Equations, and Their Practical Uses

78 TABLE 2.3

(Continued)

Cross Section

Moment of Inertia, I

R

c

b c

R

φ

Section Modulus, I/c

Radius of Gyration, κ

5 3 4 R 16

5 3 3 R 16

5 R 24

[A(12b2
a 2 )] 48

[A(12b2
a 2 )] 48 b

(12 b2
a 2 ) 48

a c = a/2 tan φ A = nb2 tan φ n = number of sides

Iy = x2A (Moment of inertia of area A about the axis Y−Y)

Y

x r

A

Ix = y2A (Moment of inertia of area A about the axis X−X)

y

Ip = (x2 + y2) A = r2A

X

X

Y FIGURE 2.17 Polar moment of inertia.

2.5.4.3

General Solution to the Transverse Wave Equation for Thin Bars 4 1 2
2 2 2 0 4 x  c o t

(1.40a)

Assuming harmonic motion, Equation 1.40a also may be written 4ξ
2  0 x 4  2c o2

(1.40b)

A solution to Equation 1.40b is of the form 4  p4 Aepx  p4 x 4

  Aepx so that Equation 1.40b may be written

p4 

CRC_DK8307_CH002.indd 78


2 0  2c o2

(1.40c)

10/29/2008 3:48:16 PM

Ultrasonic Horns, Couplers, and Tools

79

or p4 


2 0  2c o2

(1.40d)

Letting q4 = ω2/κ 2co2 , Equation 1.40d simplifies to p4 − q4 = 0

(1.40e)

The four roots of Equation 1.40e are p1 = q

p2 = −q

p3 = jq p4 = −jq

Therefore, the general solution to Equation 1.40b is  = A1eqx + A2e−qx + A3ejqx + A4e−jqx

(2.158)

or replacing the exponential functions with their trigonometric equivalents  = B1 sinh qx + B2 cosh qx + B3 sin qx + B4 cos qx

(2.159)

The phase velocity of a wave in the flexural bar is v 
c o  Because q2  v


2  2c o2

q also may be written q = ω/v Equation 2.159 is a useful general solution to Equation 1.40b that can be made specific to any uniform slender bar typical of those driven in flexure for ultrasonic power applications. The end conditions are first defined. The functions characterizing these conditions are then applied to Equation 2.159, first, at x = 0 to determine (initially) the constants Bn, and, second, at x = ℓ, to determine the allowable frequencies of resonance for the fundamental mode and the overtones. The nodal positions and positions of maximum displacement are located by a similar process, by noting the characteristics of each type of node and maximum displacement amplitude. In most cases, the solutions arrived at following this procedure are sufficiently accurate for designing ultrasonic power systems. In the case of bars with one free end, the equations lead to a first approximation for the location of the node nearest the free end that is ∼3.5% in error. The driven motion is assumed to be normal to the axis of the bars. In the present sense, a slender bar is one that (a) provides a cross-sectional dimension in the direction of motion at the coupling position that is small compared with a wavelength of sound in the material of the bar at the driven frequency and (b) exhibits negligible compressional deformation in response to the motion of the driving horn at the coupling position. The bars may be hollow or solid. (Excitation of plates and cylinders into their natural modes by horns vibrating in longitudinal modes are considered in Sections 2.6 and 2.7.) Bars vibrating in flexure typically used in ultrasonic power applications include (a) those driven initially in a free–free mode in an unloaded condition being suddenly forced against a load while the driving force remains active for a short period of time and

CRC_DK8307_CH002.indd 79

10/29/2008 3:48:17 PM

Ultrasonics: Data, Equations, and Their Practical Uses

80

(b) those driven initially in a clamped–free mode being suddenly impressed against the load while the driving force remains active for a brief period of time. The conditions at the ends, nodes, and antinodes along a bar that is vibrating in flexural resonance are explained by basic principles of strength of materials. These are summarized as follows: 1. Where the bar experiences a bending moment, M, at the point under consideration (end or boundary, node, antinode) M YI

d 2 dx 2

(2.160)

where Y is Young’s modulus of elasticity, I the moment of inertia of the cross section of the beam parallel to the direction of motion, YI the flexural rigidity of the bar,  the amplitude of displacement at position x (measured from the neutral position), and x the distance from one end of the bar. 2. The bar experiences a shear, σx, that is a function of bending moment given by x 

dM d 3 YI 3 dx dx

(2.161)

3. At all nodal positions =0 4. At free ends, nothing is located at x < 0 or at x > ℓ to cause shear or bending moment so that at x = 0, ℓ d 2 d 3  0 dx 2 dx 3

(2.162)

5. At nodal positions nearest a free end =0 but the bar does experience bending moment and shear across the nodal position so that d d 3  3 0 dx dx

(2.163)

6. At nodal positions intermediate to those nearest free ends and all other nodes between constrained ends (pinned, clamped), the bending moment at the nodal position is zero, so that both d2  = 0 and ____2 = 0 (2.164) dx 7. A sliding end is assumed to be one at which the end plane of the bar slides along a plane surface and these two planes maintain perfect parallelism during the entire vibration cycle. Under these conditions, the slope of the axis is zero (axis of the displaced bar is parallel to the axis at the equilibrium position) and the axial shear is also zero, so that d d 3  3 0 dx dx

CRC_DK8307_CH002.indd 80

(2.165)

10/29/2008 3:48:17 PM

Ultrasonic Horns, Couplers, and Tools

81

8. Equation 2.165 is also applicable at positions of maxima other than at free ends, as shear as a function of x is also zero at these points. These conditions are used to solve the equations describing the modes of bars in flexural vibration and for locating the positions of nodes and antinodes of the vibrating bar. The flexural bars of Section 2.5.4 (as well as the plates of Section 2.6) are assumed to be perfectly elastic, homogeneous, isotropic materials of uniform thickness, unless otherwise designated. The thicknesses are considered to be small compared with the wavelength and all other dimensions. 2.5.4.4

Specific Solutions to the Flexural Bar

2.5.4.4.1 Clamped–Clamped Bar Two types of nodes occur in the clamped–clamped bar. The first type is associated with the clamping conditions at each end (x = 0, ℓ), characterized by  = 0 and d/dx = 0. These are the only nodes associated with the fundamental frequency. An additional node located between the ends and characterized by  = 0 and d2/dx2 = 0 occurs for each higher overtone; that is, one node corresponding with the first overtone, two nodes with the second overtone, and so on. When x = 0  = B1(0) + B2(1) + B3(0) + B4(1) = 0 d/dx = B1q(1) + B2q(0) + B3q(1) − B4(0) = 0 Then B1 = −B3 B2 = −B4 Therefore the solution for the clamped–clamped bar may be written  = B1(sin h qx − sin qx) + B2(cos h qx − cos qx)

(2.166)

d/dx = qB1(cos h qx − cos qx) + qB2(sin h qx + sin qx)

(2.167)

and

The resonance frequencies are determined by applying the boundary conditions at x = ℓ, where  = 0, and d/dx = 0. Thus  = B1(sin h qℓ − sin qℓ) + B2 (cos h qℓ − cos qℓ) = 0

(2.166a)

d/dx = qB1(cos h qℓ − cos qℓ) + qB2 (sin h qℓ + sin qℓ) = 0

(2.167a)

B1(sin h qℓ − sin qℓ) = −B2(cos h qℓ − cos qℓ)

(2.166b)

B1(cos h qℓ − cos qℓ) = −B2(sin h qℓ + sin qℓ)

(2.167b)

and

from which

CRC_DK8307_CH002.indd 81

10/29/2008 3:48:17 PM

Ultrasonics: Data, Equations, and Their Practical Uses

82

Dividing Equation 2.166b by Equation 2.167b sinh q
 sin q
cosh q
 cos q
 cosh q
 cos q
sinh q

sin q


(2.168)

sin h2 qℓ − sin2 qℓ = cos h2 qℓ − 2 cos h qℓ cos qℓ + cos2 qℓ

(2.169)

Cross multiplying

which, after substituting for q (=ωℓ/v), reduces to cos h (ωℓ/v) cos(ωℓ/v) = 1

(2.170)

Equation 2.170 is valid only at discrete values of ωℓ/v. The function cos h θ increases from 1 to ∞ as θ increases from 0 to ∞. The function cos θ fluctuates between 1 and −1 as θ increases from 0 to ∞. Therefore, Equation 2.151 is not in a satisfactory form for determining the allowed values of ωℓ/v and, consequently, of the allowed resonant frequencies. This problem is easily solved by using the identities cosh 2 

1
tanh 2  1  tanh 2 

cos 2 

1  tan 2  1
tan 2 

(2.171)

by which Equation 2.170 is converted to tan





 tanh 2v 2v

(2.172)

The fact that tanh θ is nearly unity for all values of θ > π and that periodically tan θ will equal the value of ±tan h θ simplifies considerably identification of the discrete values of ωℓ/2v and, therefore, of the allowed frequencies. A simple means of determining these allowed values of ωℓ/2v and, therefore, the allowed frequencies is to plot ±tan h(ωℓ/2v) and tan(ωℓ/2v) to the same scale as functions of (ωℓ/2v) and locating the intercepts between the two functions (Figure 2.18). For the clamped–clamped bar, the intersections occur at



   [3.0112, 5, 7, 9,… ] 2v 2
c 4

(2.173)

2.0 tan (ωᐉ/2v) tanh (ωᐉ/2v)

1.0

0

π/2

π

−1.0 3.0112π/4

3π/2

2π

5π/2

−tanh (ωᐉ/2v)

−2.0

FIGURE 2.18 Graphical determination of allowed frequencies for flexural resonance in clamped–clamped bars.

CRC_DK8307_CH002.indd 82

10/29/2008 3:48:17 PM

Ultrasonic Horns, Couplers, and Tools

83

and, therefore
n2
2 fn
2  4 v 2n 2c

(2.173a)

where n = 1, 2, 3, 4, … is the number of the fundamental mode and the overtones. From Equations 2.173 and 2.173a fn 

c [3.01122 , 52 , 7 2 , 92 , …] 8
2

(2.174)

The nodal positions for the overtones located between those at x = 0, ℓ are determined in a manner similar to that used in determining the allowed frequencies. As stated previously, those nodes lying between the ends are characterized by  = 0 and d2/dx2 = 0. The second derivative of Equation 2.159 is d 2  q 2B1(sinh qx
sin qx)
q 2B 2 (cosh qx
cos qx) dx 2

(2.175)

The conditions characterizing the nodes, that is d2  = ____2 = 0 dx B1(sin h qx − sin qx) = −B2(cos h qx − cos qx) B1(sin h qx + sin qx) = −B2(cos h qx + cos qx) lead to the equation which applies to overtones, which is tan qx = tan h qx or tan (ωx/v) = tan h (ωx/v)

(2.176)

Equation 2.176 is valid only at discrete values of (ωx/v) and only for values of (ωx/v) appearing in quadrants where tan(ωx/v) is positive. These values of ωx/v at the nodal positions corresponding to overtones of the clamped–clamped (fixed–fixed) bar are
x   [5, 9, 13, 17 ,…] v 4

(2.177)

The positions relative to length of the bar of these intermediate nodes corresponding to overtones are determined by comparing the allowed positions (ωx/v) with the corresponding allowed frequencies (ωℓ/2v). From Equation 2.173, those quantities corresponding to the overtones,
  [5, 7 , 9, 11, … ] v 2


(2.178)

For the first overtone
x 5  v 4

CRC_DK8307_CH002.indd 83

10/29/2008 3:48:17 PM

Ultrasonics: Data, Equations, and Their Practical Uses

84

from the equation designating position (2.177) and
5  v 2
from the equation designating allowed frequencies (2.178). Then 5x 5  2
4 and x = 0.5 ℓ which is the only node occurring between the ends of the clamped–clamped bar at the first overtone (f2). The nodes corresponding to the higher-order modes are determined in a similar manner. The nodal positions, corresponding frequencies, positions of maximum displacement, and phase velocities are given in Table 2.5 for various conditions of flexural bars in resonance. The positions at which displacement is maximum for the clamped–clamped bar are characterized by  =  m, d/dx = 0, and d3/dx3 = 0. (Positions of maximum displacement and the locations of the nodes are important to the effective design of systems involving a horn coupled to a bar and driving the bar into flexural resonance by motion across its axis [usually normal to its axis].) Returning to Equation 2.166 and applying the conditions characterizing the maxima of displacement m = B1(sin h qx − sin qx) + B2(cos h qx − cos qx) d/dx = qB1(cosh qx − cos qx) + qB2 (sinh qx + sin qx) = 0 3 d /dx3 = q3B1(cos h qx + cos qx) + q3B2(sin h qx + sin qx) = 0

(2.166) (2.167) (2.179)

From Equations 2.167 and 2.179, tan h qx = −tan qx or tan h (ωx/v) = −tan(ωx/v)

(2.180)

Equation 2.180 is valid for values of (ωx/v) lying in quadrants where tan(ωx/v) is negative. The allowed values of ωx/v are
x   [3.0112, 7 , 11, 15, …] v 4

(2.181)

Following the procedure for locating the nodes, that is, by comparing Equation 2.181 with the equation for allowed frequencies (Equation 2.178), the position of maximum displacement for the fundamental frequency is determined to be
x 3.0112x 3.0112   v 2
4

(2.182)

or x = 0.5ℓ This is the only position of maximum displacement at the fundamental frequency. Positions of maxima for the overtones are determined in the same manner. Table 2.4 gives

CRC_DK8307_CH002.indd 84

10/29/2008 3:48:17 PM

Ultrasonic Horns, Couplers, and Tools

85

TABLE 2.4 Frequency, Phase Velocity, Nodal Positions, and Positions of Maximum Displacement for Uniform Bars in Flexural Resonance Under Various Boundary Conditions Clamped–clamped bar Displacement equation:  = B1[sin h ωx/v − sin ωx/v] + B2[cos h ωx/v − cos ωx/v] Frequency (fn), nodal positions (ωxn/v) other than clamped ends, and positions of maximum displacement (ωxm/v) xm = 0.3ℓ, 0.7ℓ
x m   [3.0112, 7.11, 15, …] v 4
x n   [5, 9, 13, 17 , …] 4 v xm = 0.214ℓ, 0.5ℓ, 0.786ℓ c fn  2 [1.194 2 , 2.988 2 , 52 , 7 2 , …] 8
a. n = l; f = f1; v = v1; xn = 0, ℓ; xm = 0.5ℓ b. n = 2; f2 = 2.756f1; v2 = 1.66v1; xn = 0, 0.55ℓ, ℓ c. n = 3; f3 = 5.404f1; v3 = 2.32v1; xn = 0, 0.357ℓ, 0.643ℓ

Clamped–free bar Displacement equation:  = B1[sin h ωx/v − sin ωx/v] + B2[cos h ωx/v − cos ωx/v] Frequency (fn), nodal positions (ωxn/v) other than clamped end and nearest free end, and positions of maximum displacement (ωxm/v)
x m   [3.0112, 7 , 11, 15, …] 4 v a. n = 1; f = f1; v = v1; xn = 0; xm = ℓ b. n = 2; f2 = 6.263f1; v2 = 2.50v1; xn = 0.837ℓ; xm = 0.504ℓ, ℓ c. n = 3; f3 = 17.536f1; v3 = 4.18v1; xn = 0, 0.5ℓ, 0.9ℓ; xm = 0.3ℓ, 0.7ℓ, ℓ Free–free bar Frequency (fn), nodal positions (ωxn/v) other than nearest free ends, and positions of maximum displacement (ωxm/v) fn 

c [3.01122 , 52 , 7 2 , …] 8
2


x n   [1, 5, 9, 13, 17 , …] v 4
x m   [3.00112, 7 , 11, 15, …] v 4 a. n = 1; f = f1; v = v1; xn = 0.17ℓ, 0.83ℓ; xm = 0.5ℓ b. n = 2; f2 = 2.756f1; v2 = 1.66v1; xn = 0.1ℓ, 0.5ℓ, 0.9ℓ; xm = 0.3ℓ, 0.7ℓ c. n = 3; f3 = 5.40f1; v3 = 2.32v1; xn = 0.071ℓ, 0.357ℓ, 0.643ℓ; xm = 0.215ℓ, 0.5ℓ, 0.785ℓ (continued)

CRC_DK8307_CH002.indd 85

10/29/2008 3:48:18 PM

Ultrasonics: Data, Equations, and Their Practical Uses

86 TABLE 2.4

(Continued)

Bar pinned, hinged, or simply supported at each end Frequency (fn), nodal positions (ωxn/v), and positions of maximum displacement (ωxm/v) fn 

c 2 2 2 2 [1 , 2 , 3 , 4 , …] 2
2


x n  [0, 1, 2, 3, 4, …] v
x m   [1, 3, 5, …] 2 v a. n = l; f = f1; v = v1; xn = 0, ℓ; xm = 0.5ℓ b. n = 2; f2 = 2f1; v2 = 1.414v1; xn = 0, 0.5ℓ, ℓ; xm = 0.25ℓ, 0.75ℓ c. n = 3; f3 = 3f1; v3 = 1.732v1; xn = 0, 0.33ℓ, 0.66ℓ, ℓ; xm = 0.167ℓ, 0.5ℓ, 0.833ℓ Clamped–hinged bar Frequency (fn), nodal positions (ωxn/v), and positions of maximum displacement (ωxm/v) fn 

 2 c 2 2 [5 , 9 , 13 2 , …] 32
2


x n   [0, 5, 9, 13, …] 4 v
x m   [3.0112, 7 , 11, …] 4 v a. n = 1; f = f1; v = v1; xn = 0, ℓ; xm = 0, 0.667ℓ b. n = 2; f2 = 3.24f1; v2 = 1.8v1; xn = 0, 5ℓ/9, ℓ; xm = 0.346ℓ, 0.778ℓ c. n = 3; f3 = 6.76f1; v3 = 2.6v1; xn = 0, 5ℓ/13, 9ℓ/13, ℓ; xm = 0.2316ℓ, 0.5385ℓ, 0.846ℓ Pinned–free bar Frequency (fn), nodal positions (ωxn/v) other than at pinned end and nearest free end, and positions of maximum displacement (ωxm/v) c [0, 52 , 92 , 13 2 , …] fn  32
2
x n  [1, 2, 3, …] v
x m   [1, 3, 5, …] 2 v a. n = 1; f = f1; v = v1; xn = 0.8ℓ; xm = 0.4ℓ, ℓ b. n = 2; f2 = 3.24f; v2 = 1.8v1; xn = 0, 0.444ℓ, 0.889ℓ; xm = 0.222ℓ, 0.667ℓ, ℓ c. n = 3; f3 = 6.76f1; v3 = 2.6v1; xn = 0, 4ℓ/13, 8ℓ/13, 12ℓ/13; xm = 2ℓ/13, 6ℓ/13, 10ℓ/13, ℓ

CRC_DK8307_CH002.indd 86

10/29/2008 3:48:18 PM

Ultrasonic Horns, Couplers, and Tools TABLE 2.4

87

(Continued)

Clamped–sliding bar Frequency (fn), nodal positions (ωxn/v) other than at clamped end, and positions of maximum displacement (ωxm/v) fn 

c [3.01122 , 7 2 , 112 , …] 32
2


x n   [0, 5, 9, 13, …] v 4
x m   [3.0112, 7 , 11, 15, …] v 4 a. n = 1, f = f1; v = v1; xn = 0; xm = ℓ b. n = 2; f2 = 5.504f1; v2 = 5.404f1; xn = 0, 7143ℓ; xm = 0.4302ℓ, ℓ c. n = 3; f3 = 13.345f1; v3 = 3.653v1; xn = 0, 0.4545ℓ, 0.8182ℓ; xm = 0.2738ℓ, 0.6364ℓ, ℓ Free–sliding bar Frequency (fn), nodal positions (ωxn/v), and positions of maximum displacement (ωxm/v) fn 

c 2 [7 , 112 , 152 , …] 32
2


x n   [5, 9, 13, …] 4 v
x m   [3.0112, 7 , 11, 15, 19, …] v 4 f

1.14
2

Yb2 1.14cb  2 12
12

a. n = 1; f = f1; v = v1; xn = 0.143ℓ, 0.714ℓ; xm = 0, 0.430ℓ, ℓ b. n = 2; f2 = 2.469f1; v2 = 1.571v1; xn = ℓ/11, 5ℓ/11, 9ℓ/11; xm = 0, 0.274ℓ, 0.6364ℓ, ℓ c. n = 3; f3 = 4.592f1; v3 = 2.143v1; xn = ℓ/15, ℓ/3, 3ℓ/5, 13ℓ/15; xm = 0, 0.2008ℓ, 0.4667ℓ, 0.733ℓ, ℓ

Pinned–sliding bar Frequency (fn), nodal positions (ωxn/v), and positions of maximum displacement (ωxm/v) fn 

c 2 2 2 2 [3 , 5 , 7 , 9 , …] 8
2


x n  [0, 1, 2, 3, …] v
x m   [1, 3, 5, …] v 2 a. n = 1; f = f1; v = v1; xn = 0, 0.667ℓ; xm = 0.333ℓ, ℓ

(continued)

CRC_DK8307_CH002.indd 87

10/29/2008 3:48:19 PM

Ultrasonics: Data, Equations, and Their Practical Uses

88 TABLE 2.4

(Continued) b. n = 2; f2 = 2.778f1; v2 = 1.667v1; xn = 0, 0.4ℓ, 0.8ℓ; xm = 0.2ℓ, 0.6ℓ, ℓ c. n = 3; f3 = 5.444f1; v3 = 2.333v1; xn = 0, 2ℓ/7, 4ℓ/7, 6ℓ/7; xm = 0.1429ℓ, 0.4286ℓ, 0.7143ℓ, ℓ

Fundamental resonant frequencies of wedge-shaped and conical bars rigidly mounted at the large end may be determined by the following equations: a. Wedge-shaped bar vibrating in a direction normal to parallel sides, frequency, Hz b. Wedge-shaped bar vibrating in a direction parallel to parallel sides, frequency, Hz f c.

0.85
2

0.85 cb Yb2  2 12
12

Conical bar, frequency, Hz f

1.39
2

Ya 2 1.39 ca  4 2
2

ℓ = length of bar c = bar velocity of sound b = thickness of the bar at the base in the direction of vibration a = radius of the cone at the base Phase velocity in every case is determined by (vn)2 = ωncκ.

nodal positions, corresponding frequencies, positions of maximum displacement, and phase velocities for flexural bars subjected to various boundary conditions. The overtones of none of the conditions given in Table 2.4 are harmonics. Boundary conditions shown in Table 2.4 are those corresponding to flexural modes that might be of most interest in the design of systems for ultrasonic power applications. Only the lower three modes are included, but these are sufficient for identifying the boundary conditions and characteristics of nodal and antinodal positions applicable to all overtones. The accuracy of the data of Table 2.4 as applied to real conditions of ultrasonic applications depends upon how closely the real conditions match those of the table. It may be difficult to completely satisfy certain boundary conditions. For example, it is difficult to imagine an application wherein the sliding contact end conditions leading to the listed data are exactly applicable. Those data listed are for the purpose of completeness and should sufficiently accurate for initial design purposes. 2.5.4.4.2

Rayleigh Method of Determining Fundamental Frequencies of Systems with Distributed Mass The fundamental frequency and only the fundamental frequency of an elastic system with distributed masses can be determined with good accuracy by a method derived by Rayleigh [5]. The method is based upon the fact that the total energy in a lossless vibratory system remains constant throughout each cycle of vibration. The potential energy of an elastic beam in flexure depends upon its elastic properties and the deformation it experiences at any instant of time. The kinetic energy of the beam is a function of its mass and velocity distribution at any instant. Thus, obtaining an accurate expression for the fundamental frequency of the beam requires an accurate description of the deflection curve associated with the fundamental mode of vibration of the beam. An initial curve is assumed, considering the boundary conditions, to obtain a reasonable characterization of the actual

CRC_DK8307_CH002.indd 88

10/29/2008 3:48:19 PM

Ultrasonic Horns, Couplers, and Tools

89

deflection curve. If the curve is accurate, the method gives an accurate frequency equation. If it is inaccurate, the correct curve usually can be obtained by iteration. The Rayleigh method may be explained in detail by considering the spring/mass oscillator, Equation 1.3, writing it in the form M

d 2
k  0 dt 2

(1.3a)

where M is the lumped mass attached to the end of a weightless spring (the spring can be of any form such as coil spring, cantilever, etc.),  the displacement, and k the spring constant. Equation 1.3 is recognized as a force balance equation in which the force k is what the spring exerts on the mass to accelerate it at the rate of d2/dt2 and the force Md2/dt2 is the familiar F = Ma, which is the force required to accelerate the mass at rate a. The potential energy of a body is energy of position. It is the energy possessed by a body held in such a position that it can do work upon being released. The kinetic energy of a body is energy of motion. The potential energy of a mass on a spring is the amount of work without loss required to move the mass against the force of the spring to its current position, that is P.E. = (k/2) = k2/2 The kinetic energy of the mass is given by the familiar equation K.E. = Mv2/2 Because the total energy of a lossless vibratory system remains constant, Mv 2 k 2
C 2 2

(2.183)

where v = d/dt the instantaneous velocity of the mass at position . Differentiating both sides of Equation 2.183 with respect to t gives Mv

d dv
k 0 dt dt

(2.184)

Substituting d/dt = v, d2/dt2 = dv/dt into Equation 2.184 leads to Equation 1.3a. Assuming harmonic motion, that is,  = m sin ωt, ω2 = k/M

(1.5)

Assuming simple harmonic motion, the frequency of a lossless mass-loaded spring may also be determined by the fact that the energy of the system oscillates between maximum potential energy (when kinetic energy is zero) to maximum kinetic energy (when potential energy is zero): 2 Maximum potential energy  (1/2)km

Maximum kinetic energy = (1/2) M where ωm = Vm, the maximum velocity of the mass.

CRC_DK8307_CH002.indd 89

10/29/2008 3:48:19 PM

Ultrasonics: Data, Equations, and Their Practical Uses

90

Equating these two maxima and simplifying also leads to Equation 1.5a. In this case, note that Equation 1.5a was derived by dividing the equation for maximum potential energy by the equation for maximum kinetic energy and, as these two are equal, the quotient is 1. The denominator contains the factor ω2; therefore, multiplying both sides by ω2 obtains
2 

maximum potential energy 
2 k  maximum kinetic energy M

This equation is the basis of the Rayleigh method. To determine the total potential energy of an elastic system with distributed masses, such as a flexural bar, requires an accurate identification of the deflection curve associated with the vibration. In some cases, such as that of a simply supported elastic beam of uniform cross section, it is a simple matter to obtain the correct beam deflection curve. Other systems require iteration. Using a cantilever beam with a mass load at the free end for purposes of illustration, assume the following: 1. 2. 3. 4. 5. 6.

The beam is clamped at x = 0. The length, ℓ, lies along the x-axis. Bending is restricted to within the xy plane. The mass of the beam is m. The displacement along the beam is  = (x). The mass at the end of the beam is M.

The boundary conditions are: 1. At x = 0, d/dx = 0. 2. At x = ℓ, the mass, M, is vibrating at the maximum displacement, m, at the lowest, or fundamental, mode. The potential energy of the bar is P.E.  ∫ x

d 2 dx dx 2

(2.185)

where Mx is the bending moment at x in terms of x. The general differential equation of the elastic curve of a beam, or bar, is EI

d 2  Mx dx 2

(2.186)

Therefore, the potential energy of the beam is given by 2

P.E. 

1  d 2  YI  2  dx ∫  dx  2

(2.187)

The kinetic energy of the beam is given by K.E. 

CRC_DK8307_CH002.indd 90

1 2 2 
dm 2∫

(2.188)

10/29/2008 3:48:20 PM

Ultrasonic Horns, Couplers, and Tools

91

where dm = ρS dx is the mass per unit length of the bar and S the cross-sectional area of the bar. The maximum potential energy of the bar at the lowest resonance frequency is 2

(P.E.)max 

1
 d 2  YI dx 2 ∫0  dx 2 

(2.189)

and the maximum kinetic energy of the bar is (K.E.)max 


2 2




∫0 2 dm

(2.190)

The total energy of a lossless vibratory system remains constant, that is Total Energy = Potential Energy + Kinetic Energy = Constant For the mass-loaded beam, the potential energy of the beam is converted to kinetic energy, which is shared by both the beam and the mass load, M. Therefore, the frequency equation for the mass loaded bar is 2

 d 2  YI ∫0  dx 2  dx



2 




∫0 2 dm
Mm2 2


 d 2  YI∫  2  dx 0  dx 






S x ∫  0

2

(2.191)

2 dx
Mm

Equation 2.191 is applicable to beams of any taper. For the uniform, mass-loaded cantilever, Sx = S is constant, so that


2 


 d 2  YI∫  2  dx 0  dx 


2 S∫  2 dx
Mm

(2.192)

0

The displacement curve, , usually assumed for the uniform cantilever is 

m ( 3
x 2  x 3 ) 2
3

(2.193)

This curve meets the criteria of the boundary conditions—when x = 0,  = 0, and d/dx = 0. Taking the derivatives of  required by Equation 2.192 and solving for ω2 leads to
2 

YI∫

0

S∫




0



CRC_DK8307_CH002.indd 91




2 9m (
2  2
x
x 2 )dx 6


2 m 2 (9
2x 4  6
x 5
x 6 )dx
Mm 4
6

3YI  3  33m  140
M 


(2.194)

10/29/2008 3:48:20 PM

Ultrasonics: Data, Equations, and Their Practical Uses

92

For the unloaded flexural, uniform bar (M = 0),
2 

3YI YI  12.7273 33S
4 S
4 140

(2.195)

or
 3.56753

YI S
4

(2.195a)

In Table 2.4, the lowest mode resonance frequency listed for the fixed–free (cantilever) bar is given by f11 

c (1.194)2 8


(2.196)

Equation 2.196 may be written   YI  YI
11  2f11  2  (1.194 2 )  3.5176 4  S  S
4 8
 

(2.196a)

which is within 0.074% of what is usually claimed as the exact equation. As stated previously, Equation 2.195a would have been the exact equation if the assumed bending curve had been exact. A more accurate expression than that of Equations 2.195 and 2.195a for the cantilever with a mass, M, at the free end is obtained by rewriting Equation 2.194 in the form
2 

3YI [Km
M]
3

(2.194a)

and determining the value of the constant, K, by assuming M = 0 and utilizing the more accurate Equation 2.196a as follows:
02 

3YI 3YI YI 2  
11  (3.5176)2 3 4 Km
KS
S
4

(2.196b)

K = 0.2424534 Then the frequency of the mass-loaded, uniform cantilever beam is determined by the relationship 2 
m

3YI [0.24245S

M]
3

(2.196c)

For the special case in which M = ρSℓ = m, the mass of the beam,  YI  2
m  2.4146   M
3 

CRC_DK8307_CH002.indd 92

(2.196d)

10/29/2008 3:48:20 PM

Ultrasonic Horns, Couplers, and Tools 2.5.4.5

93

Stresses in Bars in Flexure

The operating life of flexural bars used for high-intensity ultrasonic applications is a function of the maximum stresses to which the bar will be subjected during use. The maximum stresses in a uniform bar in flexure appear in the outer fibers of the bar. These stresses alternate between compression and tension, and because the frequency of vibration is ultrasonic, fatigue failure can occur rapidly if the displacement amplitude is excessive. Bending stresses in a beam are compressive on the side from the neutral axis nearest the center of curvature and tensile on the side of the neutral axis away from the center of curvature. In a uniform bar, the stress is s

My d 2  Yy 2 I dx

(2.197)

where s is the stress at y and y the distance from the neutral axis. The maximum stress occurs at the point where y = c, where c is the distance from the neutral axis to the outer fiber. Therefore, maximum stress, sm, is sm 

Mc d 2  Yc 2 I dx

(2.198)

The steps to determine the maximum stress in flexural bars under any of the conditions discussed in the following section are: 1. 2. 3. 4. 5.

Determine the equation describing displacement, . Obtain the second derivative of displacement with respect to x. Determine the distance, c, between the neutral axis and the outer fibers. Obtain Young’s modulus of elasticity for the materials of the bar. Use these quantities in Equation 2.198 to obtain an equation of stress as a function of displacement, .

The resulting equation is useful for determining the safe operating conditions (displacement) for the bar based upon its endurance limit. For example, assume that a bar is operated in a pinned–pinned mode. The equation of motion at resonance is (Table 2.4)   msin


x v

(2.199)

and d 2
2
x   sin m dx 2 v2 v

(2.200)

Assume that the cross section of the bar is rectangular with thickness, ha, parallel to displacement. The distance, c, for this bar is c = ha/2. Therefore, the equation for the maximum (outer-fiber) stress at position x is sm 

Yh
2
x m 2 sin 2 v v

(2.201)

for the uniform bar of rectangular cross section vibrating in a pinned–pinned mode.

CRC_DK8307_CH002.indd 93

10/29/2008 3:48:20 PM

Ultrasonics: Data, Equations, and Their Practical Uses

94 2.5.4.6

Coupling between Driver and Flexural Bar

Careful consideration should be given to conditions of coupling between a driving horn and a flexural bar. Conditions to consider include: (a) coupling position and (b) coupling method. Position and method of coupling should be designed so that they do not interfere with the natural, or designated, modal pattern. Interferences can be caused not only by restricting the freedom to conform to the modal pattern that the bar would exhibit if it were freely vibrating, but also by introducing distortions in stress patterns. For example, welding a horn to a flexural bar will alter the stiffness of the bar at the position affected by the weld and thus tend to alter the modal pattern. Placing this type of couple between a node and an antinode will restrict the slope of the displacement curve and thus alter the modal pattern. Bolt holes alter the stress distribution within the bar, and thus cause a change in the stiffness at the cross section in which the holes are located. The tendency is to lower the frequency by an amount that depends upon the location of the coupling—whether it is a position of high stress or low stress. The nodal positions and positions of maximum displacement given in Table 2.4 should be helpful in locating preferred positions for coupling between a horn and a flexural bar.

2.6

Plates

The plate wave equation may be written in the form ∇ 4


3(1  2 ) 2 0 Yh 2 t 2

(2.202)

where ∇4 = ∇2∇2 where ∇2 is the Laplace operator, Y the Young’s modulus of elasticity,  the displacement amplitude, h the half thickness of the plate, σ Poisson’s ratio, ρ the density of the material of the plate, and t the time. Equation 2.202 may also be written in the form D∇ 4
2h

2 0 t 2

(2.202a)

where D is flexural rigidity or D

2Yh 3 3(1  2 )

(2.203)

Letting ha = 2h (the plate thickness) and ρa = 2hρ (the mass density per unit of area of the plate), Equation 2.202a may also be written as D∇ 4
ρa

CRC_DK8307_CH002.indd 94

2 0 t 2

(2.202b)

10/29/2008 3:48:20 PM

Ultrasonic Horns, Couplers, and Tools

95

where D

Yh a3 12(1  2 )

(2.203a)

Equation 2.202a resembles the other wave equations—for example, 4 2 YI 4  2 Y  4 2 2    c       o t 2 x 4 S x 4 ρ x 4

(1.40)

which is the flexural bar equation. Rearranged  2Y

4 ξ 2
 2  0 4 x t

(1.40a)

or D

4 2
 0 x 4 t 2

(1.40b)

where D is the flexural rigidity of a uniform bar (=κ 2Y). The bar equation can be derived from the plate wave equation, Equation 2.202a. The stresses developed in plates in flexure are more complicated than those for the flexural bar. An incremental element in a plate under stress is subjected to constraints in all directions, which accounts for the term involving Poisson’s ratio in Equation 2.202. The following procedures are based upon the assumption that the motion (x, y) of a unit volume within a resonant plate is harmonic regardless of the mode of vibration, that is, the motion of the volume (or particle) at x, y is described by   (x , y )sin
t or   (x , y )cos
t

(2.204)

Either part of Equation 2.204 when inserted into Equation 2.202a leads to ∇ 4 

2h
2 0 D

(2.205)

which may be written (∇4 − k4) = 0

(2.205a)

where k4 

2h
2 3
2(1  2 )  D Yh 2

Equation 2.205a is a homogeneous linear equation. It can be factored and written in the form (∇2 + k2) (∇2 − k2) = 0

(2.205b)

The complete solution to Equation 2.205b is a combination of solutions to (∇2 + k2) = 0

CRC_DK8307_CH002.indd 95

10/29/2008 3:48:20 PM

Ultrasonics: Data, Equations, and Their Practical Uses

96 and

(∇2 − k2) = 0 following common procedures for solving homogeneous linear equations with constant coefficients. From Equations 1.45a and 1.45b, the Laplace operator may be written as follows: 1. In polar coordinates in a two-dimensional system, ∇2 

1    1  2  2 1  1  2 
2  2  r 
2  2   2
r r r r    r r r r   

(1.45)

where r and φ are the polar coordinates of distance and direction. 2. In rectangular coordinates in a two-dimensional system, ∇2 

2 2
2 2 x y

(1.45)

Morse and Ingard [10] provide solutions to Equation 2.205 in polar coordinates. Leissa [11] provides solutions in polar, elliptical, and rectangular coordinates. He also presents the Laplace operator in skew coordinates but finds no general solutions to Equation 2.205 in skew coordinates that allow a separation of variables. Timoshenko [12] and Ensminger [3] also offer solutions in polar and rectangular coordinates. The present purposes will be satisfied by including solutions in polar coordinates for circular plates and in rectangular coordinates for square or rectangular plates. Lamb wave solutions are presented in rectangular coordinates only. 2.6.1

General Solution to the Plate Equation in Polar Coordinates

Leissa [11] gives the general solution to the plate wave equation, Equation 2.205, in polar coordinates as follows: (r, ) 



∑ [Am Jm (kr)
BmYm (kr)
CmIm (kr)
DmK m (kr)]cos(m )

m0




∑

(2.206) [A *

m Jm

(kr)
B *

m Ym

(kr)
C *

mI m

(kr)
D *

m K m (kr)]sin(m )

0

where  
2  k a   D 

1/ 4

 12a
2 (1  2 )    Yh a3  

1/ 4

 3
2(1  2 )    Yh 2  

1/ 4

where ρa is mass density per unit area of the material of the plate; ha the thickness of the plate; ρ the density of the material of the plate; H the half-thickness of the plate; Jm and Ym Bessel functions of the first and second kinds, respectively; Im and Km modified Bessel functions (hyperbolic Bessel functions) of the first and second kinds, respectively; Am. and Dm constants that determine the mode shape and are solved by applying the boundary conditions; and r and φ the polar coordinates of distance and direction.

CRC_DK8307_CH002.indd 96

10/29/2008 3:48:20 PM

Ultrasonic Horns, Couplers, and Tools 2.6.2

97

General Solution to the Plate Equation in Rectangular Coordinates

The general solution to Equation 2.205 presented by Leissa [11] in rectangular coordinates is

(x, y)  ∑ [A msin k 2   2 y
Bmcos k 2   2 y m1


Cm sinh k 2
 2 y
D m cosh k 2
 2 y]sin x

* sin k 2   2 y
B * cos k 2   2 y
∑ [A m m m=1

* sinh k 2
 2 y
D * cosh k 2
 2 y]cos x
Cm m

(2.207)

where α = ω/v and v is the phase velocity in the plate. 2.6.3

Specific Solutions to the Plate Wave Equations

2.6.3.1

Circular Plate of Uniform Thickness and Radius a Clamped at the Outer Circumference [3,10]

From Equation 2.206, (a, φ) = [AJm(ka) + BYm(ka)]cos(mφ) = 0 = [AJm(ka) + BYm(ka)]sin(mφ) = 0

(2.206a)

from which  J (ka)  B A  m   Ym (ka) 

(2.208)

and Ym (ka)

d d J m (kr)  J m (ka) Ym
(kr) dr dr

(2.209)

These conditions can be satisfied only at discrete frequencies. These frequencies are fixed by the values βmn [11], from Table 2.5, where βmn = (a/π)kmn. k2 


h

3(1  2 ) Y

Therefore fmn 

h 2a 2

Y (mn )2 3ρ(1  2 )

mn n → n
-

(2.210)

m 2

where h is the plate thickness, m the number of nodal diameters, and n the number of nodal circles, including the clamped edge.

CRC_DK8307_CH002.indd 97

10/29/2008 3:48:20 PM

Ultrasonics: Data, Equations, and Their Practical Uses

98 TABLE 2.5

Values of βmn for Clamped Circular Plate β01 = 1.0174 β11 = 1.4677 β21 = 1.8799 β31 = 2.2741 β41 = 2.6568 β51 = 3.0321 β61 = 3.4018 β71 = 3.7678 β81 = 4.1288 β91 = 4.4876 β07 = 7.0019 β17 = 7.4946 β27 = 7.8752 β37 = 8.4454

β02 = 2.0074 β12 = 2.4824 β22 = 2.9274 β32 = 3.3538 β42 = 3.7677 β52 = 4,1722 β62 = 4.5694 β72 = 4.9607 β82 = 5.3472 β92 = 5.7296 β08 = 8,0016 β18 = 8.4954 β28 = 8.9779 β38 = 9.4516

β03 = 3.0047 β13 = 3.4881 β23 = 3.9477 β33 = 4.3911 β43 = 4.8224 β53 = 5.2442 β63 = 4.6584 β73 = 6.0664

β04 = 4.0030 β14 = 4.4910 β24 = 4.9590 β34 = 5.4129 β44 = 5.8556 β54 = 6.2893

β05 = 5.0027 β15 = 5.4928 β25 = 5.9668 β35 = 6.4273

β09 = 9.0015 β19 = 9.4958 β29 = 9.9803 β39 = 10.4559

From the values of βmn, Y  h f01  0.9387  2   a  (1  2 )

(2.210a)

and 2

  fmn   mn  f01  0.9661(mn )2 f01  01 

(2.210b)

Values of βmn and radii of nodal circles in the clamped-edge circular plate are independent of Poisson’s ratio. The relative radii of nodal circles, rn/a, are determined by the equation  r   r  Jm  k n  Im  k n   a  a  J m (ka) I m (ka)

(2.211)

Leissa [11] tabulates relative values of radii of nodal circles (r n/a) for the clamped plate. 2.6.3.2

Free Circular Plate of Uniform Thickness and Radius a

The vibrational modes and corresponding frequencies of resonance for a free circular plate with m nodal diameters and n nodal circles can be determined by applying the following values of βmn, from Table 2.6, to Equation 2.210. Values for relative radii of nodal circles (rn/a) for a freely vibrating circular plate (Poisson’s ratio = 0.33) are given by Leissa [11]. 2.6.3.3

Circular Plate with Fixed Center

The frequencies of vibration for circular plates with centers fixed corresponding to modes having nodal diameters are the same as those for vibrations of corresponding modes in a free plate. The values of βmn for a circular plate with its center fixed and with n nodal circles and the corresponding relative radii (rn/a) (with σ = 1/3) are listed in Table 2.7.

CRC_DK8307_CH002.indd 98

10/29/2008 3:48:21 PM

Ultrasonic Horns, Couplers, and Tools

99

TABLE 2.6 Values of βmn for Free Circular Plate of Uniform Thickness and Radius a (Poisson’s ratio = 0.33) β01 = 0.9594 β02 = 1.9763 β03 = 2.9826 β04 = 3.9884 β05 = 4.9915 β06 = 5.9940 β07 = 6.9963 β08 = 7.9959 β09 = 8.9953

β11 = 1.4419 β12 = 2.4627 β13 = 3.4724 β14 = 4.4813 β15 = 5.4847 β16 = 6.4868 β17 = 8.1266 β18 = 8.4894 β19 = 9.4886 β20 = 0.7283

β21 = 1.8899 β22 = 2.9156 β23 = 3.9501 β24 = 4.9589 β25 = 5.9618 β26 = 6.9680 β27 = 9.6915 β28 = 8.9728 β29 = 9.9728 β30 = 1.1132

β31 = 2.3154 β32 = 3.3581 β33 = 4.4118 β34 = 5.4272 β35 = 6.4327 β36 = 7.4392 β37 = 8.4415 β38 = 9.4442 β39 = 10.441 β40 = 1.4794

β41 = 2.7215 β42 = 3.8038 β43 = 4.8515 β44 = 5.8728 β45 = 6.8854 β46 = 7.8938 β47 = 8.9002 β48 = 9.9060 β49 = 10.911 β50 = 1.8313

β51 = 3.1155 β52 = 4.2108 β53 = 5.2747 β54 = 6.3054 β55 = 7.3246 β56 = 8.3395 β57 = 9.3585 β58 = 10.3683 β59 = l 1.3748

TABLE 2.7 Values of βmn and Relative Radii of Nodal Circles (rn/a) for a Circular Plate with Fixed Center

2.6.3.4

n

βmn

rn/a

0 1 2 3 4 5 6

0.6166 1.4556 2.4902 3.4956 4.5005 5.4967 6.4969

— 0.8682 0.9050, 0.5075 0.9288, 0.6447, 0.3615 0.9287, 0.7214, 0.5008, 0.2808 0.9549, 0.7719, 0.5907, 0.4100, 0.2299 0.9608, 0.8079, 0.6531, 0.4997, 0.3469, 0.1945

Circular Plates Simply Supported All Around

The boundary conditions for a simply supported circular plate are, at the circumference: 1. The displacement parallel to the axis is (a) = 0. 2. Bending and twisting moments are zero. 3. ∂2/∂θ2 = 0. These conditions, when applied to Equation 2.209, lead to AnJn(mn) + CnJn(mn) = 0         A n  J′′n (mn )
 mn  J′n (mn )
C n I′′n (mn )
 mn  I′n (mn )  0          

(2.212)

where 2  2mn 
a 2

3(1  2 ) Yh 2

(2.213)

The frequency equation is J n
1(mn ) I n
1(mn ) 2mn
 J n (mn ) I n (mn ) 1

CRC_DK8307_CH002.indd 99

(2.214)

10/29/2008 3:48:21 PM

Ultrasonics: Data, Equations, and Their Practical Uses

100

The roots of Equation 2.214 for σ = 0.3 are β01 = 0.7101 β02 = 1.7365 β03 = 2.7419 β04 = 3.7439

β11 = 1.1885 β12 = 2.2170 β13 = 3.2274 β14 = 4.2329

β21 = 1.6121 β22 = 2.6658 β23 = 3.6892 β24 = 4.7024

These values of βmn may be applied to Equation 2.210 to obtain the allowed frequencies for the various modes of vibration. The relative radii of nodal circles (rn/a) of a simply supported circular plate for σ = 0.3 are given by Leissa [11]. 2.6.4

Annular Plates of Uniform Thickness

An annular plate is a circular plate containing a concentric hole. There are nine possible basic (or simple) boundary conditions for the annular plate: Center clamped

Center simply supported

Center free

Outer circumference clamped Outer circumference simply supported Outer circumference free Outer circumference clamped Outer circumference simply supported Outer circumference free Outer circumference clamped Outer circumference simply supported Outer circumference free

One may easily visualize innumerable combinations of these basic conditions, such as center clamped over only a small portion of its circumference and the outer circumference clamped over the same segment of the plate or over a different segment of the plate, and so on. Only a few conditions have been selected for discussion for their potential benefit to ultrasonic power applications. For example, an annular plate simply supported or clamped at the internal circumference and free at the external circumference might represent a wheel driven axially at an ultrasonic frequency. The area surrounding the inner circumference of such a wheel is subjected to concentrated stresses by displacements along the axis of the center hole, causing serious fatigue problems under the influence of high-power ultrasonic energy. Fatigue damage may be minimized by using materials of high fatigue resistance and by careful geometrical design. Consistent with the terminology used in the previous discussions, the radius of the outer edge of the plate is designated a. The inner radius is designated b. Values of the frequency constant, βmn, for various ratios, b/a, for various boundary conditions are presented in Tables 2.8 through 2.16. Frequencies of the various modes may be determined by Equation 2.210. 2.6.4.1

Annular Plates Clamped on Outside and Inside

The boundaries are nodal circles characterized by (r) = d/dr = 0 at r = a and r = b. These conditions applied to Equation 2.206 lead to 2

2

2 2 2 2    r    r    r   r  r  (r)  A 1     1     ln  
B 1     1      b    b    a   a  a         

2

(2.215)

The allowed frequencies are given in Equation 2.210.

CRC_DK8307_CH002.indd 100

10/29/2008 3:48:21 PM

Ultrasonic Horns, Couplers, and Tools

101

TABLE 2.8 Frequency Parameters, βmn, for Annular Plates Clamped at the Inner and the Outer Circumferences Nodal diameters m 0 1 2 3 0 1 2 3

βmn for Values of b/a

Nodal circles n

0.1

0.3

0.5

0.7

0.9

2 2 2 2 3 3 3 3

1.6632 1.6963 1.9283 2.2776 2.7622 2.8220 3.0281 3.3687

2.1400 2.1729 2.2732 2.4657 3.5588 3.5572 3.6847 3.8330

3.0063 3.0231 3.0746 3.1671 4.9925 5.0127 5.0630 5.1227

5.0127 5.0228 5.0430 5.0930 8.3370 8.3370 8.3553 8.3855

15.0551 15.0584

24.9970 24.9970

The values of βmn for various ratios, b/a, that apply to the annular plate clamped at both the inner and the outer circumferences are given in Table 2.8. The values of n given in Table 2.8 include the nodal circles corresponding to the clamped conditions at the inner and the outer circumferences. 2.6.4.2

Annular Plate Clamped on Outside and Simply Supported on the Inside

The nodal circle corresponding to the clamped outer circumference is characterized by (r) 

d 0 dr

The nodal circle corresponding to the simple support at the inner circumference is characterized by (r) = 0 The corresponding displacement equation is 2 2 2   r 2   r   r   r   r (r)  A 1     1     ln  
B 1     1      b    a   a   a   b       

2

(2.215a)

The frequency parameters, βmn, for various ratios of b/a that apply to the annular plate clamped at the outer circumference and simply supported at the inner circumference are given in Table 2.9. The values of n listed in Table 2.9 include both the clamped outer circumference and the simply supported inner circumference. 2.6.4.3

Annular Plates Clamped on Inside and Simply Supported on the Outside

The nodal ring corresponding to the clamped inner circumference is characterized by d  = ___ = 0 dr The outer nodal ring corresponding to the simply supported circumference is characterized by =0

CRC_DK8307_CH002.indd 101

10/29/2008 3:48:21 PM

Ultrasonics: Data, Equations, and Their Practical Uses

102 TABLE 2.9

Frequency Parameters, βmn, for Annular Plates Clamped on the Outside and Simply Supported on the Inside βmn for Values of b/a m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 3

2 2 2 2 3 3 3 3

1.5132 1.5947 1.8939 2.2732 2.5781 2.6727 2.9639 3.3536

1.8478 1.9045 2.0824 2.3542 3.2461 3.2926 3.4283 3.6293

2.5445 2.5742 2.6632 2.8130 4.5240 4.5352 4.6127 4.6998

4.2108 4.2108 4.2468 4.3295 7.5191 7.5326 7.5527 7.5995

12.5319 12.5359 12.5440 12.5642 22.5169 22.5169 22.5237 22.5349

TABLE 2.10 Frequency Parameters, βmn, for Annular Plate Clamped on the Inside and Simply Supported on the Outside βmn for Values of b/a m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 3

2 2 2 2 3 3 3 3

1.3430 1.3875 1.6474 2.0132 2.4677 2.5225 2.7511 3.1074

1.7405 1.7837 1.9152 2.1448 3.1831 3.2148 3.3233 3.4869

2.4615 2.4861 2.5584 2.6821 4.4790 4.5016 4.5575 4.6237

4.1258 4.1503 4.1746 4.2348 7.4786 7.4854 7.5124 7.5527

12.4711 12.4751 12.4833 12.4954 22.4831 22.4831 22.4899 22.5012

The displacement equation is 2

2 2    r   r  r  (r)  A 1     ln  
B 1      b   a  b     

2

2   r  1     a  

(2.215b)

Frequency parameters, βmn, for various ratios, b/a, for annular plates clamped on inside and simply supported on outside are given in Table 2.10. The listed values of n include the nodal circle corresponding to the simple outer support and that corresponding to the clamped inner support. 2.6.4.4

Annular Plate Simply Supported on Inside and Outside Circumferences

The displacement of the annular plate, simply supported at inside and outside circumferences, is described by the equation (Leissa [11]) 2 2 2    r   r   r  r  (r)  A 1     ln  
B 1     1      a   b   a  b       

(2.215c)

The nodal circles corresponding to the simple supports at the inside and the outside circumferences are both characterized by =0

CRC_DK8307_CH002.indd 102

10/29/2008 3:48:21 PM

Ultrasonic Horns, Couplers, and Tools

103

Frequency constants, βmn, corresponding to various ratios, r/a, for the simply supported annular plate are given in Table 2.11. The values of n include the nodal circles corresponding to the inner and outer circumferences. 2.6.4.5

Annular Plate Clamped on Inside Circumference and Free on Outside Circumference

The one nodal circle associated with the clamped inner circumference is characterized by (r) 

d 0 dr

The displacement, (r), is described by 2 2    r   r  r  (r)  A 1     ln  
B 1      b   b  b     

2

(2.215d)

The frequency constants, βmn, corresponding to various ratios, b/a, for the annular plate clamped at the inner circumference and free at the outer circumference are given in Table 2.12. The values of n listed in Table 2.12 include the nodal circle associated with the clamped inner circumference. TABLE 2.11 Frequency Parameters, βmn, for a Simply Supported Annular Plate βmn for Values of b/a m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 3

2 2 2 2 3 3 3 3

1.2121 1.3008 1.6199 2.0132 2.2887 2.3926 2.6953 3.0976

1.4621 1.5365 1.7493 2.0629 2.8789 2.9278 3.0746 3.3080

2.0132 2.0580 2.1845 2.3820 4.0137 4.0389 4.1135 4.2348

3.3385 3.3687 3.4283 3.5158 6.6693 6.6845 6.7072 6.7748

10.0053 10.0053 10.0306 10.0558 20.0004 20.0004 20.0105 20.0257

TABLE 2.12 Frequency Parameters, βmn, for Annular Plate Clamped at the Inner Circumference and Free at the Outer Circumference βmn for Values of b/a

CRC_DK8307_CH002.indd 103

m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 3

1 1 1 1 2 2 2 2

0.6547 0.5640 0.7546 1.1209 1.6011 1.6632 1.9362 2.3217

0.8215 0.8009 0.8981 1.1595 2.0776 2.1258 2.2710 2.5084

1.1477 1.1608 1.2204 1.3691 2.9364 2.9639 3.0481 3.1831

1.9362 1.9492 1.9955 2.0776 4.9210 4.9415 4.9925 5.0630

2.2843 5.9123 5.9295 5.9720 9.9137 14.8927 14.9097 14.9301

10/29/2008 3:48:21 PM

Ultrasonics: Data, Equations, and Their Practical Uses

104 2.6.4.6

Annular Plate Simply-Supported at the Inner Circumference and Free at the Outer Circumference

The nodal circle corresponding to the simple support at the inner circumference is characterized by =0 The displacement is described by 2 2 2   r  r   r   r  (r)  A ln  
B 1    
C   1      b  b   a   b     

(2.215e)

Frequency constants, βmn, corresponding to various ratios, b/a, for the annular plate simply supported at the inner circumference and free at the outer circumference are given in Table 2.13. The values of n listed in Table 2.13 include the nodal circle associated with the simple support at the inner circumference. 2.6.4.7

Free–Free Annular Plate

Table 2.14 lists frequency constants, βmn, corresponding to various ratios, b/a, for the free–free annular plate. TABLE 2.13 Frequency Parameters, βmn, for Annular Plate Simply Supported on the Inner Circumference and Free on the Outer Circumference βmn for Values of b/a m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 3

1 1 1 1 2 2 2 2

0.5912 0.4827 0.7411 1.1209 1.4517 1.5626 1.9045 2.3173

0.5887 0.5800 0.7849 1.1299 1.7893 1.8696 2.0873 2.3969

0.6453 0.7017 0.8992 1.1910 2.4861 2.5325 2.6575 2.8524

0.7913 0.9192 1.1652 1.4412 4.1503 4.1746 4.2348 4.3295

1.3201 1.6199 2.0776 2.4942 12.4711 12.4751 12.4954 12.5238

TABLE 2.14 Frequency Constants, βmn, Corresponding to Various Ratios, b/a, for a Free–Free Annular Plate βmn for Values of b/a

CRC_DK8307_CH002.indd 104

m

n

0.1

0.3

0.5

0.7

0.9

2 3 0 1 2 3 0 1

0 0 1 1 1 1 2 2

0.7328 1.1209 0.9426 1.4412 1.8805 2.3173 1.9674 2.4450

0.7053 1.1145 0.9204 1.3617 1.8286 2.2732 2.2598 2.4408

0.6585 1.0747 0.9718 1.3201 1.7751 2.1915 3.0581 3.1237

0.6014 0.9995 1.1565 1.4930 1.9570 2.3756 5.0430 5.0630

0.5458 0.9082 1.8805 2.3756 3.0828 3.6984 15.0584 15.0652

10/29/2008 3:48:21 PM

Ultrasonic Horns, Couplers, and Tools

105

TABLE 2.15 Frequency Parameters, βmn, Corresponding to Various Ratios, b/a, for Annular Plates Clamped at the Outer Circumference and Free at the Center βmn for Values of b/a

2.6.4.8

m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 0

1 1 1 1 2 2 2 3

1.0166 1.4621 1.8696 2.2732 2.0005 2.4656 2.9069 3.0265

1.0747 1.4056 1.8146 2.2304 2.2887 2.4615 2.8292 3.6571

1.3392 1.4930 1.8006 2.1542 3.0828 3.1398 3.3080 5.0630

2.0897 2.1424 2.2843 2.4922 5.0630 5.0730 5.1227 8.3734

6.0395 6.0523 6.0813 6.1228 14.9943 14.9978 15.0146 25.0294

Annular Plate Clamped at the Outer Circumference and Free at the Inner Circumference

The displacement for this condition is described by 2 2    r   r  r  (r)  A 1     ln  
B 1      a   a  a     



2

(2.215f)

d 0 dr

The nodal circle associated with the clamped outer circumference is characterized by the frequency constants, βmn, corresponding to various ratios, b/a, for an annular plate clamped at the outer circumference and free at the inner circumference are given in Table 2.15. The values of n given in Table 2.15 include the nodal circle associated with the clamped outer circumference. 2.6.4.9

Annular Plate Simply Supported at the Outer Circumference and Free at the Center

The displacement for this condition is described by 2 2 2   r  r   r   r  (r)  ln  
B 1    
C   1      a  a   a   a     

(2.215g)

The nodal circle corresponding to the simply supported outer circumference is characterized by (a) = 0 Frequency constants, βmn, corresponding to various ratios, b/a, are given in Table 2.16. The values of n listed in Table 2.16 include the nodal circle corresponding to the simple outer support.

CRC_DK8307_CH002.indd 105

10/29/2008 3:48:21 PM

Ultrasonics: Data, Equations, and Their Practical Uses

106 TABLE 2.16

Frequency Constants, βmn, Corresponding to Various Ratios, b/a, for Annular Plates Simply Supported at the Outer Circumference and Free at the Center βmn for Values of b/a

2.6.5

m

n

0.1

0.3

0.5

0.7

0.9

0 1 2 3 0 1 2 0

1 1 1 1 2 2 2 3

0.7017 1.1867 1.6042 2.0132 1.7259 2.2053 2.6479 2.7530

0.6871 1.1388 1.5626 1.9827 1.9362 2.1542 2.5683 3.2926

0.7167 1.0841 1.5032 1.9019 2.5820 2.6613 2.8666 4.5352

0.8379 1.1608 1.5691 1.9414 4.2108 4.2468 4.3295 7.5191

1.3392 1.7347 2.2776 2.7474 12.5319 12.5440 12.5642 22.5169

Rectangular Plates

Equation 2.216 is the general solution to the plate wave equation in rectangular coordinates, that is

(x, y )  ∑ [A m sin k 2   2 y
Bm cos k 2   2 y m1


C*m sinh k 2
 2 y
D*m cosh k 2
 2 y]cos x


∑ [A*m sin k 2   2 y
B*m cos k 2   2 y m0


Cm sinh k 2
 2 y
D m cosh k 2
 2 y] sin x

(2.216)

There are 21 combinations of simple boundary conditions for rectangular plates, that is, any designated side may be clamped, simply supported, or free for its full length. Obviously, the ultrasonics engineer will encounter numerous mounting conditions that do not conform to these simple boundary conditions. Leissa [11] discusses the case of “a plate supported by (or embedded in) a massless elastic medium (or foundation).” He then modifies Equation 1.43 to include a constant K representing the stiffness of the foundation measured in units of force per unit length of deflection per unit of area of contact, as follows: D∇ 4
K
a

2 0 2

(2.217)

The parameter k then is defined by k4 

 a
2  K D

where ρa = 2ρh.

CRC_DK8307_CH002.indd 106

10/29/2008 3:48:21 PM

Ultrasonic Horns, Couplers, and Tools

107

The results pertaining to the classical plate equation also apply to the case of the mounting in an elastic medium when Equation 2.217 is used instead of Equation 2.205b, that is k4 

3
2(1  2 ) 2h
2  Yh 2 D

The case of an elastic mount having significant mass requires incorporating the conditions of the mounting coupled to the plate into the differential equation. The problem is related to impedance matching of elements in a combined resonant system. Solutions to equations for plates subjected to the various edge conditions range from fairly easy to very complicated. Assumptions in common with those made for flexural bars and circular plates used in obtaining the solutions that follow are that the materials are perfectly elastic, homogeneous, isotropic, and uniform in thickness. The thickness is considered to be small compared with all other dimensions and with the wavelength. In all of the following special solutions, the length dimension a of the plate is parallel to x and the width dimension b is parallel to y in an xy plane. For square plates, a equals b. The flexural rigidity is the same as it was for the circular plates, D

2Yh 3 3(1  2 )

(2.203)

The important stress–strain-related factors as functions of displacements in a resonant plate are as follows, in rectangular coordinates: 1. Bending and twisting moments:  2 2  M x  D  2
2  y   x  2 2 M y  D  2
2  x   y M xy  D(1  )

(2.218)

2 xy

2. Transverse shearing forces: Qx  D

 (∇ 2) x

 Qy  D (∇ 2) y

(2.219)

3. The Kelvin–Kirchoff edge reactions: Μxy y Μxy Vy  Qy
x Vx  Qx


CRC_DK8307_CH002.indd 107

(2.220)

10/29/2008 3:48:22 PM

Ultrasonics: Data, Equations, and Their Practical Uses

108

The solutions to the equation for rectangular plates depend upon the effects of the boundary conditions (clamped, simply supported, or free) on these quantities. The simple boundary conditions for rectangular plates are as follows: 1. For any side that is clamped: Assuming (a) that the edge is perfectly clamped so that the edge cannot move, (b) that the tangent plane to the deflected middle surface along this edge coincides with the initial position of the middle plane of the plate, and (c) that the x-axis coincides with the clamped edge and this clamping occurs at y = 0, ()y=0 = 0     y 

0 y0

2. For any side that is simply supported, for example, at y=0 Here, the edge is perfectly constrained so that ()y=0 = 0 The edge can rotate freely with respect to the x-axis, that is, there are no bending moments My along the edge or  2 2  (M y )y0  D  2
2   0 x   y 3. For any side that is free, at the free edge, the plate is assumed to experience no bending and twisting moments and no vertical shearing forces, therefore (My)y=0 = (Mxy)y=0 = (Qx)x=0 = 0 where Qx is vertical shearing force. Only two of these conditions are necessary for the complete determination of deflections  satisfying the plate wave equation. The two boundary conditions are: a. From the requirement that bending moments along the free edge are zero,  2 2  (M y )x0  D  2
2  0 x  x0  y

(2.221)

b. The distribution of twisting elements Mxy is statically equivalent to a distribution of shearing forces of the intensity  M xy  Qx     y  x0

CRC_DK8307_CH002.indd 108

10/29/2008 3:48:22 PM

Ultrasonic Horns, Couplers, and Tools

109

Also the edge reactions Vx = 0. From Equations 2.220, M xy   Vx   Q x  0 y  xa  These requirements lead to the second boundary condition:  3 3   3
( 2  )  0 xy 2  x0  x

(2.222)

4. For any side that is elastically supported and elastically built-in: Timoshenko [12] provides the boundary conditions for the “elastically supported and elastically built-in edge” which includes the edge x = a of a rectangular plate rigidly joined to a supporting beam. One of the boundary conditions of the plate along the edge x = a relates to the deflection curve of the beam described by the equation   4   2     2 B 4   D  2
( 2  ) 2  x  x y  xa  y  xa

(2.223)

The second boundary condition relates to the twisting of the beam leading to C

  2  2     2   D 2
2    y  xy  xa y  xa  x

(2.224)

where B is the flexural rigidity of the beam and C the torsional rigidity of the beam. The following solutions, and approximate solutions, to the plate wave equation are intended to represent those boundary conditions of most value to the designer of ultrasonic power equipment. Some boundary conditions, including reactions from mounting structures, present very difficult mathematical conditions for analysis. Finite element analysis and other methods are available to the engineer who has the need to perform such studies. For example, Donnell’s model has been used successfully in analyzing modal conditions and frequencies in plates and cylinders. This method is discussed with examples of applications in Chapter 3. Approximate methods of matching systems often prove to be satisfactory, but experimentally determined parameters may be necessary to arrive at an acceptable design, particularly in the absence of a satisfactory finite element approach. 2.6.5.1

Rectangular Plate Simply Supported on All Sides

The rectangular plate having all sides simply supported (SSSS) leads to the simplest solution to the rectangular plate equation. According to Warburton [13], this is the only boundary condition for which the frequency factor can be expressed exactly by a simple formula. Timoshenko [12] gives the displacement function for this case in the form

(x, y )  ∑



∑ mn sin

m1 n1

CRC_DK8307_CH002.indd 109

ny mx sin a b

(2.225)

10/29/2008 3:48:22 PM

Ultrasonics: Data, Equations, and Their Practical Uses

110 The boundary conditions are



2 2  0 x 2 y 2

so that   Μx  0 (for x  0, a)   Μy  0 (for y  0, b) which, when applied to Equation 2.225 leads to mn  A mn sin

ny mx sin a b

(2.226)

where Amn is an amplitude coefficient determined from the initial conditions of the problem and m and n are integers. For any mode of vibration, the nodal pattern is defined by m and n, the number of nodal lines in the x and y directions respectively, that is, the nodal lines m are normal to the x direction (side a) and nodal lines n are normal to the y direction (side b). The numbers m and n include also constrained boundaries. The frequency equation is [3]
mn   2h

Y  m2 n2 
2 2  3(1  )  a 2 b 

(2.227)

Because a equals b, the frequency equation for the square plate simply supported on all sides is [3]
mn   2h

Y  m2
n2  2  3(1  )  a 2 

(2.227a)

which, for the lowest mode, is
11 

2 2 h a2

Y 3(1  2 )

(2.227b)

The nodes for rectangular plates, in general, are straight lines parallel to the edges. In the case of square plates simply supported on all sides, the nodal lines follow the general pattern of rectangular plates. The nodal lines are always parallel to the edges. For square plates with other boundary conditions, the nodal lines may or may not be parallel to the sides depending upon the modes and method of excitation. Variations in nodal patterns are attributable to the fact that two mode shapes having the same frequency can exist simultaneously in the square plate. The relative amplitudes of these two modes depend upon the initial conditions. 2.6.5.2

Rectangular Plate Simply Supported on Two Opposite Sides and Clamped on the Remaining Two Sides

Figure 2.19 illustrates the SS-C-SS-C condition (simply supported on two opposite sides and clamped on the remaining two sides). Here the clamped edges are parallel to the x-axis (side a) and the simply supported edges are parallel to the y-axis (side b).

CRC_DK8307_CH002.indd 110

10/29/2008 3:48:22 PM

Ultrasonic Horns, Couplers, and Tools

111 F

F b

a

−F FIGURE 2.19 Plate clamped on two opposite sides and simply supported on the remaining two edges.

As stated previously, the boundary conditions at the simply supported edges are ()x=0,b = 0

(Displacement at x = 0, a)

Bending moments, Mx, along y are   2  2  (M x )x0 ,a  D  2
2   0 x   y where D

2Yh 3 3(1  2 )

and the boundary conditions along the clamped edges are    ()y0 , b    0  y  y0,b

1  k 2   2

2  k 2
 2 Applying these conditions to the general solution to the plate wave equation in rectangular coordinates (Equation 2.216) leads to the four homogeneous equations Bm + Dm = 0 Am λ1 + Cm λ 2 = 0

CRC_DK8307_CH002.indd 111

10/29/2008 3:48:22 PM

Ultrasonics: Data, Equations, and Their Practical Uses

112

Am sin λ1b + Bm cos λ1b + Cm sin h λ 2b + Dm cos h λ 2b = 0 Am λ1 cos λ1b − Bm λ1 sin λ1b + Cm λ 2 cos h λ 2b + Dm λ 2 sin h λ 2b = 0

(2.228)

where, as Leissa [11] points out, “for a nontrivial solution the determinant of the coefficients of Equations 2.228 must vanish”, that is 0

1 0

0

1 0

1

2 0 sin 1b cos 1b sinh λ 2 b cosh 2 b

1 cos 1b  1 sin 1b 2 cos h 2 b 2 sinh 2 b

(2.229)

leading to the characteristic equation 2 λ1λ 2(cos λ1b cos h λ 2b − 1) + (λ1λ 2) (sin2 λ1b − sin h2 λ 2b) = 0

(2.229a)

Various investigators have solved this problem for rectangular and square plates. Table 2.17 provides a summary of frequency constant data given in terms of ωa2√(ρh/D) for square plates for ratios of a/b.

TABLE 2.17 Frequency Parameters, λmn = ωa2√(ρh/D) for SS-C-SS-C Rectangular Plates ωa2√(ρh/D) for Values of n a/b

m

1

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0

1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6

12.139 13.718 15.692 18.258 20.824 24.080 28.946 56.25 95.28 42.58 54.743 78.975 115.6 91.7 102.21 123.3 156.0 160.7 170.3 189.23 219.20 249.43 258.5 275.85 303.6 358.0 366.8

CRC_DK8307_CH002.indd 112

2

3

4

23.65

38.7

58.63

69.32 146.03 253.6 51.65 94.584 170.1 276.8 100.23 140.19 212.85 318.0 168.98 206.6 275.18 378.4 257.5 293.8 360.0 458.8 366.0 400.9

129.1 280.13 492.0 66.3 154.77 305.325 516.4 114.35 199.8 348.3 558.0 182.63 265.2 410.85 619.2 271.0 351.1 493.425 698.4 379.25 457.4

208.4 458.33 808.4 86.15 234.58 483.98 834.4 133.78 279.63 527.63 877.2 201.73 344.6 590.63 938 289.75 429.8 672.525 1018.8 398 535.1

5 83.475

307.3 680.4 1202.8 110.95 333.93 706.74 1229.6 158.43 379.27 751.28 1272.8 226.05 443.8 814.5 1335.6 314.25 529.0 896.625 1416.4 421.5 633.7

6 113.225

425.9 947.03 1676.0 140.88 452.88 973.8 1704.4 188.05 498.5 1019.0 1748.4 252.25 563.5 1082.5 1811.6 343.75 647.9 1165.5 1893.2 450.5 752.2

10/29/2008 3:48:22 PM

Ultrasonic Horns, Couplers, and Tools 2.6.5.3

113

Other Rectangular Plate Conditions

Previously, it was mentioned that there are 21 combinations of simple boundary conditions for rectangular plates, that is, any designated side may be clamped, simply supported, or free for its full length. Additional conditions may be encountered by the ultrasonics engineer that do not conform to these simple boundary conditions for the full length of any or all sides (e.g., clamping along only a portion of one side, and so on). Also, the ultrasonics engineer is always confronted with the types of loads to which the plates are subjected (air, vacuum, liquid), method of mounting driving elements (welding, bolting, etc.) and other geometrical (thickness variations) and structural differences (anisotropy, etc.) within the plates themselves. The engineer should be aware of the different methods available for solving with sufficient accuracy the many conditions that can be encountered in ultrasonics applications. Vibrating plates will react with the supporting structures. Two of the 21 combinations of simple boundary conditions for rectangular plates have been presented, relying heavily upon the fi ne work of Leissa [11]. The material presented in this chapter has been reworked for conciseness and convenience in use. Those previously presented are (a) rectangular plate having all sides simply supported (SSSS) and (b) rectangular plate having two opposite edges simply-supported and the remaining two edges are clamped (SS-C-SS-C), which was also fairly easy to solve. The remaining combinations of simple boundary conditions are: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Two opposite sides simply supported with the remaining two sides free (SS-F-SS-F) Simply supported-clamped-simply supported-simply supported (SS-C-SS-SS) Simply supported-clamped-simply supported-free (SS-C-SS-F) Simply supported on three sides and free on the remaining side (SS-SS-SS-F) Clamped on all four sides (C-C-C-C) Clamped on three sides with the fourth simply supported (C-C-C-SS) Clamped on three sides and free on the fourth (C-C-C-F) Clamped along two adjacent sides and simply supported on the remaining two sides (C-C-SS-SS) Clamped on two adjacent sides, simply supported on the third, and free on the fourth side (C-C-SS-F) Clamped on two adjacent sides and free on the two remaining sides (C-C-F-F) Clamped along two opposite sides, simply supported along one side, and free on the remaining side (C-SS-C-F) Clamped along one side, simply supported along two adjacent sides, and free on the fourth side (C-SS-SS-F) Clamped along one side, simply supported along one side adjacent to the clamped side, and free on the remaining two adjacent sides (C-SS-F-F) Clamped along two opposite sides and free on the remaining two sides (C-F-C-F) Clamped along one side, free along the two sides adjacent to the clamped edge, and simply supported along the remaining edge (C-F-F-SS) Clamped along one edge and free along the remaining three edges (C-F-F-F) (Cantilever) Simply supported along two adjacent edges and free along the remaining two edges (SS-SS-F-F)

CRC_DK8307_CH002.indd 113

10/29/2008 3:48:22 PM

Ultrasonics: Data, Equations, and Their Practical Uses

114

18. Simply supported along one edge and free along the remaining three edges (SS-F-F-F) 19. Free along all edges (F-F-F-F) The remaining 19 combinations of boundary conditions will not be treated here, for several reasons. Should the reader have need for solutions to any of these conditions, Leissa [11] offers sets and tables of solutions that are as complete as one might find in one volume. His book, Vibration of Plates, has been revised, reprinted, and published by the Acoustical Society of America through the American Institute of Physics (1993, ISBN 1-56396-294-2). Using the same basic equations, boundary conditions, and methods presented by Leissa and other investigators, modern computer techniques and programs available commercially make possible solutions to meet specific needs of the designer of ultrasonic systems. Finite element methods and Donnell’s model (Chapter 3) are examples of techniques available to the average engineer. A more complete discussion of these problems goes beyond the objectives of this chapter. Including all of the various conditions, as one can imagine, would require a large volume.

2.7

Rings and Hollow Cylinders

The vibration principles of rings, hollow cylinders, and bells are closely related. Elementary design principles are presented in the following sections. See Chapter 3 for a better technique for designing and analyzing the more complicated modes of vibration possible in large tubular sections. A ring may be considered as an element of a hollow cylinder, although the cross section of the material from which the ring is formed may be of any geometrical shape with dimensions small compared with the radius, r, of a centerline circle identified with the crosssections about the ring. This centerline lies in a center plane normal to the axis of the ring. The material of the ring is elastic and the cross section is constant about the circumference. The possible motions in a vibrating ring are: 1. 2. 3. 4.

Radial motion (Figure 2.20) Circumferential displacements similar to those of bars Flexural motion in a radial direction Flexural motion in a direction parallel with the axis of the ring and including twist 5. Torsional or twisting action about the centerline of the cross-section of the ring

2.7.1

Pure Radial Vibration

If the ring material can be made to expand and contract in phase about its circumference, the result is a pure radial mode of vibration. The ring vibrates uniformly about the centerline of the cross-sections having a mean or equilibrium radius of r. The cross-sectional area of the ring is S and Young’s modulus of elasticity is Y, its density is ρ, and its Poisson’s ratio is µ: F

CRC_DK8307_CH002.indd 114

SYc SYc 
2r

(2.230)

10/29/2008 3:48:22 PM

Ultrasonic Horns, Couplers, and Tools

115

r +

FIGURE 2.20 Vibrating ring in pure radial motion.

As the ring vibrates in the radial direction at an amplitude of ±r it produces a strain within the material of the ring (in the circumferential direction) equal to c/2πr. The maximum force within the material of the ring occurs when the radial displacement is at its maximum or minimum value, and is where ℓ is the circumference of the centerline circle and S the cross-sectional area of the ring material. In terms of r, c = 2πr. The potential energy in the ring at maximum displacement is P.E. 

F  c SYc2 SY(2r )2   2 2
2(2r)

(2.231)

With no losses, this potential energy is converted into kinetic energy. At the instant the ring passes through the equilibrium position, the total energy within the ring is kinetic energy given by K.E. 

S(2r)  dr    2  dt 

2

(2.232)

As kinetic energy plus potential energy is a constant value during a vibration cycle (no losses), K.E. + P.E. = constant so that d d (K.E.)
(P.E.)  0 dt dt

(2.233)

Thus   dr   d 2r 2SY dr  S(2  r)
r 0    dt   dt 2 r dt 

CRC_DK8307_CH002.indd 115

10/29/2008 3:48:22 PM

Ultrasonics: Data, Equations, and Their Practical Uses

116 or

d 2r Yr  2 0 dt 2 r

(2.234)

r = A cos
t + B sin
t

(2.235)

The solution to Equation 2.234 is

where


Y r 2

(2.236)

or, the frequency of a ring vibrating in a pure radial mode is f

1 Y 2 r 2

(2236a)

When a ring vibrates in a pure radial mode, the entire mass of the ring moves in phase as the material undergoes alternately tension and compression. When the ring vibrates in a pure circumferential mode in a manner equivalent to longitudinal modes in bars with the radius, r, of the centerline circle remaining constant, no radiation occurs from any surface. The stresses also are pure compression and tension, assuming that the dimensions of the cross section of the ring are small compared with the radius, r, of the centerline circle and that the cross-sectional area, S, is continuous as a function of θ. The frequency equation of vibrations is given by
n2  (1
n 2 )

Y r 2

(2.236b)

or fn 

1 Y (1
n 2 ) 2 r 2

(2.236c)

where n is the number of wavelengths in the circumference for the fundamental frequency and all overtones (n = 1, 2, 3, … ) [5,12]. For n = 0, the equation is the same as that for pure radial modes, as in Equation 2.236. 2.7.2

Flexural Modes of Rings

If the motion is flexural and restricted to the plane of the ring containing the crosssectional axis (Figure 2.21), the frequency equation is
n2 

1 n 2 (n 2 1)2 Yrr2 4 (1
n 2 )r 4

(2.237)

where rr is the radius of the cross section (assuming a circular cross section) or
n2 

CRC_DK8307_CH002.indd 116

n 2 (n 2  1)2 Y 2 (1
n 2 )r 4

(2.238)

10/29/2008 3:48:23 PM

Ultrasonic Horns, Couplers, and Tools

117

FIGURE 2.21 Flexural mode of ring vibrating in plane containing centerline of cross section.

where κ is the radius of gyration, assuming the cross section to be uniform about the ring. Therefore
n 

n 2 (n 2  1)Y 2 (1
n 2 )r 4

(2.238a)

and fn 

1 n 2 (n 2  1)Y 2 2 (1
n 2 )r 4

(2.238b)

For a ring vibrating in flexure but in a direction normal to the plane of the ring (plane containing the axis of the cross section of the ring), the frequency equation is
n 

n 2 (n 2  1)Yrr2 4(1
σ
n 2r 4 )

(2.239)

n 2 (n 2  1)Y 2 1

n 2r 4

(2.239a)

for a ring of circular cross section or
n 

geometrical forms in which the radius of gyration, κ, is parallel to the direction of motion. Then fn 

1 n 2 (n 2  1)Y 2 2 1

n 2r 4

(2.239b)

where σ is Poisson’s ratio. 2.7.3

Hollow Cylinders

The number of potential modes of vibration of a cylinder increases as the axial length increases from the extremely short lengths usually associated with rings to lengths corresponding to several average diameters. The vibrations associated with theses modes,

CRC_DK8307_CH002.indd 117

10/29/2008 3:48:23 PM

Ultrasonics: Data, Equations, and Their Practical Uses

118

whether they appear in rings or in cylinders, fall into two classes: (a) longitudinal or extensional and (b) flexural. In the ring, the longitudinal modes involve circumferential displacements. The flexural modes involve displacements either in the plane of the ring, perpendicular to the plane of the ring, or torsionally twisting with motion parallel to the axis of the ring. In the cylinder, the longitudinal modes may also be circumferential, but they can include particle displacements parallel to the axis of the cylinder or at any angle to the axis of the cylinder. Likewise, flexural modes may correspond to waves traveling in axial, circumferential, or skewed directions. Discussion of cylinders in this section is limited to basic principles. Chapter 3 discusses and illustrates extension of these principles to predict performance of tubes and cylinders using Donnell’s model. Some special cases of cylindrical vibrations are 1. As rings—considered in previous sections. 2. In longitudinal motion equivalent to uniform bar. In this case, the radius of the outer surface is small compared with the axial length and the wavelength. Unless the wall is extremely thin, use conditions for uniform bars. 3. In flexural motion equivalent to uniform bar. The conditions are the same as in case 2. 4. Longitudinal motion regardless of wave direction in which the radial dimensions are significantly large compared with a wavelength. 5. Flexural motion comparable to case 4.

2.8

Wide and Large-Area Horns

The design of an ultrasonic system for high-power applications is usually complex. A mode of vibration is specified as being the most appropriate for a given application. The designer’s problem is to deliver the specified mode without generating deleterious spurious modes. Spurious modes of vibration rob energy from and can completely overwhelm the intended mode of operation. Energy is most efficiently transferred from the source transducer into a load when the impedance of each element is properly matched to that of each adjacent element within the active system. For example, an exponentially tapered horn coupled to a short cylindrical tool can resonate as a half-wave system in a longitudinal mode when the mechanical impedance of the horn equals that of the tool at the junction between them. The combination can be designed to vibrate at a specified frequency by using equations describing impedances of the various geometrical configurations. The impedance of the horn is always assumed to be distributed. The impedance of a long tool also is assumed to be distributed, but when the dimensions are very small compared to a wavelength, the tool may be considered as a lumped mass. Impedance matching is important to optimum performance of any high-power ultrasonic system. The various elements in the system may vary in the planned modes (e.g., longitudinally vibrating horn driving a bar, a ring, a plate, or a cylinder in flexure). Also, reflected impedances due to loading conditions are factors in the performance of the system. These should be analyzed to determine their significance to the overall design of the system. These mode and impedance considerations are especially important to the design of wide and large-area horns. For example, if the lateral dimensions of a wide or large-area

CRC_DK8307_CH002.indd 118

10/29/2008 3:48:23 PM

Ultrasonic Horns, Couplers, and Tools

119

horn are equivalent to or near a multiple of half-wavelengths at the design frequency, they present the danger of producing corresponding lateral modes of vibration that can damage products being processed. Usually these horns are designed with slots parallel to the longitudinal motion. The elements between slots are also sources of additional spurious modes of vibration. These slots maximize heat dissipation to prevent creation of thermal hot spots in operation and help control modal characteristics. The following sections include sufficient information for designing effective wide, large-area and other special types of horns without resorting to finite-element methods. Finite-element procedures and Donnell’s model for designing special horns and more complex systems such as cylinders in wall flexural modes are presented in Chapter 3. 2.8.1

Wide Blade-Type Horns

Wide blade-type horns have found their most common application in seam welding of plastic sheet or similar products. Their designs may represent arrays of various geometrical designs, such as uniform bars, stepped horns, exponentially tapered horns, wedge-shaped horns, and catenoidal horns. Figure 2.22 is a typical stepped, blade-type horn. The design procedure includes (using this design as an example): 1. 2. 3. 4.

Selecting horn material Specifying the dimensions of the two end faces Specifying the dimensions and positions of the longitudinal slots Determining impedance relationships for matching solid end elements to the elements created by the slots 5. Calculating longitudinal dimensions based upon the properties of the material of the horn and the impedances

R1 d

λ/4

ᐉ

B

R2 FIGURE 2.22 Wide blade-shaped horn.

CRC_DK8307_CH002.indd 119

10/29/2008 3:48:23 PM

Ultrasonics: Data, Equations, and Their Practical Uses

120

The dimensions of the horn of Figure 2.22 are identified as follows: • • • • • • • • • • • •

B is the total width of the horn ℓ is the total length of the horn d is the width of each slot b is the width of the portion of the faces of the horn corresponding to each element n is the number of slots S1 is the cross-sectional area at the large end of the horn (=R1B) R1b is the cross-sectional area of the larger end of the horn between slots R2b is the cross-sectional area of the smaller end of the horn between slots ρ is the density of the horn material (constant throughout the horn) c is the bar velocity of sound in the horn a1 is the distance between the end of the slot and the larger face of the horn a2 is the distance between the end of the slot and the smaller face of the horn

The nodal position at resonance is assumed to be located at the large dimension of the step of the horn. The length of each slot is less than a half-wavelength of a stepped horn by an equivalent length x of a (a1 + a2), or ℓ = λ/2 − x + (a1 + a2) x = x1 + x2

(2.240)

where x1 is a length equivalent to a1 and x2 a length equivalent to a2. From the bar impedance Equation 2.7 tan


x 1 h
a  tan 1 c b c

tan


x 2 h
a  tan 2 c b c

(2.241)

Then x

c  1  h 
a1   
a 2    1  h tan  tan   
tan  tan    c  c   
 b b

(2.242)

When a1 = a2 = a x2

c h 
a   tan1  tan     c 
b

From Figure 2.22, h = B/(n + 1) = b + d b = [B − (n + l)d]/(n + 1)

CRC_DK8307_CH002.indd 120

10/29/2008 3:48:23 PM

Ultrasonic Horns, Couplers, and Tools

121

The total length, ℓ, is, therefore


c c h h 
a   
a   
(a1
a 2 )  tan1  tan  1  
tan1  tan  2      c    2f c 
 b b

(2.243)

when a1 ≠ a2 and


c 2c 
a   h
2a  tan1  tan  2f c  
 b

(2.243a)

when a1 = a2. Because h/b = B/[B − (n + 1)d],


 c c  B
a 
(a1
a 2 )  tan1  tan 1  2f c 
  B  (n
1)d  B 
a   
tan1  tan  2     c    B  (n
1)d 

(2.243b)

when a1 ≠ a2 or


 2c B c
a 
2a  tan1  tan  c  2f
 B  (n
1)d

(2.243c)

when a = a1 = a2. Thus, ignoring the effect of the fillet at the junction between the large thickness and the smaller thickness sections of the horn, a stepped blade-type horn is the same length as one of uniform thickness for the full length when all other dimensions are the same. The distance between slots is assumed to be sufficiently small that bar velocity is applicable. The effect of Poisson’s ratio on lateral expansion and contraction cannot be ignored, however, in choosing the total width of the horn and determining the lengths of the slots. Width dimensions equivalent to multiples of half-wavelengths at frequencies close to the design frequencies are likely to lead to vibration in a lateral mode rather than the design mode. The corresponding vibrations may damage products to be processed. 2.8.2

Large-Area Block-Type Horns

Figure 2.23 is a typical rectangular block-type horn. The dimensions of the horn of Figure 2.23 are End dimensions, R and B Distances between the ends of the slots and the end surfaces, b1 (top) and b2 (bottom) Width of each slot, d (assuming the same dimensions in all directions) Length of the horn, L The dimensions of the large-area horn are such that the effect of Poisson’s ratio on the velocity of sound in the material of the horn cannot be neglected. From Equation 2.110,

CRC_DK8307_CH002.indd 121

10/29/2008 3:48:23 PM

Ultrasonics: Data, Equations, and Their Practical Uses

122

b1 L

d b2

R

B

FIGURE 2.23 Large-area block-type horn.

the corrected velocity of sound in a large area block relative to bar velocity in the same material is approximately 2

c′ 1    2 2  1   [B
R ] c 6 

(2.244)

Derks [6] applies this formula to determine a corrected value of velocity for a large area rectangular horn containing slots as follows: 2 2 2  R  n 2d   c′ 1  f   B  n1d    1 
  n
1   co 6  c   n1
1   2 

(2.245)

where c′ is the corrected value of the velocity of sound, n1 and n2 are the number of slots through sides B and R, respectively, and σ is Poisson’s ratio. The length of the horn is given by L

  c′ 1  n1d   n 2d 
b1
b2  tan1  1
 1
tan(k ′b1 )   2f k′  B  n1d   R  n1d        n1d   n 2d 
tan1  1
 1
tan(k ′b2 )    B  n1d   R  n1d    

(2.246)

where k′ = ω/c′. 2.8.3

Other Designs—Large, Cup-Shaped Horns

For applications to processes such as continuous seam welding of plastic sheet or strips, ultrasonic tools can be made to rotate while applying energy to the work through a

CRC_DK8307_CH002.indd 122

10/29/2008 3:48:23 PM

Ultrasonic Horns, Couplers, and Tools

123

D2

D3

Coupling bolt hole x ᐉ FIGURE 2.24 Horn with cylindrical shell and cavity in which the cross-sectional area function is equivalent to that of a conical horn.

wheel-like surface. The first approach considered might be to drive a disc-shaped plate in flexure using a longitudinal mode horn coupled to its axis. However, a plate driven in such a manner at an amplitude sufficient to perform its intended function is likely to experience such severe stresses near its axis that it will fail in fatigue within a very short time. There are means of applying ultrasonic energy to work on a continuous basis using a rotating horn. Figure 2.24 illustrates one such tool. This horn was chosen to illustrate a method. Other designs may be used as well. The horn of Figure 2.24 is designed so that the area as a function of position x corresponds to that of a solid cone. The dimensions of the horn are described by d1 is the smallest diameter of the cavity located in the plane of the heavy end of the horn d2 is the diameter of the cavity at the large, open end d3 is the outside diameter of the horn (d3 ≤ 0.4λ) ℓ is the length of the horn x = 0 is located at the large diameter end of the cavity For illustrative purposes, assume that the vertex of the cavity coincides with the plane of the large area end, that is, d1 = 0. Because the horn is assumed to be equivalent to a conical horn, solutions to the horn equation should be similar. However, the outer diameter of the horn is sufficiently large that effects of Poisson’s ratio on the velocity of sound in the material of the horn should be considered. A close approximation for the velocity, c′, can be obtained using Equation 1.61, that is, 2 1  d′  c′ 1 
d′   1   1   4  co 4  2c o 

2

(1.61)

where co is the bar velocity of sound in the material of the horn, c′ the corrected velocity of sound, and d′ the major diameter (=d3).

CRC_DK8307_CH002.indd 123

10/29/2008 3:48:23 PM

Ultrasonics: Data, Equations, and Their Practical Uses

124

Using the previous assumptions, the area relationship as a function of x (measured from the open end of the horn) is Sx 

(

)

2

2  2 x  d 3  d 22
d 3  d 32  d 22   2 (
 x) d 23  d 22
d 3x    
 4 4


(2.247)

Using the solution to the horn equation for a cone, one can write the velocity relationship as  (d 3  d 32  d 22 )V2   
d 3

c′ v  cos (x 
)
sin (x 
)  
c′ c′  (
 x) d 23  d 22
d 3x   (d 3  d 32  d 22 ) 

(2.248)

where c′ is the corrected value of velocity of sound for the horn, V2 the amplitude of particle velocity at the heavy end of the horn, and ℓ the length of the horn. The half-wave resonant length is determined by

tan  c′ ω 2
2d

(


c′ d 3  d 32  d 22 3

(

)

2

d 32  d 22
c′ 2 d 3  d 32  d 22

)

(2.249)

2

The velocity node occurs where tan




d3 (
 x)  c′ c′ d 3  d 23  d 22

(2.250)

The amplification factor is 2 2  V1 1 d3

c ′  d 3  d 3  d 2 

sin cos    2 2 2 2 V1  2 c′ c′ d3  d2

d 3  d 2

(2.251)

where 1 is the displacement amplitude at x = 0 and 2 the displacement amplitude at x = ℓ. The stress, s, at x is given by s j

(d  3

)

d 32  d 22 V2Y


(
 x) d 23  d 22
d 3x   

2

{(

)

d 3  d 32  d 22 (x 
)cos


(x 
) c′

  c′ 2


2d 3 d 32  d 22
2 d 3  d 32  d 22

d 3x  sin (x 
)   c′ c′  d 3  d 32  d 22
 

(

(2.252)

)

The position of maximum stress is given by tan


(x 
)  c′

4
2d 3

2 2

d 3 4c ′  (
 x) d 23  d 22
d 3x  d 3  d 23  d 22  (x 
)
   
c′ d 3  d 23  d 22    2 4c ′ 2 d 23  d 22
6
d 23x  6
d 3 d 23  d 22 x 
2 (d 23  d 22 )
2
d 22x  (d 23  d 22 ) x 2
2 d 3  d 23  d 22   


(2.253)

CRC_DK8307_CH002.indd 124

10/29/2008 3:48:23 PM

Ultrasonic Horns, Couplers, and Tools

125

The horn of Figure 2.24 can be adapted to a rolling contact type of operation, which can be done in variations of either of two ways: 1. Machining a bevel at the working (or open) end of the horn to contact the workpiece surface 2. Forming a raised section (wheel-rim) of dimensions suitable for coupling to the work surface The work coupling surface of the first method is an element of a cone of either method. In either case, the overall length of horn and tool section must be adjusted so that the combination resonates at a frequency that matches the frequency of the driving transducer. The system is broken down into the horn section and the tool section. The impedance of the horn at the junction between the two sections must match that of the tool at that position. The impedance at x of the horn of Figure 2.25 is Z = F/v = sS/v or

Z j

 (
 x) d 23  d 22
d 3x  Y   x d 3  d 32  d 22  (x 
)    
d c
′ 3 4

2 
tan (x 
)
c′  d 3  d 32  d 22 



{

  c′ 2

2d 3 d 23  d 22

2 d 3  d 23  d 22
d 3
x  tan (x 
) 
c′ c′  d 3  d 23  d 22  

(

)

(2.254) where F is the force across the cross-sectional area at x. In designing a horn, as previously discussed, to drive a wheel-like tool section, the critical design dimensions are first specified—that is, d1, d2, d3, frequency, material, and

Fillet

Dw

D3

D2

Coupling bolt hole ᐉT L FIGURE 2.25 Wheel-loaded rolling-type ultrasonic horn.

CRC_DK8307_CH002.indd 125

10/29/2008 3:48:24 PM

Ultrasonics: Data, Equations, and Their Practical Uses

126

dimensions of the wheel section, dw, and L, where dw is the outer diameter of the wheel portion, d1 is assumed to equal zero, and L is the axial length of the cylinder at the outer diameter of the wheel. Because one objective in selecting the horn design is to amplify the longitudinal displacement over that at the driven end, the enlargement will appear at the open end of the horn. This is the end corresponding to x = 0 in the design equations previously presented. The next step is to determine the length, ℓ, of the horn as a half-wave resonator without the wheel section, using Equation 2.254. The final total length, ℓT, of the horn plus wheel section will be ℓT = ℓ − X + L

(2.255)

where ℓ is the length calculated using Equation 2.249 and X is a length equivalent to that consumed by the wheel section. The value of X is determined by equating the mechanical impedance of the wheel to that of the horn at the junction between them and substituting X for x. Because the wheel is much shorter than a wavelength, its impedance may be determined fairly accurately by assuming that the wheel section is a lumped mass, that is, 2 Zm  j
M  j
L (d w  d 22 )

 4

(2.256)

Because X is measured from the origin and is <λ, the two impedances are equal in magnitude and sign. The length of the total horn including the wheel section and the remainder of the horn is determined by equating the impedance of the section represented by X, determined by Equation 2.256, to the horn impedance given by Equation 2.254 to determine the equivalent length of X. This equivalent length is subtracted from the length of the pure horn (i.e., the resonant length of the horn without the mass of the wheel portion attached). When this is done, the only unknown is X, which is determined by iterative methods, as the equation includes a trigonometric function of x on one side and an algebraic function of x on the other side. Wheel dimensions to avoid are: 1. Those that permit ring-type resonances at or near the specified design frequency. The upper limit on diameter is that given by the fundamental radial mode in Equation 2.236. 2. Those that lead to resonances that are subharmonics of the design frequency. These are avoided by restricting the diameters to values below those determined by 1. Outer diameters of the wheel on the order of 0.3λ or less are practical.

2.9

Additional Design Factors

The previous discussions include design equations that describe distribution of amplitudes of displacement, velocity, and acceleration in various vibratory configurations. They also provide means for matching elements in a complex ultrasonic system and for determining the stress distribution in an ideal element, such as an ultrasonic horn.

CRC_DK8307_CH002.indd 126

10/29/2008 3:48:24 PM

Ultrasonic Horns, Couplers, and Tools

127

These are basic factors in a design problem. Theoretically ideal conditions leading to some of the previous solutions to vibratory elements might not be realistic in practice. For example, reactions of the clamping mechanisms on a rectangular plate will change the results relative to modal conditions, especially at ultrasonic frequencies. At low frequencies, these ideal conditions might be approached in large plates embedded in heavy concrete at the edges. It is also difficult to visualize a plate simply supported all around, where the boundary conditions assumed are correct. In both of these cases, the assumed boundary conditions at extremely low amplitudes are probably applicable, but at practical amplitudes at high intensities their validity is weakened. They do, however, provide probably the best starting point for a design where such elements are involved. Finite element methods described in Chapter 3 might be used in designing many types of systems where the simpler theories of this chapter are inadequate. Many other practical problems in meeting design criteria for high-intensity applications must be considered. These include (a) preventing fatigue failure, (b) meeting conditions imposed by high temperatures, and (c) protecting against dissolution of tools irradiating harsh chemical mixtures. 2.9.1

Stress Concentration Factors

Fatigue failure in ultrasonic vibrators is most often traceable to stress concentration. It is least likely to occur in ultrasonic horns of continuous tapers (such as uniform bars, exponential horns, etc.) if these are operated at stress levels below the fatigue limit of the horn material. Stress concentrators include such imperfections as pits, scratches, machine marks, and the like. They also include fillets or geometrical discontinuities between elements of different dimensions, such as the junction between quarter-wave sections of a double-cylinder type of horn. The first type of stress concentration can be eliminated by careful machining and final polishing. The effects of the latter type can be minimized by proper design and eliminated by operating safely below the fatigue limit of the material of the horn. Design horns to meet these criteria for long life (see Section 2.3.2.1). 2.9.2

Treating Materials at High Temperature

The largest number of commercial applications for high-intensity ultrasonic energy are those that offer a minimum of design problems. Standard alloys—usually titaniumbased—are widely accepted for these applications. However, it is possible to meet harsh temperature requirements that are associated with certain processes to which it might be desirable to apply ultrasonic energy. The design criteria include selection of a horn material that will maintain suitable elastic properties to the required temperatures and provision for protecting against chemical attack, such as oxidation. As an example, experiments have been conducted to study the effects of ultrasonic irradiation to improve removal of gas bubbles (seeds) from molten glass. Certain molybdenum alloys maintain good elastic properties at temperatures exceeding 1400°C. These alloys must be protected from oxidation at these temperatures by an inert atmosphere such as nitrogen [14]. 2.9.3

Treating Materials in a Harsh Chemical Environment

Highly acidic materials are especially difficult to treat using ultrasonically activated horns or tools in direct contact with the acids. Metallic tools that seemingly resist attack by

CRC_DK8307_CH002.indd 127

10/29/2008 3:48:24 PM

Ultrasonics: Data, Equations, and Their Practical Uses

128

such chemicals under static conditions may rapidly deteriorate under the effects of ultrasonic cavitation. Resistance to chemical attack of this type of alloy usually can be traced to a tough oxide coating. Persistent cavitation erodes the protective coating, exposing the base material, which is susceptible to rapid chemical attack under these conditions. One specific need to conduct chemical processing in a strong acidic material has been met by using a tantalum horn. Tantalum is essentially inert to the acid under ultrasonic conditions. Tantalum is a very expensive material to use for horns. It is very malleable and its fatigue limit is low in comparison with conventional materials, such as titanium alloys. However, it has withstood ultrasonic intensities that are sufficiently high to produce the desired chemical results.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

R. J. Roark and W. C. Young, Formulas for Stress and Strain, 5th ed., McGraw-Hill, NY, 1975. W. P. Mason and R. F. Wick, J. Acoust. Soc. Am., 23, 209–214, 1951. D. Ensminger, Ultrasonics, 2nd ed., Marcel Dekker, NY, 1988. D. Ensminger, J. Acoust. Soc. Am., 32, 194–196, 1960. Baron Rayleigh (John William Strutt), The Theory of Sound, Vol. 1, Dover, NY, 1877, pp. 251–252. P. L. L. M. Derks, The Design of Ultrasonic Resonators with Wide Output Cross-Sections, Philips, Eindhoven, The Netherlands, 1984. W. J. McGonagle, Nondestructive Testing, 2nd ed., Gordon and Breach, NY, 1961, p. 295. R. T. Beyer and S. V. Letcher, Physical Ultrasonics, Academic Press, NY, 1969, 44ff, 242ff. R. C. Weast, Editor-in-Chief, Handbook of Chemistry and Physics, 66th ed., CRC Press, Boca Raton, FL, 1985–1986, F-96. P. M. Morse and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, NY, St. Louis, San Francisco, MO, 1968. A. W. Leissa, Vibration of Plates, Acoustical Society of America through A.I.P., Woodbury, NY, 1993. S. Timoshenko and D. H. Young, Vibration Problems in Engineering, D. van Nostrand, NY, 1955. G. B. Warburton, The vibration of rectangular plates, Proc. Inst. Mech. Eng., Ser. A, 168(12), 1954, 371–384. E. D. Spinosa and D. Ensminger, Sonic energy as a means to reduce energy consumption during glass melting, Ceram. Eng. Sci. Proc., 17(3–4), 1986, 410–425.

CRC_DK8307_CH002.indd 128

10/29/2008 3:48:24 PM

3 Advanced Designs of Ultrasonic Transducers and Devices Using Finite Element Analysis Foster B. Stulen and Robert B. Francini

CONTENTS 3.1 Introduction ........................................................................................................................ 130 3.2 Finite Element Analysis .................................................................................................... 133 3.2.1 The Finite Element ................................................................................................. 133 3.2.2 Theory...................................................................................................................... 134 3.2.3 Solution.................................................................................................................... 136 3.2.4 Piezoelectricity ....................................................................................................... 139 3.2.5 Interpolation in the Finite Element ..................................................................... 142 3.2.6 Boundary Conditions ............................................................................................ 145 3.2.7 Symmetry ................................................................................................................ 145 3.2.8 Beyond Linear Elastic Isotropic ........................................................................... 145 3.2.9 Pre- and Postprocessors ........................................................................................ 146 3.3 Theory of Plates and Shells .............................................................................................. 146 3.3.1 Introduction ............................................................................................................ 146 3.3.2 Free Plate Vibrations.............................................................................................. 147 3.3.3 Shell Vibrations ...................................................................................................... 148 3.4 Design Example: Flat Plate ............................................................................................... 151 3.5 Design Examples: Cylindrical Shells .............................................................................. 153 3.5.1 Design 1: Applying Donnell’s Model .................................................................. 153 3.5.2 Design 2: Comparing Donnell’s Model with Finite Element Analysis ............................................................................... 158 3.6 Design Example: Large Horn ........................................................................................... 162 3.7 Design Example: Ultrasonic Driver ................................................................................ 167 3.7.1 Conventional Design Approach .......................................................................... 168 3.7.2 Finite Element Analysis Approach...................................................................... 170 3.7.2.1 Resonances and Displacements ............................................................ 170 3.7.2.2 Transient Analysis .................................................................................. 173 3.7.2.3 Comments on Damping and Power Loss ............................................ 179 3.8 Commercial Software Packages ...................................................................................... 180 3.9 Summary............................................................................................................................. 181 References .................................................................................................................................... 183

129

CRC_DK8307_CH003.indd 129

10/27/2008 8:45:28 AM

Ultrasonics: Data, Equations, and Their Practical Uses

130

3.1

Introduction

Ultrasonic converters, horns, and nondestructive inspection transducers can be accurately designed using the one dimensional (1D) wave equation. The segments that form the resonant power driver and the backing and matching layers and the piezoelectric element of an inspection transducer are modeled by applying the wave equation for each segment. Equilibrium of stresses and continuity of displacements at the interfaces are used to join the segments together. But as the need to scale up ultrasonic processes for a variety of applications continues to grow, larger drivers that are more complicated in shape must be designed. Models based on the 1D wave equation are no longer sufficient. In scale-up applications, large plates and shells are vibrated by longitudinal drivers to greatly expand the driven surface and the treated volume. These ultrasonic systems can be designed using either theories for the vibrations of plates and shells or fi nite element analysis (FEA). If the system has the form of a simple engineering shape, such as a plate, disk, or cylinder, then plate and shell theories such as those described in Chapter 2 can be used effectively. For more complicated shapes and systems that may have stiffeners or unusual boundary conditions and attachments, FEA is used. FEA, just as any analytical method, is used by engineers to model, analyze, and predict the performance of real physical systems. The accuracy of the model depends not so much on the particular analytical technique, but on the assumptions made in modeling the physical system. The accuracy of the model must be validated by benchmarking its predictions to experimental data and results from simpler problems. Results from the computer or any analysis should never be assumed to be correct just because one can “turn the crank” on the problem. A physical problem is defined by its geometry, material properties, boundary conditions, and the external loads, which can be distributed in time and space. Sometimes these parameters for a particular problem are not that well understood. Material properties are often not accurately known, may have to be extrapolated from reference values, or may have a relatively high degree of variability. In such cases, once the model is known to be an accurate representation, the engineer performs parametric studies to evaluate the effects of these variations to understand their impact on the results. A flow chart of the FEA process is shown in Figure 3.1. FEA begins with either the existing physical system or its candidate design. The next step is to develop descriptions of individual features that define the system. These include the basic nature of the system, its geometry, the materials that compose it, the external loading imposed upon it, and the boundary conditions that constrain it. The physical system is basically defined by its overall geometry. FEA tools allow geometries to be represented in Cartesian, cylindrical, or polar coordinates. The geometry can be obtained from electronic design files, as-built drawings, or physical measurements. Because a finite element, the basic building block, has a single material property, the geometric description must include descriptions of interfaces between materials. The nature of the problem should be considered as well, because the type of analysis controls several of the parameters of an FEA model. Common analyses include static, dynamic, large displacements, and nonlinear problems. The development of an FEA model becomes more involved as the complexity of material, details of the geometry, and level and type of analysis increase. The system is divided into elements and the resulting assemblage of elements is known as the mesh. An element encompasses a single material and its geometry is defined by nodes located at its corners, edges, and faces. Adjacent elements share the same nodes. The types of elements are selected based on the physical system. A system is generally modeled

CRC_DK8307_CH003.indd 130

10/27/2008 8:45:32 AM

Advanced Designs of Ultrasonic Transducers and Devices

131

Physical system

System design and modifications

Description of problem: Parametric studies

Geometry Material properties Loading Boundary conditions

Finite element model

Finite element analysis: Types of elements Mesh density Loading Boundary conditions Solution parameters

Experimental data

No Modify/ Correct

RUN ? Intuition

Yes

No Simplified model results

Accurate ? Yes

FIGURE 3.1 The finite element method.

with more than one type of element. For example, a thin cantilever would be treated as a beam made up of beam elements, and the foundation it is embedded in would be modeled with 3D brick elements. The density of elements or mesh density is an important issue in the design of the FEA model. A model that is too coarse does not produce accurate results, and a mesh that is too fine requires much more computer time and may be more than needed for the variations in stress and strains. The density of the mesh is generally not uniform throughout a structure. Where gradients in stresses and strains are expected to be low, the density is low. Where gradients are high, such as in the vicinity of stress concentrations, the density of the mesh is high. It is generally best to base the mesh on the most demanding case to be considered. For example, if the dynamic problem is to be analyzed but the static case is of initial interest, it may be more efficient to initially mesh the system for the dynamic problem, which requires a denser mesh with more elements.

CRC_DK8307_CH003.indd 131

10/27/2008 8:45:32 AM

132

Ultrasonics: Data, Equations, and Their Practical Uses

Each finite element requires a unique material property assigned to it. The material law— that is, the constituent law—is a description of how the stresses and strains are related. The simplest case is linear elastic behavior, which is limited to relatively small stresses and strains. Other material behaviors include plastic behaviors and ultimate strengths. These can depend on the current temperature and pressure, or on loading history, as in the case of creep. Loading or some type of forcing function drives the system. Loads can be static, which require only a static solution, or the loads can be dynamic, which require an eigenvalue solution or direct time integration and a set of initial conditions of the system. Loads can be forces, stresses, and pressures in mechanical problems, or heat inputs in thermally driven problems. Loads are applied at nodes. Boundary conditions are constraints placed on the system by its design and environment. Boundary conditions in an FEA model are applied to nodes of an element that correspond to the physical locations where the constraints are applied. In a 2D analysis, fixing both the x and y displacements and rotation corresponds to a fixed or rigid condition. Fixing only the x and y displacements allows for rotation and is a pinned or simple boundary condition. Fixing one dimension and rotation allows the node to slide and is a roller condition. In any structural analysis, at least one node is rigidly fixed, so that free translation and rotation of the system is removed. The most anxious step in the process is the initial running of the FEA model by itself. It can fail from problems associated with either the size-exceeding limits imposed by the software and hardware, or numerical overflows, because of singularities introduced by erroneous assignment of elements, nodes, and their degrees of freedom. At this point, the FEA model must be reviewed and corrected in the indicated areas by compiling or execution errors reported by the FEA program. This is often an iterative procedure in which one error is corrected, after which another is uncovered further into the run. Once the first run is completed, the correctness of the model must be checked by asking two critical questions. The first question is, “Does the model make intuitive sense?” That is, does the system behave as you know it should or in accordance with the observed behavior? The second question is, “Are the results on the right order of magnitude?” One can compare measured displacements at specific locations from the physical system with the FEA results. If one is in the design stage or does not have that information, then the FEA results could be compared with a more simplified analytical model that should bound the displacements. For example, assume that the structure under analysis is basically a beam but with complex geometry. Then the engineer could use a standard equation for a beam of roughly its length and mean thickness to calculate the displacements. These displacements are then compared with the FEA results to determine if the latter are of the correct order. The FEA results obtained prior to this step are just numbers, not answers. The next step is to refine or modify the model. One can vary the mesh size, the type of elements, and the solution parameters until comfortable with the accuracy of the results and with the turnaround time for a computer run. At this point, the FEA model becomes a powerful tool for the engineer. With this tool, parametric studies can be performed, in which the assumptions are varied over a reasonable range to represent variations in material properties, dimension tolerances, or loading and boundary conditions. A very typical analysis is to increase the load until some internal stresses go beyond the elastic limit to set safety limits or to identify areas that need design modifications. The next level is to change the physical system to compare different designs or to obtain an optimal design. This division between parametric studies and alternative designs of the physical system is somewhat artificial. The division

CRC_DK8307_CH003.indd 132

10/27/2008 8:45:32 AM

Advanced Designs of Ultrasonic Transducers and Devices

133

between the two is where the physical system is changed to a degree that a modified or different FEA model is required. FEA is an everyday computer tool for solving complex problems in structural engineering. Its versatility and effectiveness have provided the motivation to produce similar analyses and computer codes for electromagnetics, heat transfer, and fluid dynamics (known as computational fluid dynamics [CFD]). Commercial packages are becoming more powerful, allowing coupled problems that involve several different fields of physics. These problems are known as multiphysics problems. The focus of this chapter is to present the basics of FEA of structures, including piezoelectric elements. Enough detail is given so that the reader will become familiar with most aspects of FEA. Design examples are given with comparisons of FEA model predictions with results from bar, plate, and shell theories and with data obtained from actual systems. This chapter is only an introduction, and there are entire books devoted to FEA, such as the one authored by Bathe [1].

3.2 3.2.1

Finite Element Analysis The Finite Element

The finite element is the basic unit of analysis in an FEA problem. The entire structure under consideration is an assemblage of finite elements with continuity of displacements and equilibrium of stresses maintained at the nodes belonging to a number of individual finite elements. The solution of the problem becomes one of matrix algebra, which provides efficient numerical calculation. A finite element is defined by the number of nodes, the allowed displacements of the nodes and the volume it encompasses. There are different types of elements for different engineering problems. The 3D element is the most general, but there are other types of elements for specific engineering structures and geometries. There are elements to represent trusses and beams, thin and thick plates and shells, and axisymmetric structures. There are also transition elements that are used to join the different types of elements (e.g., a transition from a shell to a solid). Any problem can be analyzed using 3D elements, but in practice it is important to reduce the dimensionality of the problem by using the specialized elements. For example, some large-scale ultrasonic devices have been constructed with conventional ultrasonic drivers, such as those used for welders to drive structures such as beams, plates, and shells. The drivers are attached at the antinodes of the plate or other structure. The driver’s longitudinal displacements and stresses correspond to the normal displacements and stresses of the other structure. In an FEA model, the driver is designed by using solid or axisymmetric elements (see the example in Section 3.7), and the beam, plate, or shell is modeled with its specific finite element type. The most common 3D elements are the 8-node and 20-node bricks. The 8-node brick element has a node at each corner of the element, which allows linear interpolation between nodes. The 20-node brick, shown in Figure 3.2, includes the 8 corner nodes and adds 12 nodes for the centers of each edge of the brick. There are three points along any edge, which allows quadratic interpolation. Thus for the same volume, the 20-node element produces a more accurate representation than the 8-node element. The dimensionality of the problem has increased 2.5 times, but fewer 20-node elements are required than 8-node elements to achieve the same accuracy.

CRC_DK8307_CH003.indd 133

10/27/2008 8:45:32 AM

Ultrasonics: Data, Equations, and Their Practical Uses

134

15

7

18

8

19 16

14 11

3

4 13

6

5

12

10 17

20

9

2

1

FIGURE 3.2 Finite element analysis, 3D 20-node solid element.

3.2.2

Theory

The formulation of FEA is based on the principle of virtual work. It allows the equilibrium equation to be expressed in an integral form, with the integrals taken over volumes and surfaces. This approach allows the entire volume to be divided into smaller elements, and hence is a key step in the derivation of the finite element method. The mathematical details of this approach are beyond the scope of this chapter, but can be found in a number of texts. The following is intended to give the reader a background in the underlying procedures and processes that comprise FEA. In actual use of computer applications, these are transparent to the user. The mathematical statement of the principal of virtual work without piezoelectricity is

∫ ∫ ∫ ST [c]S dV  ∫ ∫ ∫ uT [F  u  u ] dV
∫ ∫ uS T T dS1
uiTRic 1

V

V

(3.1)

S1

The following conventions are used. A bold letter indicates a column vector, and a superscript T on the bold letter denotes its transpose, which is a row vector. A bold letter in brackets denotes a matrix, and a superscript T indicates its transpose. The δ’s indicate that the quantity following it is a variational quantity. The displacement components as a function of position in the volume are denoted by the vector, u. In engineering mechanics, strain is typically denoted by epsilon, ε, but here the vector of strains is denoted by S. Epsilon is reserved to denote the relative dielectric constants when Equation 3.1 is extended to include piezoelectricity. The factor [c] is the stiffness matrix, so that the product of [c] and S is the vector of stresses in the volume. The left side of Equation 3.1 relates to the potential energy stored in the volume. The volume interval on the right side contains the body forces including the bulk force, F, acting on the volume, the inertial forces from the mass, and the internal damping forces. The surface integral contains the traction forces, T, acting over the surface, S1, which is a subset of the entire surface. The remaining surfaces either have no traction forces or have displacements prescribed. The last term accounts for concentrated loads, Ric, applied at a single point in the volume or on the surface.

CRC_DK8307_CH003.indd 134

10/27/2008 8:45:33 AM

Advanced Designs of Ultrasonic Transducers and Devices

135

The integral representation of Equation 3.1 allows the body to be divided into a number of volumes. The integrals are performed over these individual volumes, and their sum is taken to obtain the overall integral. The individual volumes are the finite elements. Quantities within an element are defined by interpolation functions operating on the values at the nodes um (x, y, z)  [Hm u (x, y, z)] U

(3.2)

U is the vector of global displacement at the nodes, and for solid elements, is a vector of dimension 3N. There are three displacements at each node, and N is the number of nodes for that element. The continuous displacements within the mth finite element are denoted by u m(x, y, z) and are defi ned in the local (x, y, z) coordinates. The interpolation matrix, [Hm u (x, y, z)], is therefore seen to estimate the displacements within the mth element from the displacements at the nodes. The subscript, u, denotes that the matrix is associated with displacements. This form is desirable for the inclusion of interpolation functions for electrical potentials later. The strains are also defined in the local coordinate system Sm (x, y, z)  [Bm u (x, y, z)]U

(3.3)

The estimate of strain, Sm(x, y, z), in the mth element is obtained from the nodal displacements, U with the transformation matrix [Bm u (x, y, z)]. This strain-displacement matrix is composed of derivatives from combinations of select components in [Hm u (x, y, z)]. Equations 3.2 and 3.3 have been written for the real but unknown displacements and strains. Similar expressions apply to the virtual quantities as well. These elemental definitions for the real and virtual displacements and strains are substituted into Equation 3.1 to obtain     T m m T m T m m  m  U T ∑  ∫ ∫ ∫ [Bm u ] [c][Bu ] U dV   U ∑  ∫ ∫ ∫ [Hu ]  F  [Hu ] U  [Hu ]  U  dV    m  m   Vm  Vm (3.4)   m T i T  dS
U T ∑  ∫ ∫ [Hm
U R ] T c u 1 m  m   S1

where δUT is a full vector of global variational nodal displacements. In Equation 3.1, δuS1T and δuiT are just subsets of δUT. Therefore in the nodal form of Equation 3.4, δUT is pulled outside of all the integrals and summations, because it contains only the global nodal displacements, which are independent of the element definitions. This leading factor of δUT then cancels out from all terms in the equation. The vectors of global displacement, velocity, and acceleration, U, U˙, and Ü, are independent of the individual finite elements. Therefore, they can be pulled out of both the integrals and the summations in Equation 3.4. What remains is a summation of integrals for each element as given in Equation 3.5. The integration of each individual integral can be performed over any convenient coordinate system, which provides the power of FEA. This equation can be reduced to the final form of FEA formulation given in Equation 3.6a. [M], [C], and [K] are the consistent mass, damping, and stiffness matrices for the assemblage of elements, respectively. R is a vector of the sum of all equivalent loads acting at the nodes, which includes the surface tractions, concentrated loads, and body forces. Their definitions are given in Equations 3.6b through 3.6e.

CRC_DK8307_CH003.indd 135

10/27/2008 8:45:33 AM

Ultrasonics: Data, Equations, and Their Practical Uses

136

The integral terms in Equations 3.6b through 3.6e are defined for the local coordinate system of each finite element. The summations in the equations are done over all finite elements to yield the global mass, damping, and stiffness matrices for the system being modeled.

3.2.3

Solution

Equation 3.6a is a second-order differential matrix equation, so that the solution to an FEA reduces to standard approaches for this type of equation. There are basically two approaches to the solution. One is direct numerical integration of the equation using a variety of methods, and the second is modal superposition. In the design of power ultrasonic systems, the resonant frequency and vibrational modes are of most importance, which correspond to the eigenvalues and eigenvectors of the eigenproblem. The eigenvalues and eigenvectors are used in modal superposition to determine the forced response. The eigenproblem starts by assuming that the damping is light, which allows the term [C]U˙ to be ignored. The unforced portion of Equation 3.6a is shown in Equation 3.7.       T m m m T m m m T m  U ∑  ∫ ∫ ∫ [Bm u ] [c][Bu ] dV   ∑  ∫ ∫ ∫ [Hu ] F dV   U ∑  ∫ ∫ ∫ [H u ] [H u ] dV   m  Vm   m  m   Vm  Vm     U ∑ ∫ ∫ ∫m [Hmu ]T [Hmu ] dVm   m   V   T m i
∑  ∫ ∫ [Hm ] T dS u 1
Rc (3.5)   m  S1m 

CRC_DK8307_CH003.indd 136


[C]U 
[K]U  R [M]U

(3.6a)

  T m m [M]  ∑  ∫ ∫ ∫ [Hm u ] [Hu ]dV   m   Vm

(3.6b)

  T m m [C]  ∑  ∫ ∫ ∫ [Hm u ] [Hu ]dV   m   Vm

(3.6c)

  T m m [K]  ∑  ∫ ∫ ∫ [Bm u ] [c][Bu ]dV   m   Vm

(3.6d)

    T m m T m i  dV dS R  ∑  ∫ ∫ ∫ [Hm ] F
[ H ] T  ∑ ∫ ∫m u u 1
Rc  m  m  S1 m   V 

(3.6e)

[M]Ü + [K]U = 0

(3.7)

10/27/2008 8:45:33 AM

Advanced Designs of Ultrasonic Transducers and Devices

137

The solution is assumed to be of the form U = φ sin[ω(t − t0)]

(3.8)

[K]φ = ω2[M]φ

(3.9a)

([K] − λ[M])φ = 0

(3.9b)

and Equation 3.7 becomes

or

where λ denotes the eigenvalue, which equals the radian frequency squared. Equation 3.9b has a solution only if its determinant equation equals zero, which defines the characteristic polynomial of ([K] − λ[M]) given by p(λ) = |[K] − λ[M]| = 0

(3.10)

Given that the matrices have order n, there are “n” roots to the characteristic polynomial, and each root has a corresponding eigenvector. The solution is n pairs of eigenvalues and eigenvectors: (ω12, φ1), (ω22, φ2), …, (ωn2, φn). Although not shown here, the eigenvectors are known to be orthogonal and are M-orthonormal as well. Thus the product of the transpose of an eigenvector, the matrix [M], and an eigenvector is unity when the two eigenvectors are the same, and the product is zero when they are not the same. Using the properties of orthogonality, the set of n solutions can be represented by [K][Φ] = [M][Φ][Ω2]

(3.11)

where [Φ] is a matrix of the eigenvectors where each eigenvector is a column, and [Ω2] is a diagonal matrix with the elements equal to the eigenvalues corresponding to the position of its eigenvector. The basic form of the solution is given by [U(t)] = [Φ] [X(t)], where [X(t)] is the vector of modal generalized displacements. This form is introduced because it leads to a set of decoupled equations that are readily solved. When this form is substituted into Equation 3.6a and each term is then premultiplied by [Φ]T, the governing set of equations become 
[]T [C][]X
[]2 X  [ T ]R X

(3.12a)

with initial conditions, individual forcing function are given by X0 = [Φ]T[M]U0

(3.12b)

 X 0  []T [M]U 0

(3.12c)

r1(t )  iTR(t)

(3.12d)

There are several things to note regarding Equation 3.12a through Equation 3.12d. First, this development has been totally general and, in fact, it is equally valid for lumped parameter systems and plate and shell theories, as well as for FEA. Second, the damping matrix, [C], is included in Equation 3.12a. If it is assumed to vanish, then the set of equations of Equation 3.12a are completely decoupled and the equations can be solved

CRC_DK8307_CH003.indd 137

10/27/2008 8:45:34 AM

138

Ultrasonics: Data, Equations, and Their Practical Uses

individually. When it is included, the factor [Φ]T [C] [Φ] is most generally a full matrix and the equations are not decoupled. Damping is difficult to calculate on an element basis, but there are two convenient methods to include the overall effects of damping. One method is to use modal damping in the modal superposition, assuming that [Φ]T [C][Φ] = 2
iij

(3.13)

This expression represents a diagonal matrix of damping terms that keeps the governing set of equations decoupled. If direct integration is to be performed, then the full matrix can be used. This is conveniently done using Rayleigh damping that assumes that the [C] matrix is a linear combination of matrices [M] and [K]. That is, [C] = a [M] + b [K], where a and b are chosen to best represent the modal damping. The trick in all of this is to find the eigenvalues and eigenvectors. For matrices with order greater than four, iterative means are used, because general analytical solutions for higher polynomials do not exist. It is possible to solve for either the eigenvalue or the eigenvector with standard approaches. If the eigenvalue is found first, the eigenvector is solved by using Equation 3.9b. If the eigenvector is found first, then the eigenvalue is determined using the following equation, which is readily derived from the previous equations:

i  iT [K] i

(3.14)

An individual pair of eigenvalue and eigenvector represents only one mode of vibration for the structure under consideration, but this is not the time-dependent solution to a particular forcing function and set of initial conditions. Before proceeding with the solution of the dynamic equations, the solution of static problem is given here for completeness. The static equilibrium equation, Equation 3.6a, with zero mass and damping matrices, [M] and [C], becomes [K]U = R

(3.15)

Equation 3.15 is a set of simultaneous linear equations. This class of problem is frequently solved using the Gauss elimination technique. The details of Gauss elimination can be found in any textbook on engineering analysis or FEA. It involves making a number of transformations on the set of equations until the stiffness matrix is upper-triangular. This means that the elements on the main diagonal and above it are generally nonzero, and the elements below the main diagonal are zero. Then the solution for the last displacement is reduced to a simple identity in the last equation. The next displacement is solved for using the next to the last equation and the last displacement. This process continues until the first displacement is obtained. This is the basis for most FEA computer codes. Iterative methods can also be used to solve Equation 3.15. In iterative methods, current estimates of displacements are used in a procedure to find a better estimate. The process continues until the solution—that is, the set of displacements—converges. Iterative methods can be more effective for large FEA problems compared with Gauss elimination techniques, but the number of iterations required for convergence cannot be estimated accurately a priori. Therefore, the computer time required for solution cannot be estimated until at least one run of the FEA model is made with this method. The solutions to the set of dynamic equilibrium equations are obtained by direct integration or modal superposition. Direct integration schemes are further divided into two classes: implicit and explicit integration. Implicit integration uses the equilibrium equation at the current time to calculate the current displacements. Explicit integration

CRC_DK8307_CH003.indd 138

10/27/2008 8:45:34 AM

Advanced Designs of Ultrasonic Transducers and Devices

139

uses the equilibrium equation from the immediate previous time step. Either approach may be used, and both have criteria to select the appropriate size of time step for the integration routine. Modal superposition is a method that sums the equivalent responses of individual modes to obtain the overall displacement. Its effectiveness comes from the fact that the equations of equilibrium have been transformed from the set of coupled equations in U of Equation 3.6a to a set of uncoupled equations in X in Equation 3.12a. Thus there are independent equations for the xi’s of X that are readily solved using Duhamel’s equation. They are then simply summed over all modes to obtain U, which results in n

U  ∑ ix i i1

(3.16)

The solution for an individual xi with damping included is t

xi 

1 ri ()ei
i ( t ) sin
di (t  ) d
ei
i t (A i sin
di t
Bi cos
di t)
di ∫0
di 
i (1 i2 )

(3.17a) (3.17b)

where Ai and Bi are constants determined from the initial conditions of xi and ẋ i. These initial conditions are determined from the initial conditions of U and U̇ as defined in Equations 3.12b and 3.12c. The forcing function is ri(t), which is given by Equation 3.12d. The solution in Equation 3.17a is composed of two terms. The first is the forced solution that depends on the forcing function, and the second is the free response that depends on the initial conditions. The exponential factor in each term accounts for the damping, and ωdi denotes the damped natural frequency, which depends on the damping coefficient, ξ i. Recall in this development that the damping matrix, [C], is especially constructed to obtain Rayleigh damping to maintain the independence of the individual equations in Equation 3.12a. When the matrix [C] is full, then direct integration methods must be used. Damping must be included in the time-dependent solution. Otherwise, the response may grow unbounded or continue to vibrate long after the forcing function has been removed.

3.2.4

Piezoelectricity

The development presented thus far is for structural analysis only and does not include the specific piezoelectric materials. However, some commercial packages to have some capabilities in terms of piezoelectricity through the constitutive relation in a piezoelectric material. The inclusion of piezoelectricity presented here is based on the development given by Allik et al. [2,3]. Piezoelectric materials have their electric and mechanical properties coupled through the following equations (Holland and EerNisse [4]):

CRC_DK8307_CH003.indd 139

E Tij  c ijkl S kl  emijEm

(3.18a)

S D n  enklS kl
mn Em

(3.18b)

10/27/2008 8:45:34 AM

Ultrasonics: Data, Equations, and Their Practical Uses

140

Equations 3.18a and 3.18b are in tensor notation, where each index runs from 1 to 3. Skl is the strain tensor, Tij is the stress tensor, Dn is the electric displacement, and Em is the E electric field. The coefficients cijkl are the elastic stiffness constants, e is the piezoelectric S tensor that couples the mechanical and electrical properties of the material, and εmn are the E dielectric constants. The superscript on cijkl indicates that the stiffness matrix coefficients S are determined under constant field, and the superscript on εmn indicates that the dielectric constants are determined under constant strain. The constitutive relations also can be written in terms of other combinations of independent and dependent pairs of T, S, E, and D. Structural analysis using FEA is based on independent displacements, and electrical potentials (i.e., voltages) are commonly used to describe the drive to piezoelectric materials. Strains are derivatives of displacements, and potentials are derivatives of electric fields. Therefore, the previous form of constitutive relations is most appropriate for the development of FEA with piezoelectric elements. Because of symmetry, the previous tensor form can be reduced to an engineering matrix form where it replaces double indices with a single index running from 1 to 6 T  [c E ]S  [em ]Em

(3.19a)

S D n  [en ]TS
[ nm ]Em

(3.19b)

It is easily seen that if piezoelectric matrix [e] is zero, then Equation 3.19a reduces to the more familiar matrix form of the generalized Hook’s law for elastic materials, and Equation 3.19b reduces to the relation for dielectric materials. A 6 × 6 matrix is sufficient to define the constitutive relation of an elastic material. For a piezoelectric material, the general matrix is a partitioned 9 × 9 matrix. The upper left 6 × 6 partition contains the elastic E S constants, cvμ . The lower-right 3 × 3 partition is a matrix of the dielectric constants εmn . The lower-left 3 × 6 and the upper right 6 × 3 partitions are the piezoelectric matrix [emv] and its transpose [enµ]T. The constitutive relations is reduced to a single matrix equation c11 c  12 c13  c [T]   14     c15 [D] c  16 e  11 e12 e  13

c12 c13 c14 c15 c16 e11 e12 e13  c 22 c 23 c 24 c 25 c 26 e21 e22 e23  c 23 c 33 c 34 c 35 c 36 e31 e32 e33   c 24 c 34 c 44 c 45 c 46 e 41 e 42 e 43  [S]  c 25 c 35 c 45 c 55 c 56 e51 e52 e53     c 26 c 36 c 46 c 56 c 66 e61 e62 e63  [E] e21 e31 e 41 e51 e61 11 0 0   22 0  e22 e32 e 42 e52 e62 0 0 e23 e33 e 43 e53 e63 0 33 

(3.20a)

or [T]  [c]66 [e]63  [S]      [D] [e]T36 []33  [E]

CRC_DK8307_CH003.indd 140

(3.20b)

10/27/2008 8:45:35 AM

Advanced Designs of Ultrasonic Transducers and Devices

141

The superscripts indicating constant strain and constant field are dropped for notational convenience in the remainder of this section. As seen by comparing Equations 3.20a and 3.20b, the constituent matrix can be partitioned into four matrices: a purely elastic matrix, [c]; piezoelectric coupling matrix and its transpose, [e]; and the dielectric matrix, [ε]. The size of each submatrix is shown as a subscript in Equation 3.20b. The effects of piezoelectricity are added to Equation 3.1 for the principle of virtual work to yield Equation 3.21. The new terms include the matrix of piezoelectric constants, [e], vector of electric fields, E, and electric potential, φ. Body charge, σ, surface charge, σ′, and point charge, Q, are analogous to the body force, F, surface tractions, T, and concentrated forces, Ric, respectively.

∫ ∫ ∫ ST [c]S  ST [e]E  ET [e]TS  ET []E  dV  ∫ ∫ ∫ [uTF  uTu   ]dV V

V


∫ ∫ uS1T T dS1  ∫ ∫  ′ dS 2 S1


u

iT

S2

R ic

  Qic

(3.21)

The electrical potential and electrical fields within a finite element are defined by similar relations to the displacement and strains given in Equations 3.2 and 3.3, respectively. The potential and field are given by

m (x, y, z)  [H m (x, y, z)]

(3.22)

Em (x, y, z)   [B m (x, y, z)]

(3.23)

m m as in the case of [Bm u ], the matrix [Bφ ] is calculated by differentiating elements of [Hφ ]. The development continues similar to that as described in Section 3.2.2. Additional matrices are obtained and defined by

CRC_DK8307_CH003.indd 141

  T m m [K u ]  ∑  ∫ ∫ ∫ [Bm u ] [e][B ]dV   m   Vm

(3.24a)

  m [K u ]  ∑  ∫ ∫ ∫ [B m ]T [e]T [Bm u ]dV   m   Vm

(3.24b)

  [K

] ∑  ∫ ∫ ∫ [B m ]T [][B m ]dV m   m   Vm

(3.24c)

    i  Q  ∑  ∫ ∫ ∫ [H m ]T dV m 
∑  ∫ ∫ [H m ]T ′ dS m 2
Qc   m m  m  m  V   S2 

(3.24d)

10/27/2008 8:45:35 AM

Ultrasonics: Data, Equations, and Their Practical Uses

142

When these definitions are substituted into Equation 3.21, the following expression is obtained 
[C]U 
[K ]U
[K ]  R]
 T [[K ]U
[K ]   Q]  0 U T [[M]U uu u

u

u

(3.25)

The matrix [Kuu] is the same matrix [K] as defined in Equation 3.6d, and the subscript uu has been added to denote the elastic stiffness matrix and to be consistent with the nomenclature used with the other matrices. The matrices, [Kuφ] and [Kφu], are the piezoelectric stiffness matrices, and [Kφφ] is the dielectric stiffness matrix. As seen in Equations 3.6d and 3.24a through 3.24c, there is a stiffness matrix corresponding to each of the submatrices in Equation 3.20b. Thus the stiffness matrices tie the appropriate constituent material properties to the geometry of the finite element on an element by element basis. There are two variational quantities, δUT and δΦT, in Equation 3.25, and each is a leading factor to a sum of terms. It is a property of variational functionals that for Equation 3.25 to be true in general, each bracketed sum of terms must be equal to zero. So the reduction of Equation 3.25 leads to two coupled matrix equations   [C] 0   U   [K ] [K ]  U   R  [M] 0   U uu u

 0 0    
 0 0    
[K ] [ K ] ⌽   Q 

       ⌽    ⌽   u

(3.26)

which is the basic formulation for FEA including piezoelectricity. Equation 3.26 includes structural damping due to material losses but does not include internal losses due to dielectric heating.

3.2.5

Interpolation in the Finite Element

As seen in Sections 3.2.2 and 3.2.4, one of the most important functions in the finite element method is the interpolation matrix, [H]. In an isoparametric formulation, both positions and displacements in a finite element are expressed by interpolation functions from their values at the nodes. The interpolations are defined by a natural coordinate system, denoted by (r, s, t), where each coordinate runs from −1 to 1. The positions and displacements for a 3D element are expressed by q

x  ∑ h ix i i1

q

u  ∑ h iui i1

q

y  ∑ hiyi i1

q

v  ∑ h i vi i1

q

z  ∑ h i zi

(3.27a)

i1

q

w  ∑ hiwi

(3.27b)

i1

Each interpolation function is a function of the natural coordinates r, s, and t. The number of interpolation functions is equal to the number of nodes defining the element. Here, the interpolations functions are given for the local element coordinate system. Note, that the same formulation applies to 2D and 1D elements with the appropriate number of summations. To illustrate the use of interpolation functions for the formation of the [H] and [B] matrices, a four-node, 2D rectangular element is used as shown in Figure 3.3. The interpolation

CRC_DK8307_CH003.indd 142

10/27/2008 8:45:35 AM

Advanced Designs of Ultrasonic Transducers and Devices

y, v

(x2, y2)

143

s (x1, y1)

[−1, 1] [1, 1]

[−1, −1] r

(x3, y3) [1, −1] [r, s] Natural coordinates (x, y) Global or local coordinates

(x4, y4) x, u

FIGURE 3.3 A four-node, 2D, isoparametric element.

functions are 1 h1  (1
r)(1
s) 4 1 h 2  (1  r)(1
s) 4 1 h 3  (1  r)(1  s) 4 1 h 4  (1
r)(1  s) 4

(3.28)

This form is the same as that for a standard bilinear interpolation. The interpolation matrix [Hum] is assembled from the individual interpolation functions. Its transpose is given in Equation 3.29. 1 (1
r)(1
s) 0 4 1 (1
r)(1
s) 0 4 1 (1  r)(1
s) 0 4 1 (1  r)(1
s) 0 4 m T [Hu ]  1 (1  r)(1  s) 0 4 1 0 (1  r)(1  s) 4 1 (1
r)(1
s) 0 4 1 0 (1
r)(1  s) 4

CRC_DK8307_CH003.indd 143

(3.29)

10/27/2008 8:45:35 AM

Ultrasonics: Data, Equations, and Their Practical Uses

144

The strains for the 2D problem (whether plane strain or plain stress), are in general derivatives of displacements as shown in Equation 3.30. The four individual strains are derivatives with respect to r and s. Using the defi nitions given in Equation 3.29, the strains are given in Equation 3.31. The matrix, [J], has been introduced, which denotes the Jacobian operator. Without the inverse Jacobian operator, the derivatives of u and v on the left side would be with respect to r and s. This is necessary, because although the natural coordinate system allows the integrals to be readily calculated, the real strains must be used. The strain-displacement matrix, [B], can now be assembled and is given in Equation 3.32.  ∂u        ∂x  xx     ∂v        yy     ∂y      xy   ∂u ∂v  
  ∂y ∂x 

(3.30)

 ∂u   ∂x     1 [ J]−1 1
s 0 (1
s) 0 (1  s) 0 1  s 0  u 1
r 0 1  r 0 (1  r) 0 (1
r) 0   ∂u  4     y ∂    ∂v   ∂x     1 [J]1 0 1
s 0 (1
s) 0 (1  s) 0 1  s  u 0 1
r 0 1  r 0 (1  r) 0 (1
r)  ∂v  4     y ∂  

(3.31)

 ∂u     ∂x    ∂v   [B]   ∂y     ∂u ∂v  
  ∂y ∂x  0 (1

s) 0 (1  s) 0 1 s 0  1
s 1 1   1
r 0 1 r 0 (1  r) 0 (1
r) u  [ J]  0 4 1
r 1
s 1  r (1
s) (1  r) (1  s) (1
r) 1  s  (3.32)

CRC_DK8307_CH003.indd 144

10/27/2008 8:45:36 AM

Advanced Designs of Ultrasonic Transducers and Devices

145

The Jacobian for the 2D case is given as  ∂   ∂x  ∂r   ∂r    ∂   ∂x  ∂s   ∂s

∂y   ∂  ∂r   ∂x   ∂y   ∂  ∂s   ∂y 

(3.33)

∂ ∂  [ J] ∂r ∂x The integrations for the individual elements are performed over the natural coordinate system, but the determinant of the Jacobian is used to scale the volume integral as indicated by dV = det[ J] dr ds dt 3.2.6

(3.34)

Boundary Conditions

The development thus far has dealt only with an unsupported structure, which is sufficient for dealing with the vibrational modes of the structure itself. For static problems, clearly one must introduce some boundary conditions to control rigid body motions. Otherwise a model might analytically float away. A system that is not sufficiently analytically restrained can lead to overflows and other run time errors. There are two types of boundary conditions: geometric and forced. The forced boundary conditions, which are also known as natural boundary conditions, are simply introduced through the reaction vector, R. The geometric boundary conditions are prescribed at individual nodes for a given degree or degrees of freedom. For example, a pinned condition in a 2D problem would fix the node in space but allow for rotation. FEA codes provide the functionality to set the boundary conditions on a node-by-node basis. 3.2.7

Symmetry

One of the most important considerations in the design of an FEA model is the use of symmetry to reduce the size of the problem, thereby significantly reducing the computational time. Although computer speeds continue to improve, the complexity of models also continue to grow, and the need to both iterate on designs and perform parametric studies increases. Therefore, it is still important to take advantage of symmetry. In the design example of a large ultrasonic horn that follows, three planes of symmetry were used to achieve a one-eighth model for the entire horn. Some preprocessors allow the building of a basic section such as the one-eighth model, and its tools, include the ability to reflect, duplicate, and join copies to assemble the entire structure. By prescribing symmetry, some modes of vibrations are eliminated. For example, pure torsion in our large horn example is eliminated. Therefore, the type of information being sought in the FEA analysis may dictates whether to use symmetry conditions. 3.2.8

Beyond Linear Elastic Isotropic

A discussion of all the capabilities of today’s FEA codes is beyond the scope of this chapter and book. The development presented earlier deals only with small displacements of

CRC_DK8307_CH003.indd 145

10/27/2008 8:45:36 AM

Ultrasonics: Data, Equations, and Their Practical Uses

146

linear elastic isotropic materials. These conditions do generally apply to the design of ultrasonic systems. However, the capabilities of FEA codes include large displacement models and nonlinear material properties. In addition to large displacements, there are also gap elements that allow two structures to be free until forced together. It is therefore possible to develop models that study the effect of bringing an ultrasonic welder in contact with the materials to be welded or staked. Material properties include orthotropic, anelastic, elastic–plastic, user-specified stress–strain curves, creep, and viscoelasticity. Some FEA codes include piezoelectricity as a material property that allows drivers to be modeled from their electric inputs to the displacements at the end of the horn. 3.2.9

Pre- and Postprocessors

FEA is the analytical engine that calculates strains and stresses. Other programs, called pre- and postprocessors, are used to input the model and manipulate and display the results. Many FEA packages have these built into their own programs, so that they are native tools. There are also commercial codes that are strictly pre- and postprocessors. A preprocessor is a tool that aids the user in developing the FEA model. These are graphical tools that allow the designer to manipulate elements and nodes and to build entire models accurately and quickly. Models can be viewed from various angles to check for overall correctness. Erroneous geometric descriptions are readily apparent and corrected before the model is submitted to the FEA program for solution. Preprocessors automatically mesh the model once the basis geometry is defined. The mesh may need refined based on results. A postprocessor operates on the results from the FEA analysis. It manipulates the results, displays the desired outputs, such as stresses and displacements, as colored contours and displaced meshes. Graphical tools are available to view the results at different angles or at specific slices of the model. In the case of vibrational modes and time-dependent solutions, the results can be animated to depict the motion. In integrated software suites, a solid model of a given component is used as input. This model is then meshed for FEA using a preprocessor. The FEA is run, and the results are displayed to the designer. The design is iterated to improve performance both in terms of achieving the design objectives and the accuracy of the FEA model itself. Once the designer is satisfied with the design, the solid model program is then used to generate the code for computer numerical controlled machining tools to build the component or for a stereo lithography machine to create a plastic model. This integrated approach significantly decreases the time and cost to achieve new designs.

3.3 3.3.1

Theory of Plates and Shells Introduction

Vibrations of thin plates and shells are analyzed by considering the stresses acting on a differential element. The equilibrium of the forces and moments are then used to form a set of linear differential equations that are the solution to the problem. The characteristic equation of the set yields a polynomial in frequency. The roots of the equation are the resonant frequencies of the system. These frequencies and the corresponding nodal patterns are used to design the system. Seminal references in this area are two books in which Leissa compiled and reviewed the theories of plates and shells. Now reprinted by the Acoustical Society of America, these are entitled Vibration of Plates [5] and Vibration of

CRC_DK8307_CH003.indd 146

10/27/2008 8:45:36 AM

Advanced Designs of Ultrasonic Transducers and Devices

147

Shells [6]. The theories of plates and shells are generally covered in most texts on propagation of elastic waves including books by Achenbach [7], Fahy [8], and Graff [9]. A more accurate approach is to treat the plate or shell as an elastic medium in 3D space with appropriate boundary conditions applied at the surfaces. With this approach, plate waves or Lamb waves are predicted as well as the higher-frequency surface waves or Rayleigh waves. Plate and shell theories are low-frequency approximations to the elastic solution, which is itself only an approximation to a real structure and material. Both approaches produce useful results for the design of large ultrasonic power systems, but the theories of plate and shell vibrations yield both accurate results and design insights without getting overly involved in mathematical manipulations to obtain a solution as is the case in elastic wave theory. The basic equations for plate and shell vibrations are derived under a number of assumptions: 1. 2. 3. 4.

The plate or shell wall is thin. Displacements are small. The normal stress is negligible. Plane sections remain plane and the normals to the midplane remain normal.

There is no specific criterion for thinness. Using these theories, one is not interested in waves propagating across the thickness. So a general rule is that the thickness should be less than a fifth of a quarter-wavelength in the bulk material. For shells, the thickness should be less than a tenth of the radius of the shell. The designer can use these “thinness” rules as guidelines for the applicability of the plate and shell theories. The other three assumptions are met basically by any practical ultrasonic device. The second assumption evokes Hook’s law for linear elastic constitutive relations, and allows the derivations of the equations to be referenced to the midplane. The third and fourth assumptions further simplify the solution. 3.3.2

Free Plate Vibrations

Solutions exist for a variety of shapes and boundary conditions. The most common shapes of plates are circular, elliptical, annular, triangular, and rectangular, where a square plate is a special case of a rectangular plate. Boundary conditions include combinations of free (F), simply supported (SS), and clamped (C). The type of boundary condition should be selected to best match the mounting configuration used to support the plate in the actual hardware. The out of plane displacement, w, is given by Dx

∂ 4w ∂ 4w ∂ 4w ∂ 2w ∂ 2w ∂ 2w ∂ 2w
2D
D
  N
N

q(x, y, t ) N xy y x xy y ∂x 4 ∂x 2 ∂y 2 ∂y 4 ∂t 2 ∂x 2 ∂x∂y ∂y 2 (3.35)

where x and y are the orthogonal directions of the plate; t denotes time; Nx, Ny, and Nxy are the traction forces; and q is the transverse (or normal) loading. Equation 3.35 is written for orthotropic materials. The D’s describe the material properties of the plate and are given by Dx 

CRC_DK8307_CH003.indd 147

Ex h 3 12(1   x  y )

(3.36a)

10/27/2008 8:45:36 AM

Ultrasonics: Data, Equations, and Their Practical Uses

148

Dy 

Ey h 3 12(1   x  y )

(3.36b)

Gh 3 12

(3.36c)

D xy 

For an isotropic plate with no traction forces, Equations 3.35 and 3.36 reduce to where the D’s reduce to D∇ 4 w
h D

∂ 2w q ∂t 2

(3.37)

Eh 3 12(1  )

(3.38)

Equation 3.37 is fourth order in x and fourth order in y. Thus a total of eight boundary conditions, two per edge, must be described to obtain a solution. The three types of edge conditions (SS,C,F) are commonly analyzed in plate and shell theories and each represents a set of two boundary conditions at an edge. There are a total of 21 unique combinations of the three types of edges at the four sides of the rectangular plate, which are SS-SS-SS-SS SS-C-SS-F C-C-C-C C-C-SS-SS C-SS-C-F C-F-C-F SS-SS-F-F

SS-C-SS-C SS-SS-SS-F C-C-C-SS C-C-SS-F C-SS-SS-F C-F-SS-F SS-F-F-F

SS-C-SS-SS SS-F-SS-F C-C-C-F C-C-F-F C-SS-F-F C-F-F-F F-F-F-F

Leissa [5,6] reports eigenvalues and eigenvectors for each combination of edge conditions in the forms of equations, tables, and graphs. The results for the SS-C-SS-C case are given in Chapter 2, with much greater detail in the development of the plate and shell theories. 3.3.3

Shell Vibrations

A particularly useful model for the vibrations of a cylindrical shell is the dynamic form of Donnell’s model. The development presented here is based on that given by Kraus [10]. The model is expressed as three coupled equations ∂ 2 u x 1   ∂ 2 u x 1
 ∂ 2 u   ∂w 1   ∂ 2 u x


  2 0 ∂x 2 ∂t 2a 2 ∂ 2 2a ∂x∂ a ∂x E

(3.39a)

1
 ∂ 2u x 1   ∂ 2u 1 ∂ 2u 1 ∂w 1   ∂ 2 u 


  2 0 ∂t 2a ∂x∂ 2 ∂x 2 a 2 ∂ 2 a 2 ∂ E

(3.39b)

1   2 ∂ 2w  ∂u x 1 ∂u  w h 2 4
2
2
 w
 2 0 ∂t a ∂x a ∂ a 12 E

(3.39c)

where a is the radius of the shell, ν Poissons’s ratio, E Young’s modulus, ρ density, and h half thickness.

CRC_DK8307_CH003.indd 148

10/27/2008 8:45:37 AM

Advanced Designs of Ultrasonic Transducers and Devices

149

m=1

m=3 m = 5, n = 4

m=5

n=2

n=4

n=6

FIGURE 3.4 Nodal checkerboard pattern modes of a right circular cylinder used in Donnell’s model.

The three orthogonal displacements are ux in the axial direction, uθ in the circumferential direction, and w in the radial direction, and time is denoted by t. To solve the free vibration problem, solutions corresponding to known deformation patterns for a cylinder are assumed. These are depicted in Figure 3.4 for clamped end conditions. In the circumferential direction, the displacements are an integral number of sine or cosine cycles. The integral number ensures that there is continuity of displacements and slopes as a complete encirclement is made. In the axial direction the form depends on the end conditions, which can be free, simply-supported, or clamped. The axial form is a sum of sines, cosines, and hyperbolic sines and cosines. The displacement functions for clamped ends are given by ᐉ ᐉ    u x  A  sin   x
k sinh   x  cos n cos
t     a 2 a 2 

(3.40a)

ᐉ ᐉ    u  B cos   x
k cosh   x  sin n cos
t     a 2 a 2 

(3.40b)

ᐉ ᐉ    w  C cos   x
k cosh   x  cos n cos
t     a 2 a 2 

(3.40c)

The axial position along the cylinder is denoted by x with a range of 0 to ℓ. A, B, and C are arbitrary amplitude coefficients, ℓ is the length of the cylinder, and, k, is given by k

CRC_DK8307_CH003.indd 149

sin(ᐉ/2a) sinh(ᐉ/2a)

(3.40d)

10/27/2008 8:45:37 AM

Ultrasonics: Data, Equations, and Their Practical Uses

150

The variable, µ, is determined from the transcendental equation, tan(µℓ/2a) + tanh(µℓ/2a) = 0

(3.40e)

which has roots of µℓ/a = 1.506π, 3.5π, 5.5π, 7.5π, … The characteristic equation as a function of frequency is given by ∆ 3  R 2∆ 2  R 1∆  R 0  0

(3.41a)

where ∆ is given by  a 2 (1   2 )


2 E

(3.41b)

As seen from Equations 3.41a and 3.41b, the characteristic equation is cubic in frequency squared. Thus, three different frequencies produce the same nodal pattern. The lowest frequency is generally an order of magnitude less than the other two, and it corresponds to nearly pure radial motion. The other two frequencies correspond to the in-plane axial and circumferential displacements, which do not radiate ultrasound into the surrounding medium. Thus, the lowest solution of Equation 3.41a is the most important one in driver design. The frequency of the radial displacement mode is found from the standard solution to a cubic equation given by  R   2 1/ 2 cos
2 3 3

(3.42a)

where 3  1  R2       R 1  2   3    27 

1/2

 1  RR 2R 3     cos1   R 0  1 2
2   27   3  2 

(3.42b)

(3.42c)

The coefficients, R0, R1, and R2, are given by the following equations:  3  2 2 1  4 (1  )  2  2   R2   1

1
n
 
n 4
2 2n 2 2   2 1   1  2  2 R1 

1   4 1   2 4  1  1  2 2   n

(  3)(
1) 2   2n 2
n 
2 2 2 1   2 4  (1
)(1  2)  2  2 1  1 1    2  2 3   2 2  
 n 
 1
  1     2 2 2 1  2  2       4
n 4
2 2n 2 2  1  

CRC_DK8307_CH003.indd 150

(3.42d)

(3.42e)

10/27/2008 8:45:37 AM

Advanced Designs of Ultrasonic Transducers and Devices 2    4  (  1)2  2  2 2 1    2 2  R0    n
1         4 1  2   1  

151

(3.42f)

1   4 1  2 4   4 1           1
(
1)(  3) 2   2n 2

n  
n 4
2 2n 2 2      2 4 1  2 2 1    where 1 h2   12a 2 1  2

1
k2 2a  ᐉ  1  k 2
sin    a ᐉ

(3.42g)

(3.42h)

If referring to Kraus [10], be aware that there are errors in the published equations for R’s.

3.4

Design Example: Flat Plate

This example is used to illustrate some of the interesting features of working with flat plates. The basic governing equation for a linear, isotropic, thin plate was given in Equation 3.12. The solution is readily derived assuming a solution of separation of variables of the form w(x, y, t) = X(x) Y(y) e−j
t

(3.43)

The simple separation of variables is effective for a plate only when simply supported conditions apply to opposing sides. The totally free plate is the one most easily implemented in an experiment, and one of the more difficult cases to handle analytically. In this case, mode shapes cannot be represented as a single product of X and Y functions, but rather, as summations of products. Yet for the purposes of this illustration, it is assumed that a mode shape can be represented by a product of cosine functions in the x- and y-directions. The simple form of Equation 3.43 does not support the completely free boundary conditions, but nevertheless does provide a reasonable approximation to the mode shapes of interest, especially at higher orders. Now with this simplifying approximation, the frequency equation reduces to f

  n2 m2  D
2 2  a 2 b  h

(3.44)

where f is the cyclic frequency, n the number of half-wavelengths in the x direction, m the number of half wavelengths in the y direction, a the length of the plate on the x side, b the length of the plate on the y side, D the flexural rigidity of a plate (see Equation 3.38), ρ the density of the plate material, and h the thickness of the plate. The typical mode shapes for a given direction (i.e., x or y) for the correct solution approximate cosine functions except near the sides. Here the length from the last node to the free edge is shorter than a quarter-wavelength of a cosine. Therefore Equation 3.44

CRC_DK8307_CH003.indd 151

10/27/2008 8:45:38 AM

Ultrasonics: Data, Equations, and Their Practical Uses

152

overestimates the actual frequency. The correct approach is to calculate the frequencies based on the equations developed by Gorman [11]. A 0.25 in. thick steel plate is used to create an active area of 2 ft. by 3 ft. It is desired to excite a mode that has a rectilinear nodal pattern with a commercial ultrasonic system designed to operate in the vicinity of 20 kHz. An initial experimental approach is taken in which the commercial system is attached to the center point of the plate and driven through a range of frequencies centered at 20 kHz. The plate is painted with an aluminum oxide slurry to visualize the modes. One of the modes of interest is m = 22, n = 8 shown in Figure 3.5a. Its observed frequency is 19,500 Hz. The frequency predicted by Equation 3.44 is 20,700 Hz, which is higher than the actual frequency as expected. A neighboring mode is excited at a different frequency and with a different location for the driver. Its nodal pattern is shown in Figure 3.6a. As can be inferred by the complex nodal patterns, the displacements in the x- and y-directions are highly coupled. Such a mode shape cannot be predicted by the oversimplified solution of Equation 3.43. An FEA model of the same plate is performed for the vibrational modes in the vicinity of 20 kHz. Two predicted modes are shown in Figures 3.5b and 3.6b, which correspond to the photographs in Figures 3.5a and 3.6a. The FEA results presented in these plots have

(a)

(b) FIGURE 3.5 The m = 22, n = 22 mode of a 0.25 × 24 × 36 cu. in. rectangular plate: (a) experimental nodal pattern and (b) finite element analysis nodal pattern.

CRC_DK8307_CH003.indd 152

10/27/2008 8:45:38 AM

Advanced Designs of Ultrasonic Transducers and Devices

153

(a)

(b) FIGURE 3.6 A neighboring mode to the one shown in Figure 3.5: (a) experimental nodal pattern and (b) finite element analysis nodal pattern.

been manipulated using the postprocessor to resemble the experimental results using the aluminum oxide slurry. The displacements near zero amplitude are assigned the color white. Any greater displacements of either sign are colored gray. This gives an appearance to the FEA results of a plate that has been analytically painted with an aluminum oxide slurry. The view is also manipulated with the postprocessor to have nearly the same perspective as the photographs in Figures 3.5a and 3.6a. As seen, the nodal patterns from the FEA results and the experimental observations are in excellent agreement. The predicted resonant frequencies are 19,250 and 19,275 Hz in good agreement with the observations.

3.5 3.5.1

Design Examples: Cylindrical Shells Design 1: Applying Donnell’s Model

A flow-through ultrasonic device is to be designed to strip out dissolved gases from a liquid by the process of rectified diffusion. The basic design includes a cylindrical shell constructed from Schedule 40, nominal 4 in. diameter pipe installed into existing piping using

CRC_DK8307_CH003.indd 153

10/27/2008 8:45:39 AM

Ultrasonics: Data, Equations, and Their Practical Uses

154

bolted flanges. Several factors need to be considered in the design. Operating pressure, material compatibility, and existing piping and flow requirements enter into determining the diameter, thickness, and material of the pipe. The length of the pipe is determined by the total volume being treated and the residence time required to achieve the desired effect. The frequency of the ultrasonic system is selected based on the ultrasonic mechanism or action that is to be imposed on the material. Although these considerations drive the individual design factors, they are interrelated. Selection of the best combination of factors leading to a good design is accomplished with parametric studies using FEA or an analytical model, such as Donnell’s. In this application, the size of piping and the wall thickness (Schedule 40, nominal 4 in. pipe diameter) are set by the existing piping and the maximum operating pressure of the process. The length of pipe is selected based on the residence time at a nominal flow rate of 10 gpm. A 4 ft. length provides a 16 s residence time that is assumed to be sufficiently long to achieve the desired effect at an operating frequency of nominally 20 kHz. Donnell’s model is used to evaluate the resonant frequencies of the shell. Clamped-end conditions are assumed, which allow Equations 3.39 through 3.42 to be used. This should be a good approximation, because the heavy flanges and their bolted assembly add significant stiffness and mass to the ends of the cylinder. The first step in the design is to calculate the vibrational frequencies of the shell using Equations 3.42a through 3.42h. The results are shown in Figure 3.7. The solid lines are plots of resonant frequency versus circumferential wave number for a family of axial half-wavelengths. The number of axial wavelengths increases from the bottom curve to the top curve. A solid curve is drawn for the resonances of each axial order of half-wavelengths, but the resonances exist only at points corresponding to the integral number of circumferential wavelengths. A horizontal line, representing a specific frequency, drawn through these curves indicates the modal density in the neighborhood of that frequency. The modal density is sparse at the lower frequencies and quite dense at the higher frequencies. The results from Donnell’s model were used to design and construct an actual device to strip out dissolved gases from liquids by a process known as rectified diffusion. A power ultrasonic driver (transducer), designed and constructed for nominal 20 kHz operation, 30 Donnell's model Data

25

Frequency (kHz)

Finite element analysis model 20

15

10

5

0 0

2 4 6 Number of circumferential waves (n)

8

FIGURE 3.7 Resonant frequencies of checkerboard modes for the flow-through ultrasonic processor.

CRC_DK8307_CH003.indd 154

10/27/2008 8:45:40 AM

Advanced Designs of Ultrasonic Transducers and Devices

155

FIGURE 3.8 Numerous shell modes observed in the electrical impedance spectrum in the vicinity of the resonance of the high-Q ultrasonic driver.

was attached to the center of the pipe to which the flanges had already been welded. The mounted resonance, determined by measuring the electrical impedance of the driver, was nominally 18.5 kHz. The impedance of an unmounted driver has well-defined resonance and antiresonance positions, but the impedance is much more complicated when the driver is mounted to a shell or a plate. The effect of the load impedance reflected to the driver (transducer) combined with the impedance of the driver control the resonance and antiresonance of the vibratory system. This can be observed in the frequency spectrum of the electrical impedance for a horn mounted on a cylinder as seen in Figure 3.8. The large trough is the resonance of the mounted driver, and the many smaller peaks and valleys are the resonances and antiresonances of pipe modes superimposed on it. Therefore, it is possible to excite many shell modes even with a highly resonant (high-Q) driver. In this example, the various modes were identified by painting the cylinder with a slurry of water and aluminum oxide powder, as shown in Figure 3.9. The driving frequency was then swept slowly through a narrow range about the nominal resonant frequency of the driver. The modal patterns were identified visually by the fact that the vibratory forces caused the aluminum oxide powder to migrate to the nodal positions corresponding to each mode. The observed and predicted values of the desired mode are shown in Figure 3.7 as open and closed circles, respectively. An FEA analysis was also performed for the finished device. The bolt holes in the flange were ignored in this analysis so that an axisymmetric mesh of eight-node shell elements could be developed. This simplification is justified, because the flanges add significant mass and stiffness to allow fixed conditions to be assumed for the ends of the cylinder. If this assumption is valid, then the flange should not vibrate significantly and the bolt holes would not significantly affect the result. It was further assumed that there were two planes of symmetry: one at the axial midplane and one at the circumferential midplane. The two planes and the desire to find only the corncob modes allowed a quarter model to be used. The modal pattern of the FEA model for the flow-through processor is shown in Figure 3.10. The FEA analysis predicts a lower frequency than Donnell’s model, and it is also in much better agreement with the measured frequency as indicated in Figure 3.7.

CRC_DK8307_CH003.indd 155

10/27/2008 8:45:40 AM

Ultrasonics: Data, Equations, and Their Practical Uses

156

(a)

a

(b) FIGURE 3.9 Aluminum oxide powder on the ultrasonic flow-through liquid processor at its design frequency: (a) nodal pattern on the ultrasonic degasser and (b) close-up of the nodal pattern.

FIGURE 3.10 Finite element analysis prediction of the ultrasonic flow-through liquid processor.

CRC_DK8307_CH003.indd 156

10/27/2008 8:45:41 AM

Advanced Designs of Ultrasonic Transducers and Devices

157

0.05 0.045

m=3

0.04 Stretching

Energy factor

0.035

Bending

0.03

Total 0.025 m=2

0.02 0.015

m=1

0.01 0.005 0 2

3

4

5

6

7

Number of circumferential waves, n FIGURE 3.11 Energy factor for bending, stretching, and total as a function of frequency for the checkerboard modes of a right circular cylinder.

The results demonstrate that a given driver design can excite a number of modes in the neighborhood of the design frequency. Two questions now arise: (1) which mode is the best one to drive and (2) how to avoid driving the other modes. The first question is answered from energy considerations. A solution to the second question is to use multiple drivers to force the desired mode. The total strain energy in the shell is the sum of two types of energy, as depicted in Figure 3.11. One is the stretching energy as the shell expands and contracts, and the other is bending energy that deforms the shell. Most of the energy at low values of circumferential and axial wavelengths goes into stretching the shell. As the number of circumferential and axial wavelengths increase more energy goes into bending energy. At some intermediate circumferential order for a given axial order, the total energy is a minimum. The bending energy is nearly independent of axial order, whereas stretching energy strongly depends on the axial order. Given a choice, the designer should choose a circumferential order that is at or near the minimum energy trough for a given axial order. Once the circumferential order is selected, multiple drivers can be located to minimize the number of neighboring modes that are excited. For this design, the circumferential order was selected to be 3, and three phase-matched drivers were attached at the axial center of the pipe at a spacing of 120°. With this configuration, nodal patterns with circumferential orders of multiples of 3 can be excited and other modes with different circumferential orders are suppressed. The performance is further enhanced by placing drivers at in-phase antinodes located an equal distance on either side of the center array. The additional drivers provide more power to achieve higher displacements and to further limit the number of nodal patterns that can potentially be excited. The axial placement of the drivers is predicted by Donnell’s model, but the nodal pattern in the axial direction is sensitive to end conditions especially

CRC_DK8307_CH003.indd 157

10/27/2008 8:45:43 AM

Ultrasonics: Data, Equations, and Their Practical Uses

158

FIGURE 3.12 Ultrasonic flow-through liquid processor installed in a test loop at a petroleum plant.

near the end. Therefore, the axial locations of the in-phase antinodes should be determined experimentally using an aluminum oxide slurry to map the actual nodal pattern. A picture of the finished flow-through liquid processor is shown in Figure 3.12. The liquid processor is installed in a flow loop to strip out methane from condensate. The nine drivers can be seen in the picture. 3.5.2

Design 2: Comparing Donnell’s Model with Finite Element Analysis

An ultrasonic shell is to be designed using nominal 12 in. diameter, Schedule 40 pipe, which has an outside wall diameter of 12.75 in. and a wall thickness of 0.406 in. The target design frequency is 20 kHz. Donnell’s shell model and FEA both are used to obtain solutions, and their results are compared. The shell is nominally 36 in. long and is constructed with welded slip-on 150# ANSI flanges. As in the previous example, the combination of the mass and increased radius of the flanges are assumed to approach clamped-end conditions. The modal predictions using Donnell’s model are shown in Figure 3.13. The plot is the same as Figure 3.7 and includes the results from the FEA analysis discussed shortly. As seen in the figure, the axial order can be as high as 27, and the circumferential order can be as high as 15 for frequencies up to 20 kHz.

CRC_DK8307_CH003.indd 158

10/27/2008 8:45:43 AM

Advanced Designs of Ultrasonic Transducers and Devices

159

30

Frequency (kHz)

25

Donnell's model

Finite element analysis (FEA) model

Data

2 FEA results for one mode

20

15

10

5

0 0

2

4

6 8 10 12 Number of circumferential waves, n

14

16

FIGURE 3.13 Predictions using Donnell’s model, fi nite element analysis, and observed values for a 12 in. diameter, 3/8 in. thick, and 36 in. long stainless steel cylinder.

Five modes were determined by painting a slurry of aluminum oxide on the shell and counting the axial and circumferential orders in the nodal pattern at a resonant frequency. The observed frequencies of the modal patterns are shown in Figure 3.13 with the corresponding theoretical values. The predictions from Donnell’s model parallel the observed data but are consistently higher. A commercial package is used to perform an FEA analysis of the as-built cylindrical driver. The differences between its physical model and Donnell’s model mainly concern the end conditions. For Donnell’s model, the flanges were assumed to approach the clampedend conditions. In the FEA, the flanges are treated as any other structural feature. Another difference is that the actual axial length of 36.125 in. between the inside steps of the flanges is used in the FEA instead of the design length of 36 in. The use of symmetry greatly reduces the amounts of both the time to build the model and the computer time to obtain a solution for the eigenproblem. Seeking symmetric modes in the axial direction permits the cylinder to be divided in half in the axial direction. Also the circumferential symmetry permits the cylinder to be divided in half in the circumferential direction. Therefore, only a quarter of the cylinder needs to be considered. The flanges are not truly axisymmetric, because of the bolt holes. This is when good engineering judgment is used. Does the incorporation of the holes in the FEA model justify the additional time to set up the model and the additional computational time to exercise the model? For this analysis, the holes are not included. Rather, the thickness of a solid annulus of steel is adjusted to have mass equal to that of the actual annulus with the holes. Three sample outputs from a postprocessor are shown in Figures 3.14 through 3.16, corresponding to low-circumferential order and high-axial order, the selected design of circumferential order 12 and axial order 17, and high-circumferential order and low-axial order, respectively. The alternating patterns have opposite polarities of displacements. The flange is seen to undergo little vibration in Figures 3.14 through 3.16. The lack of flange vibration validates the assumption made to use the Donnell’s model with an assumed

CRC_DK8307_CH003.indd 159

10/27/2008 8:45:44 AM

160

Ultrasonics: Data, Equations, and Their Practical Uses

FIGURE 3.14 Low circumferential, high axial order of a 12 in. diameter, 3/8 in. thick, 36 in. long stainless steel cylinder.

FIGURE 3.15 Mode shape of the design objective for a 12 in. diameter, 3/8 in. thick, 36 in. long stainless steel cylinder.

solution for fixed end conditions. Clearly, if the flange is not vibrating, especially in the vicinity of the of the pipe wall, it acts to hold the pipe fixed. The lack of vibration also supports the assumption that the inclusion of the bolt holes do not significantly affect the frequency of vibration and the nodal pattern. Figure 3.17 is a picture of the nodal pattern of a prototype of this device, which used a 12 in. diameter, Schedule 40 section of stainless steel pipe. Three conventional ultrasonic drivers are mounted internally at its center and are spaced 120° apart.

CRC_DK8307_CH003.indd 160

10/27/2008 8:45:44 AM

Advanced Designs of Ultrasonic Transducers and Devices

161

FIGURE 3.16 High circumferential, low axial order of a 12 in. diameter, 3/8 in. thick, 36 in. long stainless steel cylinder.

FIGURE 3.17 Nodal pattern of design objective on a 12 in., Schedule 40 stainless steel pipe.

CRC_DK8307_CH003.indd 161

10/27/2008 8:45:46 AM

Ultrasonics: Data, Equations, and Their Practical Uses

162 TABLE 3.1

A Comparison of Modal Frequencies from Donnell’s Model, Finite Element Analysis (FEA), and Experimental Observation Mode: n, m (n = Full Circumferential; m = Half Axial)

Donnell’s Model (kHz)

FEA (kHz) and Flange Vibration

15, 3 15, 7 14, 11 1, 15 13, 15 12, 17 11, 19 10, 21 6, 25

22.09 22.77 22.09 21.38 22.51 22.00 21.93 22.28 21.77

19.33

7, 25 0, 27 1, 27

22.91 21.60 21.68

2, 27

21.93

3, 27

22.35

Observed Frequency (kHz) 18.49

19.48 19.61 19.43 19.33 19.57 19.22 radial flange 19.30 torsional flange 19.51 19.18 bending 19.22 torsional 19.34 bending 19.38 bending 19.47 torsional 19.50 radial and torsional

18.27 17.87

18.20

18.27

The frequencies for the modes obtained from Donnell’s model, the FEA model, and experimental observations are included in Table 3.1. These are the same values plotted in Figure 3.13. It is clear from the data that the FEA predictions agree much better with the observed values as compared with the values predicted from Donnell’s model. Generally, Donnell’s model overestimates the values by 20%, which cannot be attributed solely to differences in assumed geometries and material properties compared to the actual device. Rather, it is an indication that Donnell’s model represents a much stiffer condition than what actually exists in a cylinder. There are three FEA predictions that correspond to the three observed modes. The FEA frequency predictions are within 9% and are also higher than the observed values. Though some of the difference may be due to the differences in assumed and actual material properties and geometries, the FEA model also appears to be stiffer than the actual cylinder. This suggests that a finer mesh or tighter tolerances may be needed to obtain an acceptable solution. Another interesting result is that although the amplitudes of the flange vibrations are much lower than the cylinder vibration, they do modify slightly the resonant frequencies. Note in several entries in the table, the FEA predicts two nearly equal frequencies for the same circumferential and axial orders. On closer examination of the modes, the primary difference is found to be in the vibration of the flange as noted in the table.

3.6

Design Example: Large Horn

Classical 1D techniques are very accurate for designing and analyzing slender horns with lateral dimensions less than a quarter-wavelength at the operating frequency. Under these conditions, it is easy to isolate only one mode of vibration without interference from any

CRC_DK8307_CH003.indd 162

10/27/2008 8:45:47 AM

Advanced Designs of Ultrasonic Transducers and Devices

163

other modes. The design process becomes more complicated for large horns with lateral dimensions greater than a quarter-wavelength. These larger dimensions provide the potential for resonating in the lateral directions and other modes (such as flexure) depending on the dimensions, how the horn is driven, how it is assembled, and how it is loaded. Under operation, the mode can hop from the desired one to an ineffective, and often deleterious, mode. The alignment of the horn with the driver (transducer) is critical because any offset is likely to pump energy into lateral modes. Large area horns are commonly designed with slots cut in the axial direction, that is, the direction of motion of the ultrasonic driver. The slots limit the number of lateral modes that can be excited in the neighborhood of the desired frequency, because the lateral dimensions of the remaining material between the slots is again less than a quarterwavelength. The design of the slotted horn approaches that of bars vibrating with common end masses, and bar rather than bulk velocity is the effective velocity of the ultrasonic energy. An additional benefit provided by the slots is the greater surface area for dissipating heat during operation. While a slotted design permits the fabrication of effective large area horns, there are tradeoffs. For example, the slots introduce stress concentrations at the fillets that can limit the operating life of the horn. The slender elements located between slots may vibrate in flexure. With the slots, strong lateral modes can be excited in the continuous segments at the axial ends of the horn, when the lateral dimensions correspond to resonances near the operating frequency. These concerns can be eliminated by using good design and fabrication techniques. FEA provides effective means of designing and analyzing the potential performance of the horns with large cross-sectional areas. In this section, FEA is used to compare differences of a solid block of material and the same block that is then slotted to improve ultrasonic performance. The block is assumed to be a cube that is 5 in. long in each dimension. The material is assumed to be 410 stainless steel with a speed of sound 1.98 × 105 in./s. The design process for an ultrasonic device does not begin by specifying all dimensions as done earlier. Rather, the axial dimension is calculated so that the device resonates precisely at the desired frequency for its given geometry and material properties through the frequency equation, as discussed in Chapter 2. The cube is first meshed. The mesh is designed so that it will be easy to introduce the slots for the purposes of this example. The model is then run in the vicinity of 20 kHz, and the modal density examined. There are three modes at 17.3 kHz corresponding to the half wave resonances across the three pairs of opposing faces. The basic longitudinal mode is shown in Figure 3.18 at full contraction and full extension. The center of the cube bulges in contraction and shrinks in expansion due to Poisson’s ratio. The corners are pulled in on expansion and pushed out on contraction. The displacements across the faces of the horn are not uniform. This is readily apparent from the edges of the cubes in Figure 3.18. The design is next modified to include four slots as indicated in Figure 3.19. The horn is envisioned as two plates supported by nine pillars of materials. Two slots run parallel in one lateral direction and the remaining two run parallel in the other lateral direction. The largest axial dimension is still 5 in. But now, the greatest dimension in the lateral direction of each pillar is less than a sixth of a wavelength. Full extension and full compression of the slotted block at the desired mode as predicted with the FEA model are shown in Figure 3.20. The horn is vibrating at 18 kHz. Four neighboring modes, shown in Figure 3.21, have quite complex mode shapes and do not produce the desired axial motion of the end face.

CRC_DK8307_CH003.indd 163

10/27/2008 8:45:47 AM

Ultrasonics: Data, Equations, and Their Practical Uses

164

(a)

(b)

FIGURE 3.18 Longitudinal mode of a cube of material: (a) full extension and (b) full compression.

FIGURE 3.19 Design of a slotted horn.

CRC_DK8307_CH003.indd 164

10/27/2008 8:45:47 AM

Advanced Designs of Ultrasonic Transducers and Devices

165

(a)

(b) FIGURE 3.20 (a) Full extension and (b) full compression of a slotted 5 in. cube of stainless steel operating at its first longitudinal mode.

The displacement amplitudes at the faces of the two horns at full extension are compared in Figure 3.22. The solid cube produces a peak displacement in the center of the face that tapers off toward the edges and corners. On the other hand, the slotted horn produces very uniform displacement across its face. There is only a slight decrease in displacement amplitude at the edges compared with its center, and some small variations are observed in the areas of the pillars. The overall ultrasonic performance of the slotted cube is vastly improved over that of the solid cube. To improve the uniformity of displacements, some designs have the outer perimeter near the working face undercut, which essentially makes the edges less stiff and thus allows

CRC_DK8307_CH003.indd 165

10/27/2008 8:45:48 AM

Ultrasonics: Data, Equations, and Their Practical Uses

166

(a)

(b)

(c)

(d)

FIGURE 3.21 Modes of a slotted stainless cube in vicinity of the longitudinal mode: (a) 17,495 Hz; (b) 17,295 Hz; (c) 18,023 Hz; and (d) 18,241 Hz.

them to vibrate with greater displacements, further improving uniformity. The effects of the size of the undercut and other variations in geometry can be easily evaluated using FEA. It is important to note that power ultrasonic devices are used under load against target materials. If the perimeter is undercut to improve displacement uniformity in the unloaded model, it may be lost when the horn is loaded against the material. A complete FEA model could be developed to include the horn, driver, materials, and the loading against a work piece or anvil. The FEA model of a slotted large horn was constructed quickly for the purposes of this section using simple geometry. A view of the stresses at the corners of the slots do not indicate unusual stress concentrations even though the displacements are extreme.

CRC_DK8307_CH003.indd 166

10/27/2008 8:45:50 AM

Advanced Designs of Ultrasonic Transducers and Devices

(a)

167

(b)

FIGURE 3.22 Variations in the displacements across the faces of the (a) solid and (b) slotted cubes at full extension.

Based on engineering mechanics, the corners are known to be stress concentrators. For the design of an actual horn, one would include fillets in the FEA model to significantly reduce the stress concentrations. To capture the stress concentrations, a much denser mesh of elements is needed in the vicinity of the corners/fillets of the slots. This refined FEA model can then be used to predict the high-stress concentrations. The maximum stress observed in the model could then be compared with an S–N diagram of the material to predict life.

3.7

Design Example: Ultrasonic Driver

Ultrasonic drivers are designed with an assemblage of segments that in total vibrate at the desired frequency. In commercial systems, horns are designed to be interchangeable with the driver, so that the system resonates at the correct nominal frequency. High-Q mechanical systems, such as plastic welders, are generally designed with piezoelectric elements that supply the transduction from electrical energy to mechanical energy. Information and data pertinent to the design of piezoelectric transducers is presented in Chapter 4. Systems that require lower Q for greater bandwidth or that operate in harsher environments, such as some ultrasonic cleaners, often use magnetostrictive materials for the transduction. Information and data pertinent to the design of magnetostrictive transducers is presented in Chapter 5. In this section, a driver using piezoelectric ceramics is designed using both FEA and 1D analysis for comparison. The driver used in this example is designed to excite a mode of a pipe at a nominal frequency of 18,500 Hz. The design assumes that the components will be machined from 410 stainless steel and that high-Q, high-power piezoelectric ceramic, PZT-4 (a Morgan ElectroCeramics designation) will be used. The volume of the element is selected so that it can deliver the desired power of ~200 W using a procedure discussed in Chapter 4. Two piezoelectric elements are used back-to-back, and the design is to force the mechanical

CRC_DK8307_CH003.indd 167

10/27/2008 8:45:52 AM

Ultrasonics: Data, Equations, and Their Practical Uses

168

1.25

0.5

2.25

2.431

3.437 0.375

0.375 0.125

All dimensions in inches All cross sections circular

FIGURE 3.23 The basic design of the ultrasonic driver used for the flow-through ultrasonic processor.

nodal plane to the interface of these two elements. The interface also is the electrical, high potential driving point, and the opposite faces are electrically grounded through the steel segments. The basic design is shown in Figure 3.23. Bolts (not shown) are used to hold the assembly together and supply sufficient bias stress to maintain compression in the elements throughout a cycle as discussed in Chapter 4. 3.7.1

Conventional Design Approach

The driver in Figure 3.23 is composed of six elements: two piezoelectric disks, two flanges, a cylindrical end mass, and a conical end mass. The design is to force the nodal displacement plane of a half-wave resonator to the interface of the two piezoelectric elements. This keeps the disks in the region of maximum strain where they are most effective. This design requirement allows the driver to be divided into two sections, each having three segments. The three segments of each section act as a quarter-wave resonator. The fixed end is located at the interface between the piezoelectric elements, and the free end is at the end of either end mass. Flange bolts are used to keep the disks in compression and are ignored in this example. In actuality, dynamic axial stresses are distributed across the piezoelectric disks and the bolts. For accurate predictions, the presence of the bolts must be represented in the analysis. The thickness and diameter of the piezoelectric elements are specified as discussed earlier. The thickness and diameter of the flange segments are designed to accommodate the bolts. The diameter at the end of the cone is to be less than a quarter-wavelength in either the circumferential or axial directions of the shell mode that the driver is to excite. This allows the assumption to be made that the driver acts against the shell in a region where its displacement field is relatively constant over the face of the driver. The only design parameters remaining are the lengths of the cylindrical and conical end masses for each quarter-wave resonator. The modeling approach is to use the differential equation that governs 1D, longitudinal, plane-wave propagation for each of the three segments. Continuity of displacements and equilibrium of forces at the interfaces between segments are required. Boundary conditions are required to completely define the problem. The end of the end mass is assumed

CRC_DK8307_CH003.indd 168

10/27/2008 8:45:53 AM

Advanced Designs of Ultrasonic Transducers and Devices

169

to be free (i.e., unloaded), and the end of the disk at the nodal plane is assumed fixed. For a three-segment assembly, the following set of equations are obtained E1r12

d1 (0 )  0 dx

 1(ℓ1) =  2(ℓ1) E2r22

d 2 d (ᐉ 1 )  E3r32 3 (ᐉ 1 ) dx dx  2(ℓ2) =  3(ℓ2)

E2r22

d 2 d (ᐉ 2 )  E3r32 3 (ᐉ 2 ) dx dx  3(ℓ3) = 0

(3.45a) (3.45b) (3.45c) (3.45d) (3.45e) (3.45f)

where  , is the displacement of the ultrasonic wave, and Ei and ri are the modulus of elasticity and the radius of the ith segment. The ℓ1’s are the cumulative lengths of the segments from the free end. In the case of the cone, r0 is the small end, and r1 is the large end. The form of the displacements for a cylindrical segment is given by 
 
 i  A i sin  c x
Bi cos  c x  i   i 

(3.46a)

The form of the displacement for a conical segment is more complicated than for the cylindrical segment, and is given by i 


ᐉ idi
(di  di1 )x  
ᐉ idi
(di  di1 )x   (di  di1 )/
  Bi sin   A i cos     di  di1 ᐉidi
(di  di −1 )x  di  di1  ci   ci  (3.46b)

where Ai and Bi are the coefficients of the solution to the differential equations, ci the speed of sound in the ith segment, and ℓI the length of the ith segment. The diameters of the large end and the small end of the cone are denoted by di and di−1, respectively. Equation 3.46b is the displacement form of Equation 2.55 given in Chapter 2, and is obtained by dividing Equation 2.55 by jω, where j2 = −1. The derivatives with respect to the spatial dimension in Equations 3.45a, 3.45c, and 3.45e are readily obtained by differentiating Equations 3.46a and 3.46b. The matrix equation is formed by moving all terms in the set of equations of Equation 3.45 to the left-hand side. Each equation is then a summation of terms that equals zero, and the terms have factors of the unknown coefficients (Ai and Bi for i = 1, 2, 3). As the segments are mechanically coupled through their interfaces, the solutions to their differential equations are coupled through their coefficients. The set of equations can then be written as a product of the matrix and a vector of the unknown coefficients set equal to the null vector. A resonance of the system occurs when the determinant of the matrix equals zero. The lowest frequency at which this occurs corresponds to the quarter-wavelength for a given geometry. The matrix equation that results by combining Equations 3.45 and 3.46 was implemented in a computer spreadsheet for this example. Two versions were written: one included three

CRC_DK8307_CH003.indd 169

10/27/2008 8:45:54 AM

Ultrasonics: Data, Equations, and Their Practical Uses

170 1

Normalized displacement

0.8 Conical end

0.6 0.4 0.2 0

Cylindrical end

−0.2 −0.4 −0.6 −0.8 0

1

2

3 4 Position (in.)

5

6

FIGURE 3.24 Displacements as a function of position along the ultrasonic driver at its resonant frequency.

cylindrical segments and the other contained a conical segment and two cylindrical segments. The matrix components were calculated at the desired frequency of 18,500 Hz for the specified materials and geometries. The axial length of the cone in one version or the cylinder in the other had to be assumed to obtain numerical values of the matrix components. A manual iterative process was used to vary the lengths and observe the sign changes in the value of the determinant, which was calculated by a native function in the spreadsheet. This was done until the variation in each of the two lengths was <0.001 in., which is on the order of typical machining tolerances. These values were then selected for the design that are shown in Figure 3.23. The displacement as a function of position at the resonant frequency is readily obtained. Following the same procedure using the continuity of displacements and equilibrium of forces at the interfaces between segments, the matrix equation for all six segments is assembled. The coefficients are determined by assuming a value of 1.0 for one coefficient and evaluating the matrix elements at the resonant frequency. The vector of coefficients represent the mode shape or eigenvector of that mode. The vector of coefficients and the equations for the displacements within an individual segment are combined to plot displacement as a function of position along the driver. The displacement plot for the driver is shown in Figure 3.24. The displacements are normalized by the maximum value, which occurs at the free end of the conical segment. By designing each three-segment assembly as a quarter-wavelength resonator, the nodal plane occurs at the interface between the two piezoelectric elements. The gain from the conical section is immediately obvious by comparing the magnitude of the displacement at the free end of the cylindrical segment to that of the displacement of free end of the conical segment. The overall gain is 1.8.

3.7.2 3.7.2.1

Finite Element Analysis Approach Resonances and Displacements

For comparison to the 1D approach, the driver shown in Figure 3.23 is analyzed using FEA. A full model is developed so that in addition to a comparison of the half-wave resonator

CRC_DK8307_CH003.indd 170

10/27/2008 8:45:54 AM

Advanced Designs of Ultrasonic Transducers and Devices

171

frequency, the neighboring modes are observed. The model is built with 2,880 elements and 12,211 nodes and is shown in Figure 3.25. The frequency of the desired mode predicted by FEA is 18,250 Hz, which differs by ∼1% from 18,500 Hz, which is the value obtained with the 1D approach. Three phases of displacement are shown in Figure 3.26: full extension,

FIGURE 3.25 Finite element analysis model of the ultrasonic driver shown in Figure 3.23.

(a)

(b)

(c) FIGURE 3.26 Predicted response of the driver at its longitudinal half wavelength: (a) compressed, (b) neutral, and (c) expanded.

CRC_DK8307_CH003.indd 171

10/27/2008 8:45:54 AM

172

Ultrasonics: Data, Equations, and Their Practical Uses

FIGURE 3.27 Neighboring flexural mode.

FIGURE 3.28 Neighboring torsional mode.

neutral, and full compression. The displacements are greatly magnified compared with the dimensions of the driver. The longitudinal displacements are readily seen, and in addition, a hint of Poisson’s effect in the area of the flanges can also be observed. The FEA model also predicts a number of modes in the frequency range from 18,300 Hz to 19,300 Hz. Figure 3.27 shows a flexural mode that occurs at 20,100 Hz. This mode appears to be far enough from the driving frequency that it will not be excited. However, several factors could potentially make the driver operate in this mode or hop to it. These factors include dimensional changes due to heating, poor axial alignment in the assembly, sudden application of normal loads, or eccentric loading against the workpiece, and lateral loads applied to the driver. A robust design attempts to eliminate other modes in the vicinity of the desired one or to have the greatest frequency gap between them. One interesting mode shown in Figure 3.28 is a torsional mode that occurs at a predicted frequency of 18,940 Hz. Although this is a valid eigensolution, it is unlikely to be excited by the thickness-mode piezoelectric elements. The flexural and torsional modes shown in Figures 3.27 and 3.28 demonstrate that the use of symmetry may not always be an advantage. If an axisymmetric model is used to describe the driver, then any displacement would be independent of angular position. In this case, neither of these two modes would have been indicated. Two other cases are analyzed using the basic model of Figure 3.25 to demonstrate the power of FEA. The first case assumes that a chip of piezoelectric ceramic was missing at the periphery of the element and is simulated by eliminating an element. The second case assumes that the two-disk stack is offset from the center by 5 mils and is simulated by offsetting all the nodes by 5 mils, which was readily handled by the preprocessor. The first case might occur when the driver has been mishandled during operation, and the second case might occur during assembly. The results for the two cases are similar and are shown in Figures 3.29 and 3.30, respectively. The resonant frequencies are nearly equal and differ only slightly from the base case.

CRC_DK8307_CH003.indd 172

10/27/2008 8:45:55 AM

Advanced Designs of Ultrasonic Transducers and Devices

173

(a)

(b)

(c) FIGURE 3.29 The effect of a missing chip at the edge of the piezoelectric disk, simulated by eliminating a fi nite element analysis element: (a) full compression, (b) neutral, and (c) full extension.

However, there is substantial lateral movement at the tip, which in practice would degrade performance. The tip displacements for four of the five cases analyzed with FEA are summarized in Figure 3.31. 3.7.2.2

Transient Analysis

To have a reasonably sized transient problem including the piezoelectric drive, another driver model is implemented as shown in Figure 3.32. It has a double-tapered rectangular horn and the same lengths and basic areas as the driver of Figure 3.25. It has only 88

CRC_DK8307_CH003.indd 173

10/27/2008 8:45:56 AM

Ultrasonics: Data, Equations, and Their Practical Uses

174

(a)

(b)

(c) FIGURE 3.30 The effect of a 5 mil offset of the piezoelectric disks, simulated by shifting the nodes in the fi nite element analysis model: (a) full compression, (b) neutral, and (c) full extension.

1 Axial mode Flexural mode Chiped horn Offset element

Out of plane displacement

18,940 Hz 0.75

0.5

18,173 Hz 18,245 Hz

0.25

18,246 Hz

0 0

0.25

0.5

0.75

1

Axial displacement FIGURE 3.31 A comparison of the tip displacements of the different cases evaluated using the finite element analysis model shown in Figure 3.25.

CRC_DK8307_CH003.indd 174

10/27/2008 8:45:57 AM

Advanced Designs of Ultrasonic Transducers and Devices

175

y x

FIGURE 3.32 Finite element analysis model for the transient analysis shown in compression.

elements and 207 nodes, which is much smaller compared with the previous driver. The ultrasonic driver is to be excited at its resonance with a sine wave of a 100 V. Of particular, interest is to examine how the tip displacement builds up when the driver is first energized. This model is used only for purposes of this section and does not represent an existing driver. Recall that the eigenvalue solution ignores damping when it is known to be light and when one is primarily interested in resonant frequencies and mode shapes. In a simulation without damping, the driver continues to build its vibration level when the power is on and continues to vibrate after the electrical signal is turned off. Damping causes the driver to reach some nominal level of vibration and to eventually stop vibrating when the power is removed. Therefore, this FEA model includes damping through the mass and stiffness matrices as described in Section 3.2. To obtain a desired benefit in operation, the driver transfers energy to the load, which sets the overall power requirement. There are two methods of solution for linear elastic problems. Direct integration that solves the FEA problem for each time step, and modal superposition that calculates the participation of each mode for a given forcing function. The direct integration method is used for this example, but before proceeding, modal superposition is discussed briefly. In any modal superposition problem, the theoretical solution is defined by the infinite sum over all modes. The question becomes: for a given input signal, how many eigenvalues and eigenvectors need to be included for an accurate prediction? The answer is that all modes should be included with resonant frequencies in the frequency spectrum of the drive signal. But in a transient problem where a signal is abruptly turned on and off, the frequency spectrum of the drive signal itself extends to infinity. The approach is to apply engineering judgment to obtain a solution with a reasonable number of modes. Then a few more modes can be included to determine whether there is a significant change in the solution. If there is not a significant change, the solution is accepted. If there is a significant change, then more modes are included until the solution converges. But note that as higher frequency modes are added, smaller finite elements become necessary. Consider, for example, that a system is to be driven by a windowed sine wave that is turned on at −τ/2, extends over an integer number of half cycles, and turns off at +τ/2.

CRC_DK8307_CH003.indd 175

10/27/2008 8:45:58 AM

Ultrasonics: Data, Equations, and Their Practical Uses

176

It is well known that the Fourier decomposition of a single windowed sine wave in the one-sided frequency domain is given by V(f) = A τ sinc[(f − fc)τ]

(3.47)

where A is the amplitude of the voltage sine wave driving the piezoelectric elements. The sinc function is centered at the fundamental frequency, fc, which for driver design is assumed to be at a resonance. The zeros in the sinc function occur at frequencies of (fc ± n/τ) where n is a nonnegative integer. The initial frequency range for this type of problem can be based on the two frequencies where the peak amplitude of Equation 3.47 is <10% of its maximum. However, solution convergence should still be checked by including the next highest mode(s). Direct integration numerically integrates the equations of motion using a specified time step starting with a set of initial conditions. In this problem, the driver is assumed to start at rest and it responds to a sine wave that is turned on at time = 0. The frequency of the signal is 17,303 Hz, which is the half-wave resonance as determined by the free vibration problem for the model shown in Figure 3.32. The voltage drive signal to one of the elements is shown in Figure 3.33. The other element is driven with the same signal but out of phase by 180°, because of the assumed polarity of the elements in the model. At any time, the elements are either both expanding or both contracting. The resulting current through one element is shown in Figure 3.34. As anticipated, the current increases in time as the amplitude of vibration builds. The current is quite small in this example, because the driver is operating with no load and relatively low material losses are included. In practice, the internal losses are much higher and quiescent current is 100 80 60

Input voltage (v)

40 20 0 −20 −40 −60 −80 (×10−4)

−100 0

1.6 0.6

3.2 2.4

4.8 4

6.4 5.6

8 7.2

Time (s) FIGURE 3.33 Drive voltage to each piezoelectric element.

CRC_DK8307_CH003.indd 176

10/27/2008 8:45:59 AM

Advanced Designs of Ultrasonic Transducers and Devices

177

(×10−7) 5 4 3

Output current (amp)

2 1 0 −1 −2 −3 −4 (×10−4)

−5 0

0.6

1.6

2.4

3.2

4

4.5

5.6

6.4

7.2

8

Time (s) FIGURE 3.34 Resulting current through the piezoelectric elements.

normally on the order of tenths of amperes or more, depending on the size of the driver, its operating frequency, and the piezoelectric materials and metals used in its construction. The displacements at the tip of the front of the driver and at the back are shown in Figure 3.35. There are a number of features to be observed in this figure. First, the amplitude builds up with time until the run is halted. Otherwise, the amplitude would continue to build to a steady-state level for the drive signal of a 100 V. The steady-state response is the harmonic solution that can be obtained from the harmonic analysis which is shown in Figure 3.36. The steady-state response for the tip is 2.7 µm and for the back is 1.3 µm. If the transient analysis is allowed to continue, the vibration amplitudes would level off at these values. Second, the gain from the back of the driver to its tip is readily calculated and is 2.1, which is the same value that can be obtained from the peak responses in Figure 3.36. Third, the mode of vibration is a half-wave resonance, and the tip and back displacements should be out of phase by 180°. This feature is most obvious in the transient response in Figure 3.35 after the vibration initially builds. The most interesting feature in the displacements is their early response, which is shown in detail in Figure 3.37. Both the tip and back displacements are delayed relative to the start of the voltage drive signal, and the onset of vibration at the back proceeds the onset at the tip. This is easily understood from the propagation times from the nodal plane to the tip and back. Using just the lengths of stainless steel and a nominal bar velocity of 200,000 in./s, the onset at the back should occur at 14 µs and the onset at the tip should occur at 19 µs. These are the approximate times that are observed in the displacements shown in Figure 3.37. In the figure, it is also apparent that it takes about two cycles before the tip and back displacements are out of phase by 180°.

CRC_DK8307_CH003.indd 177

10/27/2008 8:45:59 AM

Ultrasonics: Data, Equations, and Their Practical Uses

178 (×10−6) 2.4 2 1.6

FRONT

Displacement (m)

1.2 −6 −4 0 −0.4 BACK

−0.8 −1.2

(×10−4)

−1.6 0

0.6

1.6

2.4

3.2

4

4.8

5.6

6.4

7.2

8

Time (s) FIGURE 3.35 Predicted tip and back displacements for the transient problem. ( 10−6) 4 3.6 3.2

Displacement (m)

2.8 2.4 2 1.6 1.2 0.8 FRONT 0.4 0

BACK ( 10−6) 1600 1900 2100 2000 1800 1700 1850 1750 1950 1650 2050 Frequency (Hz)

FIGURE 3.36 Displacements predicted by steady-state harmonic analysis.

CRC_DK8307_CH003.indd 178

10/27/2008 8:46:00 AM

Advanced Designs of Ultrasonic Transducers and Devices

179

(×10−4) 4.8 4 FRONT 3.2

Displacement (m)

2.4 1.6 0.8 0 −0.8 BACK

−1.6 −2.4 −3.2

(×10−4) 0

0.8

0.4 0.2

0.6

1.6

1.2 1

1.4

2 1.8

Time (s) FIGURE 3.37 Expanded view of the early response of the tip and back displacements.

3.7.2.3

Comments on Damping and Power Loss

Purely elastic models, whether analytical or FEA, accurately predict resonant frequencies, mechanical gains, and positions of nodal planes. Therefore, they are very effective tools for the design and analysis of ultrasonic drivers. The absolute displacements are set by voltage and current, and damping determines voltage for a current drive or current for a voltage drive. Damping is responsible for quiescent power losses for the unloaded system. The power losses represent heat that raises the temperature of the driver. The elevated temperatures might be sufficient to produce failures of the driver via a number of mechanisms, accelerated aging of the piezoelectric materials, or unacceptable thermal shift of the resonant frequency. The elevated temperatures could also cause the driver to fail regulatory requirements that apply to its field of application, such as medical devices. In industrial applications, air or water cooling can be supplied to minimize the temperature increase. But in other applications, the heating may limit its application or make the ultrasonic option less attractive than its competitors. Therefore, it is important to evaluate damping losses in a design. Although damping is a key parameter of a system, it is the most difficult parameter to actually include in a model a priori based on material constants. A common approach in structural FEA is to measure the decay times of vibration of the structure either in the laboratory or in situ. These times are then used to calculate Rayleigh damping for inclusion in the FEA model.

CRC_DK8307_CH003.indd 179

10/27/2008 8:46:00 AM

Ultrasonics: Data, Equations, and Their Practical Uses

180

Losses in a piezoelectric driver include the internal viscoelastic losses of vibrations in a material, frictional losses at the interfaces between the elements, and the resistivity losses in the piezoelectric materials. Viscoelastic losses in most metals are relatively low, but increase with square of the product of strain and frequency. Therefore these losses might become a significant portion in high gain designs operating at higher frequencies. The resistivity losses are due to the fact that the piezoelectric ceramic is not a pure dielectric. The resistivity losses in the elements also create additional viscoelastic losses through the coupled constitutive relations for piezoelectricity. But these losses are generally very low. In practice, infrared images of active drivers indicate that the heat is primarily being generated in the piezoelectric elements and at their interfaces. Manufacturers’ data for the losses in piezoelectric materials are obtained from a radial mode. The losses observed in a stack of elements operating in a longitudinal mode are much higher. Therefore, one should never assume that FEA models will be accurate using the published data alone. Damping needs to be measured on an existing prototype system or similar previous design and then appropriately included in the FEA model.

3.8

Commercial Software Packages

FEA has been used for more than 40 years to solve various structural, thermal, and acoustic problems. In the past, specialty codes were written to solve specific problems using the FEA methodology. Over the years, commercial products have incorporated more and more capabilities, so there is little need to write custom code. The commercial software producers are also very quick to incorporate new developments in areas such as solution algorithms and mesh generation schemes. Commercial FEA codes can accept inputs from CAD files and solid model packages and can import and export to other FEA packages. Many, but not all, include piezoelectric materials, so that ultrasonic drivers and transducers can be modeled from the electrical inputs to the mechanical/acoustic outputs. The size of the problem and the solution speed in FEA have increased in proportion to improvement in computer power in terms of memory size, computer speed, and distributed computing. Most recently, codes known as multiphysics packages, integrate the physics of many fields, so that highly coupled problems are readily solved. The coupling of piezoelectricity, structural vibrations, and heat generation are readily implemented in these packages. As a result, very complicated ultrasonic systems can be evaluated quickly to push an application forward. For example, the deleterious consequences of internal losses are impossible to assess using just a structural FEA package. With current multiphysics codes, the analysis is straightforward. The strain and stress in the structural problem are used to calculate local internal volumetric heating that is equal to the in-phase component of the product of strain and stress times frequency squared. The in-phase component arises from a loss factor that is entered into the structural problem as an imaginary part of Young’s modulus. The internal heating is used as input to the heat transfer problem to calculate temperatures. The local temperature is then used to change temperature-dependent material constants such as Young’s modulus and piezoelectric constants in the structural problem. The surface temperatures of an ultrasonic handpiece used in medical applications is required to be <50°C. The power loss in the transducer is the heat that raises the handpiece temperature. Therefore the desire to pack as much piezoelectric material into as

CRC_DK8307_CH003.indd 180

10/27/2008 8:46:01 AM

Advanced Designs of Ultrasonic Transducers and Devices

181

small a handpiece as possible is in conflict with keeping the surface temperature below a limit. Multiphysics packages can be used to predict the heat loss in the transducer and the heat removed by conduction, convection, and radiation into the surrounding air. Therefore design strategies to mitigate or manage heat can be analyzed quickly. Commercial packages have integrated the physics of many fields so that entire ultrasonic systems can be modeled. Even the nonlinear mechanical mechanisms, such as nonlinear wave propagation and cavitation, are now starting to be analyzed with commercial codes. Though the FEA analyses of these nonlinear effects are the subject of current research, advances are expected to be rapid. Commercial FEA packages Adina R & D, Inc. ADINA ALGOR, Inc. ALGOR Altair Engineering, Inc. Hyperworks ANSYS, Inc. ANSYS COMSOL, Inc. MULTIPHYSICS Dassault Systemes SIMULIA (Abaqus and CATIA) SolidWorks (and COSMOS) Livermore Software Technology Corp. LS-DYNA MSC Software Corporation MSC Nastran Dytran Noran Engineering, Inc. NeiNastran Parametric Technology Corp. (PTC) Pro/Engineer Mechanica UGS, Corp. NX Speciality FEA codes with piezoelectric elements Micromechatronics, Inc (US Distributor) ATILA Weidlinger Associates, Inc. PZFlex

3.9

www.adina.com www.algor.com www.altair.com www.ansys.com www.comsol.com www.3ds.com www.simulia.com www.solidworks.com www.lstc.com www.mscsoftware.com

www.nenastran.com www.ptc.com www.ugs.com

www.micromechatronicsinc.com www.wai.com www.pzflex.com

Summary

The development of the finite element method including piezoelectric elements has been presented. An overview of the capabilities of FEA was presented and demonstrated with several design examples. The FEA results were compared with analytical solutions for shells

CRC_DK8307_CH003.indd 181

10/27/2008 8:46:01 AM

Ultrasonics: Data, Equations, and Their Practical Uses

182

Transducer Waveguide

End-effector

Total displacement of distal end Transverse motion of distal end

0

Position along ultrasonic blade FIGURE 3.38 Advanced performance of an ultrasonic surgical instrument obtained with fi nite element analysis. (Courtesy of Ethicon Endo-Surgery.)

and plates and with experimental data from the actual ultrasonic systems being modeled. The predictions from FEA, shell and plate theories, and experimental results agreed very well. FEA predictions must always be checked against results from simplified models or experiments to ensure predictions represent reality. For analyzing ultrasonic systems, models for elastic waves in rods, shells, and plates are useful in this regard. However, the usefulness of basic analytical models is quickly lost when more advanced and complicated systems are being designed and analyzed. In these cases, FEA is not only a valuable tool, but perhaps the only tool. Consider an advanced ultrasonic blade used in the Harmonic™ ultrasonic surgery system from Ethicon Endo-Surgery (Cincinnati, Ohio), as shown in Figure 3.38 [12]. This advanced design is possible only with FEA. The end effector or blade is a combination of a curved tip and geometric features (not visible at the scale of the model given in the figure). The curved blade is important for surgeon visibility and for access to the targeted tissue. The offset of mass of the curved blade from the overall centerline of the instrument produces transverse motion from the longitudinal motion. So the motion of the tip is a controlled combination of both transverse and longitudinal vibrations. The geometric features squelch the transverse vibrations in the remainder of the ultrasonic drive train, so that the ultrasonic motion is primarily longitudinal in the majority of the system. This behavior is shown in the graph in the figure for the last three half wavelengths of the instrument. The system is very well behaved at very high amplitudes of vibration. This level of sophistication is achieved only with FEA.

CRC_DK8307_CH003.indd 182

10/27/2008 8:46:01 AM

Advanced Designs of Ultrasonic Transducers and Devices

183

References 1. K.-J. Bathe, Finite Element Procedures, Prentice Hall, Englewood Cliffs, NJ, 1996. 2. H. Allik and T. J. R. Hughes, Finite element method for piezoelectric vibration, Int. J. Num. Meth. Eng., 2, 151–157, 1970. 3. H. Allik and K. M. Webman, Vibrational response of sonar transducers using piezoelectric finite elements, J. Acoust. Soc. Am., 56(6), 1782–1791, 1974. 4. R. Holland and E. P. EerNisse, Design of Resonant Piezoelectric Devices, Research Monograph No. 56, the MIT Press, Cambridge, MA, 1969. 5. A. Leissa, Vibration of Plates, American Institute of Physics, Woodbury, NY, 1993 (edition by Acoustical Society of America—originally issued by NASA, 1973). 6. A. Leissa, Vibration of Shells, American Institute of Physics, Woodbury, NY, 1993 (edition by Acoustical Society of America—originally issued by NASA, 1973). 7. J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1987. 8. F. Fahy, Sound and Structural Vibration: Radiation, Transmission and Response, Academic Press, London, 1985. 9. K. F. Graff, Wave Motion in Elastic Solids, Ohio State University Press, Columbus, OH, 1975. 10. H. Kraus, Thin Elastic Shells, John Wiley & Sons, Inc., New York, NY, 1967. 11. D. J. Gorman, Free vibration analysis of the completely free rectangular plate by the method of superposition, J. Sound Vib., 57(3), 437–447, 1978. 12. Ethicon Endo-Surgery, 4545 Creek Rd., Cincinnati, Ohio, USA.

CRC_DK8307_CH003.indd 183

10/27/2008 8:46:01 AM

CRC_DK8307_CH003.indd 184

10/27/2008 8:46:01 AM

4 Piezoelectric Materials: Properties and Design Data Dale Ensminger

CONTENTS 4.1 Introduction ........................................................................................................................ 185 4.1.1 Symbols and Direction Notations ....................................................................... 186 4.2 Definitions........................................................................................................................... 187 4.2.1 Mechanical Data .................................................................................................... 193 4.2.2 Thermal Data .......................................................................................................... 194 4.3 Mathematical Relationships ............................................................................................. 194 4.3.1 Electromechanical Coupling Coefficient, Stored Energy, and Sensitivity ........................................................................................................ 194 4.4 Properties of Piezoelectric Materials .............................................................................. 196 4.5 Selection and Design Guidelines ..................................................................................... 196 4.5.1 Fundamental Piezoelectric Action ...................................................................... 196 4.5.2 Transducer Assembly ............................................................................................ 227 4.5.2.1 Sandwich Types for Power Applications ............................................. 227 4.5.2.2 Transducer Structural Elements ........................................................... 229 4.5.2.3 Soldering to Electrodes .......................................................................... 231 4.5.3 General Technical Considerations in Piezoelectric Transducer Design ................................................................................................. 231 4.5.3.1 Selection of Piezoelectric Elements According to Properties and Vibrational Modes........................................................ 231 4.5.4 Electrode Types ...................................................................................................... 232 4.6 Aging ................................................................................................................................... 232 4.7 Uses for Piezoelectric Transducers .................................................................................. 232 References .................................................................................................................................... 233

4.1

Introduction

Piezoelectric elements are the heart of most ultrasonic transducers commercially available today. Their physical characteristics make them readily adaptable to active devices (sources of ultrasonic energy), passive devices (receivers), and combination devices (such as pulse-echo, as used in nondestructive testing [NDT] and medical diagnosis).

185

CRC_DK8307_CH004.indd 185

10/27/2008 10:44:20 AM

Ultrasonics: Data, Equations, and Their Practical Uses

186

Piezo comes from the Greek word, piezein, meaning “to press.” The term piezoelectric refers to producing electricity by applying pressure to a material. Pressure applied to certain crystalline materials produces electric charges on specific surfaces of the material. Conversely, an electric field applied between these surfaces causes dimensional changes in the materials. Three classes of materials exhibiting piezoelectric characteristics are of interest to ultrasonics: ceramic materials, single crystals, and polymeric materials. Properties of materials falling into each of these classes are given in Tables 4.1 through 4.4. Piezoelectric effects are attributable to a specific alignment of molecular dipoles in the materials (crystals) such that the sum of stresses applied to the element produces a net change in separation of the centers of gravity of the positive and negative charges of these dipoles. Piezoelectric crystals have no center of symmetry. This condition is necessary for any combination of applied uniform stresses to produce the necessary change in relative position of positive and negative charges of the dipoles to allow the crystal to be piezoelectric. Most commercially available ultrasonic transducers use piezoelectric ceramic materials for energy conversion. In their natural condition, the dipoles in a ceramic element are oriented randomly. The elements are polarized to make them piezoelectric by heating slightly above the Curie point and maintaining a high-voltage DC electric field between terminals as they cool slowly toward room temperature. The dipoles tend to reorient parallel to the applied electric field and to become locked in place as the temperature is lowered. Some ceramic materials may be polarized at room temperature by applying adequate DC voltages across opposite faces of elements composed of the materials. Piezoelectric materials have a variety of uses in addition to ultrasonic applications. Some of the oldest uses are in the areas of sensors (hearing aids, microphones), speakers, and selective filter circuits. The objective of this chapter is to provide data and information useful to the design and application of devices employing piezoelectric elements. The various properties of piezoelectric elements are defined and explained in Section 4.2. The mathematical relationships for use in arriving at an optimized transducer design using these properties are presented in Section 4.3. Properties of available piezoelectric materials are tabulated in Section 4.4. Selection and design guidelines for various types of applications are offered in Section 4.5. 4.1.1

Symbols and Direction Notations

To properly defi ne for any purpose the properties of piezoelectric materials, it is necessary to recall the property that renders a material piezoelectric. A material must be anisotropic to be piezoelectric. Its crystal structure can have no center of symmetry. It is made to be piezoelectric by the presence of electric dipoles, that is, pairs of positive and negative electric charges separated by a finite distance. The lack of a center of symmetry allows combinations of uniform stresses to produce changes in the relative position of the positive and negative charges corresponding to the dipoles or of an applied electric field to produce a change in dimensions by its effect on the dipoles. Because piezoelectric materials are anisotropic, their properties vary according to the direction in which they are measured, that is, their electrical, mechanical, and electromechanical properties differ along different axes or directions. Crystallographers use an orthogonal axis. The same orthogonal axes are applied to piezoelectric crystals to provide a standardized means of identifying directions. A common practice is to identify components along these axes by numerals such as 1, 2, and 3, corresponding respectively

CRC_DK8307_CH004.indd 186

10/27/2008 10:44:22 AM

Piezoelectric Materials: Properties and Design Data

187

Poling direction

3

z

6 2

y 5 4 x

1

FIGURE 4.1 Axial notations used to defi ne directions in piezoelectric elements.

to the x-, y-, and z-axes of the crystallographer (see Figure 4.1). The direction of positive polarization in piezoelectric ceramic plates is generally considered to be the z-axis, that is, the 3 direction. Ceramics have complete symmetry about the polar axis so that the 1 and 2 axes are located arbitrarily but always perpendicular to each other. Subscripts 4, 5, and 6 denote stress or strain around the 1(x), 2(y), and 3(z) axes, respectively. Because piezoelectric materials are anisotropic, it is necessary to identify the axis along which a force or action is imposed and that corresponding to the resulting action or force from which the constant is determined. This is done by adding the subscripts corresponding to these axes to each symbol. The use of these symbols and subscripts will be illustrated with the definitions of properties that follow.

4.2

Definitions

Anisotropy. The characteristic of exhibiting unique properties corresponding to different directions along which the properties are measured; showing different properties in different directions. Capacitance. The ratio of charge that appears on the conducting elements (plates) constituting a capacitor to the voltage between the elements, the capacitance being determined by the dielectric constant of the material between elements, the area of the conducting elements, and the distance (uniform) between these elements. Crystal axes. Piezoelectric materials are anisotropic. Therefore, the properties of piezoelectric materials vary according to the direction in which they are measured, that is, for piezoelectric materials, electrical, mechanical, and electromechanical properties differ along different axes or directions. Crystallographers use an orthogonal axis. The same orthogonal axes are applied to piezoelectric crystals to provide a standardized means of identifying directions. A common practice is to identify components along these axes by numerals such as 1, 2, and 3, corresponding respectively to the x, y, and z axes of the crystallographer (see Figure 4.1). The direction of poling in piezoelectric ceramic plates is identified as the 3 direction. In the plane perpendicular to the 3 axis, the ceramics are

CRC_DK8307_CH004.indd 187

10/27/2008 10:44:22 AM

Ultrasonics: Data, Equations, and Their Practical Uses

188

nondirectional. In these materials, the 1 and 2 axes are located arbitrarily, but always perpendicular to each other. Subscripts 4, 5, and 6 denote stress or strain around the 1(x), 2(y), and 3(z) axes, respectively. Because piezoelectric materials are anisotropic, it is necessary to identify the axis along which a force or action is imposed and that corresponding to the resulting action or force from which the constant is determined. This is done by adding the subscripts corresponding to these axes to each symbol. The use of these symbols will be illustrated with the definitions of properties that follow. Curie temperature. The temperature at which piezoelectric or ferroelectric materials and ferromagnetic materials change to paraelectric and paramagnetic phases and vice versa. The ferroelectric phase is characterized by a natural polarization of the electric fields in a crystal lattice producing exceptional piezoelectric and elastic properties. Similarly, the ferromagnetic phase is characterized by a spontaneous organization of internal magnetic moments in a common direction producing magnetic permeability considerably greater than that of a vacuum. In the paraelectric phase, the electric dipoles in a crystal lattice are randomly oriented and the material is nonpiezoelectric. Piezoelectric ceramic materials, such as barium titanate, lose their piezoelectric properties permanently at temperatures exceeding the Curie temperature. Their activity degrades rapidly at temperatures below but near the Curie point. In the paramagnetic phase, the magnetic permeability is only slightly greater than unity and varies only to a small extent with the magnetizing force, as in aluminum or platinum. Therefore, at temperatures above the Curie point magnetostrictive materials, such as nickel, lose their magnetostrictive qualities but regain them when the temperature returns to below the Curie point (or temperature). Rochelle salt (NaKC4H4O6-4H2O) has two Curie temperatures, 18 and 24°C. It is piezoelectric between these two temperatures but nonpiezoelectric outside this range. Dielectric constant. A constant denoting the effect of an isotropic dielectric material on the capacitance of a capacitor. Absolute dielectric constant, ε. See permittivity. Dielectric displacement or charge density/electric field. Relative dielectric constant, εr. See relative permittivity. The ratio of the capacitance of the capacitor filled with the material to that of the same capacitor with only vacuum located between electrodes; the ratio of absolute dielectric constant, ε, to the dielectric constant of a vacuum, ε 0. [ε 0 = 8.854 × 10−12 (F/m)] For most piezoelectric materials used in transducers, the field and the resulting dielectric displacement are along the same axis. For these materials, a single subscript is sufficient to describe both the direction of the field and the dielectric displacement, such as ε1 for ε11 referring to the dielectric constant measured in the x direction or ε 3 for ε 33 referring to the dielectric constant measured in the z direction. The mechanical boundary conditions affect the dielectric constant, that for an element that is completely free to vibrate being higher than one that is mechanically restrained. Superscripts are used to indicate the boundary conditions. If there are no constraints, the material experiences constant stress and superscript T (viz. εT3 ) is used to indicate this condition. If constraints are applied in such a manner that they allow no mechanical deformation whatsoever (which is very difficult to achieve), the material experiences constant strain and superscript S (viz. ε 0S) is used to denote this condition.

CRC_DK8307_CH004.indd 188

10/27/2008 10:44:22 AM

Piezoelectric Materials: Properties and Design Data

189

If one could clamp a piezoelectric crystal so hard that it could not expand or contract under the influence of an electric field, the field would produce no additional mechanical energy in the crystal. A measurement of the dielectric constant under this condition would give the clamped dielectric constant. Removing all external pressure would allow the crystal to expand or contract under the influence of an electric field, thus producing an increased mechanical form of energy and a correspondingly higher dielectric constant, the free dielectric constant. The difference between these two constants is determined by the electromechanical coupling factor, or coefficient, k. Dielectric displacement. Electric displacement; electric field intensity multiplied by the permittivity. Dielectric loss factor, tan δ. 1/QE, relative dissipation factor at 1 kHz, at low electric field. Dipole. In physics, any object or system having equal but opposite electric charges or magnetic poles at two points separated by a very small distance. In chemistry, a polar molecule in which centers of positive and negative charge are separated. Efficiency. The ratio of useful energy output from a conversion system to the energy supplied to it. The efficiency of a transducer includes the capability of the active element (piezoelectric, magnetostrictive, or other) to convert electrical energy to mechanical energy or vice versa and the losses associated with the entire transducer assembly. Elastic compliance, smn. Strain/stress; displacement of a linear mechanical system under a unit force. In a piezoelectric plate, elastic compliance may be given in terms of compliance at constant electric field or compliance at system charge density. Elastic compliance is defined for piezoelectric materials according to the axial directions of strain and stress, according to the condition of the electric field as to whether it is constant with electrodes shorted or whether electric charge density is constant with open circuit at the electrodes, and with the condition that all other external stresses are constant. Thus, E S11 is the elastic compliance of a piezoelectric element or ratio of strain in the 1(x) direction to stress in the 1(x) direction, provided that there is no change in stress in the 2(y) and 3(z) directions and the electric field is constant. E S33 elastic compliance or ratio of strain in the 3(z) direction to the stress in the 3(z) direction, provided that there is no change in stress in the 1 and 2 directions and the electric field is constant. E S44 is the ratio of strain about the 1(x) axis to the stress about the 1(x) axis, provided that there is no change in stress in the 2 and 3 directions or about these axes and the electric field is constant. E S12 is the ratio of strain in the 2(y) direction to stress in the 1(x) direction, provided that there is no change in stress in the 2 and 3 directions and the electric field is constant. E S13 is the ratio of strain in the 3(z) direction to stress in the 1(x) direction, provided that there is no change in stress in the 2 and 3 directions and the electric field is constant. D S11 is the ratio of strain in the 1(x) direction to stress in the 1 direction, provided that there is no change in stress in the 2 and 3 directions and the electric charge density is held constant. D S33 is the ratio of strain in the 3(z) direction to stress in the 3(z) direction, provided that there is no change in stress in the 1 and 2 directions and the electric charge density is held constant.

CRC_DK8307_CH004.indd 189

10/27/2008 10:44:22 AM

190

Ultrasonics: Data, Equations, and Their Practical Uses

D S44 is the ratio of strain about the 1(x) axis to stress about the 1(x) axis, provided that there is no change in stress in the 2 and 3 directions or about the 2 and 3 axis and the electric charge density is held constant. D S12 is the ratio of strain in the 2 direction to the stress in the 1 direction, provided that there is no change in stress in the 2 and 3 directions and the electric charge density is held constant. D S13 is the ratio of strain in the 3 direction to the stress in the 1 direction, provided that there is no change in stress in the 2 and 3 directions and the electric charge density is held constant.

Electrical conductivity. The ratio of electric current density to the electric field in a material; specific conductance (millimhos per centimeter; mho per meter). Electrical resistivity, ρel. Reciprocal of electrical conductivity; electrical resistance to current flow times the cross-sectional area of current flow and per unit length of current path (RA/L) (ohm meters). Electric field. Electric force per unit test charge between a charged body (source) and another charged body tending to attract or repel each other. Electric field intensity, E. The force on a stationary positive charge per unit charge at a point in an electric field. Electromechanical coupling coefficient, k. An expression for the ability of a piezoelectric material to exchange electrical energy for mechanical energy or vice versa. It is a constant relating strain produced along a given axis of a crystal by an electric field along the same axis (or an axis normal to it) to the applied field. The square of the electromechanical coupling coefficient, k2, equals the transformed electrical energy causing mechanical strain along a given axis when all external stresses are constant divided by the electrical energy input to electrodes on faces perpendicular to the same or different axes. The same coupling factor is applied for conversion of mechanical to electrical energy, that is, k2 also is equal to the transformed mechanical energy causing electrical charge to flow between connected electrodes on opposite faces normal to an axis of the crystal divided by the mechanical energy associated with stress applied along the same or another axis, the axial directions being identified by appropriate subscripts to k. The directions of the electrical field and the mechanical strain are indicated by subscripts, such as k31, in which the first subscript (number 3 corresponding to axis z) designates the direction of the electrical field and the second subscript (number 1 corresponding to axis x) refers to the direction of mechanical strain. Planar coupling factor, kp. Relates coupling between an electric field in the 3 (or z) direction in ceramic elements (or plates) and the simultaneous mechanical action in the 1 (or x) and 2 (or y) directions. The planar coupling coefficient is especially important, because it can be measured very accurately and easily, providing a simple measure of the effectiveness of poling in ceramic test specimens [1]. Radial coupling factor, kr: equal to kp. The mechanical actions in the 1 and 2 directions cause radial vibration, hence the term radial coupling. Transverse or lateral coupling factor, k31. Relates coupling between mechanical strain in the 1 (or x) direction and an electric field in the 3 (or z) direction. Longitudinal coupling factor, k33. Relates coupling between an electric field in the 3 (or z) direction and mechanical strain in the 3 (or z) direction. Shear coupling factor, k15. Relates coupling between an electric field in the 1 (or x) direction and mechanical strain about the y-axis.

CRC_DK8307_CH004.indd 190

10/27/2008 10:44:22 AM

Piezoelectric Materials: Properties and Design Data

191

Thickness coupling factor (laterally clamped), kt. Coupling coefficient applicable to ceramic plates or discs vibrating in a thickness mode at frequencies above the resonant frequencies corresponding to the length and width dimensions of the element. In this situation, the inertia of the piezoelectric material effectively prevents lateral vibrations (as though they were rigidly clamped in those directions). The longitudinal coupling factor, k33 , does not define the electromechanical coupling under these conditions, because the qualification that all external stresses are constant is not met. Electrostrictive effect. A second-order effect demonstrated by all crystals and all solid insulators, which is distortion due to a voltage applied between two surfaces of the material. The effect is very small, except in those materials that are ferroelectric. The resulting stress does not change direction with a change in direction of the applied field but is proportional to the square of the applied field strength. Ferroelectric crystals. Ferroelectric crystals derive their name from the fact that their electrical characteristics are much like the magnetic behavior of ferromagnetic materials, in that the plot of polarization versus electric field includes a hysteresis loop. They belong to the pyroelectric family of crystals. The direction of spontaneous polarization can be reversed by an electric field, that is, their electric dipoles (or direction of polarization) can be reversed by applying an electric field of sufficient strength. All ceramic piezoelectric materials are ferroelectric, having been made ferroelectric by proper polarization. The basic ingredients of ferroelectric (ceramic) materials are metallic oxides. Frequency constant, N. In piezoelectric crystals and the like, the frequency constant for a given geometry of a material (in the form of bars or plates) is a value relating resonance characteristics and corresponding dimensions by which the resonance frequency of a plain, unloaded element will resonate in the direction specified for the constant. For example, a plate 0.25 in. (6.35 × 10−3 m) thick for which the thin plate frequency constant is 1900 Hz m will resonate in the thickness mode at ~299,213 Hz, that is, f = N3t/6.35 × 10−3 m is the product of the mechanical resonant frequency under specified electrical boundary conditions (short circuit or open circuit) and the dimension that determines that resonant frequency. N1 is the frequency constant of a thin bar polarized perpendicular to length and measured with the polarizing electrodes connected together, fRℓ, hertz meters. N3a is the frequency constant of a long, slender bar poled along the length and electroded on the ends with electrodes open-circuited, faℓ, hertz meters. N3t is the frequency constant of a thin plate electroded on the opposite faces, fRt, hertz meters. Impedances. Characteristic impedance, acoustic. Product of density of a medium and the velocity of sound in the medium; ρc. Standing alone, ρc is a pure resistance. Mechanical impedance, acoustic. Ratio of force, F, impressed on an increment of an acoustical system to the resultant velocity of motion, v, imparted to the increment in proportion to F, that is, Zm = F/v. In correspondence to electrical impedance, Ze, Zm = Rm + jXm, where Rm is the resistive component of the impedance of the vibrating system. The exact form of the mechanical impedance equation depends upon geometrical, material, and mechanical factors relative to the acoustic system under consideration. Mechanical impedance is important to the design of transducers.

CRC_DK8307_CH004.indd 191

10/27/2008 10:44:23 AM

192

Ultrasonics: Data, Equations, and Their Practical Uses

Isotropy. Characteristic of a material exhibiting equal properties in any direction in which they are measured. Permittivity, ε. Relative dielectric constant times the permittivity of empty space, where the permittivity of empty space, ε 0 , is a constant appearing in Coulomb’s law, having the value of 8.854 × 10−12 F/m (in MKS units). Relative permittivity, ε/εo (or relative dielectric constant). The ratio of the capacitance of a condenser with dielectric material between the plates to that with only a vacuum between the plates. Relative permittivity (or relative dielectric constant) is given for piezoelectric elements in the free state and the clamped state, the two values differing from each other due to the effects of stress. Piezoelectric effect. The generation of electric charges on specific faces of certain crystalline materials subjected to pressure. Conversely, an electric field applied between the specific faces produces mechanical stresses (and consequently, strains) in the piezoelectric crystals. When an alternating electric field is applied to the crystal, it expands and contracts dynamically at a rate determined by the frequency of the applied field. This is the basis for the design of active piezoelectric transducers. An alternating mechanical stress across the piezoelectric crystal produces an alternating electric field having amplitude and frequency determined by the magnitude and rate of the applied stress. This is the basis for the design of passive piezoelectric transducers, such as microphones, and receiving elements used in the NDT of materials and in medical diagnostic systems. Piezoelectric crystals have no center of symmetry. Piezoelectricity. Electricity produced by pressure applied to a piezoelectric material. Piezoelectric strain (or charge) constants (d). (a) The ratio of strain developed along or around a specified axis to the field applied parallel to a specified axis, when all external stresses are constant and (b) the ratio of short-circuit charge per unit area of electrode flowing between connected electrodes that are perpendicular to a specified axis to the stress applied along or around a specified axis, when all other external stresses are constant [2]. d constants commonly observed or used: d31. The direction of induced strain is along the 1 (or x) axis when the electrical field is applied along the 3 (or z) axis. The charge developed is due to applying stress along the 1 axis flows between shorted electrodes normal to the 3 axis. d33. Induced strain is along the same axis (3) as the applied electrical field. d11. Induced strain is along the same axis (1) as the applied electrical field. d14. Induced strain is torque about the x-axis due to applied electrical field along the 1 (or x) axis. d15. Induced strain is about the y-axis due to electrical field along the x-axis. d22. Applied electrical field and resultant strain are along the y-axis. d36. Applied electrical field in the 3 (or z) direction induces a torque about the z-axis. dh. Hydrostatic piezoelectric charge constant. Short-circuit charge per unit area of electrode flowing between connected electrodes that are perpendicular to a specified axis of a piezoelectric crystal to equal stresses applied along all three axes (hydrostatic stress). This constant is applicable to piezoelectric ceramics and lithium sulfate. Piezoelectric stress (or voltage) constants (g). (a) The ratio of field developed along a specified axis to the stress applied along or around a specified axis when all other external

CRC_DK8307_CH004.indd 192

10/27/2008 10:44:23 AM

Piezoelectric Materials: Properties and Design Data

193

stresses are constant and (b) the ratio of strain developed along or around a specified axis to the electric charge per unit area of electrode applied to electrodes that are perpendicular to a specified axis [2]. g31. The electric field in the 3 (or z) direction per unit applied stress in the 1 (or x) direction or the induced strain in the 1 (or x) direction per unit dielectric displacement in the 3 (or z) direction. g33. The electric field in the 3 (or z) direction per unit stress in the 3 (or z) direction. g15. The electric field in the 1 (or x) direction per unit stress about the 5 (or y) axis. Similar definitions apply to stress constants g11, g14, g22, and g36, corresponding to strain constants d11, d14, d22, and d36 in which the word strain in the definition of the d constants is replaced by the word stress in the definition of the g constants. Using the previous strain (d) and stress (g) constants to predict the behavior of a piezoelectric transducer can be complicated by the conditions under which the piezoelectric elements must perform. Applying static or low-frequency field (i.e., low frequency compared with half-wave resonance) to piezoelectric elements permits the assumption that stress, strain, electric field, and dielectric displacement involved in the constants are uniform and the change in dimensions of the element can be determined by applying the appropriate constants times the applied voltage equals the change in length field times the appropriate dimension (e.g., ∆ℓ = d times applied field times ℓ for change in length of a piezoelectric bar). However, when an element is driven at frequencies at or very near half-wave resonance, stress and strain are not uniform throughout the element. To apply the same constants under these conditions requires integration of incremental elements throughout the piezoelectric device to determine the actual effect to be expected. At resonance, other factors also enter into the use of these constants and distribution of stress, strain, and other factors that affect their use. For example, the elastic properties of the system are major factors that determine the amplitude of vibrations to be obtained and the system (transducer) design determines the stress distribution across the piezoelectric elements at system resonance. These conditions will be discussed more fully as they are needed in this chapter in the sections dealing with transducer design. Q, quality factor. The Q of an electrical inductance is Q = ωL/R, where ω = 2πf, L is inductance in henries, and R is resistance in ohms. For the capacitor, Q = 1/ωCR, where C is the capacitance in farads. In a mechanical resonance system, Q is equal to 2π times the average energy stored in the system divided by the energy dissipated per cycle. Mechanical quality factor ( for radial mode, etc.) (QEm, etc.). Equivalent of Q of an electrical component (inductance or capacitance), such as ωL/R, where ω = 2πf, the angular frequency, and R is the resistance of the element. Mechanical quality factor is a measure of the energy stored in a periodically vibrating system to the energy dissipated per cycle. Time constant. Time required for a physical quantity to rise from 0% to 63.2% of its steady state or to fall to 36.8% of its initial value when it varies with time. 4.2.1

Mechanical Data

Compressional strength. Maximum stress that a material can withstand in compression before failure. Density, ρ. Mass per unit volume. Modulus of elasticity, bulk: YB. Ratio of compressive or tensile force applied to a substance per unit surface area to the change in volume of the substance per unit volume.

CRC_DK8307_CH004.indd 193

10/27/2008 10:44:23 AM

Ultrasonics: Data, Equations, and Their Practical Uses

194

Modulus of elasticity, Young’s: Yo. Ratio of simple stress in tension applied to a material within its elastic range to the resulting strain parallel to the tension. Poisson’s ratio, σ: Ratio of strain in the lateral direction to strain in the longitudinal direction in a rod under the influence of stresses imposed by forces applied at the ends of the rod and parallel to the axis of the rod. Strain. Change in length per unit of undisturbed length in the same or another defined direction due to applied stress, electric field, or other; for example, ∂ε/∂x. Tensile strength. Maximum stress that a material can withstand in tension before failure. 4.2.2

Thermal Data

Specific heat. Ratio of the amount of heat required to raise a mass of material one degree in temperature to the amount of heat required to raise an equal mass of a reference material, usually water, 1° in temperature, both measurements being made at a reference temperature, usually at a constant pressure or constant volume, assuming that no phase or chemical change occurs during the measurement. Thermal conductivity. Heat flow across a surface per unit area per unit of time divided by the negative of the rate of change of temperature with distance in a direction perpendicular to the surface.

4.3

Mathematical Relationships

Piezoelectric elements have many uses in technology. Manufacturers of ceramic elements provide data relevant to the design and use of products incorporating their materials. The objective in presenting the mathematical relationships that follow is to assist the designer in making most effective use of the data presented by the manufacturers in arriving at an optimized design, regardless of the intended application. The applications may range from low-intensity and passive devices (such as diagnostic transducers and receiving devices or sensors) to high-intensity ultrasonic sources and high-voltage generators (such as ultrasonic plastic welders and disintegrators). 4.3.1

Electromechanical Coupling Coefficient, Stored Energy, and Sensitivity

Ultrasonics infers the existence of a form of energy that is particularly dependent upon the elastic properties of the media through which it propagates or upon which it operates. Because a piezoelectric transducer transforms electrical energy to mechanical energy and vice versa, it must of necessity exhibit both electrical properties (viz. capacitance) and mechanical properties (viz. elasticity and mass). According to the previous definition, the transformed electrical energy causing mechanical strain when all external stresses are constant divided by the electrical energy input to electrodes on faces perpendicular to the same or different axes equals the square of the electromechanical coupling coefficient, k2. The same factor applies whether energy is converted from electrical to mechanical or vice versa. Thus, k2 = Wm(out) / We(in) = We(out) / Wm(in) As piezoelectric plates consist of materials of high-dielectric constants, the conversion may be related by comparing the electrical energy stored in a condenser to the elastic

CRC_DK8307_CH004.indd 194

10/27/2008 10:44:23 AM

Piezoelectric Materials: Properties and Design Data

195

energy stored in the piezoelectric plate corresponding to the stored electrical energy. For example, 2 k 33  Wm3(out) / We3(in)  We3(out) / Wm3(in)

where Wm3(out) is the mechanical energy out by stress in the 3 direction across a plate, We3(in) the energy by means of an electrical field between electrodes on the surfaces normal to the 3 direction, We3(out) the electrical energy out from electrodes on the surfaces normal to the 3 direction, and Wm3(in) the mechanical energy due to pressure in the 3 direction across the plate at frequencies far below resonant frequency. In this simplest form, the electrical energy stored in a capacitor is We = CE2/2 (J)

(4.1)

where C is the capacitance in farads and E is the voltage in volts across the electrodes of the condenser. The capacitance, C, of a condenser is C

8.854r A  1014 (F)


Therefore, the electrical energy stored in a capacitor is We =

E2  8.854r A × 10 −14   ( J) 2 


(4.2)

where εr is the relative dielectric constant, A the area of the electrodes on each surface in square centimeters, and ℓ the distance between electrodes (or thickness of the plate) in centimeters. The mechanical energy stored in an elastic material subjected to a force, pA, is  p 2 A
 7 Wm     10 ( J)  2Y 

(4.3)

where p is pressure in dynes per square centimeter, ℓ is length in centimeters, and Y is the appropriate modulus of elasticity in dynes per square centimeter. Combining Equations 4.1 and 4.3 in accordance with the definition of k2 gives k2 

8.854E2r Y  107 p2
2

(4.4)

from which the sensitivity of the element is E/p = 1.062748 kℓ(εrY)−1/2 × 103 (V/dyn/cm2)

(4.5a)

E/pN = 1.062748 kℓ(εrYN)−1/2 × 104 (V/N/m2 or V/Pa) = g × ℓm (V/Pa)

(4.5b)

or

CRC_DK8307_CH004.indd 195

10/27/2008 10:44:23 AM

Ultrasonics: Data, Equations, and Their Practical Uses

196

where ℓ is the length in meters (distance between electrodes) and YN the modulus of elasticity of the material in newtons per square meter. A relationship that applies to all piezoelectric materials is k2 = 8.854 εYNg210−12

(4.6)

where ε is the dielectric constant of the material; g the piezoelectric stress constant for the material, in volt meters per newton; and k the electromechanical coupling coefficient. The value of k can never be as high as 1.0. The bulk modulus, YB, of a solid is (1 − σ)Y0 YB = ______________ (1 + σ)(1 − 2σ) The following relationships between piezoelectric constants are true: k2 = gYd

(4.7)

g = d/8.854 × 10−12εr

(4.8)

always noting axial directions by subscripts and maintaining consistency: for example, g33:d33, g31:d31, and so on.

4.4

Properties of Piezoelectric Materials

Tables 4.1 through 4.4 include data provided by manufacturers of piezoelectric materials. They are grouped according to manufacturer. All data are published by permission of the manufacturers as identified. Additional properties of Kynar Piezo film, as provided by Pennwalt, are given in Table 4.5.

4.5 4.5.1

Selection and Design Guidelines Fundamental Piezoelectric Action

When a voltage is applied to electrodes properly placed on a piezoelectic crystal, the crystal changes dimensions. When a force is applied across these same terminals, a voltage is generated. The actions are fixed on a natural crystal. By applying voltages (or forces) across appropriate sets of electrodes, the crystal may be made to expand and contract (or produce voltages), shear, or develop a torsion depending upon the location of the electrodes. A ceramic piezoelectric element can be shaped to expand the number of modes—plates, bars, cylinders, hemispheres, and spheres. The objective at this point is to show various available shapes of piezoelectrics, piezoceramics, the modes of vibrations that can be produced or the voltages that might be generated, and some common uses for the products. Some manufacturers will make and blend transducers to the customer’s desires, provided that the desired materials are practical.

CRC_DK8307_CH004.indd 196

10/27/2008 10:44:23 AM

CRC_DK8307_CH004.indd 197

ε33/εo

2750

3250

1250

1000

APC 850B

APC 855

APC 840

APC 880

Lead zirconate titanate APC 850 1750

Materials

ε11/εo

Free Dielectric Constant k33

0.30

0.35

0.39

0.36

0.65 [kp = 0.56]

0.70 [kp = 0.62]

0.75 [kp = 0.65]

0.73 [kp = 0.64]

0.36 0.72 [kp (= kr) =0.63]

k31

Coupling Factor

−95

−125

−270

−230

−175

d31

250

300

590

500

450

d33

Piezoelectric Strain Constant (10−12 m/V)

Piezoelectric Materials and Their Properties

TABLE 4.1

−11.0

−10.5

−8.8

−9.0

−12.4

g31

25.0

26.0

9.5

21.0

26.0

g33

Piezoelectric Stress Constant (10−3V m/N)

1000 [Np = 3150; N31 = 2323]

400 [Np = 3307; N31 = 2362]

70

75

80

3270

3110

3230 [Np = 2992; N31 = 2323]

2990 [Np = 3228; N31 = 2362]

3150 [radial Np = 2953; longitudinal N31 = 2362]

7.65

7.6

7.5

7.5

7.5

7.8

6.8

6.0

6.0

6.3

325

340

195

210

360

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point kg/m3) (1010 N/m2) Q Thin Disk (°C) (Hz m)

0.35

0.4

2.0

2.0

1.6

Dissipation Factor (% 102 tan δ)

(continued)

American Piezo Ceramics Inc. (APC), Duck Run Road, PO Box 189, Mackeyville, PA 17750 American Piezo Ceramics Inc. (APC), Duck Run Road, PO Box 189, Mackeyville, PA 17750 American Piezo Ceramics Inc. (APC), Duck Run Road, PO Box 189, Mackeyville, PA 17750 American Piezo Ceramics Inc. (APC), Duck Run Road, PO Box 189, Mackeyville, PA 17750 American Piezo Ceramics Inc. (APC), Duck Run Road, PO Box 189, Mackeyville, PA 17750

Source

Piezoelectric Materials: Properties and Design Data 197

10/27/2008 10:44:24 AM

CRC_DK8307_CH004.indd 198

ε33/εo

570

625

1350

600

1300 Navy IV

1750

2600

3200

5500 Navy II

5600 Navy V

5700 Navy VI

2950

2400

1775

Lead zirconate titanate 5400 1300 1475 Navy I

1450

1350

Barium titanate 300 1250

ε11/εo

Free Dielectric Constant k33

0.40 [kp = 0.06]

k31

Coupling Factor

−0.37 0.72 [k15 = 0.65; kp = 0.62]

−0.36 0.73 [k15 = 0.68; kp = −0.62]

−0.37 0.73 [k15 = 0.71; kp = −0.62]

−0.36 0.71 [k15 = 0.72; kp = −0.60]

−0.18 0.45 [k15 = 0.45; kp = −0.30]

−0.16 0.39 [k15 = 0.39; kp = −0.27]

−0.19 0.46 [k15 = 0.46; kp = −0.32]

0.05

(Continued)

Lead metaniobate APC 810 250

Materials

TABLE 4.1

145 [d15 = 245]

82 [d15 = 150]

145 [d15 = 245]

90

d33

−250 550 [d15 = 690]

−225 505 [d15 = 670]

−185 400 [d15 = 625]

−135 300 [d15 = 525]

−56

−33

−58

−16

d31

Piezoelectric Strain Constant (10−12 m/V)

12.2 [g15 = 19.1]

16.8 [g15 = 29.8]

13.1 [g15 = 20.5]

33.0

g33

−8.8

−9.8

19.4 [g15 = 26.4]

22.0 [g15 = 31.5]

−11.9 25.8 [g15 = 40.0]

−11.7 26.1 [g15 = 40.5]

−4.7

−6.8

−5.2

−4.2

g31

Piezoelectric Stress Constant (10−3V m/N)

65

70

75

500

600

1200

450

15

1980 [Np = 1980; N31 = 1400]

1900 [Np = 1980; N31 = 1420]

1980 [Np = 1980; N31 = 1400]

2000 [Np = 2210; N31 = 1500]

2690 [Np = 3150; N31 = 2290]

2640 [Np = 3150; N31 = 2290]

2690 [Np = 3150; N31 = 2290]

2360

7.4

7.5

7.6

7.55

5.55

5.4

5.5

6.0

>350

>24

>190

Y11E = 6.4 Y33E = 5.2 Y11E = 6.2 Y33E = 5.1 Y11E = 6.2 Y33E = 4.8

>115

Y11E = 11.9 Y33E = 11.3

>300

>140

Y11E = 11.6 Y33E = 11.0

Y11E = 8.2 Y33E = 6.5

>115

500

Y11E = 11.7 Y33E = 11.1

3.8

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point (°C) kg/m3) (1010 N/m2) Q Thin Disk (Hz m)

2.0

2.0

2.0

0.4

0.8

0.3

0.8

1.0

Dissipation Factor (% 102 tan δ)

Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111

Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111

American Piezo Ceramics Inc. (APC), Duck Run Road, PO Box 189, Mackeyville, PA 17750

Source

198 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:24 AM

CRC_DK8307_CH004.indd 199

1050

5804 Navy III

1220

640

EC-55

EC-57

1300

1725

2125

1100

1050

2750

EC-64

EC-65

EC-66

EC-67

EC-69

EC-70

Lead zirconate titanate EC-63 1250

1170

EC-31

Barium titanate EC-21 1070

1100

5800

1340

1400

−105 240 [d15 = 382]

−107 245 [d15 = 390]

0.38

−32

87

−95 220 [d15 = 330]

0.37 0.74 −230 490 [d15 = 670] [k15 = 0.67; kp = 0.63]

0.31 0.62 [k15 = 0.55; kp = 0.5]

0.33 0.66 −107 241 [d15 = 362] [k15 = 0.59; kp = 0.56]

0.36 0.72 −198 415 [d15 = 626] [k15 = 0.68; kp = 0.62]

0.36 0.72 −173 380 [d15 = 584] [k15 = 0.69; kp = 0.62]

0.35 0.71 −127 295 [d15 = 506] [k15 = 0.72; kp = 0.60]

0.34 0.68 −120 270 [d15 = 475] [k15 = 0.69; kp = 0.58]

0.15

0.19 0.46 −58 150 [d15 = 245] [k15 = 0.48; kp = 0.31]

0.19 0.48 −59 152 [d15 = 248] [k15 = 0.49; kp = 0.32]

0.17 0.38 −49 117 [d15 = 191] [k15 = 0.37; kp = 0.26]

0.32 0.66 [k15 = 0.59; kp = 0.54]

−0.32 0.67 [k15 = 0.60; kp = 0.55]

16.2

900

−10.9 24.8 [g15 = 28.7]

−9.8 20.9 [g15 = 35.0]

75

960

80

−10.6 23.0 [g15 = 36.6]

−10.2 23.7 [g15 = 28.9]

100

400

500

600

550

400

1400

1050

1100

−11.5 25.0 [g15 = 38.2]

−10.7 25.0 [g15 = 39.8]

−10.3 24.1 [g15 = 37.0]

−5.5

−5.6 14.3 [g15 = 20.1]

−5.8 14.8 [g15 = 20.4]

−5.2 12.4 [g15 = 15.7]

−11.3 25.8 [g15 = 32.2]

−11.0 25.2 [g15 = 31.5]

1727

2181

2141

1752

1778

2026

2069

2845

2868

2845

2818

2110 [Np = 2310; N31 = 1570]

2110 [Np = 2260; N31 = 1570]

7.45

7.5

7.5

7.45

7.5

7.5

7.5

5.3

5.55

5.55

5.7

7.55

7.55

6.3

9.9

9.3

6.2

6.6

7.8

8.9

12.5

11.6

10.7

220

300

300

270

350

320

320

140

115

115

130

>300

Y11E = 8.6 Y33E = 7.1

11.4

>300

Y11E = 8.6 Y33E = 7.1

2.0

0.3

0.3

2.0

2.0

0.4

0.4

0.6

0.5

0.7

0.5

0.4

0.4

(continued)

Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115

Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115 Edo Western, 2645 South 300 West, Salt Lake City, UT 84115

Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111 Channel Industries, Inc., 839 Ward Drive, Santa Barbara, CA 93111

Piezoelectric Materials: Properties and Design Data 199

10/27/2008 10:44:24 AM

CRC_DK8307_CH004.indd 200

236

Lead titanate EC-97

ε11/εo

1150

HS-21

k33

d31

d33

Piezoelectric Strain Constant (10−12 m/V)

0.35 0.72 −312 730 [d15 = 825] [k15 = 0.67; kp = 0.61]

0.01 0.53 −3 68 [d15 = 67] [k15 = 0.35; kp = 0.01]

k31

Coupling Factor

2100

250

1700

2600

1800

G-1500

G-1512

HST-41

Lead metaniobate G-2000 250

2200

1000

G-1408

0.04 0.38 −10 80 [d15 = 115] [k15 = 0.33; kp = 0.066]

0.35 0.66 −157 325 [d15 = 625] [k15 = 0.69; kp = 0.59]

0.37 0.72 −232 500 [d15 = 680] [k15 = 0.78; kp = 0.63]

0.35 0.66 −120 280 [d15 = 360] [k15 = 0.59; kp = 0.57] 1250 0.26 0.60 −80 200 [d15 = 315] [k15 = 0.60; kp = 0.50] 1700 0.34 0.69 −166 370 [d15 = 540] [k15 = 0.65; kp = 0.58]

1350

1300

HDT-31

0.29 0.60 −84 190 [d15 = 300] [k15 = 0.64; kp = 0.50]

0.16 0.45 −30 86 [d15 = 125] [k15 = 0.42; kp = 0.26] 1050 0.18 0.51 −50 148 [d15 = 225] [k15 = 0.48; kp = 0.30]

570

Lead zirconate titanate G-53 720 960

600

Barium titanate HD-11

Lead magnesium niobate EC-98 5500

ε33/εo

Free Dielectric Constant

(Continued)

Materials

TABLE 4.1

g33

−4.5 [g15 = 50]

−11 [g15 = 37]

−9.3 [g15 = 35]

−11 [g15 = 29] −9.0 [g15 = 29] −11 [g15 = 36]

−13 [g15 = 36]

−5.5 [g15 = 25] −5.2 [g15 = 25]

36

22

20

25

22

23

30

16.0

16.0

−6.4 15.6 [g15 = 17.0]

−1.7 32 [g15 = 33.5]

g31

Piezoelectric Stress Constant (10−3V m/N)

—

>1200

15–20

70

70

—

—

—

—

—

>500

80

—

— Y33E = 13.0 —

1803

2210

140

300

>800

70

950

>5.8

Y11E = 4.0 Y33E = 4.7

>400

>270

Y11E = 7.0 Y33E = 5.9 >7.6

1.5

>360

0.6

2.2

1.8

0.3

>300

>240

0.6

2.2

1.5

0.6

2.0

0.9

Dissipation Factor (% 102 tan δ)

>330

Y33E = 6.3 Y33E = 5.4

Y11E = 8.1 Y33E = 6.7 Y11E = 9.0 Y33E = 8.2 Y11E = 6.3 Y33E = 4.9

>330

>125

Y11E = 11.1 Y33E = 11.5 Y11E = 8.1 Y33E = 6.5

>135

170

240

Y11E = 13.7

6.1

12.8

>7.4

>7.6

>7.5

>7.6

>7.6

−5.6

>5.6

7.85

6.7

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point (°C) kg/m3) (1010 N/m2) Q Thin Disk (Hz m)

Gulton Industries Inc., Metuchen, NJ 08840

Gulton Industries Inc., Metuchen, NJ 08840 Gulton Industries Inc., Metuchen, NJ 08840 Gulton Industries Inc., Metuchen, NJ 08840 Gulton Industries Inc., Metuchen, NJ 08840 Gulton Industries Inc., Metuchen, NJ 08840

Gulton Industries Inc., Metuchen, NJ 08840

Gulton Industries Inc., Metuchen, NJ 08840 Gulton Industries Inc., Metuchen, NJ 08840

Edo Western, 2645 South 300 West, Salt Lake City, UT 84115

Edo Western, 2645 South 300 West, Salt Lake City, UT 84115

Source

200 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:24 AM

CRC_DK8307_CH004.indd 201

1050

1000

1160

1800

3000

2600

Sonox P5

Sonox P51

Sonox P52

—

—

—

Type materials not identified Sonox P2 480 800

Sonox P8

Lead zirconate titanate Sonox P4 1300 1480

Barium titanate Sonox P-1 960

−180

−210

−230

0.71 −0.33 [kp = −0.60]

0.71 −0.33 [kp = −0.60]

−45

530

510

400

125

−95 215 [d15 = 290]

−130 280 [d15 = 450]

−40 125 [d15 = 205]

−0.33 0.70 [k15 = 0.67; kp = −0.59]

−0.26 0.60 [k15 = 0.65; kp = −0.46]

−0.31 0.60 [k15 = 0.50; kp = −0.50]

−0.31 0.68 [k15 = 0.64; kp = −0.55]

−0.15 0.43 [k15 = 0.42; kp = −0.25]

−10.0

−8.0

−11.0

−11.0

−11.0 [g15 = 28]

−11.0 [g15 = 34]

−5.0 [g15 = 22]

23

19

25

30

25

25

14

70

100

90

400

1000

400 (radial)

350 (radial)

1320

1400

1500

1650

1600

1570

2300

7.5

7.4

7.7

7.8

7.7

7.8

5.3

360

340

200

220

Y11E = 6.3 Y33E = 5.3

Y11E = 6.7 Y33E = 5.3

Y11E = 6.7 Y33E = 5.3

300

Y11E = 9.1

Y11E = 10.0 Y33E = 7.7

325

120

Y11E = 7.7 Y33E = 5.9

Y11E = 11.1 Y33E = 11.1

1.5

2.0

2.0

0.5

0.2

0.5

0.7

(continued)

Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148 Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148 Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148 Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148

Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148 Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148

Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148

Piezoelectric Materials: Properties and Design Data 201

10/27/2008 10:44:25 AM

CRC_DK8307_CH004.indd 202

570

1070

140

300

175

Sonox P6

Sonox P62

K15

K81

K83

—

—

—

—

ε11/εo

425

1300

K180

K270

—

—

Lead-zirconate titanate K85 800 —

ε33/εo

Free Dielectric Constant k31

0.38

0.15

—

0.33 [k15 = 0.71; kp = 0.58]

0.30 [k15 = 0.68; kp = 0.52] −11 [g15 = 38]

270 −120 [d15 = 490]

0.70

—

—

−15.0 [g15 = 50]

180

65

85

—

—

−8.0

g31

26

41

27

42

−7.0

14.5

—

20

g33

Piezoelectric Stress Constant (10−3V m/N)

180 60 [d15 = 350]

—

—

≥18

205

100

−40

—

d33

d31

Piezoelectric Strain Constant (10−12 m/V)

0.67

0.43 — [kt =0.35; kp = 0.35]

[kt = 0.30; kp = <0.10] −15 [kt = 0.47; kp = 0.10]

— [kp = 0.06]

−0.25

k33 —

Coupling Factor

−0.19 [k15 = 0.39; kp = −0.34]

(Continued)

Materials

TABLE 4.1

400

600

15

650

32

100

2200

1100

3150

3100

2600

4250

<15

3230

—

1880

7.5

7.7

5.7

4.5

2360

7.2

8.0

7.7

—

—

—

—

6.2

325

350

300

200

−400

0.5

3.0

2.5

3.5

<1.0

1.0

>600 10

0.1

320

0.4

Y11E = 11.1

Y11E 400

Dissipation Factor (% 102 tan δ)

= 11.1

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point kg/m3) (1010 N/m2) Q Thin Disk (°C) (Hz m)

Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268 Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268

Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268

Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148 Hoschst Ceramtec North America Inc., 171 Forbes Boulevard, Mansfield, MA 02048-1148 Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268 Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268 Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268

Source

202 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:25 AM

1100

—

CRC_DK8307_CH004.indd 203

3000

170

215

K550

NOVA 7A

NOVA 3B

—

—

—

—

1300

1473

1500

PZT-4

PZT-4D

PZT-4S-1

—

—

1475

Lead zirconate titanate PZT-2 450 990

Modified barium-titanate Ceramic B 1200 1300

2700

K500

Lead-zirconate-titanate K350 1700 —

K278

—

0.515

0.375 [kp = 0.640]

—

0.626 −0.28 [k15 = 0.701; kp = −0.47] 0.70 −0.334 [k15 = 0.71; kp = −0.58] — −0.354 [kp = 0.602]

37

−160

360

−12.1

−11.17

−145.4

27.1

27.26

−11.1 26.1 [g15 = 39.4]

289 −123 [d15 = 496] 355.1

−15.1 38.1 [g15 = 50.3]

−5.5 14.1 [g15 = 21.0]

—

40

152 −60.2 [d15 = 440]

149 −58 [d15 = 242]

70

—

—

0.476

62

−9 20 [g15 = 26.5]

550 −274 [d15 = 740]

0.75

20

−9 [g15 = 25]

500 −260 [d15 = 700]

0.75

25

25.5

−11 [g15 = 37]

−10.9

380 −170 [d15 = 575]

240 −95 [d15 = 325]

0.70

0.64

0.48 −0.194 [k15 = 0.48; kp = −0.33]

— [kp = 0.01]

— [kp = 0.035]

0.39 [k15 = 0.67; kp = 0.65]

0.37 [k15 = 0.65; kp = 0.63]

0.35 [k15 = 0.65; kp = 0.60]

0.30 [k15 = 0.55; kp = 0.51]

300

642

500

680

400

300

300

70

70

75

1000

7.5

7.61

7.7

Np = 2183 Np = 2120

7.6

5.55

7.65

7.5

7.8

7.6

7.7

7.5

2000

2090

2740

3300

3230

3150

3070

3070

3150

Y11E = 7.35

315

315

328

Y11E = 8.1 Y33E = 6.5 Y11E = 7.69

370

115

350

350

180

220

360

300

Y11E = 8.6 Y33E = 6.8

Y11E = 11.6 Y33E = 11.0

—

—

—

—

—

—

0.4

0.25

0.4

0.5

0.6

1.5

2.0

2.0

2.0

2.0

0.4

(continued)

Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146

Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146

Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268 Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268 Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268 Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268 Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268

Keramos Inc., 5460 West 84th St., Indianapolis, IN 46268

Piezoelectric Materials: Properties and Design Data 203

10/27/2008 10:44:25 AM

CRC_DK8307_CH004.indd 204

ε33/εo

1350

1700

3400

1050

890

460

425

1301

1000

1050

1150

PZT-4S-2

PZT-5A

PZT-5H

PZT-6A1

PZT-6A2

PZT-6B

PZT-7A

PZT-7D

PZT-8

PZT-8M

PZT-8D

—

—

1290

—

840

475

—

—

3130

1730

—

ε11/εo

Free Dielectric Constant k31

k33

0.47

0.330 [kp = 0.550]

−0.30 [k15 = 0.55; kp = −0.51] 0.340 [kp = 0.580] −110

−110

0.70

0.685

315

295

225 −0.97 [d15 = 330]

−10.9

−11.9

30.9

31.8

−10.9 25.4 [g15 = 28.9]

23.5

−9.4

−107

271

−15.9 39.9 [g15 = 48.8]

19.2

150 −60 [d15 = 362]

−7.5

−6.85 18.0 [g15 = 30.8]

151

71 −27 [d15 = 130]

−59.5

20.4

−8.5

−80

189

−9.11 19.7 [g15 = 26.8]

27.5

g33

593 −274 [d15 = 741]

−12.3

g31

−11.4 24.8 [g15 = 38.2]

330

d33

Piezoelectric Stress Constant (10−3V m/N)

374 −171 [d15 = 584]

−148

d31

Piezoelectric Strain Constant (10−12 m/V)

0.64

0.375 −0.145 [k15 = 0.377; kp = 0.25] 0.66 −0.30 [k15 = 0.67; kp = 0.51] — −0.293 [kp = 0.499]

−0.207 [kp = 0.35]

0.705 −0.344 [k15 = 0.685; kp = −0.60] 0.752 −0.388 [k15 = 0.675; kp = −0.65] 0.54 −0.248 [kp = 0.42]

—

Coupling Factor

0.367 [kp = 0.625]

(Continued)

Materials

TABLE 4.1

1250

1000

1000

600

600

1300

550

450

65

75

350

7.6

7.7

7.6

Np = 2290 Np = 2290

7.75

Np = 2283

2070

7.6

7.55

7.45

7.45

7.5

7.75

7.7

2100

2225

2090

2140

2000

1890

Np = 2140

335

≈350

≈350

Y11E = 8.93 Y33E = 7.69 Y11E = 11.1 Y33E = 10.7 Y11E = 9.35 Y33E = 7.19

Y11E = 9.0 Y33E = 6.9

Y11E = 8.6

290

305

300

335

Y11E = 9.35 Y33E = 7.69

Y11E = 8.7 Y33E = 7.4

193

Y11E = 6.06 Y33E = 4.83

325

365

Y11E = 6.1 Y33E = 5.3

Y11E = 8.6

315

= 7.46

Y11E

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point kg/m3) (1010 N/m2) Q Thin Disk (°C) (Hz m)

0.4

0.4

0.4

0.49

1.7

0.9

2.0

2.0

2.0

2.0

0.4

Dissipation Factor (% 102 tan δ)

Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146

Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146 Morgan Matroc Inc., 232 Forbes Road, Bedford, OH 44146

Source

204 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:26 AM

1750

2800

2800

600

700

PXE-4

PXE-5

CRC_DK8307_CH004.indd 205

PXE-5

PXE-6

PXE-7

1000

—

1800

1800

—

0.31 [k15 = 0.66; kp = 0.52]

0.19 [kp = 0.32]

0.37 [k15 = 0.66; kp = 0.58]

0.37 [k15 = 0.66; kp = 0.58]

0.32 [kp = 0.55]

220 −86 [d15 = 370]

−14.0 35.4 [g15 = 42.0]

≈ 80

≈1000

0.70

—

−44

—

−8.0

80

−10.9 22.0 [g15 = 32.5]

370 −190 [d15 = 515]

0.69

—

80

500

−10.9 22.0 [g15 = 32.5]

18.8

370 −190 [d15 = 515]

−8.9

0.69

292

−138

0.64

7.7

7.7

7.75

NPE = 2460

NPE = 2200

7.7

NPE = 2000

NPE = 2000

7.5

NPE = 2300 [NPE = radial mode]

Y11E = 8.0 Y33E = 6.33

Y11E = 9.80

Y11E = 6.49 Y33E = 5.29

Y11E = 6.49 Y33E = 5.29

Y11E = 8.47 Y33E = 6.06

320

370

285

285

265

2.0

0.8

2.0

2.0

0.6

(continued)

Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477 Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477 Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477 Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477 Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477

Piezoelectric Materials: Properties and Design Data 205

10/27/2008 10:44:26 AM

CRC_DK8307_CH004.indd 206

ε33/εo

400

1750

2250

1200

PXE-11

PXE-21

PXE-22

PXE-41

1400

—

—

600

ε11/εo

Free Dielectric Constant k31

0.34 [k15 = 0.70; kp = 0.58]

0.37 [kp = 0.62]

0.37 [kp = 0.62]

−202

268 −119 [d15 = 480]

0.72

0.68

438

385

−180

d33

0.72

d31 100 −47.5 [d15 = 235]

k33

Piezoelectric Strain Constant (10−12 m/V)

0.55

Coupling Factor

0.25 [k15 = 0.65; kp = 0.43]

(Continued)

Materials

TABLE 4.1

g33

22.0

25.0

−11.6 25.2 [g15 = 38.5]

−10.1

−11.6

−13.4 28.2 [g15 = 44.0]

g31

Piezoelectric Stress Constant (10−3V m/N)

≈1000

≈80

≈80

≈270

7.75

7.90

NEP = 2000

NEP = 2200

4.5

7.75

= 3600

NEP = 2000

NPE

= 12.35 = 10.53

Y11E = 8.2 Y33E = 6.85

Y11E = 6.62 Y33E = 5.38

Y11E = 6.62 Y33E = 5.38

Y11E Y33E

315

270

270

400/ 180a

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point kg/m3) (1010 N/m2) Q Thin Disk (°C) (Hz m)

0.25

1.6

1.8

2.5

Dissipation Factor (% 102 tan δ)

Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477 Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477 Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477 Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477

Source

206 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:26 AM

CRC_DK8307_CH004.indd 207

1100

2800

1300

PXE-43

PXE-51

PXE-71

1700

—

—

—

Lead zirconate titanate (PZT) EBL#1 1300 —

1300

PXE-42

0.36 [k15 = 0.72; kp = 0.60]

0.35 [k15 = 0.66; kp = 0.6]

0.39 [kp = 66]

0.30

0.34 [kp = 0.58]

0.69

— −147 [d15 = 500]

—

295 −127 [d15 = 506]

480

−234

0.72

210

285

−95

−130

0.63

0.68

19.3

25.0

25.0

−10.7 25.0 [g15 = 39.8]

−12.8 — [g15 = 33.2]

−9.5

−10.7

−11.0

400

≈80

≈50

1000

≈750

7.70

7.75

NEP = 2050

NEP = 2050

7.50

7.70

NEP = 2350

2026

7.70

NEP = 2200

—

Y11E = 6.67

Y11E = 6.9 Y33E = 5.62

Y11E = 8.85 Y33E = 7.94

Y11E = 7.87 Y33E = 6.54

320

270

220

300

325

0.4

2.0

1.6

0.2

0.25

(continued)

Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/East, Hartford, CT 06108

Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477 Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477 Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477 Philips, Ferroxcube, Division of Amperex Electronic Corporation, Subsidiary of North America Philips, 5083 Kings Highway, Saugerties, NY 12477

Piezoelectric Materials: Properties and Design Data 207

10/27/2008 10:44:27 AM

CRC_DK8307_CH004.indd 208

ε33/εo

1725

3450

1050

425

2750

EBL#2

EBL#3

EBL#4

EBL#5

EBL#6

—

—

—

—

—

ε11/εo

Free Dielectric Constant k31

0.39 [k15 = 0.67; kp = 0.63]

0.30 [k15 = 0.68; kp = 0.52]

0.31 [k15 = 0.55; kp = 0.52]

0.38 [k15 = 0.68; kp = 0.64]

−10.5 24.5 [g15 = 28.9]

−16.0 [g15 = 50]

−9.8 20.9 [g15 = 35.0]

220 −95 [d15 = 330]

150 −60 [d15 = 360]

480 −260 [d15 = 670]

0.62

0.67

0.74

41.0

−8.6 19.1 [g15 = 28.9]

g33

583 −262 [d15 = 730]

g31

0.75

d33 −11.5 25.0 [g15 = 38.2]

d31

Piezoelectric Stress Constant (10−3V m/N)

380 −173 [d15 = 39.8]

k33

Piezoelectric Strain Constant (10−12 m/V)

0.72

Coupling Factor

0.36 [k15 = 0.69; kp = 0.62]

(Continued)

Materials

TABLE 4.1

75

600

960

65

100

1727

2050

2181

1765

1778

7.45

7.70

7.50

7.45

7.50

—

—

—

—

—

220

350

300

190

350

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point (°C) kg/m3) (1010 N/m2) Q Thin Disk (Hz m)

2.0

2.5

0.4

2.0

2.0

Dissipation Factor (% 102 tan δ)

Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/ East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/ East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/East, Hartford, CT 06108

Source

208 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:27 AM

1100

1450

4300

3800

1800

EBL#7

EBL#9

CRC_DK8307_CH004.indd 209

EBL#16

EBL#23

EBL#25

—

—

—

—

—

0.30 [kp = 0.6]

0.44 [k15 = 0.68; kp = 0.75]

0.40 [kp 0.70]

0.34 [k15 = 0.60]

—

—

−135

−299

−320

−179

—

0.71

0.78

0.75

0.70

350

650

802

315

240

−11.0

−9.0

−8.5

−10.5

—

24.2

19.0

23.0

24.6

24.8

80

30

70

600

900

2050

2030

1870

1990

2141

—

7.80

7.82

7.60

7.50

—

—

—

7.5

—

350

250

195

320

300

1.8

2.4

2.8

0.4

0.4

(continued)

Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/ East, Hartford, CT 06108

Piezoelectric Materials: Properties and Design Data 209

10/27/2008 10:44:27 AM

CRC_DK8307_CH004.indd 210

ε33/εo

ε11/εo

Free Dielectric Constant

1725

3450

1050

EBL#2

EBL#3

EBL#4

k31

Coupling Factor

[k15 = 0.69; kp = 0.62]

(Continued)

Lead zirconate titanate tubes EBL#1 1300

Materials

TABLE 4.1

k33 295

d33

−95

−262

220

583

−173 380 [d15 = 39.8]

−127

d31

Piezoelectric Strain Constant (10−12 m/V)

25.0

g33

−10.2

−8.6

23.7

19.1

−11.5 25.0 [g15 = 38.2]

−10.7

g31

Piezoelectric Stress Constant (10−3V m/N)

960

65

—

400

Mode: length = 2717 thickness = 3386

Mode: length = 2087 thickness = 2717

Mode: length = 2087 thickness = 2756

Mode: length = 2520 thickness = 3110

7.5

7.45

7.5

7.5

8.5

6.3

6.3

8.1

300

190

350

320

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point kg/m3) (1010 N/m2) Q Thin Disk (°C) (Hz m)

—

—

—

—

Dissipation Factor (% 102 tan δ)

Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/ East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/ East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/ East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/East, Hartford, CT 06108

Source

210 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:27 AM

CRC_DK8307_CH004.indd 211

—

200

700

#2LM

#4LM

VP-A40

1300

1475

Modified lead titanate EBL#1LT 190 —

—

—

Lead metaniobate #1LM 300

—

—

—

0.700 −0.330 [k15 = 0.710; kp = 0.58]

[k15 = 0.50; kp = 0.035]

—

—

—

68

160

65

—

285 −122 [d15 = 495]

—

—

85

38.0

26.0

40.0

15

−10.6 24.9 [g15 = 38.0]

—

—

—

35.0

500

—

15

500

1575

5.6

4.5

—

2000

7.6

Mode: 7.7 longitudinal 2108

1650

2670

6.0

Y33 = 6.45 Y11 = 8.13

—

—

285

450

325

>300

400

2.0

1.0

—

<1.5

2.0

—

—

(continued)

Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/ East, Hartford, CT 06108 Valpey-Fisher Corporation, A Subsidiary of Matec, 75 South Street, Hopkinton, MA 01748

Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/ East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/ East, Hartford, CT 06108 Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/ East, Hartford, CT 06108

Piezoelectric Materials: Properties and Design Data 211

10/27/2008 10:44:27 AM

CRC_DK8307_CH004.indd 212

ε33/εo

1700

3400

425

1000

VP-A50

VP-A55

VP-A70

VP-A80

1290

840

3130

1730

ε11/εo

Free Dielectric Constant

−16.0 41.0 [g15 = 50.0]

−10.9 25.4 [g15 = 28.9]

153 −60 [d15 = 360]

225 −97 [d15 = 330]

0.670 −0.300 [k15 = 680; kp = 0.520]

0.640 −0.300 [k15 = 0.550; kp = 0.510]

g33

−9.1 19.7 [g15 = 26.8]

g31

593 −274 [d15 = 741]

d33

0.750 −0.390 [k15 = 0.680; kp = 0.650]

d31

Piezoelectric Stress Constant (10−3V m/N)

−11.4 24.8 [g15 = 38.2]

k33

Piezoelectric Strain Constant (10−12 m/V)

374 −171 [d15 = 585]

k31

Coupling Factor

−0.34 0.710 [k15 = 0.690; kp = 0.60]

(Continued)

Materials

TABLE 4.1

1000

600

65

75

2070

2100

2000

1890

7.6

7.7

7.5

7.7

365

195

350

300

Y33 = 5.32 Y11 = 6.09

Y33 = 4.87 Y11 =6.1

Y33 = 7.19 Y11 = 9.35

Y33 = 7.41 Y11 = 8.70

Young’s Curie Frequency Density, Mechanical Constant, N3 ρ (103 Modulus, Y Point kg/m3) (1010 N/m2) Q Thin Disk (°C) (Hz m)

—

—

—

—

Dissipation Factor (% 102 tan δ)

Source Valpey-Fisher Corporation, A Subsidiary of Matec, 75 South Street, Hopkinton, MA 01748 Valpey-Fisher Corporation, A Subsidiary of Matec, 75 South Street, Hopkinton, MA 01748 Valpey-Fisher Corporation, A Subsidiary of Matec, 75 South Street, Hopkinton, MA 01748 Valpey-Fisher Corporation, A Subsidiary of Matec, 75 South Street, Hopkinton, MA 01748

212 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:27 AM

CRC_DK8307_CH004.indd 213

800

VP-M58

—

—

—

65 −9.7 [g15 = 22.0]

180 −10.0 [g15 = 27.0]

0.380 −15 0.130 [k15 = 0.250; [d15 = kp = 0.10] 54]

0.430 −65 [d15 = 160]

0.130 [k15 = 0.30; kp = 0.30]

85 −7.0 [g15 = 31.0]

0.100 0.350 −15 [k15 = [d15 = 0.275; kp = 105] 0.100]

27.0

42.0

32.0

15

650

15

1675

2740

1525

5.7

4.5

6.2

N/A

N/A

N/A

400

200

400

—

—

—

Valpey-Fisher Corporation, A Subsidiary of Matec, 75 South Street, Hopkinton, MA 01748 Valpey-Fisher Corporation, A Subsidiary of Matec, 75 South Street, Hopkinton, MA 01748 Valpey-Fisher Corporation, A Subsidiary of Matec, 75 South Street, Hopkinton, MA 01748

Quote from Piezoelectric Ceramics Application Book by Ferroxcube Corporation, Saugerties, New York, “In PXE11 there is a transition from the ferroelectric orthorhombic to the ferroelectric tetragonal phase at 180°C. If the material passes through this temperature in either direction then it must be repoled.”

175

VP-M38

a

300

VP-M18

Piezoelectric Materials: Properties and Design Data 213

10/27/2008 10:44:27 AM

CRC_DK8307_CH004.indd 214

10.65

—

CdTe Cadmium telluride

10.33

ε33/ε0

CdSe Cadmium selenide

Sodium-ammonium tartrate [NaNH4C4-H4O6+4H2O] CdS Cadmium sulfide

Ammonium tartrate [(NH4)2C4H4O6]

ADP Ammonium dihydrogen phosphate Synthetic crystal [NH4H2PO4] (Free of water of crystallization)

Materials

11.0

9.70

9.35

ε33/ε0

Free Dielectric Constant

Piezoelectric Properties of Crystals

TABLE 4.2

[k14 = −0.023]

0.083 [k15 = 0.130; kt = 0.124]

0.119 [k15 = 0.188; kt = 0.154]

k31

0.194

0.262

k33

Coupling Factor

[d14 = 1.5]

−3.92 [d15 = −10.51]

−5.18 [d15 = −13.98]

[d14 = 5.65; d15 = −14; d24 = −8.5] [d11 = 56; d25 = −150; d36 = 28.3]

[d14 = 4.35; d36 = −137] 1.8

2.4

d31

7.87

10.32

−26.2

d33

Piezoelectric Strain Constant (10−12 m/V) g31

g33

Piezoelectric Stress Constant (10−3 V m/N)

Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite]

3

3

3, 4

Source and References

214 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:28 AM

CRC_DK8307_CH004.indd 215

Lithium gallate [LiGaO2]

EDT Ethylene diamine tartrate Synthetic crystal [C6H14N2O6] (Free of water crystallization) Monoclinic spheroidal crystal [Meets requirements necessary for filter crystals] DKT [K2C4H4O6+(1/2)H2O] Dipotassium tartrate Monoclinic spheroidal crystal [Meets requirements necessary for filter crystals] KDP Potassium dihydrogen phosphate ferroelectric [KH2PO4] Modified lead metaniobate Kezite K-81 Lead metaniobate Lithium sulfate (0° Y-cut) Lithium niobate [LiNbO3] (0°, Z-cut)

84.6

29.1

7.2

— 9.0 84.0

225.0 10.3 30.0

8.8

—

6.44

6.49

300.0

5.0

6.0

−0.12 [kt = 0.25]

— — [kt = 0.17]

—

0.42 0.33

−7.0

32.0

6.0

9.7

[d15 = 69.2; d22 = 20.8] −2.1

85 16 6.0

−0.85

— −4.0 —

[g15 = 91; g22 = 28]

— — −4.0

42.5 175.0 23.0

4 (continued)

7 3, 4, 7 Information sheets from Crystal Technology, Inc., Palo Alto, CA, June 1993. See also Ref. 8

6

3

[d14 = 3.85; d36 = −62.0]

85

5

[d14 = −25; d16 = 6.5; d34 = 29.4; d36 = −66.0]

−15

5

[d16 = −25; d22 = 20]

Piezoelectric Materials: Properties and Design Data 215

10/27/2008 10:44:28 AM

CRC_DK8307_CH004.indd 216

−0.73

—

350.0

−0.10 −0.14 −0.08

—

0.14

k31

—

—

0.10

0.49

k33

Coupling Factor

4.5 4.5

4.5

Quartz [SiO2] X-cut

Y-cut AT-cut [M.P. = 1750°C] Rochelle salt, 45° X-cut [NaKC4H4O6-4H2O] (most strongly piezoelectric at room temperature) [ferroelectric]

—

Lithium tantalate [LiTaO3]

≈13a

≈500a

Lithium iodate [LiIO3]

[k14 = 0.07; k15 = 0.615; kt = 0.498]

ε33/ε0

Free Dielectric Constant ε33/ε0

(Continued)

Materials

TABLE 4.2

[d14 = 1000; d25 = −138; d36 = 35.6]

−275

−3.4

−2.0 [d11 = −6.9; d14 = 1.7]

−3.0

[d14 = 7.3; d15 = 49.3]

3.7

d31

—

−4.6

2.3

5.7

46.0

d33

Piezoelectric Strain Constant (10−12 m/V)

−90

−50

[g15 = 58; g22 = 15]

−6.0

g31

—

58.0

21.0

g33

Piezoelectric Stress Constant (10−3 V m/N)

3, 5

4, 5, 7 4

5, 7

Information sheets from Crystal Technology, Inc., Palo Alto, CA, June 1993

Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite]

Source and References

216 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:28 AM

CRC_DK8307_CH004.indd 217

a

At 1.0 kHz.

Zinc telluride ZnTe

—

Zinc selenide ZnSe

9.25

8.3

8.58

8.0

—

—

—

Zinc sulfide (Cub) [ZnS—cubic]

Tourmaline [only one piezoelectric coincides with the optical axis] Zinc sulfide (Hex) [ZnS—hexagonal]

Silver gallium selenide [AgGaSe2]

[d14 = 1.1]

[d14 = 0.9]

[k14 = 0.026]

[k14 = 0.017]

−1.1

[d15 = −2.8; dh = 1.1] [d14 = 3.1]

0.127

−52.4

[k15 = 0.052] [k14 = 0.079]

0.39

—

[d14 = 9.0; d36 = 3.7]

3.2

1.93

33.0

—

Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite]

Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite]

Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] 3, 9

Piezoelectric Materials: Properties and Design Data 217

10/27/2008 10:44:28 AM

CRC_DK8307_CH004.indd 218

1.8

5.67

5.849

1.538

CdSe Cadmium selenide

CdTe (Cadmium telluride)

EDT Ethylene diamine tartrate Synthetic crystal [C6H14N2O6] (Free of water of crystallization) Monoclinic spheroidal crystal (Meets requirements necessary for filter crystals)

Frequency Constant, N3 (Hz m)

ADP Ammonium dihydrogen phosphate Synthetic crystal [NH4H2PO4] (Free of water of crystallization) Ammonium tartrate [(NH4)2C4H4O6] Sodium-ammonium tartrate [NaNH4C4-H4O6+4H2O] CdS Cadmium sulfide

Materials

Mechanical Q Thin Disk

Mechanical Properties of Piezoelectric Crystals

TABLE 4.3

4.28

4.83

Density, ρ (103 kg/m3)

[=1/sE11]

Young’s Modulus, Y (1010 N/m2) Curie Point (°C)

Dissipation Factor (%)

Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] 5

3

3

3, 4

Source and References

218 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:28 AM

CRC_DK8307_CH004.indd 219

2.65 2.65 2.65

2870

Quartz [SiO2] X-cut Y-cut AT-cut [M.P. = 1750°C] 106

7.45

—

575

610

1142.3

[C11 = 20.3] [C33 = 24.5; C55 = 7.5]

4.65 4.64

6.8

550 — 1150

>400

— — 7.5

—

121

5.8 2.06 4.7

Lithium tantalate [LiTaO3]

1400 2730

11 —

6.2

4.19 4.487

2559

<15

1.988

Lithium gallate [LiGaO2] Lithium iodate [LiIO3] (at 1.0 kHz)

DKT [K2C4H4O6 + (1/2)H2O] Dipotassium tartrate Monoclinic spheroidal crystal (Meets requirements necessary for filter crystals) KDP Potassium dihydrogen phosphate ferroelectric [KH2PO4] Modified lead Metaniobate kezite K-81 Lead metaniobate Lithium sulfate (0° Y-cut) Lithium niobate [LiNbO3] (0°, Z-cut)

—

— —

<1.0

5, 7 4, 5, 7 4 (continued)

4 Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] Information sheets from Crystal Technology, Inc., Palo Alto, CA, June 1993

7 3, 4, 7 Information sheets from Crystal Technology, Inc., Palo Alto, CA, June 1993. See also Ref. 8 6

6

3

5

Piezoelectric Materials: Properties and Design Data 219

10/27/2008 10:44:28 AM

CRC_DK8307_CH004.indd 220

(Continued)

5.262

5.636

ZnTe Zinc telluride

4.087

3.1

3.82

1.77

Density, ρ (103 kg/m3)

ZnSe Zinc selenide

Frequency Constant, N3 (Hz m)

4.088

Mechanical Q Thin Disk

Zinc sulfide (Cub) (ZnS—Cubic)

Tourmaline (Only one piezoelectric axis coincides with the optical axis) Zinc sulfide (Hex) (ZnS—hexagonal)

Rochelle salt, 45° X-cut [NaKC4H4O6-4H2O] (most strongly piezoelectric at room temperature)[ferroelectric] Silver gallium selenide [AgGaSe2]

Materials

TABLE 4.3

4.167

5.43

5.438

—

—

Young’s Modulus, Y (1010 N/m2)

—

—

−18, +24

—

Dissipation Factor (%)

Curie Point (°C)

Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite] Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite]

3, 9

Staveley Sensors Inc., EBL Product Line, A Subsidiary of Staveley NDT Technologies Inc., 91 Prestige Park Circle/East, Hartford, CT 06108, 3, 5 Information sheets from Cleveland Crystals, Inc., Cleveland, OH, May 1992 [Original source: Clevite]

Source and References

220 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 10:44:28 AM

CRC_DK8307_CH004.indd 221

11.3 — — — — — —

13 12 13

12

2.9 8.5 3.3 3.3 3.2

3.6

PVF2 PVF2 (LE) PVF2 (Z-cut)

Polytrifluorovinyl acetate Polyvinyl chloride Polyvinyl flouride Polysulfene Nylon 11 Polymethyl methacrylite PAN

—

—

— — — — —

—

0.20 — —

1.5

1.3 1.3 0.35 0.50 0.43

106.4

20 20 20

−24 20–25

−23

—

— — — — —

—

−31 — —

−39 −20–22

21

Piezoelectric Strain Constant (10−12 m/V) d31 d33

30.8

48.9 17.3 13.2 17.1 14.3

174 190 174

180 230

230

—

— — — — —

—

−265 — —

−280 219

210

Piezoelectric Stress Constant (10−3 Vm/N) g31 g33

—

— — — — —

—

1.78 1.76 1.78

1.785 1.8

1.78

Density, ρ (103 kg/m3)

—

— — — — —

0.11–0.24 (TD) 0.25 — 0.20–0.50 (MD) 0.24–0.28 (TD) —

0.10–0.24 (TD) 0.28 0.15–0.3 (MD)

0.15–0.31 (MD)

Young’s Modulus,Y (1010 N/m2)

a

Note: MD, machine direction to film orientation; TD, transverse direction to film orientation; LE, length extensional mode. PVF2 is polyvinylidine fluoride. b Values depend upon thickness and preparation history.

0.102 0.14 0.102

— — −0.09–0.15 —

15 12 + 1

PVF2 PVF2b

0.12

12

Kynar film PVF2a

Materials

Free Dielectric Coupling Factor Constant, ε33/ε0 k31 k33

Piezoelectric Polymers and Their Properties

TABLE 4.4

≈70

Curie Point (°C)

2.0

1.5–2

Factor (% 102 tan δ)

13

13 13 13 13 13

13

10 11 12

2 Battelle Columbus Laboratories Compilation, 1977

Pennwalt information publication, Kynar Piezo Film

Dissipation-Source

Piezoelectric Materials: Properties and Design Data 221

10/27/2008 10:44:28 AM

Ultrasonics: Data, Equations, and Their Practical Uses

222 TABLE 4.5

Additional Properties of Kynar Piezo Film—PVF2 Thickness (×10−6 m) Property

9

Surface conductivity metallized film Aluminum Nickel Pyroelectric coefficient Shrinkage in machine direction 60°C 80°C Volume resistivity Tensile strength at yield MD TD Tensile strength at break MD TD Elongation at break MD TD Melting point

— 25 27 2 4 1013

28

52

110

27

2 10 25

mho/m 23

μC/m2/K

2 4 1013

2 4 1013

2 4 1013

Percentage after annealing for 100 h ohm m

— 40–110

106 N/m2

160–330 30–55

106 N/m2

25–40 230–430 165–180°C (329–356°F) 44 0.13 2.5 1.4 1.5–2.2

Flammability Thermal conductivity Specific heat Thermal expansion coefficient Sound velocity

Units

%

% O2 W/m/K MJ/m3/K 10–4/K 103 m/s

Note: MD, machine direction (1); TD, transverse direction to film orientation (2). Source: Pennwalt data sheet, Penwalt, 900 First Avenue, PO Box C, King of Prussia, PA.

t

d

ᐉ

ᐉ

w FIGURE 4.2 Parallel longitudinal mode.

The first figure to be examined is the parallel longitudinal mode (Figure 4.2). The vibration is along the 3(z) direction (see Figure 4.1), the direction of applied voltage. The length, ℓ, is >3d, 3w, and 3t, frequency is N33/ℓ, capacitance is AKT3 ε 0/ℓ, generated or applied voltage ℓg33F/A, and displacement is d33V, where A = πd20/4 for the round bar and A = wt for the rectangular bar. As the width of the rectangular bar is increased (Figure 4.3) so that (a) the length, ℓ, is >3w which is also >3t or (b) the width is >3t but <0.3ℓ, and the 3(z) surfaces are electroded, the displacement is in the (a) ℓ, that is, transverse length mode, or (b) w, that is, transverse width mode.

CRC_DK8307_CH004.indd 222

10/27/2008 10:44:28 AM

Piezoelectric Materials: Properties and Design Data

223 t

z ᐉ

ᐉ w

x y

t

ᐉ

w (a)

(b)

FIGURE 4.3 Transverse (a) length and (b) width modes.

ᐉ t

w

3 FIGURE 4.4 Parallel width mode.

1 t

ᐉ

3 w FIGURE 4.5 The thickness shear mode.

For (a), the frequency is N1/ℓ, generated or applied voltage is Fg31/w, and the displacement is d31ℓV/t. For (b), the frequency is 1.06N1/w, the voltage is Fg31/ℓ, and the displacement is Vd31w/t. The capacitance is the same for either case, that is, C = KT3ℓεw/t. The parallel width mode is achieved by poling the plate in a direction normal to the long edges (Figure 4.4). The 3(z) direction is, therefore, normal to the plane of the edge. The width, w, is <0.3ℓ and >3t (the thickness). The frequency is N3/w, the capacitance is KT3 ε 0 ℓt/w, the generated or applied voltage is wFg33/ℓt, and the displacement is d33V0. Another mode of vibration of interest in a rectangular plate is the thickness shear mode (Figure 4.5). In Figure 4.5, t is less than both 0.2w and 0.2ℓ.

CRC_DK8307_CH004.indd 223

10/27/2008 10:44:29 AM

Ultrasonics: Data, Equations, and Their Practical Uses

224

t t

d d FIGURE 4.6 Round disk.

2 h

t 1

3 3

d0 (a)

2 (b)

FIGURE 4.7 Ring modes.

The frequency is N3/t, the capacitance is KT1ε 0 ℓw/t, the applied voltage or voltage generated is g15F/w, and the displacement is d15V. The flat, round plate (Figure 4.6) can produce either (a) a planar mode or (b) a thickness extensional mode. For the condition (a) the frequency is given by Nr/d0, the generated voltage is not applicable, and the displacement (d31d/t)V0. For (b) the frequency is Nt/t, the generated or applied voltage is (4t/πd2)g33F, and the displacement is d33V. The capacitance is KT3 ε 0/4t. Figure 4.7 illustrates the ring modes. In one case, the wall thickness is the 3(z) direction and the walls, inside and outside are electroded. In the second, the 3(z) direction is axial and the flats on the ends are electroded. The motion and the frequency are the same in each case, both referring to a radial vibration pattern. The outer diameter is greater than either 3h or 3t. The frequency is Nc/dm, where dm is the mean diameter. For (a), the capacitance is 2πKT1ε 0h(d0/d1), the generated or applied voltage is (d0/2)g31P, and the displacement is (d31dm/t)V. For (b), the capacitance is (π/4)[KT3 ε 0(d20 – d21)]/h, generated or applied voltage is g31h(d0/ℓ) P/2, and the displacement is (d31dm/h)V. Figure 4.8 shows that the thin wall cylinders can vibrate like the ring (a). However, they also exhibit a vibration in the axial direction (b). The capacitance is the same for both. For (a), the frequency is Nc/dm, the generated or applied voltage is (g31d0P)/2, and the displacement is (d31dm/t)V. For (b), the frequency is N1/ℓ, the generated or applied voltage is g31F/πdm, and the displacement is (d31ℓ/t)V.

CRC_DK8307_CH004.indd 224

10/27/2008 10:44:29 AM

Piezoelectric Materials: Properties and Design Data

225

t 1 1

h

2

3

t 2

ᐉ

3 d0 (a)

d0

(b)

FIGURE 4.8 Thin hollow cylinders.

FIGURE 4.9 Striped hollow cylinder.

2

1

FIGURE 4.10 Torsional quartz crystal.

The striped hollow cylinder, in which d0 is greater than 8t, frequency is N3c/dm, the capacitance is KT3 ℓε 0 t2r /πdm, the generated or applied voltage is (πd0/n) (g33/ln(d0/d1))P, and the displacement is nd33V, is shown in Figure 4.9. Figure 4.10 shows a torsional mode that can be excited in quartz. Four electrodes are applied to the sides of the crystal and connected interchangeably. The frequency is given by f = (1/2ℓ)(μ/ρ)1/2 = (1/2ℓ)(c66/ρ)1/2

CRC_DK8307_CH004.indd 225

10/27/2008 10:44:30 AM

Ultrasonics: Data, Equations, and Their Practical Uses

226

where μ is the shear elastic constant for an isotropic body and c66 is the corresponding shear elastic constant for shears around the z-axis in dynes per square centimeter. Density, ρ, is in gram per square centimeter. Uses for this configuration are minimal, but have been applied to measuring shear constants of solids and to measuring shear elasticities and shear viscosities of liquids. Figure 4.11 shows a hemisphere. The diameter, d0, is greater than 8t. The frequency is 1.24Nsp/dm, the capacitance is πKT3 ε 0dm2 /2t, the generated or applied voltage is (g31d0/2)P, and the displacement is (d31dm/t)V. A similar unit consists of a hollow sphere with a hole in it. This type is used primarily as a listening device. Its frequency is given by Nsp/dm, its capacitance is πKT3 ε 0dm2 /t, its generated voltage is (g31d0/2)P, and its displacement is (d31dm/t)V. Some additional designs include unimorphs (Figure 4.12), bimorphs (Figure 4.13), and multimorphs (Figure 4.14), which are used for a number of applications such as stereo phonograph cartridges, microphones, headphones, and dingers (attention-getters). The thin piezoelectric polymer films are used primarily for passive devices. Another very important class of ceramic elements is the type used for NDT and for medical diagnosis. These are mounted so that they will die down rapidly; that is, they are highly damped. This design allows for a sharp pulse that increases the sensitivity or improves accuracy of the unit. The more specific details of the designs of transducers will be given as their uses are discussed in later chapters.

2

Ceramic element

t 3

d0

1

Metal element FIGURE 4.12 Unimorph.

FIGURE 4.11 Hemisphere.

Ceramic element

Grounded outer surface

Ceramic element

Electroded inner surfaces

Metal element FIGURE 4.13 Bimorph.

CRC_DK8307_CH004.indd 226

FIGURE 4.14 Multimorph.

10/27/2008 10:44:30 AM

Piezoelectric Materials: Properties and Design Data 4.5.2 4.5.2.1

227

Transducer Assembly Sandwich Types for Power Applications

Figures 4.15 and 4.16 illustrate two designs of piezoelectric transducers for power applications. These transducers are assemblies of piezoelectric elements and metal elements that are bolted together and designed to resonate at predetermined frequencies. The pressure Piezoelectric plates

H.V.

Output horn

Ground

Coupling washer

Coupling bolts Backup horn

ξ1

ξ2

FIGURE 4.15 Two-element piezoelectric ultrasonic transducer, externally clamped. H.V. Backup horn

Piezoelectric elements Output horn

Coupling washer

Clamping bolt

ξ1

ξ2

FIGURE 4.16 External-head clamping bolt.

CRC_DK8307_CH004.indd 227

10/27/2008 10:44:31 AM

Ultrasonics: Data, Equations, and Their Practical Uses

228

applied by the bolts is also a predetermined value. The clamping, or bias, pressure is that about which the stress varies during operation. This bias pressure (a) holds the transducer assembly together, (b) provides optimum coupling between elements, (c) prevents negative pressure from occurring across the sandwich structure during operation, and (d) leads to maximum operating efficiency. These practical transducer designs for power applications illustrate the fact that d, g, and k are applicable only at frequencies well below resonance. The transducer of Figure 4.15 consists of a soft brass electrode between two piezoelectric plates sandwiched between flanged metallic elements. The piezoelectric elements are poled in opposite directions. The assembly is clamped together by bolts spaced evenly about the flanges on the metallic elements. The electrode is located at a velocity node where a lead may be attached with minimum potential for fatigue failure in operation. 4.5.2.1.1 Stress Developed in a Bolt Due to Applied Torque A calibrated torque wrench is a means of determining initial bolt load and, therefore, the force that the individual bolt contributes to the overall bias force across the piezoelectric plates. However, variations in coefficient of friction between the nut and the bolt (if a nut is used), between the threaded elements (i.e., tapped threads and mating bolt threads), and the bolt head or nut and the abutting surface will cause errors in determining bolt load calculated on the basis of torque applied. The bolt tension or clamping load, P, due to torque, T, on a bolt or nut on a bolt is given by P = T/KD

(4.9)

where K is a torque coefficient and D the nominal diameter of the bolt in meters. In Equation 4.9, P is kilograms and T kilogram meters. An average value for K with dry surfaces and unlubricated bolts is ~0.2. Lubrication reduces this value. A more nearly correct value of K is obtained by the equation K

 BR B R T   Tsec B
tan C 
  D D  1   Tsec B tan C 

(4.10)

where μB is the coefficient of friction at the bearing face of the nut or bolt, μT the coefficient of friction at the thread contact surfaces, RB the effective radius of action of frictional forces on the bearing face in inches, RT the effective radius of action of frictional forces on the thread surfaces in inches, B the thread half-angle, and C the helix angle of the thread. Measured average K with American standard hexagon nuts on bolts in sizes from 1/2–20 to 1–14 range from 0.243 to 0.161 for high-point torque and 0.267 to 0.167 for mid-point torque. Higher values correspond to the small thread sizes and lower values correspond to the larger thread sizes. Transducers like those of Figures 4.15 and 4.16 are assembled initially with all elements in position according to the design. As the bolts are tightened, the pressure across the piezoelectric elements produces a corresponding voltage. The open-circuit voltage is a good indicator of stress level across the elements. This voltage buildup is measured starting at the lowest stress meter or oscilloscope display. The voltage corresponding to this stress level is predetermined according to the equation V

CRC_DK8307_CH004.indd 228

g 33F3T r 2

(4.11)

10/27/2008 10:44:31 AM

Piezoelectric Materials: Properties and Design Data

229

where V is volts; g33 the stress constant for the piezoelectric material in the z direction, that is, across the thickness of the piezoelectric element in (volts per meter)/(newtons per square meter); F3 the force in the 3 direction (thickness) in newtons; T the thickness of the plate in meters; r the radius of the plate in meters. To determine the correct value of voltage corresponding to the designed bias stress for transducers represented by Figures 4.15 and 4.16, note that the two piezoelectric elements are electrically in parallel. Therefore, assuming that the elements are identical, the value of V is determined accurately by using dimensions T and r for one piezoelectric element only. Example 4.1 Assume that a transducer is to include two solid piezoelectric plates for which g33 = 24.5 × 10−3 V m/N. Each of the plates is 0.1 in. thick and 1.0 in. in diameter. The design bias pressure is 10,000 lb/in.2. Therefore, in Equation 4.11, the quantity F3/πr2 may be replaced by the SI units equivalent to 10,000 lb/in.2, that is, 6.895 × 107 N/m2 and the thickness, T, by the equivalent 2.54 × 10−3 m. Then

V = (24.5 × 10−3 V m/N)(6.895 × 107 N/m2) × (2.54 × 10−3 m) = 4290 V 4.5.2.2

Transducer Structural Elements

The basic components of the transducers illustrated in Figures 4.15 and 4.16 are (a) a quarter-wave section consisting of one of the piezoelectric plates, sometimes a soft copper coupling element, and a metal extension to the left of the high-potential electrode completing the quarter-wavelength at the design resonance frequency and (b) a section to the right of the high-potential electrode designed to produce a complete half-wave, oneand-a-half wavelength, or additional lengths to produce a transducer resonant at the design frequency. The elements to the right of the high-potential electrode include the second piezoelectric plate, sometimes a soft copper coupling element, and the solid metal extension that completes the transducer to resonate with whatever number of chosen half-wavelengths at a design frequency. The designer is free to select the geometry of the resonant elements as long as they meet the requirements for good matching. Refer to Chapter 2 for geometrical design guidelines and to Chapter 7 for properties of metals. Some practical guidelines in construction and assembly: 1. The metals are selected to meet the stress requirements of the resonant system. 2. The metal elements are selected to be compatible with the environment of the application. 3. Machined parts are highly polished and free of voids and scratches, fillets with tiny radii, or of any other high-stress-concentration conditions that can cause early fatigue failure. 4. All elements contributing to the resonance of the transducer as a unit must be matched acoustically for maximum efficiency in operation. 5. Mating surfaces should be as flat and smooth as possible to obtain optimum coupling between elements. Because the properties of the materials will vary between cold and equilibrium in a typical power application of a piezoelectric transducer, one can also expect a corresponding

CRC_DK8307_CH004.indd 229

10/27/2008 10:44:32 AM

Ultrasonics: Data, Equations, and Their Practical Uses

230

change in acoustic properties of its structural elements. Therefore, precise calculations for matching mating components may not be practical; however, experience has shown that care in this area of matching is still important to producing transducers of highest efficiency. 4.5.2.2.1 Matching Elements of a Transducer When several elastic elements combine to form a longitudinally resonant system, each element supplies a share of the overall resonance dimension in wavelengths or in fractions of wavelengths determined by its dimensions, density, and elastic properties. The transducer of Figure 4.15 uses a backup element, which consists of a short cylindrical section and a flange for bolts. This element combined with the piezoelectric plate and the coupling film (soft copper), if one is used, supply one-quarter of a wavelength at the resonance frequency of the transducer. The part contributed by the piezoelectric element is determined using the velocity of sound in the element. For this transducer, the velocity of sound to be used is twice the value of the frequency constant, N3t. Let f be the design frequency and fc the natural frequency of the piezoelectric element, that is, the fundamental frequency at which the element would resonate in the thickness mode when it is freely suspended. The frequency, fc = N3t/T, where T is the thickness of the piezoelectric element. The corresponding velocity, cc, is cc = 2Tfc = 2N3t A quarter-wavelength of the piezoelectric material at the design frequency, f, would be Lc = cc/4f or cc = 4fLc The percentage of λ/4 at the design frequency, f, occupied by one piezoelectric element is (T/Lc)100%, where T is the thickness of the piezoelectric element and Lc would be the thickness of the element if it were λ/4 thick. The axial length of the metal element, assuming the coupling layer has negligible effect on the performance of the transducer, is (1 − T/Lc)λ/4 based upon the velocity of sound in the metal, where cm = λ mf and λ m/4 = cm/4f. Example 4.2 Assume that a sandwich type of piezoelectric transducer is to be designed according to Figure 4.16 to operate at a frequency of 50 kHz. It is designed to use two piezoelectric elements, each 0.75 in. in diameter and 0.1 in. thick, and two horn sections, one for the backup member and one for the delivery section, that are 0.75 in. in diameter, ignoring the flanges. The frequency constant of the piezoelectric elements is 2000 cycle m/s and the horn materials are titanium 6Al4V, in which the velocity of sound is 5.08 × 103 m/s. A piezoelectric element, λ/4 in length at 50 kHz is Lc = (2000 cycle m/s)/(4 × 50 × 103/s) = 0.01 m = 1 cm = 0.3937 in. The thickness of the piezoelectric element is 0.10 in. and, therefore, only 25.4% of a quarter of a wavelength at the design frequency.

CRC_DK8307_CH004.indd 230

10/27/2008 10:44:32 AM

Piezoelectric Materials: Properties and Design Data

231

The titanium section must make up the remaining 74.6% of the λ/4. At 5.08 × 103 m/s and 50 kHz, a quarter of a wavelength is 2.54 cm, or 1.0 in. Therefore, the length of the metallic section is 0.746 in. to complete the quarter-wavelength backup section. The combined length of piezoelectric element plus horn section making up the backup λ/4 is 0.846 in. The delivery metal horn section is λ/2 + 0.746 in. = 2.746 in. Adding the second piezoelectric element, the entire length from electrode to active end is 2.846 in. and the total length of the transducer is λT = 2.846 in. + 0.846 in. = 3.692 in.

4.5.2.3

Soldering to Electrodes

Leads may be soldered to the electrodes by using a low-temperature solder, such as 62/36/2 (tin/lead/silver solder). The surface to be soldered must be prepared prior to attempting the solder joint. This can be done conveniently by using a soft lead pencil eraser to remove any silver oxides. The pretinned lead is dipped into a mild noncorrosive flux. A small quantity of solder is melted onto a soldering iron from 30 W to 50 W. The lead is positioned onto the surface to be soldered and the soldering iron is brought under pressure into a normal position to melt the solder. The shorter the duration of the applied heat the better—certainly, it must be <5 s to prevent excessive alloying of the silver electrode into the solder. 4.5.3

General Technical Considerations in Piezoelectric Transducer Design

To present design details of piezoelectric transducers for all potential applications is beyond the purpose of this present chapter. The designs of Figures 4.15 and 4.16 are illustrative only. The present objective is to provide guidelines, rules, and data for use in designing piezoelectric transducers for the majority of practical applications of ultrasonic energy and other applications for which piezoelectric transducers are useful, both present and future. In a piezoelectric transducer, the piezoelectric element is the heart of either a source or receiver of mechanical stress such as continuous waves, impulses, or any conceivable stress type or drive to which it may be subjected. Its total function is dependent upon a combination of the physical and electrical characteristics of the piezoelectric element, the physical and mechanical characteristics of the assembly of which it is a part, and the mechanical feedback characteristics of the load to which the assembly is subjected. Therefore, there are a wide range of current and potential design modes ranging from completely embedding elements in a potting material to mounting so as to allow complete freedom of vibration to be controlled only by the load (such as atmosphere in which it is located), distribution of the electrode material, and the manner of mounting (such as clamping around or along the outer periphery of a plate or resting a plate on edge supports without clamping or bonding). Electrical and mechanical properties of piezoelectric materials are presented in Tables 4.1 through 4.4. 4.5.3.1

Selection of Piezoelectric Elements According to Properties and Vibrational Modes

The design of a piezoelectric transducer must begin with determining the manner in which the active element responds either to the voltage or the stress to which it is subjected. A wide selection of properties may be obtained from commercially available

CRC_DK8307_CH004.indd 231

10/27/2008 10:44:32 AM

Ultrasonics: Data, Equations, and Their Practical Uses

232

piezoelectric materials. Morgan Matroc, a major manufacturer of piezoelectric ceramic materials, describes the ceramic materials as being arranged in three groups: hard materials, soft materials, and custom materials [14]. Hard materials are those that are used for power applications and that can withstand high levels of electrical and mechanical stress. Soft materials, featuring high sensitivity and permittivity, are “used in various pick-ups and sensors, low-power motor-type transducers, receivers, and low-power generators.” High voltages or driving conditions cause these elements to heat beyond their operating temperature. Custom materials are ceramic materials formulated to meet particular needs of specific applications. In addition to the wide range of piezoelectric sensitivities and mechanical properties, piezoelectric elements can be made or cut to vibrate in any variety of modes. The basic modes are thickness (i.e., expansion and contraction) and shear. These modes are controlled by the poling direction and the location of the electrodes. 4.5.4

Electrode Types

Silver electrodes are applied to most piezoelectric materials. The silver is in layers 0.0001–0.001 in. thick. Unless otherwise specified by the purchaser, coverage is from edge to edge, that is, full surface coverage. The adhesive strength is ~3500 psi. Purchasers may also have the elements plated with margins, wraparounds, special patterns, or stripes. Nickel electrodes are also used. They are much thinner than the silver electrodes and are almost always used on the faces of shear elements. Nickel electrodes are applied to faces perpendicular to the 1 axis of a plate that has been prepared. This step prevents depolarization that would otherwise occur with high-temperature electrode application.

4.6

Aging

After polarization, most of the properties of piezoelectric ceramics begin a gradual change over time. These changes are very nearly logarithmic with time, so it is convenient to express aging as a percentage of change per time decade. The longer the period after polarization, or other event such as exposure to high temperatures or high stress, the more stable the material becomes. Some transducer manufacturers intentionally subject an assembled transducer to heat for a brief period of time to approach a more uniform condition in the piezoelectric elements.

4.7

Uses for Piezoelectric Transducers

The list of uses for piezoelectric transducers is very extensive. The list is organized by category for simplification: • High-voltage generators for ignition purposes include gas appliances, cigarette lighters, fuses for explosives, flash bulbs, small gasoline motors, and so on. • Ultrasonic cleaning both for industrial and domestic appliances, sonar (both low-power and high-power), echo or depth sounding, underwater telephone, highstatic-motion transducers, high-dynamic-motion transducers, ultrasonic welding

CRC_DK8307_CH004.indd 232

10/27/2008 10:44:32 AM

Piezoelectric Materials: Properties and Design Data

• •

• • • • •

233

of plastics and metals, ultrasonic drilling and machining of brittle materials, ultrasonic soldering, atomization of liquids, pulverization, and so on. Microphones, telephones, intruder alarm systems, remote controls, loudspeakers such as tweeters, audio tone generators in signaling devices, and so on. Pickups and receiver transducers, hydrophones for use in shallow water, hydrophones for use in deep water, vibration pickups, record players, accelerometers, detection systems in machinery such as those used with textiles, medical equipment for low-intensity use (such as diagnosis), for medium-intensity use (such as for deep heat), and for high-intensity use (such as surgery and forming of lesions), motor cars such as knock sensors, musical instruments, and so on. Resonators and filters for radio, television, remote control, and so on. Delay lines for color television, electric computers, and so on. Teleprinters, desk calculators and electronic computers, slot machines, telephones, and so on. Ink jet printers, fine movement control, flow meters, high temperature transformers, flow meters, small motors, analogue memories, and so on. NDT, flaw detection, thickness gauging, and so on.

References 1. Piezoelectric Technology Data for Designers, Clevite, Piezoelectric Division, Cleveland, Ohio, 1965, p. 11. 2. M.A. Marcus, Ferroelectricity at Pennsylvania State University, Pennsylvania State University, August 17–21, 1981. 3. T.F. Hueter and R.H. Bolt, Sonics, John Wiley & Sons, Inc., New York, 1955, p. 91. 4. H. Jaffe and D.A. Berlincourt, Proc. IEEE, 1965, pp. 1372–1386. 5. W.P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, D. Van Nostrand, New York, 1950, pp. 117–118. 6. R. Staut, APC International Ltd., personal communication, May 19, 1997. 7. J.R. Frederick, Ultrasonic Engineering, John Wiley & Sons, New York, 1965, p. 66. 8. Z.J. Shi, Ultrasonics, 1980, pp. 57–60. 9. L. Bergmann, Der Ultraschall, S. Hirzel, Verlag, Stuttgart, 1954, p. 97. 10. D.E. De Rossi, P.M. Galleti, P. Daris, and D. Richardson, The electromechanical connection: piezoelectric polymers in artificial organs, ASAIO J., 1983, 6(l), 1–11, 1983. 11. Aeronautical Analytical Rework Program, Final Report, Piezoelectric polymer transducers for detection of structural defects in aircraft. Contract NC2269-77-C-3186 for Analytical Rework/Service Life Project Office, Naval Air Development Center, Warminster, PA18974, July 22, 1977. 12. F.N. Murrayayama, R. Nakamura, H. Abara, and M. Segawa, The strong piezoelctricity in polyvinylidene fluoride (PVDF), Ultrasonics, 15–23, 1976. 13. Pennwalt Data Sheet, Pennwalt, 900 First Avenue P.O. Box C, King of Prussia, PA, 194–0018. 14. Guide to Modern Piezoelectric Ceramics, Morgan Matroc, Inc., Electro Ceramics Division, Bedford, OH 44146.

CRC_DK8307_CH004.indd 233

10/27/2008 10:44:32 AM

CRC_DK8307_CH004.indd 234

10/27/2008 10:44:32 AM

5 Magnetostriction: Materials and Transducers Dale Ensminger

CONTENTS 5.1 Introduction ........................................................................................................................ 236 5.2 Magnetic Properties of Magnetostrictive Transducers................................................. 237 5.3 Definitions ........................................................................................................................... 238 5.4 Magnetostrictive Properties of Materials ....................................................................... 241 5.5 Typical Magnetostriction Strain Curves ......................................................................... 241 5.6 Fundamental Steady-State Magnetic Relationships ...................................................... 241 5.6.1 Definition of Terms ................................................................................................ 244 5.6.2 Equivalent Circuits ................................................................................................ 245 5.6.3 Impedance of Magnetostrictive Transducers..................................................... 246 5.6.3.1 Motional Impedance ............................................................................... 247 5.6.3.2 The Impedance Circle Diagram ............................................................ 248 5.6.3.3 Mechanical Impedance .......................................................................... 248 5.6.3.4 Electrical Impedance and Efficiency .................................................... 249 5.6.4 Eddy Currents......................................................................................................... 251 5.6.5 Hysteresis Losses ................................................................................................... 251 5.7 Types of Magnetostrictive Transducers .......................................................................... 253 5.7.1 The Toroid ................................................................................................................ 253 5.7.1.1 The Mechanical Impedance of the Toroid (Ring) ................................ 253 5.7.1.2 The Electrical Impedance of the Toroid (Ring) .................................... 253 5.7.1.3 The Mutual Impedance of the Toroid ...................................................254 5.7.1.4 The Load Impedance on the Toroid ...................................................... 255 5.7.1.5 The Equivalent Circuit of the Toroid Transducer ................................ 255 5.7.1.6 The Efficiency of the Toroid Radiating into a Medium ...................... 256 5.7.1.7 Maximum Power Transfer to the Magnetostrictive Transducer Radiating into a Medium................................................... 258 5.7.2 The Magnetostrictive Bar ...................................................................................... 258 5.7.2.1 Typical Bar Designs and Corresponding Coil Positions .................... 260 5.7.2.2 Equivalent Circuit of a Magnetostrictive Bar (Rod) ........................... 261 5.7.2.3 Half-Wave Rod Free at One End and Loaded at the Other End with Mass and Resistance ................................................... 265 5.8 Design and Construction of Magnetostrictive Transducers ........................................ 266 5.8.1 Simple Uniform Stack ............................................................................................ 267 5.8.2 Closed-Loop, Uniform, Rectangular Bar Stack.................................................. 267 5.8.3 Multiple-Bar Type of Transducer Driving a Head ............................................ 268 5.8.4 Transducer with Gap for Permanent Magnet Bias ............................................ 269

235

CRC_DK8307_CH005.indd 235

10/27/2008 3:30:38 PM

Ultrasonics: Data, Equations, and Their Practical Uses

236 5.9

Combining Elements to a Transducer ............................................................................. 270 5.9.1 Silver Soldering and Brazing ................................................................................ 270 5.10 Practical Uses for Magnetostrictive Transducers......................................................... 271 5.11 The Latest Commercially Available Magnetostrictrive Material .............................. 271 References .................................................................................................................................... 272

5.1

Introduction

The history of magnetostriction studies goes back to the mid-1800s. In 1847, Joule [1] published results of experiments on the effects of magnetism upon the dimensions of iron and steel bars. By placing an iron or steel bar, fixed at one end, inside a magnifying solenoid and using a system of levers and a micrometer microscope, he was able to measure the change in length under constant stress. He measured the magnetic field by balancing the pull of an induced magnetic pole on one end of a suspended magnet by weights applied to the opposite end. Two significant conclusions from these experiments were (1) that the elongation in a given bar is proportional to the square of the magnetic intensity and (2) that the elongation is greater in an annealed than in an unannealed bar [2]. Later work by Villari [3], published in 1868, stated a reverse effect, varying the stress in a magnetized body will produce changes in the induced magnetization or a change in the susceptibility [2,3]. Later work has confirmed the studies of Joule and Villari. The later studies show that the relationships involved are complicated and differ widely from material to material and with heat treatment and history for the same material [2]. Some interest was shown in magnetostriction in the years after Joule and Villari. For example, Langevin, who is noted for using quartz crystals to produce high-intensity ultrasonic vibrations in attempts to detect German submarines during World War I, showed that he also knew about magnetostriction in his patent issued in 1920 [4]. Other scientists, such as Pierce, also showed interest in this area. Pierce used magnetostrictive transducers to control the frequency of audio and radio electric oscillations, for production of sound, and for the measurement of the elastic constants of metals [5,6]. Interest in magnetostriction increased during World War II in an effort to develop improved underwater sound devices. Some commercial spin-offs from this research included ultrasonic drills and cleaners. Ultrasonic drilling offered the ability to machine any shape of cross section to which a drill could be formed. It was, however, slow and messy, using a water slurry of boron carbide, aluminum oxide, carborundum, or other hard abrasive to erode the surface of the workpiece. The Cavitron Company was a leader in manufacturing ultrasonic drills both for industry and for dentistry in the late 1940s and throughout the 1950s. A significant breakthrough in the 1960s was the discovery of giant magnetostriction in various rare-earth alloyed materials. For example, YbFe2 was found to exhibit a saturation magnetostriction at room temperature of ~1800 ppm, which is ~55 times that of A-nickel, the material generally used in commercially available ultrasonic products of that day. Material characteristics are greatly affected by the ternary compositions which appear to be most useful: terbium–holmium–iron and terbium–dysprosium–iron [7,8]. There has been a very strong shift from the use of magnetostrictive to piezoelectric transducers in commercial products in recent years. The reasons are that, for the majority of uses, piezoelectric materials have been more efficient, less expensive,

CRC_DK8307_CH005.indd 236

10/27/2008 3:30:40 PM

Magnetostriction: Materials and Transducers

237

more easily fabricated, and require slightly less complicated electronic circuitry for driving the transducers. However, some applications for ultrasonic energy may continue to favor magnetostrictive devices. Magnetostriction offers better ruggedness, longer life, and can be water cooled to maintain high-power throughput over a period of time. Excluding the price, some of the magnetostrictive materials developed since 1971 have very competitive efficiencies. Etrema Products, Inc., is a company capita lizing on the superior performance of rare-earth materials by designing and manufacturing high-power ultrasonic transducers based upon Terfenol-D. These are used in continuous ultrasonic welding, joining, and sonochemical applications [9–11]. This chapter presents helpful data, design details, and suggested uses for magnetostrictive transducers.

5.2

Magnetic Properties of Magnetostrictive Transducers [12–14]

To properly design a magnetostrictive transducer, one must understand the basics both of the magnetic forces involved and of acoustics. As mentioned previously, magnetostriction occurs in magnetic materials. Magnetic lines are set up due to a magnetic force called the gilbert. The number of gilberts in a coil carrying a current of electricity is equal to 0.4π, or 1.26, times the product of the amperes flowing in the coil by the number of turns of wire in the coil, that is, gilberts = 0.4πNI. Setting up of magnetic lines in a material is opposed by the reluctance of the material. The standard is air, which varies little from other nonmagnetic materials, in which the reluctance is 1 for 1 cm3 of air. Therefore, the reluctance of an air gap in a magnetic circuit is ℜ = 1.0 × ℓ/S, where ℓ is the length and S the cross-sectional area of the gap. In the case of a long, uniform coil, the reluctance of the air inside the coil is ℜ = ℓ/S, where S is the inside cross-sectional area of the coil in square centimeters and ℓ the length of the coil in centimeters. The total flux lines are

T 

Magnetomotive force  0.4 NIS/
Total reluctance

where N is the number of turns on the coil and I the current in amperes. This is only the flux through a center plane normal to the axis of the coil of a long solenoid. Of this total flux, one-half leaves or enters the coil at the ends and the other half leaves through the sides along the length. It is more convenient in magnetic computations to use the reciprocal of reluctance and reluctivity, that is, permeability and permeance, where Permeability  1/reluctivity Permeance  1/reluctance The permeability of a material is the number of magnetic lines set up in 1 cm3 when the magnetomotive force (mmf) between opposite faces of the cube is 1 Gb, or 1 A turn divided by 0.4π. The permeability of magnetostrictive materials is >1, it is not constant for any magnetic material as shown by their magnetization curves.

CRC_DK8307_CH005.indd 237

10/27/2008 3:30:40 PM

Ultrasonics: Data, Equations, and Their Practical Uses

238

5.3

Definitions [20,22]

Terms used to describe the properties of magnetostrictive materials and associated circuitry are defined in the following paragraphs. Abampere. The unit of electric current in the electromagnetic centimeter-gram-second system, 1 abA equals 10 A in the absolute meter-kilogram-second system. Abvolt. The unit of electromotive force in the electromagnetic centimeter-gram-second system, 1 abV equals 10−8 V in the absolute meter-kilogram-second system. Ampere. The amount of electrical current that flows through a 1 Ω conductor subjected to 1 V across it. Ampere’s law. A conductor carrying a current and located in a magnetic field at right angles to the flux will be pushed by a force that is proportional to the flux density B to the current I and to the length of the wire ℓ. Antiferromagnetism. Giving no net total moment in zero applied magnetic field. Capacitance (C). Farad, a capacitor (condenser) consists of two conducting surfaces spaced a uniform distance apart. A voltage applied between these surfaces generates equal and opposite charges on them. The capacitance of the capacitor (condenser) is the ratio of charge on one of these surfaces to the potential difference between the two surfaces. Conductivity. The ratio of the electric current density to the electric field in a material. Coulomb. The unit of electric charge that crosses a surface in 1 s when a steady current of one absolute ampere is flowing across the surface. Current (I). Amperes, net transfer of electrical charge per unit of time. Dysprosium (Dy). A silver-white chemical element of the rare-earth metals. It is one of the most magnetic and most magnetostrictrive of all known substances. Electric charge (q). An accumulation of electrical particles (electrons and protons), more electrons than normal produce a negative charge, fewer electrons than normal produce a positive charge. Electric potential (E). Volt, the electrical force or pressure that tends to cause a current to flow in a circuit. Electromotive force (E). See electric potential. Erbium (Er). A trivalent metallic rare-earth element of the yttrium subgroup, atomic number 68 and atomic weight 167.26. Flux penetration (δ). The depth to which a significant value of magnetic flux is generated beneath the outer surface of a magnetic material under a given set of driving conditions. Gadolinium (Gd). A rare-earth element, atomic number 64 and atomic weight 157.25; highly magnetic, especially at low temperatures. Gilbert (F). Magnetic force, or mmf, in a coil of electrically conductive material equal to 0.4πNI, where N is the number of turns in the coil and I the current flowing in the coil. Hamiltonian function. A function of the generalized coordinates and moments of a system, equal in value to the sum over the coordinates of the product of the generalized momentum corresponding to the coordinate, and the coordinate’s time derivative, minus the Lagrangian function of the system. It is numerically equal to the total energy if the Lagrangian does not depend on time explicitly. The equations of motion of the system are determined by the functional dependence of the Hamiltonian on the generalized coordinates and moments.

CRC_DK8307_CH005.indd 238

10/27/2008 3:30:40 PM

Magnetostriction: Materials and Transducers

239

Hamilton–Jacobi equation. A particular partial differential equation useful in studying certain systems of ordinary equations arising in the calculus of variations, dynamics, and optics: H(q1, … , qn, ∂φ/∂Φq1, … , ∂Φ/qnt) + ∂Φ/∂r = 0, where q1, … , qn are generalized coordinates, t the time coordinate, H the Hamiltonian function, and Φ a function that generates a transformation by means of which the generalized coordinates and moments may be expressed in terms of new generalized coordinates and moments which are constants of motion. Hamilton–Jacobi theory. A theory that provides a means for discussing the motion of a dynamic system in terms of a single partial differential equation of the first order, the Hamilton–Jacobi equation. Hamilton’s equations of motion. A set of first-order, highly symmetrical equations describing the motion of a classical dynamic system; namely, qj = ∂H/∂pp , −pj = ∂H/∂qj; here qj( j = 1, 2, …) are generalized coordinates of the system, pj is the momentum conjugate to qj, and H is the Hamiltonian (also known as the canonical equations). Hamilton’s principle. A variational principle that states that the path of a conservative system in configuration space between two configurations is such that the integral of the Lagrangian function over time is a minimum or maximum relative to nearby paths between the same end points and taking the same time. Henry. The inductance of a circuit in which each ampere of current produces 108 flux linkages in that circuit. Inductance (L). Henrys; in a coil, the ratio of the electromotive force across the coil to the rate of change in the current through the coil; the property that permits a varying current in a coil to induce a voltage in the same or in a neighboring circuit (as in a transformer). Intensity of magnetization (H). Magnetic field strength, an auxiliary vector field whose curl, in the case of static charges and currents, equals the free current density vector. It is independent of the magnetic permeability of the material. Lagrangian. For a dynamical system of fields, a function that plays the same role as the Lagrangian function of a system of particles; its integral over a time interval is a maximum or a minimum with respect to infinitesimal variations of the fields, provided that the initial and final fields are held fixed. Lagrangian equations. Equations of motion of a mechanical system for which a classical (nonquantum-mechanical) description is suitable, and which relate the kinetic energy of the system to the generalized coordinates, the generalized forces, and the time (also known as the Lagrangian equations of motion). Laves phases. Alloy phases which have the general formula AB2, and the crystal structures of either MgCu2 (cubic) or MgZn2 or MgNi2 (both hexagonal). Magnetic field strength (H). Oersted, see intensity of magnetization. Magnetic flux. The total lines of flux passing through a given area, magnetic pressure divided by magnetic reluctance. Magnetic flux density (B). Gauss, the number of lines of magnetic force crossing a unit area. Magnetic induction. Flux per unit area which is perpendicular to the direction of flux. Magnetic pole strength (m). The magnitude of a magnetic pole, equal to the force exerted on the pole divided by the magnetic induction or by the magnetic field strength. Magnetostriction. A change in dimensions of a ferromagnetic material when it is subjected to an applied magnetic field.

CRC_DK8307_CH005.indd 239

10/27/2008 3:30:40 PM

240

Ultrasonics: Data, Equations, and Their Practical Uses

Metastable. In physics, a condition in which a system returns to equilibrium after small (but not large) displacements; it may be represented by a ball resting in a small depression on top of a hill. In physical chemistry, a state of pseudoequilibrium having a higher free energy than the true equilibrium. Mossbauer effect. The emission and absorption of gamma rays by certain nuclei, bound in crystals, without loss of energy through nuclear recoil, with the result that radiation emitted by one such nucleus can be absorbed by another. Neel temperature. A temperature characteristic of certain metal alloys and salts, below which spontaneous nonparallel magnetic ordering takes place so that they become antiferromagnetic, and above which they are paramagnetic. Oersted. Unit of magnetic field strength in the centimeter-gram-second electromagnetic system of units, equal to the field strength at the center of a plane circular coil of one turn and 1 cm radius, when there is a current of 1/2π abA in the coil. Paramagnetic. A material state, such as that of aluminum, having a magnetic permeability slightly greater than unity and varying to only a small extent with the magnetizing force. Permeability (µ). The ratio of flux density B to the field intensity H, equals B/H. Permendur. A magnetic alloy composed of equal parts of iron and cobalt and has an extremely high permeability when saturated. Planck’s constant. A fundamental physical constant, the elementary quantum of action; the ratio of the energy of a photon to its frequency, it is equal to 6.626075 ± 0.000004 × 10−34 J s; symbolized h (also known as quantum of action). Quantum. In physics, one of the discrete quantities of energy or momentum of an atomic system that are characteristic of the quantum theory. Quantum theory. A theory in physics that physical processes—especially changes of energy in molecules and atoms—are discontinuous, involving discrete quantities of energy called quanta. Reluctance (ℜ). The magnetic property that always opposes the setting up of a magnetic flux in a circuit. Resistance (R). Ohms; the opposing force in a conductor to the flow of electrical directcurrent voltage; voltage/current; in alternating current (AC) flow, it is the real part of the complex impedance. Samarium (Sm). A rare-earth metal, atomic number 62. Schrödinger equation. A partial differential equation governing the Schrödinger wave function Ψ of a system of one or more nonrelativistic particles; h(∂Ψ/∂t) = HΨ, where H is a linear operator, the Hamiltonian, that depends on the dynamics of the system, and h is Planck’s constant divided by 2π. Susceptibility. Ratio of electric or magnetic polarization in a material to the strength of the field producing that polarization. Terbium (Tb). A silver-gray, soft, ductile chemical element of the rare-earth metals, found in gadolinite and other minerals; in the Yttrium subgroup of the transition elements. Terfenol-D. An alloy of terbium, dysprosium, and iron, commercially available for magnetostriction transducers; with very high magnetostriction properties [Tb0.3Dy0.7Fe1.9–2]. Villari effect. A change in magnetic induction within a ferromagnetic substance in a magnetic field when the substance is subjected to mechanical stress.

CRC_DK8307_CH005.indd 240

10/27/2008 3:30:40 PM

Magnetostriction: Materials and Transducers

241

Ytterbium (Yb). A rare-earth metal of the yttrium subgroup; atomic number 70 and atomic weight 173.04; lustrous, malleable, soluble in dilute acids and liquid ammonia, reacts slowly with water; melts at 824°C; boils at 1427°C; used in chemical research, lasers, garnet doping, and x-ray tubes. Yttrium (Y). A rare-earth metal; atomic number 39 and atomic weight 88.905; darkgray, flammable (as powder), soluble in dilute acids and potassium hydroxide solution, and decomposes in water; melts at 1500°C; boils at 2927°C; used in alloys and nuclear technology and as a metal deoxidizer.

5.4

Magnetostrictive Properties of Materials

The earliest alloys used for magnetostrictive transducers were nickel, nickel and iron alloys, and permendur (an iron 49%/cobalt 49%/vanadium 2% alloy). The magnetostriction characteristics of these and other materials with much higher values of magnetostriction discovered since 1970 are given in Table 5.1. Table 5.1 shows the effects of alloying as well as a value for rating the alloys according to ∆ℓ/ℓ and to temperature. It does not rate according to abundance or cost. This table may be used to select a material for a given design, but the curve of magnetostrictive strain versus magnetization must be used for the actual design [15–18].

5.5

Typical Magnetostriction Strain Curves

According to Table 5.1, the ∆ℓ/ℓ of dysprosium far exceeds that of any other material. Dysprosium is also one of the most magnetic of all known substances. The curves of Figure 5.1 illustrate the difference between the rare-earth types and common permendur and A-nickel materials. These curves should not be taken as exact, but might be considered average. The actual curves follow different paths between increasing and decreasing field strengths due to hysteresis. The bias and the driving fields together determine the efficiency of a magnetostrictive transducer. A very high bias field and a high driving field will produce large hysteresis losses. A low bias field working in the straighter portion of the curve will produce lower hysteresis losses, provided that the driving field does not cause the total field to cross the zero line. (Magnetostriction of A-nickel is negative, whereas that of permendur and Terfenhol-D is positive [19].)

5.6

Fundamental Steady-State Magnetic Relationships

A magnetostrictive material is a magnetic material. Designing an efficient magnetostrictive transducer requires an understanding of the principles of magnetism that correspond with its geometrical configuration. It requires considering the influence of the use

CRC_DK8307_CH005.indd 241

10/27/2008 3:30:40 PM

Ultrasonics: Data, Equations, and Their Practical Uses

242 TABLE 5.1 Magnetostriction in Various Materials Materials Annealed cobalt Cast cobalt Cobalt CeAl2 CeAl2 CeAl2

Magnetostriction (∆ℓ/ℓ × 106)

Temperature (°C)

−8.4 +11.25 −62 −40 +10 +25 (cycled)

20 20 −271.2 −269 −271.2

Field Strength (H, Oersteds) 800 800 9000 9000 9000

(During cycling change is only 15 × 10−6 in./in.) Dysprosium DyFe2 DyFe2 ErFe2 ErFe2 Gadolinium Gadolinium Gadolinium HoFe2 Iron Iron 60%/nickel Nickel Nickel 62.25%/ palladium 37.5% Palladium 37.75% Permendur (Fe 49%/Co 49%/V 2%) Permendur Permendur SmFe2 SmFe2 Tb0.5Dy0.2Fe2 Tb0.3Dy0.7Fe2 TbFe2 TbFe2 TbFe2 TbFe2 (sputtered) TbFe3 TbFe17 85% Tb–15% Fe 70% Tb–30% Fe TbFe1.6Co0.4 TbFe1.6Ni0.4 Tb2Ni17 Terfenol-D (Tb0.3Dy0.7Fe1.9–2) Thallium YCo3 Yb2Co17 Yttrium/iron/garnet Yttrium/iron/garnet Yttrium/iron/garnet

CRC_DK8307_CH005.indd 242

+6000 +1260 +433 −300 −229 −180 (λ0 in single crystal) +120 (λc in single crystal) +10 (polycrystalline) +135 −5 −34 −40 +70 −63 +45

20 20 20 20 −269.2 −269.2 −269.2 20 20 0 20

800 200–400

20 200

+47.5 +50.5 −2100 −1560 +1330 +1060 +2460 +1800 (at saturation) +1753 +304 +693 +131 +539 +1590 +1467 +1151 −4 +1300

20 20 20 20 20 20 20 20 20 20 21–28

+4230 0.4 47 −8 −3.4 −2

20 20 20 −269.2 76.84 176.84

400 800 20 20 20

10/27/2008 3:30:40 PM

Magnetostriction: Materials and Transducers 700 −

Terfenol-D

243

Tb3Dy7Fe2

Tb3Ho7Fe2

Strain (ppm)

470 −

46 −

Permendur

0 − 30 −

Nickel 0

150

Magnetic bias field (kA/m) FIGURE 5.1 Typical curves of strain produced in various magnetostrictive materials by an applied magnetic field.

of the transducer, including the environment, on the total impedance reflected into the electronic source of energy as well as the reaction of the transducer to outside sources of mechanical energy. The magnetostrictive material responds to the magnitude of the electrical excitation. The magnetization curves are not straight lines so that the inductance varies with the magnitude of the driving force. For the transducer to respond at the driving frequency, a direct current (DC) or magnet may be applied as a bias, which provides a zero point on the magnetization curve about which the AC operates. In this manner, the bias controls (to a certain extent) the inductance of the magnetostriction device. Small amplitudes of AC about the bias point produce a hysteresis loop about the position where it is operating. The average slope and the area of the displaced hysteresis loop, regardless of size, as long as the total bias and driving current remain on the same side of the zero position, give a measure of the permeability and losses of the material. These determine the effective inductance and resistance of the inductor for the driving AC. The bias, or polarizing flux, in a magnetostrictive material reduces the value of Young’s modulus. This also reduces the Q of the transducer by an amount depending on the pretreatment of the material. For the magnetostrictive transducer, the inductance depends upon the metallurgical makeup of the material, the geometry of the material (including air gaps), the history of the material, and of the excitation. The common methods of applying bias current to a magnetostrictive transducer are to (1) use separate coils for each type of current (DC and AC), (2) apply both bias and AC through the same coil, and (3) apply a separate magnet into the magnetic circuitry. A magnetic field exists in the neighborhood of a permanent magnet. It also exists about a conductor carrying an electric current and, when the conductor is wrapped about a magnetic material, it produces an electromagnet. A coil of wire wound uniformly in a long helix is called a solenoid.

CRC_DK8307_CH005.indd 243

10/27/2008 3:30:41 PM

Ultrasonics: Data, Equations, and Their Practical Uses

244 5.6.1

Definition of Terms

B

B0 H S ε Λ L N S µ ‘ Φ F ℜ M A, b ρm D

χ E k K K0 ω ω0 f Q Qm δ t P P Z Zm + ZL Zm + ZL

Zmot Zmot

CRC_DK8307_CH005.indd 244

Total magnetic induction = µH (static case in incremental values) µχH (dynamic case in incremental values when eddy currents are present) Static polarizing induction Total applied magnetic field Longitudinal strain (H, B, s in same direction) Magnetostrictive coefficient 2εB0 is the normal magnetostrictive constant Inductance, henrys Number of turns in an electric coil or about a magnetic core Cross-sectional area Magnetic permeability Total length of core in centimeters The number of magnetic lines in a field The force in gilberts Reluctance of the magnetic path Mass Cross-sectional dimensions, centimeters, rectangular Density of the magnetostrictive core Z2em/(Rm + RL ) = diameter of the impedance circle obtained in the neighborhood of resonance (when Zem changes little in comparison with Zm + ZL so as to be considered constant) Eddy current factor Voltage across the input terminals of equivalent circuit for the magnetostrictive transducer 2 Electromechanical coupling coefficient = 4  /E

Stiffness = K0(1 − k2χ) for magnetostrictive materials 2πblY/a = stiffness of a nonmagnetostrictive core 2πf = the angular frequency 2πf0 = K/M = angular frequency at resonance Frequency Sharpness of resonance Sharpness of mechanical resonance, that is, Qm  MK/(Rm
RL ) Flux penetration depth Lamination thickness –ΛB + Es = the longitudinal stress in the absence of eddy currents –ΛµχH0 + (E − 4πΛ2µχ)s the longitudinal stress in the presence of eddy currents Impedance (Rm + RL) [1 + j2Qmp] where p = 1/2[(ω/ω0) – (ω/ω)] (Rm + RL) + j(ωM − K/ω) = the total mechanical impedance in the vicinity of mechanical resonance f0 where M is the equivalent lumped mass and K the equivalent lumped stiffness of the system, all constant over the interval for which the equation for (Zm + ZL) applies Motional impedance, that part of the electric impedance arising from the motion of its mechanical terminals Zi – Zc = Z2em/(Zm + ZL )

10/27/2008 3:30:41 PM

Magnetostriction: Materials and Transducers Zi

The impedance measured at the electric terminals when the transducer is free to move under mechanical load ZL The impedance measured at the electric terminals when the transducer is blocked so that no motion is possible at its mechanical terminals Electrical impedance Internal mechanical impedance (Zm = Rm + jXm = R″m + j(ωM – K/ω)) Electromechanical mutual impedances Zem = ±Zme according to the type of coupling, Zem = +Zme (electrostatic or piezoelectric coupling), Zem = −Zme (electromagnetic or magnetostrictive coupling) Internal mechanical resistance Power lost due to electric resistance of the core and associated eddy currents due to Villari effect Power lost by purely mechanical resistance Mechanical load resistance Load impedance (ZL = RL + jXL) Load reactance Internal mechanical reactance Young’s modulus without polarization, when the flux density is held constant Y(1 − k2χ), the value of Young’s modulus with polarization (Y − 4πΛ2µχ) in a constant applied field

Zc Ze Zm Zem, Zme

Rm R′mv2 R″mv2 RL ZL XL Xm Y Y′

5.6.2

245

Equivalent Circuits

All electromechanical vibratory systems may be represented by equivalent circuits. In forming an equivalent circuit, it is necessary to determine the materials that constitute it, where in the system these act, whether they are electrical or mechanical, and how the mechanical parts relate to the electrical system. Various circuit designs are possible and have been proposed for magnetostrictive transducers. However, it is possible to show all that is necessary with one master circuit that may be modified to meet various conditions. A complete equivalent circuit of a magnetostrictive transducer and load includes the electrical portion of the transducer, the mutual impedance, and the load impedances as shown in Figure 5.2. Figure 5.2 is a summary type of circuit. The various elements that make up each component shown in Figure 5.2 are illustrated in Figure 5.3. Figures 5.2 and 5.3 are four-terminal circuits in which the electrical elements are located on the left side of the transformer and the mechanical elements are located on the right side of the transformer. The transformer allows the transfer of the mechanical values to the equivalent electrical values in the circuit [14].

I

Ze

Zem

E

v

F

ZL

FIGURE 5.2 Four-terminal transducer network connected to load [14].

CRC_DK8307_CH005.indd 245

10/27/2008 3:30:41 PM

Ultrasonics: Data, Equations, and Their Practical Uses

246 I

Zi

Zc

1 to ϕ2

M

1/kc Rm″ I = v/ j

e ′ = F/j

E

ZL

ϕ = Λ/Nf FIGURE 5.3 Equivalent circuit of a magnetostrictive transducer.

Start by listing all factors entering into the operation of the transducer and of any load to develop a circuit diagram. Arrange these factors in an orderly manner according to their location in the diagram. Some items to be included are the following: Electrical Applied voltage Initial current Electrical impedances Eddy currents Hysteresis losses Coil resistances Coil inductances These are included to the left of the transformer between the electrical and the mechanical parts of the circuit: Mechanical Forces Velocities Internal friction External friction Mass Elasticity These are included to the right of the transformer: Electromechanical coupling This is represented by the transformer. Having a good circuit diagram helps in diagnosing a design and in determining its efficiency. In Figure 5.2, F is the force of the load on the radiating surface and v is its velocity at the position of the load connection. Figure 5.3 is an equivalent circuit that will be used throughout this chapter, with specific values altered to meet the need. The circuit of Figure 5.3 is good to f = fc. Deviations in resistance increase as frequency f increases above fc. The quantity, φ = jZem/Ze = Λ/Nf, is the transformation ratio of the transformer. 5.6.3

Impedance of Magnetostrictive Transducers

The transducer is meant to perform work. This means that it moves by stretching and contracting at a rate determined by the frequency of a driving current and in so doing it emits

CRC_DK8307_CH005.indd 246

10/27/2008 3:30:41 PM

Magnetostriction: Materials and Transducers

247

vibrational energy into another element or the surrounding medium such as air or water. Energy is expended in stretching and contracting the transducer material and energy is also expended through frictional losses between laminations and in the radiation of the waves from the surfaces of the radiation faces. Thus the total impedance of the transducer and load may become quite complex for many conditions of performance. The equivalent circuit of a magnetostrictive transducer includes both electrical and mechanical impedances. As mentioned previously, the electrical impedances include core impedances associated with hysteresis, eddy currents, coil resistances, and inductance. The equivalent mechanical impedances include the mass, elasticity, internal friction, and friction between laminations. These may be classed as blocked and motional impedances. The blocked impedance is what one would measure if the transducer would remain perfectly stationary with AC current passing through its coils. The motional (or dynamic) impedance is that which is added as the transducer is allowed to move under various conditions of loading—such as air or water. 5.6.3.1

Motional Impedance

The motional impedance of a transducer is that part of its electric impedance arising from the motion of its mechanical terminals. From the table of definitions, Zmot  Zi  Zc 

2 Zem (Zm
ZL )

(5.1)

where Zmot is the motional impedance of the transducer; Zi the impedance measured at the electric terminals when the transducer is free to move under mechanical load ZL; Zc the impedance measured when the transducer is blocked so that no motion is possible at its mechanical terminals; Zem the mutual impedance, that is, electrical to mechanical, of the transducer, [Zem = (E/v)I=0]; Zm the internal mechanical impedance; ZL the load impedance; E the voltage across the input terminals; v the velocity at the output terminals; and I the current at the input terminals. Letting the total mechanical impedance (internal plus load) be resonant at f0, the impedance will be Zm
ZL  (Rm
RL )
j(
M  K/
)

(5.2)

where Rm is the internal mechanical impedance, M the equivalent lumped mass, and K the equivalent lumped stiffness of the system. These quantities are constant over the interval to which Equation 5.2 applies. The mechanical Q is a measure of the sharpness of mechanical resonance or Q

MK (Rm
RL )

(5.3)

The motional impedance is 2  Zem    1 Zmot  Zi  Ze      Rm
RL   1
j 2Qp 

(5.4)

where p = 1/2((ω/ω 0) − (ω/ω)).

CRC_DK8307_CH005.indd 247

10/27/2008 3:30:41 PM

Ultrasonics: Data, Equations, and Their Practical Uses

248

Xc JXtot P

r

2ϕ D

ϕ Rc′

0′

C

ZL

Rtot Ri

FIGURE 5.4 The impedance circle diagram.

5.6.3.2

The Impedance Circle Diagram

When the impedance of the loaded transducer is plotted with increasing frequency, resistive components along the horizontal axis and the inductances along the vertical axis, the curve forms nearly a circle at frequencies surrounding the resonant point. The size of the circle depends upon the load, the diameter decreases with increasing load. In the discussion that follows, the diameter D will be ascribed certain subscripts referring to the type of load (air, water, or other). The quantities from this circle diagram may be used to determine the efficiency of the transducer (Figure 5.4). 5.6.3.3

Mechanical Impedance

The mechanical impedance consists of the effects of mass and stiffness on the driving force and resulting motion. This is K  Zm  Rm
jX m  Rm′′
j 
M   


(5.5)

where Zm is the mechanical impedance F/v of the transducer when the electrical terminals are open-circuited. For a physically realizable passive system, Ze  Re
jX e

Re  0

Zm  Rm
jX m

Rm  0

(5.6)

Zem  R em
jX em If the core were not magnetostrictive, the stiffness would be K0 

CRC_DK8307_CH005.indd 248

2b
Y a

(5.7)

10/27/2008 3:30:42 PM

Magnetostriction: Materials and Transducers

249

where Y is Young’s modulus of the material. However, Y depends on the permeability µ and, in dynamic problems, µ must be replaced by µχ to account for eddy currents ( χ is the eddy current factor). Even under static conditions, a magnetostrictive core has a Young’s modulus of Y or Y′ according to whether the induction B or the applied field H0 is held constant. Therefore, µ, in the dynamic problem, must be replaced by µχ. This results in the term K/jω in Equation 5.5 becoming complex when magnetostriction is present, because χ has an imaginary part. The resistive parts of the mechanical impedance are due to purely mechanical resistance to stresses and to the Villari effect, which is a change in inductance due to strain caused by the magnetostriction. A periodic magnetic field and the induction associated with the Villari effect produce eddy currents that dissipate power because of the electric resistivity of the core. The mechanical impedance, therefore, is in two parts: R″m is that due to purely mechanical resistance. An alternating force F produces an alternating velocity v, which causes a power loss R″mv2. R′m is that associated with the inverse Villari effect. The dissipation of power due to the Villari effect is R′mv2. R′m depends on the magnetostrictive constant Λ or the electromechanical coupling coefficient k. If there is no polarizing flux, R′m is zero. The total Rm is R′m + R″m. The results of including eddy currents in the dynamic mechanical impedance may be summarized in the following equations: K  K 0 (1  k 2) Zm  Zm′
Zm′′

(5.8)

The impedance of purely mechanical origin is K   Zm′′  Rm′′
jX m′′  Rm′′
j 
M  0  


(5.9)

The impedance due to electromechanical coupling is  k 2K0  (
j R ) Zm′  Rm′
jX m′   
 I

(5.10)

K   k 2K0   Zm  (Rm′
Rm′′ )
j(X m′
X m′′ )   ( I
j R )
Rm′′
j 
M  0    



(5.11)

Therefore

The mechanical impedances of various geometrical forms without electromagnetic fields are presented in Chapter 2. 5.6.3.4

Electrical Impedance and Efficiency

When the mechanical side of the electromechanical transducer or network is rigidly clamped, Zc  Rc
jXc Under these clamped conditions, v = 0.

CRC_DK8307_CH005.indd 249

10/27/2008 3:30:42 PM

Ultrasonics: Data, Equations, and Their Practical Uses

250

The electric input impedance with the mechanical termination is Zi  Ri
jXi  Ze


Ze2m Zm
ZL

(5.12)

The efficiency η in converting electric power into mechanical power is Zem R  L Ri Zm
ZL

2

(5.13)

The maximum of Equation 5.4 with respect to XL occurs when Rc (X m
X L )
RemX em  0

(5.14)

and that with respect to RL is when 2 Re2RL2  ( Re Rm
Rem )(Re Rm  Xem2 )
[Re (Xm
XL )
RemXem ] 2

(5.15)

The potential efficiency is the maximum efficiency that the four-terminal electromechanical network can exhibit in transferring power from the electric to its mechanical terminals or vice versa. It occurs when the value of Equation 5.4 is maximum with respect to both XL and RL. It is Potential efficiency 

2 2 Re Rm
Rem  Re Rm  X em 2 2 Re Rm
Rem
Re Rm  X em

Rmax  Rmin  Rmax
Rmin

(5.16)

where Rmax  Re
 Re
Rmin  Re   Re 

DA [1
cos 2
em ]  Re
DA cos 2
em 2 2 Rem Rm

DA [1  2 cos 2
em ]  Re  DA sin 2
em 2 2 X em Rm

Here, DA is the diameter of the impedance circle while operating in air. The condition that Equation 5.16 remains <1 is that 2 Re Rm  X em 0

The efficiency at resonance in water is res 

Dw Ri

 Dw   1  D  A

(5.17)

where Ri is the resistance at resonance in water and Dw the diameter of the motional impedance circle in water.

CRC_DK8307_CH005.indd 250

10/27/2008 3:30:42 PM

Magnetostriction: Materials and Transducers 5.6.4

251

Eddy Currents

The eddy current factor χ can be used to determine the core impedance of a coil with laminated core. The core impedance is that part of the total impedance resulting from the magnetic flux that traverses the core, which is assumed to have no gap. The total impedance consists of the core impedance plus the leakage, the latter being composed of the copper resistance and the leakage reactance arising from the flux that links the winding without entering the core. This is a convenient separation of impedances that causes insignificant error in well-designed magnetic circuits. Eddy current losses are taken into account by multiplying the permeability µ by χ. χ is a complex factor which depends on the geometry of the magnetic material and on a ________ characteristic depth δ = √ρe109/µf (cm), where ρe is resistivity in ohm-centimeters, µ magnetic permeability, and f the frequency of the applied field. A magnetic field applied tangentially to a thick sheet is reduced to e−2π times its value at the surface in penetrating through a depth δ. For a stack of large flat sheets of thickness t, insulated from one another, the eddy current factor χ depends upon the ratio δ/t. This ratio is used to define a characteristic frequency fc, where fc 

c 109 2 2t 2

(5.18)

The characteristic frequency is used to define mathematically the eddy current factor χ, that is    0 eR   R  j L 

tan h jf/f c

(5.19)

jf/f c

At low frequencies, 2

R  1 

2  f 62  f 
  15  f c  2835  f c 

1  f  17  f  L     3  f c  315  f c 

4

(5.20)

(5.21)

At high frequencies, R → L →

1 2 f/f c

(5.22)

The eddy current losses are large when the frequency f is > fc. For this reason, generally f/fc is held to less than unity for transducers designed for high efficiency. 5.6.5

Hysteresis Losses

When a completely demagnetized magnetic material is subjected to an mmf, the flux in the material increases according to the characteristic of the magnetization curve of the material. If the mmf is decreased to zero, the flux also decreases, but by a different path, leaving a residual flux in the material as the mmf reaches zero. As the mmf continues in a negative direction, the flux also continues to follow along a path of its own, out of phase with the driving mmf. As the driving mmf is again reversed, the flux curve also reverses

CRC_DK8307_CH005.indd 251

10/27/2008 3:30:43 PM

Ultrasonics: Data, Equations, and Their Practical Uses

252

direction, continuing out of phase with the driving force, tracing a curve on the side of the driving mmf opposite from the initial downward flux curve. The completed trace of the flux curve produces a hysteresis loop as shown in Figure 5.5. The magnetostrictive stack is a magnetic material. Therefore, as AC is supplied to the magnetostrictive transducer for work, a portion of its energy is lost to the hysteresis of the magnetostrictive element. The energy lost to hysteresis is proportional to the area of the hysteresis loop. Magnetostrictive transducers utilize a bias current to adjust the zero position about which the hysteresis loop forms. The bias current controls where the transducer operates on the stress–strain curve. If the bias exceeds the peak of the applied AC mmf, the frequency at which the transducer operates will be the same as that of the driving frequency. Looking at Figure 5.5, the total work done in one cycle, requiring time t, is W  S∫

Ht dB dt 108 0.4 dt

(5.23)

For a system in which the cross-sectional area of the core is uniform, the volume of the core is V = Sℓ and B

W

m V 108 ∫ H dB 0.4 B

(5.24)

m

Here ∫Bm –Bm H dB is the area of the hysteresis loop with a maximum flux density of Bm . Equation 5.24 gives the loss in joules per cycle. Multiplying by frequency gives the watts lost to hysteresis. The area of the hysteresis loop may be measured using curves obtained experimentally. This method is not always convenient. It is a matter of obtaining an area in terms of flux lines times mmf. The method involves letting a unit length along the B axis represent b flux lines per square centimeter and unit length along the mmf axis represent h gilberts per centimeter. Unit area = bh/4π. Hysteresis loss equals units measured by the measuring method times bh/4π in ergs per cubic centimeter [23]. 1 W  107 erg/s

FIGURE 5.5 Typical hysteresis loop.

CRC_DK8307_CH005.indd 252

FIGURE 5.6 The toroid.

10/27/2008 3:30:43 PM

Magnetostriction: Materials and Transducers

5.7

253

Types of Magnetostrictive Transducers

The magnetostrictive transducer designs to be discussed in this chapter include laminated: 1. 2. 3. 4.

Toroids Bars Mass-loaded bars Bars driving horns for various applications, either (1) free or (2) mass-loaded

5.7.1

The Toroid

A coil wrapped about a magnetic closed-loop of uniform cross section containing no air gaps produces the strongest field (greatest number of lines of magnetic flux) per ampere turn. It is the simplest type of magnetostrictive transducer. Because the toroid and other closedloop configurations have no open poles, they are not used as electromagnets (using DC) for attracting materials. However, the closed-loop configuration is very important in the design of transformers and transducers, which use AC. The toroid is illustrated in Figure 5.6. 5.7.1.1

The Mechanical Impedance of the Toroid (Ring)

As mentioned earlier, the mechanical impedance consists of the effects of mass and stiffness on the driving force and resulting motion. This is K  Zm  Rm
jX m  Rm′′
j 
M   


(5.25)

For the toroid in which all parts of the surface have the same radial component of velocity, M is the mass of the cylindrical shell, and K its stiffness. The mass remains constant and is M  2ab
m

(5.26)

where ρ m is the density of the magnetostrictive core, a the radial thickness, b the longitudinal thickness (the cross-sectional dimensions), and ℓ the circumferential centerline length of the path in the core. 5.7.1.2

The Electrical Impedance of the Toroid (Ring)

The blocked electric impedance of the core, that is, the impedance that the core would have if it could not move under electric excitation, consists of two parts: (1) that due to the magnetic flux within the core itself, Zc and (2) that due to the resistance of the windings and to the flux that links the windings but does not pass through the core, Zℓ. The core impedance is j
2 N 2b
 2 a

(5.27)

Z
 R

j
L


(5.28)

Zc  The leakage impedance is

CRC_DK8307_CH005.indd 253

10/27/2008 3:30:43 PM

Ultrasonics: Data, Equations, and Their Practical Uses

254

where Rℓ is the resistance of the windings and Lℓ the inductance of the windings caused by the flux that does not pass through the core. The total blocked impedance is ZT  Z

Zc 5.7.1.3

(5.29)

The Mutual Impedance of the Toroid

Mutual impedance is defined as follows: For two meshes of a network carrying AC, the ratio of the complex voltage to one mesh to the complex current in the other, when all meshes besides the latter one carry no current [20]. In the present case,  E Zem     v  I0

(5.30)

 F Zme     I  v0

(5.31)

where v is the velocity at the output terminals, E the voltage across the input terminals, F the force across the output terminals, and I the current at the input terminals. The incremental value of the stress in the core parallel to the direction of flux is P B
Es

(5.32)

where Λ is the magnetostrictive constant = εB0; B0 the polarizing induction; B the flux density, lines per square centimeters; for a blocked transducer, s = 0, and P = −ΛB. In the presence of eddy currents, B = 2NχµI/a Therefore 2NI a

(5.33)

4Nb
I a

(5.34)

P  B  The corresponding total radial force is F  2b
P  Therefore

4Nb
  F Zme      I  v0 a

(5.35)

It can be shown that this mutual impedance is equal but opposite in sign to the other mutual impedance, Zem: 4Nb
  E Zme      v  I0 a

CRC_DK8307_CH005.indd 254

(5.36)

10/27/2008 3:30:43 PM

Magnetostriction: Materials and Transducers 5.7.1.4

255

The Load Impedance on the Toroid

The toroid as a transducer is generally used in water. It is assumed that the water pressure is uniform and operation is at an intensity below the cavitation level. Further, operation at the fundamental resonance frequency is assumed. At this frequency, all portions of the outer surface move radially in synchronism. For the present situation, the inner surface of the ring is not in contact with the water, that is, the outer surface alone is subjected to the load. The ring is of uniform width w parallel to its axis. Under these conditions, the load area is the actual outer surface of the ring, S  D0b

(5.37)

where D0 is the outer diameter of the ring and b the width of the outer surface of ring in the axial direction. The load impedance is, therefore ZL  cS  cD0b

(5.38)

When the transducer is placed in water, the load impedance is the resistance placed on the radiation area by the water. When it is placed in air, the radiation resistance may be set to zero. The most noticeable effect will be a change in frequency, except when the dimensions of the transducer are large in comparison with the wavelength, making the radiation impedance almost purely resistive. 5.7.1.5

The Equivalent Circuit of the Toroid Transducer

The toroid is the simplest magnetostrictive transducer structure from the viewpoint of electromagnetic analysis. It has more limited usefulness than other types such as bars, which can drive horns, mass loads, or flexural plates and bars to do work. The toroid’s chief advantages are probably its ability to transmit and receive underwater signals and to receive sound signals in air. In the present context, the toroid is assumed to have a continuous magnetic path. All biasing is done electrically. The equivalent basic circuit of Figure 5.3 fits that of a toroidal magnetostrictive transducer. In Figure 5.3,   (/
N )e j

where Λ is the magnetostrictive constant; e –jφ a phase factor accounting for hysteresis; N the number of turns on the core, wound in series; and ω2πf, where f is the frequency. The magnetostrictive constant Λ and the transformation coefficients are related to the electromechanical coupling coefficient kc as follows: kc 

 i 0 2   Y0 N

Lc
L

 c e j

SY0 SY0

(5.39)

or 2 

CRC_DK8307_CH005.indd 255

jSkc2Y0 e j 2



Z0

(5.40)

10/27/2008 3:30:44 PM

Ultrasonics: Data, Equations, and Their Practical Uses

256 where

Z0  j
L0 The operating frequency is lower than a critical value given by fc 

c 2 21 0 2

(5.41)

where ρc is the resistivity of the core and δ the lamination thickness, in meters. In a laminated structure, the lamination thickness is twice the flux penetration depth. Flux penetration depth is given by   1.98

e ∆ f

(5.42)

where ρe is the electrical conductivity of the magnetic material (in ohm-centimeters), µ∆ the relative magnetic permeability of the magnetic material, and f the frequency (in hertz). 5.7.1.6

The Efficiency of the Toroid Radiating into a Medium

The general equation for efficiency of any device converting energy from one form into another form is 

Power output Wo  Power input Win

(5.43)

Using the impedance circle diagram for operation in air and in water, the efficiency equation of the toroid is converted to  Z  Z0   RL  Eff     i    Ri   Zm
ZL 

(5.44)

Noting that at resonance, [Zi − Z0] is equal to the diameter of the motional circle in water and [Zm + ZL] becomes [Rm + RL], Equation 5.44 can be converted to res 

Dw Ri

 Dw   1  D 

(5.45)

A

where Ri is resistance at resonance, Dw diameter of the motional circle in water, and DA diameter of the motional circle in air. The factor {Dw/Ri} in Equation 5.17 is a gross electromechanical efficiency, which may sometimes be greater than unity. The factor {1 − Dw/DA} is a pure mechanical efficiency, and its use in Equation 5.17 always makes the product <1. Equations 5.17 and 5.44 may be considered as general for any type of magnetostrictive transducer. Applying the general equations for efficiency to the toroid and simplifying by assuming leakage inductance and copper resistance are negligible, so that Ze = Zc, the efficiency of the toroid can be written 

CRC_DK8307_CH005.indd 256

2 Rc RL ( Rm2
X em ) AB
C

(5.46)

10/27/2008 3:30:44 PM

Magnetostriction: Materials and Transducers

257

where 2 A  Re (Rm
RL )
Rem 2 B  Rc (Rm
RL )  X em

C  [Rc (X m
X L )
RemX em ]2 Because Rm = R′m + R″m and Xm = X′m + X″m , where the double-primed quantities are of purely mechanical origin and the single-primed are reflections of the core impedance into the mechanical system by the magnetostrictive coupling, the efficiency of the toroid is 

2 2 RL ( Rem
X em ) DF
G

(5.47)

where 2 2 D  Rc (Rm′′
RL )
Rem
X em

F  Rm′′
RL G  Rc (X m′′
X L )2 The frequency f E of the toroid at maximum efficiency is constant and independent of magnetostrictive coupling. The frequency f R at resonance is variable and dependent upon the magnetostrictive coupling. As the magnetostrictive constant decreases, f R approaches f E. Without magnetostrictive coupling, f E is the value of f R. The relationship between f R and f E is fR 2  1  k eff fE

(5.48)

where keff is the effective coefficient of electromechanical coupling given by k eff 

( f 2  f1 ) fE

(5.49)

where f 2 and f1 are the nominal cutoff frequencies for the transducers. The coupling coefficient for the magnetostrictive material k is k

4 2 E

(5.50)

 02 R

(5.51)

For the cylindrical (or toroidal) transducer, k eff  k

In any reasonably designed cylindrical transducer, keff is nearly as large as k. For forms other than the toroid, the relation between keff and k becomes more complicated.

CRC_DK8307_CH005.indd 257

10/27/2008 3:30:44 PM

Ultrasonics: Data, Equations, and Their Practical Uses

258 Maximum efficiency is Eff max 

 RL  1 1
( f E1(Rm′′
RL ))/4 2b
 02   Rm′′
RL 

(5.52)

The Q of the mechanical system Qz including damping due to eddy currents is Qz 

5.7.1.7

Frequency of resonance Difference of frequencies at ends of diameeter perpendicular to resonance diameter

(5.53)

Maximum Power Transfer to the Magnetostrictive Transducer Radiating into a Medium

Maximum power transfer occurs when the impedance of the load equals that of the driver. Referring to Figure 5.3, at resonance the reactive components of the mechanical branch of the circuit are tuned out. By adding a tuning capacitor at the output of the generator, jωL0 also may be tuned out. This leaves Ri as the internal impedance, which must be balanced against the load impedance for maximum power transfer as follows:   (R
Z )   (R
Z ) Ri  R0
 m 2 R Zc   m 2 R
Zc       

(5.54)

R0 Rm
 2Zc R0
RmZc  Ri Rm   2Ri Rc Ri  R0  Zc

(5.55)

ZR 

for a transducer radiating from one end, where ZR = (ρc)ℓS and (ρc)ℓ are the acoustic impedance of the load medium. In using Equation 5.55, if the load medium is water or liquid, the intensity level must be lower than the level that causes cavitation. Cavitation increases the complexity of the load impedance and will disturb applications such as communications or transmission of information when it occurs. 5.7.2

The Magnetostrictive Bar

The bar, with its many combinations of forms and assemblies, is more adaptable than the toroid to industrial and medical applications. The simplest form is a uniform rod one-half wavelength long. It is generally laminated with layers designed for maximum flux penetration and for resonance at a desired frequency. The length at the resonance frequency is determined using the equation
 c/2 f where ℓ is the length of the bar, c the velocity of sound in the bar, and f the frequency at which the bar is to resonate. The velocity c is controlled somewhat by the effects of the eddy currents on the modulus of elasticity Y as discussed in Section 5.7.1.1 c  Y/

(5.56)

where ρ is the density of the material.

CRC_DK8307_CH005.indd 258

10/27/2008 3:30:45 PM

Magnetostriction: Materials and Transducers

259

FIGURE 5.7 A uniform bar-type magnetostriction transducer with resonant load made in accordance with the design of Figure 5.8d.

Adding a mass to either or both ends of the bar shortens the resonant length to accommodate the impedance of the added mass. This would happen in the case of a closed-loop magnetostrictive transducer, as shown in Figure 5.7. Here the added mass is of the same material as the bar, with no discontinuity between the bar and the mass. The dimensions are determined for a uniform bar, as shown in Chapter 2, according to the following example: 1. Let the bar velocity of sound in the material be 200,000 in./s (5,080 m/s). 2. Let the design frequency be 20,000 Hz. 3. Let the dimensions of the end masses be a = the thickness of the bar, inches. b1 = 1.125/2 in. = 0.5625 in. (1.405 cm). b2 = 1.5/2 in. = 0.75 in. (1.905 cm). r = 0.75 in. (1.905 cm), assuming that r is small enough that the load may be considered a lumped mass. 4. Let the bar be laminated for maximum flux penetration. Under these conditions, a half-wave bar would be 5 in. (12.7 cm) long. Adding the masses shortens the length of the bar. To find the amount to shorten the bar, apply the equation 
x  
M tan1    c  cS1

(5.57)

where M is the mass = ρab2r and ρ the density of the mass. This equation determines a balance position between the mass and the uniform bar. Substituting the given dimensions into Equation 5.57, 
x  2 f ( ab2r ) 2 fb2r 2(0.585) tan1      cab1 b1c 5.625  c 

CRC_DK8307_CH005.indd 259

10/27/2008 3:30:45 PM

Ultrasonics: Data, Equations, and Their Practical Uses

260

(a)

(b)

(c)

(d) FIGURE 5.8 Uniform bar-type magnetostriction transducer with resonant. (a) Uniform half-wave bar, (b) half-wave bar with lumped mass load, (c) half-wave bar loaded at each end with equivalent lumped masses, and (d) half-wave twin bars with equivalent masses at each end.

or x

147.858 in.  4.1072 in. 36

Thus the bar must be shortened by (5 − 4.1072) in. from the end to which the mass is to be added. The length of the bar plus a load at one end is x + r, which is 4.8572 in. But the objective was to design a magnetostrictive stack, which involves adding a mass of equal size at both ends of the bar. Therefore it is now necessary to subtract a similar amount from the opposite end and to add the required mass. Therefore the length of the uniform bar between the end masses is 3.2144 in. This will be the length of the slot between the masses, as shown in Figure 5.8, of a closed magnetic loop transducer. The total length of the loaded bar is 3.2144 + 2(0.75) in. or 4.7144 in. or, more practically, 4.71 in. This procedure has designed the laminations for cores for four transducers operating nominally at 20 kHz: (1) a straight, uniform bar, 5 in. long; (2) a straight uniform bar carrying a mass load at one end, 4.8572 in. total length; (3) a uniform bar carrying a mass load at each end, 4.7144 in. long; and (4) a closed-loop transducer containing a slot, 3/8 in. by 3.2144 in. long, centered in a bar 4.7144 in. long. These all vibrate nominally at 20 kHz in longitudinal half-wave resonance [21]. 5.7.2.1

Typical Bar Designs and Corresponding Coil Positions

A major difference between the toroid and the uniform bar is that the toroid forms a lumped system while the bar is a distributed system. Vibration in the lumped system is equally distributed about the ring, which occurs only at the fundamental radial frequency. In a distributed system, the vibration amplitude and stress vary from point to point along the vibrating unit. For example, the velocity and stress vary sinusoidally along a uniform bar vibrating in longitudinal half-wave resonance.

CRC_DK8307_CH005.indd 260

10/27/2008 3:30:46 PM

Magnetostriction: Materials and Transducers

261

The position of the coil in transducers, like that of Figure 5.8, is important. The bar expands and contracts at resonance, and for the half-wave resonant uniform bar, the stress is maximum across a cross-sectional area located at the midpoint between the two ends. This determines the best location for the driving coil, centered over the high-stress plane. If the uniform bar is attached to a half-wave extension such as a horn, resonant flange, or flexural element, the location of the coil will remain the same, as though the transducer bore no load. However, if the load is a lumped mass, the plain of maximum stress moves closer to the load and the coil must be moved toward the mass. In this case, if the mass is large compared with the cross section of the transducer but of a dimension such as to be considered a lumped mass, its impedance is Zm = −jωM. If the resonant load, bar or horn, and the transducer have equivalent cross sections, slight variations in frequency will cause the point of maximum displacement to move into either the load or the transducer, depending on whether the frequency has been reduced or increased around the original resonant frequency. This is especially useful, for example, in ultrasonic applications where the load element is eroded by the application or where the load element is subjected to varying impedance loads itself which cause variations in resonance frequency. Figure 5.8d is a loop formed by punching out a window in a bar making two uniform legs between two end pieces. It is a design drawing of the laminations for the transducer of Figure 5.7. It has end masses of equal size and would be used in air or to drive any resonant unit. Usually, the coil is wound continuously, that is, in series along both legs. In this case, the coil is wound to produce a continuous flux circuit; that is, the flux lines form a continuous loop. Insulation strips shaped to match the laminations of the loop are laid between the stack and the coil. Good materials for this are neoprene and Bakelite—the neoprene laid against the stack and Bakelite laid on top of the neoprene to provide a stiff layer against the stresses produced by the coil windings. Other plastic materials are useful for this purpose. The flux path is not broken and the windows set the location of the coil. Figure 5.9 is similar to Figure 5.8d but includes an air space in the opening at one end. This end is opposite the closed end and is for inserting a permanent magnet for biasing the system. Figures 5.7 through 5.9 are the basic designs most frequently used today. 5.7.2.2

Equivalent Circuit of a Magnetostrictive Bar (Rod)

The equivalent circuit for magnetostrictive bars, or rods, may be of any of several designs: T, π, L, or other. The one to be used here is similar to Figure 5.3. This transducer is to be used for specific applications at or near resonance frequency. Most power types of applications may involve cavitation, heat, chemical reactions, eroding or cutting, cleaning, bubble removal, and many others. (A magnetostrictive transducer can be used as an electrical filter but these can hardly compete with the many modern methods of electrical filtering available.) Four general load conditions may apply at separate times: 1. The load may be a lumped mass in which RL and CL are zero. 2. The load may consist of mass, resistance, and capacitance in series.

FIGURE 5.9 Closed loop with break in magnetic path for inserting biasing magnet.

CRC_DK8307_CH005.indd 261

10/27/2008 3:30:46 PM

Ultrasonics: Data, Equations, and Their Practical Uses

262

3. The load may be a lumped mass with resistance but XL is approximately zero. 4. The load may be a unit with distributed mass and capacitance but with low enough RL that the resistance may be considered zero. The load is nonmagnetic or at least not affected by the magnetic field used to drive the transducer. Condition 1 applies to a mass driven by the transducer that has dimensions much smaller than a wavelength in the material of the mass. This mass will affect the operation of the transducer by lowering the resonant frequency and by reducing the displacement amplitude below that of an unloaded unit. The rod must be shortened to bring the frequency back to its original resonance. At no point does the load ride against a rough or lossy surface. Condition 2 is a distributed elastic system such as a horn that has mass ML, resistance RL, and capacitance CL. The mass is generally conceived as the density times the volume; however, with the distributed mass, such treatment gives a mass that is excessively high. The resistance RL includes both the losses internal to the load (such as a horn) and to any external frictional loss mechanism. The capacitance refers to the elastic properties in the direction of motion K L. Obviously, it is not possible to cover all conditions of application in one brief chapter. The purpose here is to concentrate on the factors that influence transducer design and to do this to give a preview of applications that affect the performance of the transducer. In some cases, the transducer energizes an elastic unit, either horn or flexural bar or plate with essentially no resistance component, to its fullest amplitude of vibration before the end of the unit is thrust under pressure into a dead load. Then the power to the transducer is turned off immediately. This is the case in ultrasonic spot welding. The operation takes place in a matter of seconds, saves energy, and results in little heating, so that the frequency remains fairly constant. This procedure permits the use of a more economical electronic driver design. The mechanical impedances of horns are discussed with each type of horn presented in Chapter 2. The load impedance may be added to that of the mechanical impedance of the transducer as shown in Figure 5.10. Equations 2.1 through 2.7 describe the actions in a uniform bar in free–free half-wave resonance. These equations apply to bars not influenced by a magnetic field. Letting the wave number ω/c be held constant or the magnetostrictive constant be zero, the equation describing the particle displacement in a half-wave resonant bar is 
x  cos
t    m cos   c 

I

Lᐉ

Xᐉ

Zc

1 to (ϕϕ′)2

R1

(2.3)

C1

L1

i = v/j

ML E

E = f/j

CL RL

FIGURE 5.10 Approximate equivalent circuit for magnetostrictive rod with load of mass and resistance in series.

CRC_DK8307_CH005.indd 262

10/27/2008 3:30:46 PM

Magnetostriction: Materials and Transducers

263

and the particle velocity is 
x  v 
 m cos  sin(
t)  c 

(2.4)

Applying an AC field to a magnetostrictive bar brings new factors into considering its impedance and performance. A DC bias field is applied and the AC operates about this bias field. For maximum efficiency, the AC driving field does not cross zero excitation. A long and slender magnetostrictive bar will be subjected to forces and velocities by the AC through the coils. As the bar is long and slender, the demagnetization can be ignored. The induction is B  H

(5.58)

where H is the average field along the rod produced by current in the winding and by the strain in the rod, H  H e
4s

(5.59)

where He is that field due to the current in the coil, Λ the magnetostrictive constant, and s the strain. The average strain is sa 

 2  1


(5.60)

where 1 is the displacement amplitude at x = 0 and 2 the displacement amplitude at x = ℓ. The induction B is B  Be
4sa  H e


4( 2  1 )


(5.61)

The stress P in the rod is P  B
Ys  Be 

4 2( 2  1 )   
Y   x 


(5.62)

where s = ∂/∂x. The wave equation for a rod may be obtained from Equation 5.61 by differentiating P with respect to t and multiplying by (1/c) as follows: 1 P  2  2  m 2  Y 2 t x c t

(5.63)

 2 Y  2  t 2  m x 2 from which
x
x     e j
t  C1 cos
C2 sin   c c 

CRC_DK8307_CH005.indd 263

(5.64)

10/27/2008 3:30:46 PM

Ultrasonics: Data, Equations, and Their Practical Uses

264

which is similar to Equation 2.3, leading to the simple designs illustrated earlier in this chapter, which assumed that the wave number was held constant or that the magnetostrictive constant was zero. To apply a magnetic driving force to a magnetostrictive rod, start with the boundary conditions as being at x = 0 and P = F1/S d  v1 dt

(5.65a)

d  v2 dt

(5.65b)

and at x = ℓ and P = F2/S

where S is the cross-sectional area. From Equations 5.63 through 5.65b, F1  SBe 

4 2S(v2
v1 )  j(cS)m (v1 cot an k

v2 cos ec k
) j



4 2S(v2
v1 )  j(cS)m (v1 cos ec k

v2 cot an k
) F2  SBe  j



(5.66)

The induction Be, due to the externally applied field, is GI  4N  Be   I 
 S

(5.67)

where G = 4πΛSNμχ/ℓ. By neglecting demagnetization, the core impedance can be written Zc 

j
4N 2S


(5.68)

By using Equations 5.61, 5.65a, and 5.65b the voltage across the winding is E  Z
I
NSB  (Z

Zc )I
Gv1
Gv2

(5.69)

where Zℓ represents copper resistance and leakage inductance. Now Equation 5.66 can be rewritten as   G2  G2  F1 GI   j(cS)m cot an k

 v1   j(cS)m cos ec k

 v2 Zc  Zc   

(5.70)

  G2  G2  F2 GI   j(cS)m cos ec k

 v1   j(cS)m cot an k

 v2 Zc  Zc    Equations 5.69 and 5.70 are the final results for the magnetostrictive rod as a six-terminal network (see Figure 5.11).

CRC_DK8307_CH005.indd 264

10/27/2008 3:30:47 PM

Magnetostriction: Materials and Transducers

265 i 2 = v2 /j

I1 = F1 /j j (ρc σ)m tan (kL /2)

j (ρσ)mtan (kL /2)

−j (ρc σ)m cosec kᐉ e1 = F1 /j

e2 = F2 /j ZL

E

Zc ϕ = jG /Zc = Λ /Nω

FIGURE 5.11 An equivalent T-circuit for a magnetostrictive rod.

1 to ϕ2

I

1 to ϕ′2

i = v/j kcML

E

Zc

e = F /j

RL

FIGURE 5.12 Equivalent circuit for magnetostrictive rod with load consisting of mass and resistance in series.

5.7.2.3

Half-Wave Rod Free at One End and Loaded at the Other End with Mass and Resistance

This design is the most common one today in the industrial area. The load mass may vary from zero to a maximum, moving from a full-length half-wave vibrator to one that is loaded at one end and free at the other, that is, toward a quarter-wave unit as might be used in underwater sound. The load resistance may also vary from zero to a maximum, as a unit freely vibrating in air to a unit driving a forming tool in a metal bar–drawing operation. The equivalent circuit will be changed a bit for this condition. The circuit of Figure 5.12 is useful over a narrow frequency range, but should cover most of the applications here. This circuit applies to conditions in which the rod is greater than a quarter-wavelength by an amount allowing φ′ to be regarded as constant. This occurs

CRC_DK8307_CH005.indd 265

10/27/2008 3:30:47 PM

Ultrasonics: Data, Equations, and Their Practical Uses

266

Lᐉ

Rᐉ

Zc

1 to ϕ2

rc

X0

R0

FIGURE 5.13 Equivalent circuit of Figure 5.12 at resonance of series arm.

when the mechanical system without magnetostrictive coupling is resonant, that is, when j(ρcS)mtan kℓ and jkcMi are resonant, so that tan k0
 ML/m0 where k0 = 2πf0/cm is the wave number at frequency f0 and m0 = (ρS)m/k0 is the mass of a section of the rod 1/2π times a wavelength (or 1 rad). The shunt element is j(cS)m cot an k0
 j(S)m (m0/ML ) The turns ratio φ′ is M 

′  1  sec k0
 1
1   L  m 

2

0

Therefore, at f0, Figure 5.12 reduces to Figure 5.13. R0  X0 

5.8

RL [1
1
( ML/M0 )]2 (c )mm0
0 ML

Design and Construction of Magnetostrictive Transducers

The relationships dealing with the magnetic flux produced in configurations such as those represented by Figures 5.7 through 5.9 to be used in longitudinally vibrating transducers must take into consideration the level at which the coil is being energized and the bias level, whether stress maxima (or velocity nodes) are critical, and the geometrical factors involved in the total flux path.

CRC_DK8307_CH005.indd 266

10/27/2008 3:30:47 PM

Magnetostriction: Materials and Transducers 5.8.1

267

Simple Uniform Stack

Figure 5.8a represents a bar-type half-wave resonator vibrating in the longitudinal mode. For the present example, it is to resonate at 20 kHz. The cross section of the bar is 0.5 in. by 0.5 in. A-nickel will be used for this example, and one may use the approach as a guide for designing transducers of other materials. Nickel transducers were most commonly used during the early years (1940s and 1950s) of power ultrasonics. Commercially pure nickel (A-nickel) is a very satisfactory magnetostrictive material. It combines good fatigue properties and fair magnetostriction characteristics. It is readily available, which is the primary factor in its favor. Nickel oxide has high electrical resistance, so oxide-annealed laminations are sufficiently electrically insulated that they can be stacked in direct contact with one another without significant loss. The lamination thickness is twice the flux penetration depth in a laminated structure. The flux penetration depth is given very accurately by Equation 5.42. For maximum flux penetration, a typical lamination thickness of an A-nickel stack to operate at 20 kHz is 0.25 mm (0.010 in.). A typical stack preparation consists of rolling a sheet in the desired direction to the desired thickness, soft-annealing by heating in a furnace in an oxidizing atmosphere at 1400°F for 1 h, and allowing the furnace to cool very slowly before removing the material. The laminations are then punched out, stacked, clamped in fi nal form, and annealed again as before. Working tends to harden the material, therefore, it is annealed again following any forming operation to improve the magnetostriction. Preferably, the stacks are produced to nearly the design size before the fi nal annealing and are kept in the original stacks until after assembling into the fi nal form to be used. The lateral dimensions of the stack for the transducer of Figure 5.7 have already been given. It is assumed that the bar is slender enough to use the bar velocity of sound to determine its length. From Table 2.1, the velocity c0 is 4.9 × 105 cm/s for A-nickel. Since c = λf, a half-wave bar of nickel to vibrate at 20 kHz is
 c/2 f  4.9  105 cm/s /2  20, 000/s  12.25 cm ( 4.82 in.) If the transducer is to drive no load or to drive an element resonant at the same frequency, the velocity node or stress antinode appears at the center plane normal to the direction of motion. The driving coil, then, should be centered about this plane. 5.8.2

Closed-Loop, Uniform, Rectangular Bar Stack

The stack to be designed is represented by Figure 5.8d. It will operate at 20 kHz nominal frequency. The stack is 3.81 cm (1.5 in.) by 3.81 cm (1.5 in.) in cross section. Because a half-wave uniform bar of nickel is 12.25 cm (4.82 in.) long, it should be safe to call for a slot through the stack parallel to the longer dimension 7.62 cm (3 in.) long. The slot will be 1.9 cm (0.375 in.) wide and centered in the stack. This design leaves two masses of equal size, one at each end of the stack. Because they are punched from a single material, the density will not change. The first requirement is to determine the actual length of a stack of the given dimensions to vibrate at 20 kHz. This is done by equating the acoustic impedances of the uniform section containing the slot with the solid masses at each end using the following equation: Z  jc tan
x/c  j
pV

CRC_DK8307_CH005.indd 267

(5.71)

10/27/2008 3:30:48 PM

Ultrasonics: Data, Equations, and Their Practical Uses

268

where S1 is the cross-sectional area of the legs of the stack, ρ the density of the material of the stack, c the bar velocity of sound in the stack, and V the volume of each mass load. The volume V is the product of the cross-sectional area of each mass S2 and of the length of the mass ℓm as it is essentially rectangular in all plains. Therefore, Equation 5.12 reduces to S1c tan
x/c 
S2
m or tan
x/c 
S2
m/S1c

(5.72)

The term ω/c is 0.651444821/in. at 20 kHz for A-nickel. Therefore tan 0.651444821 x  0.651444821(S2/S1 )
m/in.  0.8685931
m/in. This equation is solved by iteration. The best way is to start with a dimension for ℓm that one would expect to be satisfactory and determine whether it leaves room for the preassigned length dimension of the slot. As the length of a half-wave, uniform bar has already been determined to be 12.25 cm (4.82 in.), it would appear to be safe to assign a value for ℓm of 1.9 cm (0.75 in.). Inserting this value into Equation 5.12a gives tan(0.651444821 x)  0.651444825 Therefore (0.651444821 x  (  33.1/180)  0.81621 and x  3.9362 in. There are several ways this figure may be used to find the exact length of the stack but the simplest is to realize that the x = 0 position is that for a uniform bar. Therefore, the distance from this zero point to the junction with the mass load is 3.9362 in. (10 cm). One quarterwavelength of the uniform bar is 2.41 in. This value gives the nodal position (v = 0) along the stack. Subtracting this value from 3.9362 in. gives the distance from the node to the junction with the mass load, which is 1.5262 in. (3.88 cm). Adding the mass (0.75 in.) to this length and doubling gives the half-wave length of the slotted stack, that is, 2(1.526 + 0.75) in. = 4.552 in. This calculation leaves room for a sufficient number of turns in the driving coil and the design is, therefore, fi nal. Ten-gauge stranded copper wire with Kel-F insulation is a good material to use with this design. Kel-F is tough and waterproof, so that the transducer may be immersed in water or have water spray for cooling. In addition, the insulation remains in good condition for many years. 5.8.3

Multiple-Bar Type of Transducer Driving a Head

The multiple-bar type of transducer may be used, for example, to perform functions requiring liquid baths. The liquid may present variable impedance to the transducers,

CRC_DK8307_CH005.indd 268

10/27/2008 3:30:48 PM

Magnetostriction: Materials and Transducers

269

depending upon depth and type. However, in general, liquids have lower impedances than solids and their flow properties make them more readily adapted to the driving system. One can use the transducer of the example of Section 5.8.2 to attach to the head. If left unaltered, the system would function well if the reflected impedance is from a matched resonance load or if the head were very thin. In other situations, the head would appear as a dead load and the mating end of the transducer would have to be cut back to match the load imposed by the head load. In this case, the driving current would be altered a bit by the change in magnetic reluctance due to the shorter mass at the end of the transducer. Different materials may be used for the load-head, some of which are magnetic and some are not. The reluctance of these materials must also be considered. In most cases, if everything else is acoustically matched, the difference may not be significant. Another factor relates to the resonance between transducers. It is difficult to get a number of transducers to vibrate in phase, even when they are punched using the same tool. Silver brazing or silver soldering may vary from transducer to transducer very slightly, but this slight difference is enough to cause variations in resonance. Silver soldering is a common means of attaching a transducer to another resonance system, such as a horn, or to another device. 5.8.4

Transducer with Gap for Permanent Magnet Bias

As mentioned previously, sometimes permanent magnets are used for biasing (Figure 5.9). This arrangement allows for simpler coupling circuitry, but offers only a fixed bias—not adjustments. This setup is acceptable in many on/off situations requiring only the maximum or design operating power levels. The opening path in the core affects both the length and the driving conditions due to the reluctance of the air gap. Dealing with the dimensional aspect first and using the transducer design parameters of cross section and head length used previously in the example of Section 5.8.2, that is, cross section of 3.81 × 3.81 cm, head length at the end opposite the magnet of 0.75 in. (1.9 cm), the slot width of 0.375 in., and operation at 20 kHz, most of the calculations for length have been completed. Using the length of a uniform bar vibrating at 20 kHz and adjusting for the one head remaining, the length of the transducer stack is found to be
 (3.9362
0.75) in.  4.6862 in. which is the distance to the junction with the end mass figured in the example of Section 5.8.2 plus the length of the end mass. Regarding the driving force, the reluctance across the magnet area must be combined with that of the magnetostrictive stack. The reluctance may actually be less than that of the magnetostrictive material, depending upon what the magnetic material is. If the values are close, the differences in driving current also will be small. The power capabilities of the transducers of the examples of Sections 5.8.2 through 5.8.4 are about 1000 W when they are well cooled (by water). Figure 5.7 is a photograph of a 20 kHz transducer designed according to Section 5.8.2. This transducer was designed originally for use in delaminating jig mica. It has since been used in many other applications. Jig mica is a laminated rock. Dry ground mica is used in many ways and is inexpensive. Wet ground mica is much more expensive. The transducer is attached to a double-cylinder type of horn with a flange at the load end. The elements are half-wave units resonant at 20 kHz. The reason for the flange is to throw the small

CRC_DK8307_CH005.indd 269

10/27/2008 3:30:48 PM

Ultrasonics: Data, Equations, and Their Practical Uses

270

chunks of mica into the fluid where cavitation can surround them more easily, tearing layers of laminations free. If the mica is allowed to fall to the bottom of the container without being drawn into the liquid (water), cavitation will have very little effect, drawing only from those chunks that are properly oriented on the pack. This process produced a very high grade of wet ground jig mica—very clean and glistening. There were many other uses of practical worth for this unit [21].

5.9

Combining Elements to a Transducer

There are several methods of combining resonant elements in an ultrasonic system. These include silver soldering and brazing, bolting, and welding. The most common type used for attaching laminations of a transducer permanently to a horn, plate, or flexural bar is silver soldering or brazing. Threading into a stack of laminations is impractical. A plate-like type of end piece may be attached to one end of a magnetostrictive stack by silver soldering or welding, which makes bolting very practical and makes it convenient to connect to horns, flexural bars, or plates. The bolt is centered on the axis of both the transducer and of the attachment. These are screwed on very tightly. The mating surfaces must be machined to very smooth plains or a malleable element, such as a copper or brass washer, must be used to produce uniform pressure between the surfaces of the elements. Welding may take the place of silver soldering in certain heat-related circumstances that would loosen the bond. The transducer must still be maintained at a temperature below the Curie limit during operation. In either welding or silver soldering, the bond strength must be uniform across the bond area. 5.9.1

Silver Soldering and Brazing

A permanent attachment between the transducer and the horn of Figure 5.7 may be made using silver solder. With the laminations being oxide annealed, the face mating with the stack must be cleared of oxide first, which is accomplished by carefully grinding off the oxide coating to present pure metal for silver soldering. The stack must be clamped in final assembly before grinding from the face to be bonded and remain in this form until after bonding has been completed. The grinding is done in the direction parallel to the wider portion of the lamination ends. Some material should be removed from the edges surrounding the stack to allow a small fillet to form and lower the stress concentration while the transducer is being operated. It is essential that no voids appear between the horn and the magnetostrictive stack. To accomplish this, the mating ends of the stack and the horn are coated with a layer of solder before they are brought together. They are then enclosed in a vertical position under pressure in an insulated chamber between guides to promote concentricity. They are held in this position until the heating has been removed. Various means of applying heat may be used, such as acetylene torches or induction heating coils. The transducer and horn are allowed to cool in the insulated chamber before being removed. Always apply lavish amounts of flux before silver soldering, brazing, or welding. The neoprene, the Bakelite, and the windings are added to the stack after all joining operations have been completed.

CRC_DK8307_CH005.indd 270

10/27/2008 3:30:48 PM

Magnetostriction: Materials and Transducers

5.10

271

Practical Uses for Magnetostrictive Transducers

There are many practical uses for magnetostrictive transducers. Their greatest assets are their ruggedness, ease of assembly, long life, and adaptability to adverse environments such as heat and severe chemical conditions. Cleaning is still a good use, although piezoelectric transducers are used more extensively for this purpose. Magnetostrictive transducers hold a major superiority where environmental conditions are destructive to piezoelectric transducers. Magnetostrictive transducers may be used to transfer ultrasonic energy into very hot molten materials. Molybdenum is a good material for use as a transmission line to transfer the energy for this purpose into a hot bath, but under these conditions molybdenum must be protected by a nonoxidizing atmosphere such as nitrogen. A material that is good for working in acidic or basic materials, whether hot or cold, is tantalum. Tantalum is malleable and very expensive. A rod of tantalum sufficient to produce a horn resonant at 20 kHz costs on the order of $2000. However, it is long-lasting if it is driven at an amplitude below its endurance limit. Cooling systems for these applications must be designed to protect the transducer.

5.11

The Latest Commercially Available Magnetostrictrive Material

Thanks to research performed by Naval Ordnance Laboratories (NOL), now Naval Surface Warfare Center (NSWC), materials have been developed with huge magnetostrictive properties. The interest in these materials was primarily their use in sonar. One material now being made for commercial use is Terfenol-D, which gets its name from the elements used in the alloy: Ter for terbium, Fe for iron, and NOL in recognition of the organization that discovered it. The D refers to dysprosium, which is added to reduce hysteresis with only a minimal effect on magnetostriction. The general chemical formula for Terfenol-D is TbxDy1−xFey. It is manufactured by ETREMA Products, a subsidiary of Edge Technologies.* ETREMA normally uses a composition of x = 0.30 and y = 1.92. Magnetostriction increases with terbium content. Hysteresis also increases with terbium content. Dysprosium decreases hysteresis with only a minimum effect on magnetostriction. Terfenol-D is brittle. Its magnetostrictive motion is positive, that is, it expands under electromagnetic excitation. Therefore, it is operated with the material under compression along the driven axis. It also has a very low Q, but this is overcome by its extremely high magnetostriction constant, allowing large output over a very wide bandwidth. It produces strains 40 times greater than either nickel or cobalt magnetostrictive alloys. A Terfenol-D rod routinely produces a linear motion >0.1%, that is, a 50 mm rod produces 50 µm displacement routinely. Their powerful linear displacements are accomplished with very low electrical input. The thinnest lamination cut to minimize the effects of, eddy currents, that is, to optimize flux penetration, in Terfenol-D is 1 mm. The net effect of eddy currents in both the coil and the core limits the maximum operation range of frequency to DC to 50 kHz [24]. An excellent discussion of giant magnetostriction materials is contained in a book edited by Goran Engdahl entitled Handbook of Giant Magnetostricive Materials [25].

* 2500 N. Loop Drive, Ames, Iowa, IA 50010.

CRC_DK8307_CH005.indd 271

10/27/2008 3:30:48 PM

272

Ultrasonics: Data, Equations, and Their Practical Uses

References 1. J. P. Joule, On the effects of magnetization upon the dimensions of iron and steel bars. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Ser. 3, 30, January– June 1847, 225–241. 2. The design and construction of magnetostriction transducers. Summary Technical Report of the National Defense Research Committee, Available through the Library of Congress, Washington, DC, 1946. 3. F. Villari, Feber die Denderungen des Magnetoschen Moments, Welehe der Zug und das Hindurchleiten Eines Gaklvanischer Stoms in Einen Stabe Von Stahl oder Eisen Hervorbringen, Annalen der Phyusik and Chemie, J. G. Poggnedorff, 126, 1865, 87–147. 4. P. Langevin, British Patent 145,601,1920. 5. G. W. Pierce, U.S. Patent 1,750,124,1927. 6. G. W. Pierce, Magnetostriction oscillators, Proceedings of the American Academy of Arts and Sciences, 63, 1928, 1. 7. K. Andrus and B. Luth, Magnetostriction of yttrium iron garnet, Journal of Physical Chemistry of Solids (GB), 24, April 1963, 584–586. 8. R. R. Birss and P. M. Wallis, The temperature dependence of magnetostriction, Proceedings of the Physical Society (GB), 81 (Pt. 2), February 1963, 368–370. 9. G. V. Blessing, J. R. Cullen, and S. Rinaldi, Isotropy breaking magneto-elastic behavior in highly magnetostrictive (rare-earth-iron) compounds, Physical Letters, 66A, 6, 26 June 1978, 498–500. 10. R. M. Bozorth and T. Wakiyama, Magnetostriction and anomalous thermal expansion of single crystals of gadolinium, Journal of Applied Physics (USA), 34, 4 (Pt. 2), April 1963, 1351–1352. 11. A. E. Clark, B. DeSavage, W. Coleman, E. R. Callen, and H. B. Callen, Saturation magnetostriction of single-crystal YIG, Journal of Applied Physics (USA), 34, 4 (Pt. 2), April 1963, 1295–1297. 12. A. E. Crawford, Ultrasonic Engineering, Academic Press, Inc., New York; Butterworths Scientific Publications, London, 1955. 13. M. Croft, I. Zoic, and R. D. Parks, Anisotropic magnetostriction of CeAl2 near its antiferromagnetic transition, Physical Review (B)(USA), 18, 1, 1 July 1978, 345–362. 14. D. Ensminger, Ultrasonics, Marcel-Dekker, Inc., New York, 1973. 15. Iron–Nickel Alloys for Magnetic Purposes (20% to 90%). The International Nickel Company, Inc., New York, 1949. 16. Nickel, The International Nickel Company, Inc., New York, 1951. 17. A. E. Clark, Magnetoelastic and Related Properties of Rare Earth-Earth Intermetallic Compounds from Proceedings of the National Rare-Earth Research Conference, Naval Ordnance Laboratoy, Silver Springs, MD. 18. R. M. Bosorth, Ferromagnetism, D. Van Nostrand, New York, 1951. 19. M. J. Dapino, R. C. Smith, L. E. Faidley, and A. B. Platau, A coupled structural–magnetic strain and stress model for magnetostrictive transducers. Department of Aerospace Engineering and Engineering Mechanics, Iowa State University, 2019 Black Engineering Building, Ames, IA 50011. 20. S. P. Parker, Editor-in-Chief, McGraw-Hill Dictionary of Scientific and Technical Terms, Fifth Edition, McGraw-Hill, Inc., New York, 1994. 21. E. C. Kunz and D. Ensminger, Method and Apparatus for Treating Mica, U.S. Patent No. 2,798,673, July 9, 1957. 22. V. Neufeldt and D. B. Guralnik, Editors-in-Chief, New World Dictionary of American English, Third College Edition, Webster’s New World, New York. 23. A. F. Puchstein and T. C. Lloyd, Alternating Circuit Machines, John Wiley & Sons, Inc., New York and London, 1942, p. 163. 24. T. T. Hansen, Magnetostrictive materials and ultrasonics, Chemtech, August 1996, 56–59. 25. G. Engdahl, Handbook of Giant Magnetostrictive Materials, Academic Press, San Diego, CA, 2000.

CRC_DK8307_CH005.indd 272

10/27/2008 3:30:49 PM

6 Pneumatic Transducer Design Data Dale Ensminger

CONTENTS 6.1 Introduction ........................................................................................................................ 273 6.2 Basic Principles of Whistles .............................................................................................. 274 6.3 Types of Whistles ............................................................................................................... 274 6.3.1 Common Whistles.................................................................................................. 274 6.3.2 Galton Whistles ...................................................................................................... 275 6.3.3 Basic Theory of the Hartmann Whistle .............................................................. 275 6.3.3.1 The Stem-Jet Whistle ............................................................................... 277 6.3.3.2 Construction of the Stem-Jet Whistle ................................................... 278 6.3.3.3 Modification of the Hartmann Whistle ............................................... 279 6.3.3.4 The Pin-Jet Whistle ................................................................................. 280 6.3.3.5 Some Special Applications for the Whistle ......................................... 280 6.3.4 Vortex Whistles ...................................................................................................... 282 6.3.5 Vibrating Blade ....................................................................................................... 282 References .................................................................................................................................... 283

6.1

Introduction

Several means of generating sound and ultrasound have been in existence for many years. Among the earliest were vocal chords, tongue, lips, and teeth. Man early learned to make whistles and whistle-like musical instruments, such as fifes. The modern industrial use of ultrasonics is commonly related to Galton’s development of moderately high-power ultrasonic whistles, which he described in 1883. The frequency of Galton’s whistle could be varied over a wide range, from sonic to ultrasonic, by varying the distance from a nozzle to the edge against which a jet of air or other gas from the nozzle blasted. In the late 1920s, Hartmann conducted studies, following those of Galton, which resulted in the device commonly called the Hartmann whistle. This whistle is described in an article published in 1939 [1]. Further studies with Truds Φ led to the development of the stem-jet whistle. This whistle operates over a wide range of fluid flow rates, from subsonic to supersonic. It operates best at air pressures of 3–6 atm. Additional modifications of this whistle have been studied, leading to several patents. R. M. G. Boucher patented [2] an improvement in the stem-jet whistle, involving an empirically determined cup-to-nozzle ratio for higher efficiency. In 1957, E. Brun and Boucher published an excellent article covering the design of the stem-jet whistle [3]. These units have been used for many different applications, such as breaking up foams, coating small particles separately, and drying heat-sensitive 273

CRC_DK8307_CH006.indd 273

10/24/2008 7:42:08 AM

Ultrasonics: Data, Equations, and Their Practical Uses

274

materials. In spite of the fine results obtained, the stem-jet whistle has failed to develop into a widely accepted commercial unit. However, its effectiveness in breaking up foams in commercial systems alone is considered to be justification for including it here. The efficiency of a well-designed and well-built stem-jet whistle may be as high as 14%. The sound emitted is sufficient to cause hearing damage to the unprotected ear. Therefore, these whistles usually are operated within soundproof structures and personnel within hearing distance are provided with suitable protection. Data are included to help the prospective user construct his own whistle.

6.2

Basic Principles of Whistles

In general, a whistle is a device that generates acoustic disturbances caused by a stream of air or other fluid in a medium. The character of the disturbance is determined by the nature of the fluid, its motion, and by any element that shapes or obstructs its flow. The common whistle, the Galton whistle, the Hartmann whistle, the vortex, and the jet-edge whistle will be discussed.

6.3 6.3.1

Types of Whistles Common Whistles

Whistles are common to all mankind. There are many types, but what is common to all is a jet of air or other fluid and usually a resonance element. A child’s whistle and a police whistle are examples of the simplest types (Figure 6.1). The sound emitted from such a whistle is due to the edge-tone caused by the jet of air (or gas) impinging on the knife-edge, a. The jet is split by the edge, causing some of the air to be deflected to the outside and some to follow the inside surface. As it returns to the incoming jet, the flow on the lowflow side increases the deflection of the jet to the outside of the obstruction. This deflection leaves less air to flow inside the cavity. When this reduced flow reaches the incoming jet, it exerts lower deflecting force and the jet now feeds more air to the low-flow side. The cycle is then repeated. This condition produces pulsations, which transmit sound waves, the frequency of which are dependent primarily upon the dimensions of the cavity. Air out

Air in

Cavity FIGURE 6.1 Cross-sectional view of a simple whistle.

CRC_DK8307_CH006.indd 274

10/24/2008 7:42:09 AM

Pneumatic Transducer Design Data

275

This common whistle is included here to illustrate a basic principle of whistle design. It is considered a commercial unit, because of the applications with children, police, or with situations calling for attention, such as in games. 6.3.2

Galton Whistles

Galton generated sound and ultrasound by forcing air through a slit onto the sharp edge of an object facing the slit [4,5]. The Galton whistle is a type of cavity resonator. It includes a nozzle with an annular slit located at the exit end of the nozzle and a cavity located a short distance away. Air blowing from this annular slit encounters a circular knife edge at the near end of the cavity. A micrometer is used to adjust the depth of the cavity and another adjusts the distance between the slit and the knife edge. The frequency depends on the spacing between the slit and the knife edge, a; the depth of the cavity, b; and the jet pressure, p (Figure 6.2). The Galton whistle made possible the generation of ultrasound in air over a wide range of frequencies. Its major disadvantage is a very low efficiency. Researchers began working to improve the performance of whistles in the twentieth century. 6.3.3

Basic Theory of the Hartmann Whistle

The Hartmann whistle includes a nozzle from which a jet of air is directed into a cavity or chamber (Figure 6.3). This is another case of cavity resonance. In constructing his whistle, Hartmann mounted the nozzle and cavity bodies rigidly with respect to one another and bored them out together, making the opening in the nozzle and that of the cavity of equal

Knife edge

Cavity

a

Air in

b

Piston

Nozzle FIGURE 6.2 The Galton whistle.

D1

D2

Air in

a

b

FIGURE 6.3 Modified Hartmann whistle.

CRC_DK8307_CH006.indd 275

10/24/2008 7:42:10 AM

Ultrasonics: Data, Equations, and Their Practical Uses

276

diameter. The lip of the cup was formed to a sharp edge. The spacing between the cup and the nozzle could be changed by an adjustment screw on either the cup or the nozzle, causing slight changes in the output frequencies. The maximum efficiency of this whistle was about 5%. For a whistle in which the diameters of the jet and the cavity, D, are equal, the frequency is f

c 4(b
0.3D)

(6.1)

The wavelength of the resulting sound is approximately

= 4(b + 0.3D2)

(6.2)

where b is the depth of the cavity and the quantity 0.3D2 an end correction factor. D2 is the diameter of the lip of the resonator. The axial alignment between the nozzle and the cavity is critical. According to Hartmann [1], the relationship between wavelength λ, nozzle and resonator diameter D1 = D2 = b, excess gas pressure p, and nozzle-to-resonator spacing a, is

/D1 = /D2 = 5.8 + 2.5{a/D1 − [1 + 0.041(p − 0.9)2]}

(6.3)

where p is in atmospheres and a, b, D1, and D2 are in millimeters. Hartmann shows that “a” may vary within limits a0 and am given by a0 = [1 + 0.041(p − 0.9)2] D2

(6.4)

(am − a0)/D2 = 0.13(p − 0.9)2

(6.5)

and

Within the range a0 to am, the wavelength increases nearly linearly at a rate d /da = 2.5

(6.6)

The permissible range over which the efficiency is maximum is somewhat less than that given by Equation 6.5 and is more nearly (a 2  a0 )/ D2  0.44 p  1.8

(6.7)

Hartmann claims that the radiated power, P, in watts is P  3D 22 p  0.9

(6.8)

Efficiency, η, is 

CRC_DK8307_CH006.indd 276

3D 22 p  0.9 P0  Pj Pj

(6.9)

10/24/2008 7:42:11 AM

Pneumatic Transducer Design Data

277

where Pj is the theoretical power required to maintain the jet. Cleanliness affects the efficiency. A thin film of grease, dirt, or liquid in the cup can lower the efficiency by a considerable amount. Variations of the design have been aimed at improving the efficiency and decreasing the airflow. One such variation was a change in the nozzle geometry, using a conical exit to increase the pressure exerted by the fluid at the orifice. The increased pressure causes the fluid to exit at supersonic velocity. This creates shock waves, which in the surrounding circumstances generate sonic or ultrasonic waves by resonance in the jet and the cavity. The frequency generated is a function of the diameter of the jet and that of the cavity and of the depth of the cavity. The exit angle of the nozzle must be gradual enough to enhance the increasing velocity of the emission of the fluid. An angle of approximately 30° or less with the axis should be adequate. An angle of 90° (i.e., an orifice) would create no buildup of velocity and the emerging fluid would not exceed that of the upstream flow rates. Claims for this device are that 1. It produces cavitation up to 60 kHz. 2. It produces as much as 50 W acoustic power at 20 kHz. 3. Its maximum efficiency is approximately 5%. The claim of cavitation to 60 kHz requires an explanation. Is this in water with the air blast deflected by a covering diaphragm? The logical conclusion seems to be that the whistle is aimed in a vertical direction downward into a glass beaker containing water at room temperature with a thin polymer sheet stretched across the top. The spacing between the whistle and the plastic cover at 60 kHz would be on the order of 0.5–10 cm (0.22–4 in.). Some improvement may occur by making the cavity-to-nozzle diameter >1. Boucher finds an improvement in efficiency by making the ratio D2/D1 ≥ 1.3. Ratios from 1.0 to 1.55 have proven to give good performance. 6.3.3.1

The Stem-Jet Whistle

The stem-jet whistle consists of a nozzle and a cavity connected by a stem holding the nozzle and the cavity in perfect axial alignment. The statement holding the nozzle and the cavity in perfect axial alignment is not an exaggeration. A slight misalignment will reduce the ability of the whistle to function at maximum efficiency. Any device added to correct the condition will at best interfere with the output of the whistle. The parts must be accurately machined and polished. The equations governing the design of the regular Hartmann whistle may also be applied to the stem-jet whistle. However, Boucher finds that over the interval a0 to am, dλ/da = 1.88. The stem-jet whistle operates at much lower pressures than the Hartmann whistle (output of 160 dB at a distance of 25 cm with an input pressure of 30 psi). Sonic radiation is proportional to (p − 0.3) for the stem-jet whistle. A whistle designed according to Figure 6.4 operates at approximately 10 kHz when the critical dimensions are Nozzle ID = 4.7625 mm (3/16 in.) Cup ID = 7.3152 mm (0.288 in.) Cavity depth = 7.3152 mm (0.288 in.) Cup OD = 9.525 mm (3/8 in.) Smaller diameter of stem = 2.3812 mm (3/32 in.)

CRC_DK8307_CH006.indd 277

10/24/2008 7:42:11 AM

Ultrasonics: Data, Equations, and Their Practical Uses

278

Stem Cup

Air hose connector

Body

Reflector/Nozzle

FIGURE 6.4 Adjustable stem-jet whistle.

Length of smaller diameter of stem = 30.1626 mm (1.1875 in. = 1–3/16 in.) Larger diameter of stem = 3.175 mm (1/8 in.) Total stem length = 61.9126 mm (2.4375 in. = 2–7/16 in.) 6.3.3.2

Construction of the Stem-Jet Whistle

The stem-jet whistle requires very careful and accurate machining, especially with respect to the alignment of the cavity, stem, and nozzle. The reflector and nozzle are in one solid member. The outer, large diameter is first machined to size. Then the reflecting surface is machined to the desired shape. The first step is to machine the back side of the reflector before cutting the threaded section. In Figure 6.4, the threaded section is 1.27 cm (1/2 in.) in diameter and 1.5875 cm (5/8 in.) long. It is best to thread this section before boring out the inner air passageway. The reflector may include any of several geometries. Those commonly used are the spherical section, the hyperbolic section, the elliptical section, and the catenoidal section. The general positions of the reflecting surfaces relative to the sources are a minimum distance of approximately one wavelength, according to Equation 6.2, from the center of the spherical section or less and from the foci of the elliptical and hyperbolic sections. The catenoidal reflectors have no foci, and the position of the source relative to the reflecting surfaces is determined more by judgment relative to the wavelength and the reflecting angle and the beams projected as sketched geometrically. These are all general guidelines, and the designer has some freedom in designing each type. The body consists of a cylindrical shell threaded inside to match the threads of the reflector and nozzle section. It contains a block to which the stem is screwed using fine threads (such as 5–40 threads) on the stem. The passage holes for the airflow are also drilled through this block, close to the inner wall of the body. The cup of Figure 6.4 is machined with the cavity depth equal to the cavity diameter. The small end of the stem is lightly silver-soldered from the outside end of the cup. The parts are then screwed together and the alignment is very carefully checked. A suitable air hose connector is silver-soldered to the open end of the cylindrical section. The connection is carefully machined to the diameter of the body. After tuning the system, the setscrew is screwed in snugly to prevent any movement in the parts. The whistle is now ready to test and use. The individual parts are shown in Figure 6.5.

CRC_DK8307_CH006.indd 278

10/24/2008 7:42:11 AM

Pneumatic Transducer Design Data

279

Stem

Cup Reflector Setscrew

Body

Air hose connector FIGURE 6.5 Individual parts of the basic stem-jet whistle.

Nozzle

Stem Cavity

Nozzle cavity FIGURE 6.6 Added jet-edge aspect to the nozzle of the Hartmann whistle.

The previous design procedure is given only as an example. Changes in dimensions are possible, but similar care in construction must be taken to be certain that the components are well aligned. 6.3.3.3

Modification of the Hartmann Whistle

In this case, Hartmann whistle refers to modifications of the stem-jet whistle. One such modification refers to an addition to the nozzle section, in which a recess is machined to increase the jet-edge aspect of the sound emitted (see Figure 6.6). It is hoped that this addition will increase the efficiency of the whistle.

CRC_DK8307_CH006.indd 279

10/24/2008 7:42:11 AM

Ultrasonics: Data, Equations, and Their Practical Uses

280

If the nozzle diameter and the cavity diameter are equal, the diameter of the recess might be on the order of 1.3 D. For larger cavity diameters, a good choice might be to increase the diameter of the recess by at least 1.3 times the diameter of the cavity. 6.3.3.4

The Pin-Jet Whistle

The pin-jet whistle is another modification of the Hartmann whistle claimed to give higher efficiencies. In this case, a pin extending through the resonator and into the open end of the nozzle replaces the stem. The end of the pin is beveled with an included angle of 20–30°. Again, alignment of the axes of the pin, the nozzle, and the resonator is very important for efficient operation of the whistle. However, efficiencies claimed for properly designed and assembled pin-jet whistles are higher than efficiencies obtained using the stem-jet whistles. Equations used in designing the stem-jet whistle are also applicable to the pin-jet whistle. 6.3.3.5

Some Special Applications for the Whistle

6.3.3.5.1 Coating Very Fine Particles Tubing, such as copper, may be placed through the reflector of a whistle for supplying materials for coating small particles with polymeric materials. In Figure 6.7, using the same design as that of Figure 6.4, feed tubes are located on two circular centerlines through the reflector. These centerlines are of two different diameters. The inner-diameter centerline locates the feed tubes for a liquid polymer to be atomized for coating the particles and the outer ones are for the particles to be coated. Both sets of feed tubes are aimed axially at the outer circumference of the resonant cavity. The sonic energy atomizes the fluid into a very fine mist, which surrounds and coats the suspended solid particles and enables the polymer to quickly set before the coated particles settle out of suspension. This sonic method has been used to produce individually coated particles of very fine size at a reasonable rate. An example of past experience in coating individually spherical particles, 5 μm in diameter, with a 1 μm coating of polymer (Figure 6.8). This method appears to be a unique way of accomplishing this requirement. 6.3.3.5.2 Breaking Down Foams The stem-jet whistle is effective in breaking down certain foams. Foams consist of bubbles. To break them down, the sonic intensity must produce a tension in the bubble surfaces exceeding the tension required to break each bubble. This calculation takes into account the surface tension of the liquid from which the bubble is formed and its viscosity. Pin A logical explanation of the effectiveness of ultrasonics for this purpose appears to be as follows: when an ultrasonic wave impinges on a foam, each bubble in its path experiences motion. The motion includes two types: tensile and shear. Tensile shear comes from the longitudinal motion within the wave. The bubble undergoes compression and rarefaction under the influence of this motion, which produces tension and compression within the bubble wall. These stresses Cavity Nozzle are parallel to the wall of the bubble. It also produces tension across the bubble wall, that is, normal to the FIGURE 6.7 bubble wall in the path of the impinging wave. Pin-jet whistle.

CRC_DK8307_CH006.indd 280

10/24/2008 7:42:12 AM

Pneumatic Transducer Design Data

281 Particle feed tube

Stem

Body Air hose connection

Resonant cavity Coating feed tube

FIGURE 6.8 Sonic whistle for individually coating very small solid particles with a thin layer of polymer.

Compression/rarefaction

Shear

Tension/rarefaction FIGURE 6.9 Stresses developed in bubbles in a foam by ultrasonic waves.

The shear motion is due to the movement of a bubble under the influence of the ultrasound relative to its neighbor, which otherwise may be less or not at all influenced by the ultrasound. The result is a shear force acting on the bubbles. The viscosity plays a prominent part here. Therefore, three stresses are tending to disrupt the continuity of the bubble wall, leading to the bubble’s collapse: two tensile and one shear. Collapse occurs when the total of these forces exceeds the strength of the wall. The action of these stresses is illustrated in Figure 6.9. The assumption is made that the air blast producing the sonic energy is deflected from the foam by a protective film and that only the sonic energy is acting on it. A typical output intensity of an efficient 10 kHz whistle is 150 dB. This is approximately 6600 dynes/cm2 in air. If the surface tension of the liquid exceeds the output stress of the whistle, it is unlikely that the stress of the sound wave will break up the foam. 6.3.3.5.3 Other Means of Increasing Power from Hartmann-Type Whistles The Hartmann-type whistles, including the stem-jets and the pin-jets are ideal for multipleunit applications, that is, two or more whistles mounted in parallel to increase the total output power.

CRC_DK8307_CH006.indd 281

10/24/2008 7:42:12 AM

Ultrasonics: Data, Equations, and Their Practical Uses

282 6.3.4

Vortex Whistles

Vortices are problems that engineers must consider in many areas of industry such as aircraft, turbines, and wind tunnels. The theory will be simplified by considering vortices of two types: forced and free. Forced vortices are the type observed in a cylindrical tube filled with a fluid when it is spun. Free vortices are the type observed in the surface of a flowing river or in a bathtub when it is being drained. A vortex whistle uses the free vortex. A vortex whistle is designed to produce a vortex in a gas or liquid and to take advantage of the sound it produces. Figure 6.10 shows schematically the construction of such a device. A free vortex is formed when a fluid escapes from a larger volume through a small opening. It is maintained by the conservation of angular momentum. Angular momentum is the product of the momentum of a particle by its distance from the axis of rotation. Momentum is defined as the mass times velocity, that is, Mv. The angular momentum is therefore Mvr, where M is the mass of a particle, v its velocity, and r the distance from the axis of rotation. To preserve its angular momentum, as a particle draws nearer to the axis, it increases speed. Velocity and radius are the only variables. Variations in design from Figure 6.10 to increase the output intensities of the vortex whistles are possible. One method might be to use a conical or exponential shape from the inlet volume to the exit (Figure 6.11). The main advantage of a vortex whistle appears to be the possibility of using this design with either gases or liquids. 6.3.5

Vibrating Blade

A system has been commercially available based upon creating cavitation in a liquid using a flexural resonant blade. A narrow jet of fluid impinges on the sharp edge of the blade, causing it to vibrate in flexural resonance. The basics of this design are shown schematically in Figure 6.12. The fluid is ejected from a slit in a nozzle. The slit is narrow and long enough that the fluid impacts the full height of the blade. The slit should be perfectly parallel with the blade. Spacing between the blade and the nozzle may range from as small as the width of the slit to about four times the thickness of the blade. A blade, either one-quarter wavelength or

Air in Sound out

Hollow cylinder FIGURE 6.10 Vortex whistle. Air in Sound out

Hollow cylinder

Funnel

FIGURE 6.11 Conical vortex whistle.

CRC_DK8307_CH006.indd 282

10/24/2008 7:42:12 AM

Pneumatic Transducer Design Data

283 Slotted nozzle

Resonant blade

Fluid in

a. Top view, quarter-wave plate

Resonant blade Fluid in λ /4

λ /4

b. Top view, full-wave unit Resonant blade

Fluid in

λ/4 c. Side view, quarter-wave plate FIGURE 6.12 Liquid blade whistle.

one full wavelength long, is rigidly mounted parallel to the direction of the fluid’s flow. The quarter-wavelength blade is mounted at the first nodal position. The mounting must impede the flow of the fluid in a minimal amount. The full-wavelength blade is mounted at the two nodal positions by heavy pins. In either case, the mounting must be strong enough to resist the flow pressures and the forces of vibration. Typically, the source pressures are 12.5–20.7 kg/cm2 (150–250 psi) at flow rates of 45–60 m/s (150–200 ft./s). The design of the blade is critical not only from the standpoint of resonance characteristics (low damping), but also for fatigue life. A well-designed blade type of whistle should be useful for emulsification and homogenization on an industrial scale.

References 1. T. Hartmann, Journal of Scientific Instruments, 16, 1939, 146. 2. R. M. G. Boucher, U.S. Patent 2,800,100, July 23, 1957. 3. E. Brun and R. M. G. Boucher, Research on the acoustic air-jet generator: A new development, Journal of the Acoustical Society of America, 29(5), 573, May 1957. 4. J. Blitz, Fundamentals of Ultrasonics, Butterworths, London, 1963, p. 72. 5. L. Bergmann, Ultrasonics, John Wiley & Sons, Inc., Newyork, 1938, p. 2.

CRC_DK8307_CH006.indd 283

10/24/2008 7:42:12 AM

CRC_DK8307_CH006.indd 284

10/24/2008 7:42:13 AM

7 Properties of Materials Dale Ensminger

CONTENTS 7.1 Introduction ........................................................................................................................ 286 7.2 Significance of Physical Properties of Tables 7.3 through 7.5 ...................................... 286 7.3 Ultimate Stress and Fatigue Limit................................................................................... 299 7.4 Materials Properties for Processing by Ultrasonics......................................................300 7.4.1 Friction and Wear ...................................................................................................300 7.4.2 Welding and Staking ............................................................................................. 302 7.4.2.1 Properties Necessary for Ultrasonic Welding and Forming of Materials ...................................................................... 302 7.4.2.2 Joint Designs for Efficient Welding .......................................................304 7.4.3 Tools for Ultrasonic Welding ................................................................................308 7.4.4 Staking .....................................................................................................................309 7.4.4.1 Hydraulic Speed Control ........................................................................ 310 7.4.5 Five Basic Staking Designs .................................................................................... 310 7.4.5.1 The Standard Rosette Profile Stake ....................................................... 311 7.4.5.2 The Dome Stake ....................................................................................... 311 7.4.5.3 The Hollow Stake ..................................................................................... 311 7.4.5.4 The Knurled Stake ................................................................................... 312 7.4.5.5 The Flush Stake ........................................................................................ 312 7.5 Stud Welding ...................................................................................................................... 313 7.6 Insertion .............................................................................................................................. 313 7.7 Swaging and Forming ....................................................................................................... 314 7.8 Spot Welding ...................................................................................................................... 315 7.9 Degating .............................................................................................................................. 316 7.10 Scan Welding ...................................................................................................................... 316 7.11 Bonding and Slitting ......................................................................................................... 317 7.11.1 Ultrasonic Bonding ................................................................................................ 317 7.11.2 Ultrasonic Slitting .................................................................................................. 317 7.12 Other Tool Designs and Application Methods .............................................................. 317 7.12.1 Fixed-Strap Bonding .............................................................................................. 318 7.12.2 Continuous Rolling Contact ................................................................................. 318 7.13 Metals Properties for Choice of Horns [1–3,6] ............................................................... 318 7.13.1 Horn Materials........................................................................................................ 318 7.13.2 High-Strength Low-Alloy Steels .......................................................................... 319 7.13.3 Ductile (Nodular) Iron ........................................................................................... 319 7.13.4 Hard-Facing Wear Surfaces .................................................................................. 319 7.13.5 Stainless Steels (ASM Hdbk)................................................................................. 320 285

CRC_DK8307_CH007.indd 285

10/27/2008 2:27:59 PM

Ultrasonics: Data, Equations, and Their Practical Uses

286

7.14 Important Refractory Metals............................................................................................ 320 7.15 Comments and Conclusions ............................................................................................ 321 References .................................................................................................................................... 321

7.1

Introduction

Data are presented in this chapter for selecting materials to be used in the wide variety of applications of ultrasonic energy. These properties include moduli, Poisson’s ratio, densities, fatigue limits and ultimate stresses of the materials, sonic velocities, characteristic acoustic impedances, melting points, boiling points, electrical resistivities, thermal conductivities, compatibilities between materials, wear resistances, effects of oxide protective coatings, toughness or brittleness, resistances to acids and bases, corrosion resistance, protective atmospheres, chemical resistance, effects of stress risers, weldability, solderability, and effects of positions of welds and solders in an ultrasonically activated structure. Table 7.1 lists the values and symbols of prefixes used in weights and measures recommended by the National Bureau of Standards [1]. These symbols will be used in all later references to these quantities. Table 7.2 gives the atomic numbers of the elements listed in alphabetical order [2,6]. This table will be useful in locating various elements in Table 7.3 as needed. Table 7.3 gives the atomic numbers, atomic weights, densities, melting points, and boiling points of the elements listed according to their atomic numbers [2,6]. Tables 7.4 and 7.5 list the physical properties of engineering materials useful in the design of ultrasonic apparatus [2–4].

7.2

Significance of Physical Properties of Tables 7.3 through 7.5

The equations of Chapter 2 demonstrate the importance of the modulus of elasticity, Poisson’s ratio, density, velocity of sound, and the characteristic acoustic impedance of materials. These are the significant quantities needed in the design of ultrasonic horns and systems used generally in the high-intensity applications of ultrasonic energy. TABLE 7.1 Prefixes of Weights and Measures

CRC_DK8307_CH007.indd 286

Prefix

Symbol

Multiple

Tera Giga Mega Kilo Hecto Deka Deci Centi Milli Micro Nano Pico

T G M k h dk d c m µ n p

1012 109 106 103 102 10 10−1 10−2 10−3 10−6 10−9 10−12

10/27/2008 2:28:00 PM

Properties of Materials

287

TABLE 7.2 Alphabetical Order and Atomic Numbers of Elements Element Actinium Aluminum Americium Antimony (stibium) Argon Arsenic Astatine Barium Berkelium Beryllium Bismuth Boron Bromine Cadmium Calcium Californium Carbon Cerium Cesium (caesium) Chlorine Chromium Cobalt Copper (cuprum) Curium Dysprosium Einsteinium Erbium Europium Fermium Fluorine Francium Gadolinium Gallium Germanium Gold (aurum) Hafnium Helium Holmium Hydrogen Indium Iodine Iridium Iron (ferrum) Krypton Lanthanum Lawrencium Lead (plumbum) Lithium Lutelium Magnesium Manganese Mendelevium Mercury

Symbol Ac Al Am Sb Ar As At Ba Bk Be Bi B Br Cd Ca Cf C Ce Cs Cl Cr Co Cu Cm Dy Es Er Eu Fm F Fr Gd Ga Ge Au Hf He Ho H In I Ir Fe K La Lr Pb Li Lu Mg Mn Md Hg

Atomic Number 89 13 95 51 18 33 85 56 97 4 83 5 35 48 20 98 6 58 55 17 24 27 29 96 66 99 68 63 100 9 87 64 31 32 79 72 2 67 1 49 53 77 26 36 57 103 82 3 71 12 25 101 80

Element Molybdenum Neodymium Neon Neptunium Nickel Niobium (columbium) Nitrogen Nobelium Osmium Oxygen Palladium Phosphorus Platinum Plutonium Polonium Potassium (kalium) Praesodymium Promethium Protoactinium Radium Radon Rhenium Rhodium Rubidium (wolfram) Ruthenium Samarium Scandium Selenium Silicon Silver (argentum) Sodium (natrium) Strontium Sulfur Tantalum Technetium Tellurium Terbium Thallium Thorium Thulium Tin (stannum) Titanium Tungsten Unipentium Unnihexium Unnilhexium Uniquadrium Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium

Symbol

Atomic Number

Mo Nd Ne Np Ni Nb N No On O Pd P Pt Pu Po K Pr Pm Pa Ra Rn Re Rh Rb Ru Sm Sc Se Si Ag Na Sr S Ta Tc Te Tb Tl Th Tm Sn Ti W Unp Uns Unh Unq U V Xe Yb Y Zn Zr

42 60 10 93 28 41 7 102 76 8 46 15 78 94 84 19 59 61 91 88 86 75 45 37 44 62 21 34 14 47 11 38 16 73 43 52 65 81 90 69 50 22 74 105 106 107 104 92 23 54 70 39 30 40

Source: Dean, J.E., Lange’s Handbook of Chemistry, McGraw-Hill, Inc., New York, 1994; Weast, R.C., CRC Handbook of Chemistry and Physics, CRC Press, Inc., Boca Raton, FL, 1985.

CRC_DK8307_CH007.indd 287

10/27/2008 2:28:00 PM

CRC_DK8307_CH007.indd 288

Hydrogen (H) Helium (He) Lithium (Li) Beryllium (Be) Boron (B) Carbon (C) Nitrogen (N) Oxygen (O) Fluorine (F) Neon (Ne) Sodium (Na) Magnesium (Mg) Aluminum (Al) Silicon (Si) Phosphorus (P) Sulfur (S) Chlorine (Cl) Argon (Ar) Potassium (K) Calcium (Ca) Scandium (Sc) Titanium (Ti) Vanadium (V) Chromium (Cr) Manganese (Mn) Iron (Fe) Cobalt (Co) Nickel (Ni) Copper (Cu) Zinc (Zn) Gallium (Ga) Germanium (Ge) Arsenic (As)

Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Atomic Number 1.00794 4.0026 6.941 9.012 10.81 12.011 14.007 16 18.998 20.179 22.9898 24.305 26.9815 28.0855 30.9738 32.06 35.45 39.95 39.098 40.08 44.96 47.88 50.942 51.996 54.938 55.847 58.94 58.69 63.546 65.39 69.72 72.59 74.92

Atomic Weight lb/in.

g/cm

3

0.8376 × 10−4 1.76 × 10−4 0.5259 1.68213 2.2794 2.23 1.163 × 10−3 1.33 × 10−3 1.67 × 10−3 9.0 × 10−4 0.9688 1.74 2.699 2.233 1.821 2.071 3.214 × 10−3 1.782 × 10−3 0.8581 1.5501 2.985 4.540 6.007 7.197 7.4182 7.8611 8.8575 8.9129 9.8683 7.1414 5.9789 5.3145 5.7297

Density

3.03 × 10−6 6.36 × 10−6 0.019 0.0668 0.0887 0.0802 0.042 × 10−3 0.0516 × 10−3 0.06 × 10−3 3.25 × 10−5 0.035 0.063 0.098 0.084 0.0658 0.0748 0.116 × 10−5 0.644 × 10−4 0.031 0.056 0.1078 0.164 0.217 0.26 0.268 0.284 0.32 0.322 0.324 0.258 0.216 0.192 0.207

3

Physical Properties of Elements Listed According to Atomic Number

TABLE 7.3

°F −434.6 −458 356.97 2,332.4 3,812 6,605.6 −345.8 −361.12 −363.32 −415.6 208.06 1,202 1,220.4 2,570 111.38 235.04 −149.76 −308.56 145.85 1,542.2 2,802.2 3,020 3,434 3,374.6 2,271.2 2,795 2,723 2,647.4 1,982.1 787.2 85.6 1,719.3 1,497

−259.14 −272.2 180.54 1,278 2,100 3,826 −209.9 −218.4 −219.6 −248.2 97.8 650 660.37 1,410 44.1 112.8 −100.98 −189.2 63.25 839 ± 2 1,539 1,660 ± 10 1,929 ± 10 1,857 ± 20 1,244 ± 3 1,535 1,495 1,453 1,083.4 ± 2 419.58 29.78 937.4 603 (@28 atm)

°C

Melting Point

2,500 5,000 4,622 8,730 −320.4 −297.3 −306.7 −410.9 1,621.2 2,030 4,442 4,271 536 832.41 −30.28 −302.26 1,420 2,703.2 5,129.6 6,395 5,430 4,841.6 3,863.6 4,982 6,420 4,949.6 4,652.6 1,664.6 4,357.4 5,126 1,135.4

°F

1,371 2,771 2,550 4,832 −195.8 −183 −188.14 −246.05 882.9 1,110 2,518 2,355 280 444.7 −34.6 −185.7 771 1,484 2,832 3,535 2,999 2,672 2,129 2,750 3,549 2,732 2,567 907 2,403 2,830 613

°C

Boiling Point

288 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 2:28:01 PM

CRC_DK8307_CH007.indd 289

Xenon (Xe) Cesium (Cs) Barium (Ba) Lanthanum (La) Cerium (Ce) Praseodymium (Pr) (α-form) Neodymium (Nd) Promethium (Pm) Samarium (Sm) Europium (Eu) Gadolinium (Gd) Terbium (Tb) Dysprosium (Dy)

Selenium (Se) Bromine (Br) Krypton (Kr) Rubidium (Rb) Strontium (Sr) Yttrium (Y) Zirconium (Zr) Niobium (Nb) (Columbium) Molybdenum (Mo) Technitium-95 (Tc) Ruthenium (Ru) Rhodium (Rh) Palladium (Pd) Silver (Ag) Cadmium (Cd) Indium (In) Tin (Sn) Antimony (Sb) Tellurium (Te) Iodine (I) 131.29 132.91 137.33 138.906 140.12 140.91 144.24 (145) 150.3 151.96 157.25 158.93 162.5

60 61 62 63 64 65 66

95.94 (98) 101.07 102.906 106.42 107.868 112.48 114.82 118.71 121.75 127.60 126.91

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

78.96 79.904 83.80 85.468 87.62 88.91 91.224 92.906

34 35 36 37 38 39 40 41

0.2533 0.26084 0.270 0.1895 0.287 0.2793 0.3085

2.08 × 10−4 0.06787 0.13 0.223 0.244 0.2339

0.369 0.3974 0.4498 0.4495 0.434 0.379 0.313 0.264 0.264 0.245 0.225 0.178 (@25°C)

0.174 0.113 1.355 × 10−4 0.554 × 10−1 0.0954 0.1616 0.2355 0.310

7.01 7.22 7.52 5.244 7.9441 8.23 8.54 (@ 25°C)

5.61 × 10−3 1.8785 3.598 6.1726 6.770 6.475

10.2139 11.0 12.45 12.4421 12.0131 10.4967 8.6638 7.3075 7.3075 6.65 6.228 4.94

4.8163 3.12 3.749 × 10−3 1.532 2.64 4.472 6.52 8.5807

1,850 1,976 1,961.6 1,511.6 2,391.8 2,480 2,568.2

−183.2 83.12 1,337 1,688 1,468.4 1,707.8

4,742.6 3,941.6 4,190 3,569 2,829.2 1,763.5 609.62 313.9 449.54 1,167.33 841.1 236.3

422.6 19.04 −249.88 102 1,416.2 2,773.4 3,365.6 4,474.4

1,010 1,080 1,072 822 1,311 ± 1 1,360 ± 4 1,409

−111.9 28.4 725 920 798 ± 2 931

2,617 2,204 2,310 1,965 1,554 961.93 320.9 156.61 231.97 630.74 449.5 113.5

217 −7.2 −156.6 38.89 769 1,523 ± 8 1,852 2,468 ± 10

5,660.6 4,460 3,232.4 2,906.6 5,851.4 5,505.8 4,235

−160.8 1,236.74 2,984 8,000 5,894.6 5,813.6

8,333.6 8,810.6 7,052 8,100 7,200 4,013 1,409 3,776 4,118 2,620 2,530 363.83

1,264.8 137.8 −242.1 1,266.8 1,523.2 6,038.6 29,030 5,970

(continued)

3,127 2,460(?) 1,778 1,597 3,233 3,041 2,335

−107.1 669.3 1,640 4,427 3,257 3,212

4,612 4,877 3,900 4,482 3,982 2,212 765 2,080 2,270 1,438 1,388 184.4

685 58.78 −152.3 686 1,384 3,337 4,999 3,299

Properties of Materials 289

10/27/2008 2:28:01 PM

CRC_DK8307_CH007.indd 290

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

Osmium (Os) Iridium (Ir) Platinum (Pt) Gold (Au) Mercury (Hg) Thallium (Ti) Lead (Pb) Bismuth (Bi) Polonium (Po)

Astatine (At) Radon (Rn) Francium (Fr) Radium (Ra) Actinium (Ac) Thorium (Th)

Protactinium (Pa) Uranium (U) Neptunium (Np) Plutonium (Pu) Americium (Am)

Atomic Number

(Continued)

Holmium (Ho) Erbium (Er) Thulium (Tm) Ytterbium (Yb) Lutetium (Lu) Hafnium (Hf) Tantalum (Ta) Tungsten (W) Rhenium (Re)

Element

TABLE 7.3

231.036 238.029 237.048 244 (243)

(210) (222) 223 226.025 227.028 232.038

190.2 192.23 195.08 196.967 200.59 204.38 207.2 208.98 (209)

164.93 167.26 168.934 173.04 174.967 178.49 180.948 183.85 186.21

Atomic Weight lb/in.

0.687 0.740 0.700 0.4285

0.422

0.18

0.3956

0.813 0.813 0.775 0.698 0.4896 0.428 0.4097 0.354 — 0.339

2.45 0.330 0.336 0.161 0.356 0.473 0.600 0.697 0.765

3

15.37 19.0161 20.5 19.4 12

22.5038 22.5038 21.4519 19.32 13.5521 11.847 11.3405 9.7987 9.196α 9.398β 7.0 9.73 g/L 5.20 g/L 4.98 10.07 11.6509

3

g/cm 6.18 9.15 9.31 4.472 9.85 13.0926 16.6079 19.2929 21.1751

Density °F

2,912 2,070 1,184 1,229 1,821.2

575.6 −95.8 80.6 1,292 1,922 3,182 (Approximately)

4,900 4,370 3,221.6 1,948 −37.966 578.3 621.5 530.34 489.2

2,678 2,776.6 2,813 1,515.2 3,012.8 4,040.6 5,424.8 6,170 5,756

1,575 1,134 637 ± 1 640 1,176 ± 4

302 −71 (27) 700 1,050 1,758

3,033 2,449 1,769 1,064.4 −38.87 303.5 327.5 271.3 254

1,470 1,522 1,545 ± 15 824 ± 5 1,656 ± 5 2,227 ± 20 3,020 3,410 3,189

°C

Melting Point °F

6,904.4 7,055.6 5,849.6 4,724.6

638.6 −79.24 1,250.6 2,084 5,792 8,654

4,928 5,550 3,140.6 2,179.4 5,999 9,700 9,797 10,220 10,160.6 (Est) 9,080.6 9,600 7,970 5,380 673.8 2,654.6 3,164 2,590 1,763.6

3,818 3,902 3,232 2,011

337 −61.8 677 1,140 3,200 4,790

5,027 5,316 4,410 2,971 356.6 1,457 1,740 1,421 962

2,720 2,510 1,727 1,193 3,315 5,371 5,425 5,660 5,627

°C

Boiling Point

290 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 2:28:01 PM

CRC_DK8307_CH007.indd 291

98 99 100 101 102 103 104 105 106 107

96 97 (251) (252) (257) (258) (259) (260) (261) (262) (263) (262)

(247) (247) 0.18

0.31

7 14.78α 13.25β 4.9824 8.84

2,444 983 840 820 (1,527) (827) (827) (1,627)

1,340 ± 40

Source: Dean, J.E., Lange’s Handbook of Chemistry, McGraw-Hill, Inc., New York, 1994; Ensminger, D., Ultrasonics, Marcel Dekker, Inc., New York, 1988; Weast, R.C., CRC Handbook of Chemistry and Physics, CRC Press, Inc., Boca Raton, FL, 1985.

Note: Equivalent values: F = (9C/5) + 32. C = 5(F − 32)/9. 1.0 in. = 2.54 cm. l.0 lb = 0.45359237 kg. 1 kg = 2.204622622 lb. 1 lb/in.2 = 70.306956 g/cm2 = 0.070306956 kg/cm2. 1.0 kpsi = 70.30695796 kg/cm2. 1 lb/in.3 = 27.67990471 g/cm3. 1 g/cm2 = 10 kg/m2. 1 L = 1000 cm3 = 0.001 m3 = 0.3048 gal. 1 gal = 3.785411784 L. 1 qt. = 0.946352946 L. Atomic number = the number of protons in an atomic nucleus. Atomic weight = the relative mass of an atom based on a scale in which a specific carbon atom (carbon-12) is assigned a value of 12. Molecular weight = the total of the atomic weights of the elements in a molecule. Fatigue limit (endurance limit) = maximum value of repeated stress which will not produce failure regardless of the number of applied cycles.

Californium (Cf) Einsteinium (Es) Fermium (Fm) Mendelevium (Md) Nobelium (No) Lawrencium (Lr) (Uniquadrium) (Unq) (Unnipentium) (Unp) (Unnihexium) (Uns) (Unnilhexium) (Unh)

Curium (Cm) Berkelium (Bk)

Properties of Materials 291

10/27/2008 2:28:01 PM

CRC_DK8307_CH007.indd 292

Aluminum Aluminum 2SO 17ST 2024-T4 ALCLAD 2024-T4 2025-T6 6061-T6 7075-T6 XAPOO1 Antimony Arsenic Barium Beryllium Bismuth Boron Brass 70–30 Naval Bronze (phosphor 5%) Bronze (7.5% In) Bronze (7.7% Al) Cadmium Calcium Carbon Cerium Chromium Cobalt

Materials 0.703 0.724

0.731 0.745 0.724 0.724 0.7944 0.7733 0.127 2.95 0.3234

0.914 0.914

0.5624 0.211 0.049 2.5308

10.3

10.4

10.6 10.3 10.3 11.3 11 1.8 40–44 4.6

13 13

8 3 0.7

36 30

kg/cm (×106)

2

10

psi (×10 )

6

Young’s Modulus

5 5

4 4 4 4 4

4 4

6

0.352 0.352

1.85

0.024–0.030

0.35 0.331 0.33 0.35

0.33 0.33 0.33

0.33

0.33 0.33

0.33

Poisson’s Ratio

0.281 0.281 0.281 0.281 0.281

0.281 0.281

kg/cm2 (×106)

Shear Modulus psi (×10 )

Physical Properties of Engineering Materials

TABLE 7.4

1.55 2.22 6.92 7.197 8.86

6.70 5.73 3.626 33–51 9.80 2.297 8.50 8.60 8.10 8.86

2.77 2.77 2.70 2.79

2.71 2.80

2.7

Density g/cm3

844

12 8.60

1,476

490–1,406

7–20 7–20 7–20 21

2,320–3,586

2,883 1,266 949 1,687 37

1,406

20 41 18 13.5 24

562.5–1,266

kg/cm

2

8–18

kpsi

Fatigue Limit

60 35

64 58 45 82 2,601

22

kpsi

4,218 2,461

4,500 4,078 3,164 5,765

1,5467

kg/cm2

Ultimate Strength

292 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 2:28:02 PM

Columbium (niobium) Constantan (60% Cu, 40% Ni) Copper Annealed Rolled Duralumin 17S Gadolinium Gallium Germanium German silver Gold, pure Hafnium Hastelloy X Hastelloy C Hydrogen Inconel X Wrought Indium Iridium Iron Electrolytic Armco Iron Cast Wrought Lanthanum Lead Pure Annealed Rolled Antimony 6% Lithium Magnesium Am35 Drawn Annealed

CRC_DK8307_CH007.indd 293

1.1248

0.0703 0.8014

1.406

0.1104 5.2725 2.0036

0.3515 0.1828

0.1195 0.35

17

1 11.4

10.8 20

1.57 75 28.5

5.0

2.0

1.7

6.4

1.0545

15

0.35

0.35

0.40–0.45

1.74

11.30 11.40 11.40 10.90 0.526 1.738 1.74

7.20 7.80 6.17

0.28 0.28

0.42

8.90 8.93 8.93 2.79 7.94 5.98 5.31 8.40 19.32 13.09 8.23 8.94 3.026 × 10−6 8.30 8.25 7.31 22.40 7.80–7.90 7.90 7.85

0.35

8.80

37

7–17 37

6–18

12–17 12–17 12–17

2,614

492–1,195 2,601

422–1,266

844–1,195 844–1,195 844–1,195

18

(continued)

1,266

Properties of Materials 293

10/27/2008 2:28:02 PM

CRC_DK8307_CH007.indd 294

— 21.3 14.0

Phosphorus Platinum Plutonium (alpha phase) Polonium

5.624

— 80

1.4763

1.1951

2.109

30

17

3.515

40–50

Palladium

1.6169

kg/cm (×106)

2

23

psi (×10 )

6

Young’s Modulus

(Continued)

Manganese Manganin (84% Cu) Manganin (12% Mn, 4% Ni) Mercury Molydenum Monel Metal (67% Ni, 29.2% Cu, 1.7% Fe, 1.0% Mn) Monel Wrought Neodymium Neon Neptunium Nickel, pure Nickel Silver (64% Cu, 17% Zn, 18% Ni) Nitrogen Osmium Oxygen

Materials

TABLE 7.4

psi (×10 )

6

kg/cm2 (×106)

Shear Modulus

0.39 0.15–0.21

0.31

0.32 0.315

Poisson’s Ratio

9.196α 9.398β

1.82 21.45 19.0–19.7

1.163 × 10−3 0.813 1.3286 × 10−3 12.013

8.83 7.01 0.8999 g/L 20.2 8.8–8.9

13.582 10.3 8.90

7.41 8.40

Density g/cm3

80 20–50

kpsi

2

5,625 1,406–3,515

kg/cm

Fatigue Limit

20–24 60

120–200

kpsi

1,406–1,687 4,218

8,437–14,060

kg/cm2

Ultimate Strength

294 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 2:28:02 PM

Potassium Radium Rhenium Rhodium Rubidium Ruthenium Samarium Scandium Selenium Silicon Silver, pure Silver–nickel 18% Sodium Steel 1% Carbon 1% C hardened HY80 Mild Stainless 302 Stainless 304 Stainless 304L Stainless 347 Stainless 410 Stainless 416 Steel 4130 Steel 4340 Strontium Sulfur α γ Tantalum Tellurium Terbium Thallium

CRC_DK8307_CH007.indd 295

0.5905 1.1248 0.7733

0.0914

8.4 16 11

1.3

— 1.8981 0.4219 0.0844

—

27 6

1.2

28

5.2725 3.7962

0.0352

0.5 — 75 54

0.35

0.298

0.29 0.29

0.28–0.29

0.37

0.9688 7.70 7.84 7.84 7.76 7.85–7.90 7.9 7.9 8.03 7.90 7.67 7.70 8.0 8.0 2.64 2.07 1.92 16.60 6.24 8.3 11.85

0.858 5.5 21.175 12.442 1.532 12.45 7.52 2.985 hex 4.82 2.325 10.40–10.50 8.75

50–145

3,515–10,195

87

25–75

18

(continued )

6,117

1,758–5,273

1,266

Properties of Materials 295

10/27/2008 2:28:02 PM

CRC_DK8307_CH007.indd 296

2.5308 1.1810

6 16.8

2.0879 1.2935

0.8436 0.7733

29.7 24 18.4

12

11

psi (×10 )

6

kg/cm2 (×106)

Shear Modulus

0.21 0.21

0.28

0.322 0.322 0.323 0.3

0.27

Poisson’s Ratio

19.30 19.30 19.1 18.97 6.01 5.761 g/L 6.90 4.472 7.14 7.10 6.52

19.10–19.25

7.30 4.50–4.85 4.43 4.54 4.85

11.68

Density g/cm3 kpsi

2

kg/cm

Fatigue Limit

16–18

56

18–600

200

32

kpsi

1,125–1,266

3,937

1,265–42,184

14,061

2,250

kg/cm2

Ultimate Strength

Source: Bauccio, M., ASM Metals Reference Book, ASM International, Materials Park, OH, 1993; Dean, J.E., Lange’s Handbook of Chemistry, McGraw-Hill, Inc., New York, 1994; Ensminger, D., Ultrasonics, Marcel Dekker, Inc., New York, 1988; Weast, R.C., CRC Handbook of Chemistry and Physics, CRC Press, Inc., Boca Raton, FL, 1985.

3.515

50

14.8

0.8014

kg/cm (×106)

2

7–10

psi (×10 )

6

Young’s Modulus

(Continued)

Thorium (induction melt) Tin Titanium 6Al4V Ti150A B120VCA (aged) Tungsten Tungsten Annealed Drawn Uranium D-38 Vanadium Xenon Ytterbium Yttrium Zinc Rolled Zirconium

Materials

TABLE 7.4

296 Ultrasonics: Data, Equations, and Their Practical Uses

10/27/2008 2:28:03 PM

Properties of Materials

297

TABLE 7.5 Physical Properties of Engineering Materials Velocity of Sound Bulk cB Materials Aluminum Aluminum 2SO 17ST Al 2024-T3 6061-T6 7075-T6 Antimony Arsenic Barium Beryllium Bismuth Boron Brass 70–30 Brass (naval) Bronze (Phosphor 5%) (Sn 7.5%) (Al 7.7%) Cadmium Calcium Carbon Cerium Chromium Cobalt Columbium (niobium) Constantan (Cu 60%, Ni 40%) Copper Annealed Rolled Curium Duralumin 17S Gallium Germanium German silver Gold Hafnium Hastelloy X Hastelloy C Inconel X Inconel wrought Indium Iridium Iron Iron alnico Iron, cast Iron electrolytic Iron wrought

Characteristic Impedance ρco (106 g/(cm2−s))

Bar co

(105 cm/s)

Shear cs

5.06

2.655

1740

5.10 5.08

6.35 6.25

3.10 3.10

1.73 1.75

5.05 5.07 3.40

6.27 6.35

3.08 3.10

1.70 1.78

12.75–12.87 1.79

12.80–12.89 2.18

8.71–8.88 1.10

2.33–2.41 2.14

3.40–3.48 3.49

4.37–4.70 4.43

2.10 2.12

3.70–4.04 3.61

3.43

3.53

2.23

3.12

2.40

2.78

1.50

2.40

4.30

5.24

2.64

4.60

3.60–3.71 3.81 3.75

4.60–4.80 4.76 5.01

2.26–2.33 2.33 2.27

4.10–4.25 4.25 4.47

5.15

6.32

3.13

1.76

Electrical Resistivity µohm cm

Thermal Conductivity Btu/h/ft2/in./°F

1320 1080 39.0 35.0 50.0 5.9 106.8 1.8 × 1012

131

1100 58 1015 697

6.83 3.43 1,375.0 78.0 13.0 6.24 13.1

412 470 639 871 165 464 479

153 1.673

2730

2,444.0 56.8 60 × 104 3.58 2.03

4.76 3.24

2.16 1.20

4.00 6.26

— — — 5.08

5.79 5.84 5.94 7.82

2.74 2.90 3.12 3.02

4.77 5.22 4.93 6.45

4.79 5.17–5.18 5.20 3.0–4.7 5.12

— 5.96 5.96 3.5–5.6 5.95

— 2.05 3.24 2.2–3.2 3.24

— 4.68 4.69 2.5–4.0 4.70

2.19 32.4

765 232

2060

91 8.37 9.71

175 406 498

420 (continued)

CRC_DK8307_CH007.indd 297

10/27/2008 2:28:03 PM

Ultrasonics: Data, Equations, and Their Practical Uses

298 TABLE 7.5

(Continued) Velocity of Sound Bulk cB

Materials Lanthanum Lead Pure Annealed Rolled Antimony 6% Lithium Magnesium Magnesium Am 35 Magnesium drawn, annealed Manganin (Cu 64%, Zn 17%, Ni 18%) Mercury Molybdenum Monel metal Monel wrought Nickel Nickel–silver (Cu 64%, Zn 17%, Ni 18%) Nitrogen Osmium Oxygen Palladium Phosphorus Platinum Plutonium Potassium Radium Rhenium Rhodium Selenium Silicon Silver Sodium Steel Steel, 1%C Steel, 1%C hardened Steel HY80 Steel, mild Steel Stainless 302 Stainless 304 Stainless 304L Stainless 347 Stainless 410 Stainless 416

CRC_DK8307_CH007.indd 298

Bar co

(105 cm/s)

Shear cs

Characteristic Impedance ρco (106 g/(cm2−s))

Electrical Resistivity µohm cm

Thermal Conductivity Btu/h/ft2/in./°F

59 1.20–1.25 1.19 1.21 1.37

2.16–2.40 2.16 1.96 2.16

0.70–0.79 0.70 0.69 0.81

2.46–2.72 2.46 2.23 2.36

20.65

4.90 5.00

5.74 5.79

3.08 3.10

0.99 1.01

4.46

4.94

5.77

3.05

1.00

3.83

4.66

2.35

3.90

5.40–5.45 4.40 4.52 4.79–4.90 3.83

6.25–6.29 5.35 6.02 5.60–6.04 4.62

3.35 2.72 2.72 2.96–3.00 2.32

6.35 4.76 5.31 4.90–5.38 4.03

241

494

138

94.1 5.17

58 950 144

6.84

430 217

0.147 9.5

2.80

3.26–3.96

1.67–1.73

6.98–8.46

10.8 1017 9.83 150 6.15 21.0 4.5

2.64–2.68

3.60–3.70

1.59–1.70

3.80–3.90

5.05–5.17

5.85–6.10

3.23

4.56

5.18 5.07

5.94 5.854

3.22 3.15

4.66 4.59

— 5.05–5.20

5.88 5.96–6.10

— 3.24

4.56 4.68–4.82

4.90 4.92 4.93 5.00 5.03 5.20

5.66 — 5.64 5.79 5.76 6.02

3.12 — 3.07 3.10 2.99 3.23

4.55 — 4.45 4.57 4.42 4.64

1.59 4.2

0.191 468 494 60 697

1056 3 581 2400 929

315

10/27/2008 2:28:03 PM

Properties of Materials TABLE 7.5

299

(Continued) Velocity of Sound

Bar co

(105 cm/s)

Shear cs

Characteristic Impedance ρco (106 g/(cm2−s))

5.04 —

5.83 5.85

— 3.24

— 4.82

3.35

4.10

2.90

5.48

— 2.73–2.74 5.08 5.08 5.08 4.31–4.60 4.62 4.32 —

2.94 3.32–3.38 5.99–6.07 6.23 6.10 5.17–5.46 5.22 5.41 3.37

1.56 1.61–1.67 3.12–3.125 — 3.12 2.62–2.87 2.89 2.64 1.98–2.02

3.32 2.42–2.47 2.70–2.73 — 2.77 9.98–10.42 10.07 10.44 6.30

3.81 3.85 —

4.17 4.21 4.65

2.41 2.44 2.22–2.30

2.96 2.99 1.32–3.01(8.0)

Bulk cB Materials Steel 4130 Steel 4340 Sulfur Tantalum Tellurium Thallium Thorium Tin Titanium Titanium 6Al4V Titanium 150A Tungsten Tungsten Tungsten drawn Uranium Vanadium Zinc Zinc rolled Zirconium

7.3

Electrical Resistivity µohm cm

Thermal Conductivity Btu/h/ft2/in./°F

2 × 1023 12.4 2 × 105 18.0 18.6 11.5 47.8

1.83 377 41 348 204 464 198

5.5

1100

29.0

168

26.0 5.92

240 780 784 2.98–41.0(8.0)

Ultimate Stress and Fatigue Limit

Because ultrasonic waves are stress waves, the ultimate stress and fatigue limit are important in the design of the vibratory elements of an ultrasonic system. Ultimate stress is that stress at which rupture occurs. Therefore, every precaution is taken to avoid these intense stress levels in the design of an ultrasonically vibrating system. The fatigue limit is especially important. It is the limit of stress below which a vibrating element may operate continuously without failure. The figures given for these values in Table 7.4 apply to standard low-frequency measurements. Ultrasonically, the cycles accumulate at a much higher rate and therefore the figures given in Table 7.4 must be used as relative values for endurance. The value used depends upon the application. Elements used to construct transducers, a transducer attached to a driven part, horns and horn designs, flanges on horns, or flexural bars are all subjected to stresses that may lead to failure or disconnection. Bolts used in the assembly of transducers are also subjected to stress. The stresses at which these elements will be subjected at a given location and amplitude of vibration may be determined fairly accurately using the equations of Chapter 2. Stress risers play an important part in determining the fatigue life of a part vibrating at ultrasonic frequency. Pits, scratches, embedded defects, and small radius fillets at the junction between two sectors of different diameters of a double cylinder are stress risers. A machine scratch within the fillet of a double-cylinder or similar structure may lead to failure even when the fillet radius itself would otherwise be acceptable for the applied stress.

CRC_DK8307_CH007.indd 299

10/27/2008 2:28:03 PM

Ultrasonics: Data, Equations, and Their Practical Uses

300

12

11

Stress PSI × 104

10

Curve of steady-state stress Curve adjusted to compensate for peak

14 Peak 14

15 Peak 15

9

(11) (10)

8 7

(7) (8) (9)

(13)

6 5 4 3 2 1 0 104

105

106

107

108

109

1010

Cycles

FIGURE 7.1 S–N curve for SNC-631 steel alloy at 38.6 kHz.

Ultrasonic waves are stress waves. A half-wave resonant bar vibrating in the longitudinal mode represents a standing stress wave. Stress is distributed along the bar in accordance with the amount the bar stretches at each point of the length to maintain the amplitude of displacement at the opposite ends. As shown in Chapter 2, the stress wave is dependent upon the geometry of the bar. Weld and silver solder joints should be located at positions where the stress level will never exceed the strength of the joint. Careful attention should be given to the location of welds and silver solder joints with respect to stresses in all designs of resonant systems. One method of determining the fatigue limit of a material ultrasonically is to prepare a series of identical specimens, such as well-machined and polished double-cylinder horns. The fillet itself is a stress riser, and by using the equations for calculating the stress rise (Equations 2.16 through 2.18), the actual stress at that point for a given amplitude of vibration may be determined accurately. The transducer and horn are mounted in a rigid structure, so that a capacitance type of displacement gage can be used to monitor the displacement at the free end of the horn. The horns are then driven at different discrete levels of intensities to failure in a noncorrosive atmosphere. From the displacement amplitude at the end and the distance to the position of the break, one can now calculate the stress level to which the horn was subjected at that point. The time-on and time-off and the temperature (held at a constant value during the test) are monitored. Figure 7.1 was obtained at a frequency of 38.6 kHz in this manner for special steel alloy, Japanese SNC-631.

7.4 7.4.1

Materials Properties for Processing by Ultrasonics Friction and Wear

Friction and wear are consequences of two mating surfaces rubbing together. Friction is the resistance to relative motion between surfaces of objects in contact with each other under pressure. The coefficient of friction is the ratio of this resistive force to the

CRC_DK8307_CH007.indd 300

10/27/2008 2:28:04 PM

Properties of Materials

301

force normal to the surfaces of contact. Various lubricants may be used to minimize or control friction between moving surfaces. Kragelskii states that the interaction between surfaces is of a dual molecular-mechanical nature. The molecular interaction is due to the attraction between the two solids, their adhesion; the mechanical interaction is due to the mutual inter-penetration of localized regions on the compressed surfaces [5]. In ultrasonic applications, the compatibilities of materials, the effects of relative velocity, temperature, and pressure, surface conditions or contours, relative hardness, and of material toughness are important factors. Relative velocity and pressure affect the temperature between two mating surfaces. Acoustic vibration is an oscillating movement. The velocity of an ultrasonic tool is a function of the frequency and the amplitude of vibration. As the pressure increases, the amount of mechanical energy transformed into heat increases under constant velocity. Under constant pressure, the heating rate increases as the velocity increases. The increased temperature changes the mechanical properties of the rubbing materials and the films forming at the surfaces. These are conditions of friction between mating surfaces whether the motion is due to acoustic vibration or not. Two aspects of motion must also be considered: (1) a system in which one surface is sliding against an ultrasonically driven surface like a belt moving against an ultrasonic tool in a fixed position and (2) a system in which one surface is stationary and the other is driven ultrasonically. What happens in each case depends upon the pressure, the amplitude of vibration, the dwell time, and the physical and mechanical properties of each surface. All surfaces have asperities, large or small. The size of asperities in the surfaces relative to the amplitude of the movement are important when ultrasonic energy is applied. Wear is the erosion of surfaces by frictional forces. These frictional effects are not restricted to the mating surfaces. They are a means of coupling shear energy into the materials as well. By special design, the absorption of energy can be localized within certain regions of the mating parts to produce the types of welds and forms desired in thermoplastic materials. These principles lead to a wide range of uses for ultrasonic energy. A polymeric material must be thermoplastic to be molded, extruded, or bonded (welded) ultrasonically. The following terms are defined with regard to their importance in the effectiveness of ultrasound in welding, molding, extruding, or forming of plastics or metals: Friction. As defined previously, the resistance to relative motion between surfaces of objects in contact with each other under pressure. Weld. The localized coalescence of material wherein the coalescence is produced by heating to suitable temperatures, with or without the application of pressure, and with or without the use of filler material. Welding. Joining two materials by applying heat to melt and fuse them, with or without filler material. Weldability. The capacity of a material to be welded under the fabrication conditions imposed with a specific, suitably designed structure and to perform satisfactorily in the intended service. Bond. The joining of the weld material and the base material, or the joining of the base material parts when weld material is not used. Glue. An adhesive material made of crude, impure, amber-colored form of commercial gelatin of unknown detailed composition produced by the hydrolysis of animal collagen; gelatinizes in aqueous solutions, and dries to form a strong adhesive layer.

CRC_DK8307_CH007.indd 301

10/27/2008 2:28:04 PM

Ultrasonics: Data, Equations, and Their Practical Uses

302

Adhesive. A substance used to bond two or more solids so that they act or can be used as a single piece; examples are resins, formaldehydes, glue, paste, cement, putty, and polyvinyl resin emulsions. Adhesive bond. The forces such as dipole bonds that attract adhesives and base materials to each other. Adhesive bonding. The fastening together of two or more solids by the use of glue, cement, or other adhesive. 7.4.2 7.4.2.1

Welding and Staking Properties Necessary for Ultrasonic Welding and Forming of Materials *

Plastic materials are of two types: thermoplastic and thermosetting. Thermoplastic materials soften or become liquid as they are heated. This makes them capable of being bonded together by fusion. Thermosetting plastics are hard, brittle materials that char or degrade when they are subjected to intense heat. They cannot be welded ultrasonically [6]. 7.4.2.1.1 Compatibility of Materials It is necessary that any two plastic materials to be welded together be chemically compatible. If they are not compatible, there can be no chemical bonds, even when they melt at the same temperature. Similar thermoplastic materials, such as two acrylonitrile butadiene styrene (ABS) parts, may be welded together. Two parts of incompatible materials, such as polyethylene and polypropylene, cannot be welded together, although both have a similar appearance and many common physical properties. Two materials of similar molecular structure may be bonded if their melt temperatures are within 40°F (6°C) of each other. Generally speaking, only similar amorphous polymers have an excellent probability of being welded to each other. The chemical properties of any semicrystalline material make each one compatible with itself. Other factors that affect the weldability of plastic parts include hygroscopicity, mold release agents, lubricants, plasticizers, fillers, flame retardants, regrind, pigments, and resin grades. 7.4.2.1.2 Hygroscopicity Hygroscopicity is the tendency of a material to absorb moisture. Resins such as polyamide (nylon), polycarbonate, polycarbonate/polyester alloy (xenoy), and polysulfone are hygroscopic (i.e., they absorb and retain moisture from the air). When moist parts are welded, the water inside the materials boils off when the temperature reaches the boiling point. This process creates a foamy condition at the joint interface, making it difficult to achieve a hermetic seal and giving the assembled parts a poor cosmetic appearance. The bond strength is also weakened. 7.4.2.1.3 Mold Release Agents Mold release agents are usually sprayed directly on the mold cavity surface and are used to make parts eject from the mold cavity more readily by reducing friction between the part and the cavity walls. Unfortunately, mold release agent on the molded parts reduces surface friction in the joint interface between the parts when they are being welded.

* Much of the following information has been obtained by permission from the Guide to Ultrasonic Plastics Assembly by the Dukane Ultrasonics Corporation.

CRC_DK8307_CH007.indd 302

10/27/2008 2:28:04 PM

Properties of Materials

303

Because the ultrasonic assembly process depends on surface friction, the use of mold release agents can be detrimental to weldability. Furthermore, the chemical contamination of the resin by the release agent can inhibit the formation of the desired bond. Some agents can be removed from parts with a preassembly cleaning operation using suitable solvents. If a release agent must be used, paintable/printable grades that permit painting and silk screening are preferred, because they interfere the least with ultrasonic assembly and often require no preassembly cleaning. Use of zinc stearate, aluminum stearate, fluorocarbons, and silicones should be avoided if possible. 7.4.2.1.4 Lubricants Lubricants such as waxes, zinc stearate, stearic acid, aluminum stearate, and fatty esters are added to resins to improve flow characteristics and enhance resin processability. Internal lubricants reduce the coefficient of friction. They cannot be removed. Therefore, they have a negative effect on ultrasonic welding. 7.4.2.1.5 Plasticizers Plasticizers are used to increase the flexibility and softness of a material and have a tendency to migrate or return to the joint of a welded part after a period of time, resulting in a weakened bond or joint. FDA-approved plasticizers are preferred to the metallic plasticizers. Experimentation is advised before setting up for production by means of ultrasonic assembly. 7.4.2.1.6 Fillers Fillers, such as glass fiber, talc, carbon fiber, and calcium carbonate, are added to resins to alter their physical properties. For instance, a glass filler might be added to a resin to improve its dimensional stability or material strength. Common mineral fillers, such as glass or talc, can actually enhance the weldability of thermoplastics, particularly semicrystalline materials, because they improve the resin’s ability to transmit vibrational energy. A glass content of 10–20% can substantially improve the transmission properties of a resin. The increase of the ratio of fillers to the increase in weldability exists below a certain prescribed level. Levels exceeding 10–20% may cause the accumulation of filler at the joint to be so severe that there may not be enough resin in the joint interface to form an acceptable weld. The accumulation at the joint interface is known as agglomeration or filler enrichment. If the amount of the filler in the joint exceeds 40%, there is more unweldable material there than weldable material. The weldability is more difficult to achieve consistently and overall assembly strength suffers. Filler contents >20% can cause excessive horn and fixture wear that may require special tooling. Because of the presence of particles at the resin surface, heat-treated steel or carbide-faced titanium horns may need to be used. Higher-powered ultrasonic equipment may also be required to create sufficient heat at the joint. 7.4.2.1.7 Flame Retardants Flame retardants are used to alter the combustible properties of plastics. Retardants such as antimony, boron, halogens, nitrogen, and phosphorous are added to resins to keep temperatures below a combustion level or to prevent a chemical reaction between the resin and oxygen or other combustion-aiding gases. Flame retardants can directly affect thermoplastic weldability by reducing the strength of the finished joint. High-power equipment, operating at higher than normal amplitudes, is often required so that the parts can be overwelded to achieve adequate strength.

CRC_DK8307_CH007.indd 303

10/27/2008 2:28:05 PM

Ultrasonics: Data, Equations, and Their Practical Uses

304

7.4.2.1.8 Regrind Regrind is the term for plastic material that has been recycled or reprocessed and added to the original resin. Ultrasonic assembly is one of the few processing methods that permits regrinding of parts, as no foreign substance is introduced into the resin. Providing that the percentage of regrind is not excessive, and the plastic has not been degraded or contaminated, few problems should arise. However, for best results, it is advisable to keep the regrind percentage as low as possible. 7.4.2.1.9 Colorants Colorants—liquid or dry—or pigments which have very little effect on weldability, unless the percentage of colorant to resin is exceedingly high. White and black parts often require more pigments than other colors and may cause some problems. Different colors of the same part may result in different setup parameters. Experimentation is recommended prior to full production. 7.4.2.1.10 Resin Grade Resin grade can have a significant effect on an application’s weldability. Resin grade is important, because different grades of the same material can have very different melt temperatures, resulting in poor welds or apparent incompatibility. Whenever possible, materials of the same grade should be used in the ultrasonic assembly process. Table 8.7 lists several thermoplastic materials that can be bonded ultrasonically, giving ratings for the effectiveness of the various types of welding operations. 7.4.2.2

Joint Designs for Efficient Welding

The Dukane Corporation has published an excellent treatise (referenced earlier) on ultrasonic bonding, which contains joint designs for ultrasonic welding. The joint design of the mating pieces is critical in achieving optimum assembly results. The joint design of a particular part depends upon factors such as type of plastic, part geometry, and the requirements of the weld. There are many different joint designs, each with its own advantages. Some of these designs are discussed later in this section. There are three basic requirements in joint design as follows: 1. A uniform contact area 2. A small initial contact area 3. A means of alignment A uniform contact area means that the mating surfaces should be in intimate contact around the entire joint. The joint should also be in one plane, if possible. A small initial contact area should be established between the mating halves. Doing so means less energy, and therefore less time, is required to start and complete the meltdown between the mating parts. A means of alignment is recommended, so that the mating halves do not misalign during the welding operation. Alignment pins and sockets, channels, and tongues are often molded into parts to serve as ways to align them. It is best not to use the horn or the fixture to provide part alignment. The need to follow the basic requirements for any joint design can be demonstrated using a flat butt joint. Only the high points will weld on a flat butt joint, resulting in erratic, inconsistent welds. Extending the weld time to increase the melt simply enlarges the original weld points and causes excessive flash outside of the joint. When one of the surfaces is brought to a point, it may produce a weld with a good appearance but having

CRC_DK8307_CH007.indd 304

10/27/2008 2:28:05 PM

Properties of Materials

305

little strength, or, if good strength is obtained, it produces excessive flash and thus ruins the appearance of the weld. 7.4.2.2.1 The Energy Director An energy director is a triangular-shaped bead molded into the part interface. It typically runs around the entire joint perimeter. The energy director was developed to provide a specific volume of material to be melted so that good bond strength could be achieved without excessive flash. It is the joint design that is generally recommended for amorphous polymers. When ultrasonic energy is transmitted through the part under pressure and over time, the energy concentrates at the apex of the energy director (i.e., where the apex of the triangular-shaped bead contacts the other mating surface), resulting in a rapid buildup of heat that causes the bead to melt. The molten material flows across the joint interface, forming a molecular bond with the mating surface. The energy director meets two of the three basic requirements of a joint design: it provides (1) a uniform contact area and (2) a small initial contact area. The energy director itself does not provide a means of alignment or provide a means to control material flash. These requirements must be incorporated into the part design. The basic energy director design for an amorphous resin is a right triangle with the 90° angle at the apex and the base angles each at 45° (Figures 7.2a and 7.2b). This makes the height one-half the width of the base. The size of this energy director can range from 0.005 in. (0.127 mm) to 0.030 in. (0.762 mm) high and from 0.10 in. (0.254 mm) to 0.060 in. (1.53 mm) wide. For polycarbonate, acrylics, and semicrystalline resins, the energy director is an equilateral triangle, with all three angles being 60°. This design makes the height 0.866 times the base width. The base width can range from 0.010 in. (0.254 mm) to 0.050 in. (1.27 mm). The most common and basic joint design is the butt joint with an energy director. The width of the base of the energy director is between 20% and 25% of the thickness of the wall (i.e., B = W/4 to W/5). When the wall is thick enough to produce an energy director

α

β

α = 90°

β = 60°

A B A = height of the energy director B = width of the energy director

B

B = W/4 to W/5

At the base = W/4 to W/5 W

(a)

(b)

FIGURE 7.2 Butt joint configurations: (a) energy director for amorphous resins and (b) energy director for semicrystalline resins.

CRC_DK8307_CH007.indd 305

10/27/2008 2:28:05 PM

306

Ultrasonics: Data, Equations, and Their Practical Uses

larger than the maximum size, two smaller parallel energy directors should be used. The height at the apex of the energy director is either half the base or 0.866 times the base, depending on the material. This design produces a weld across the entire wall section with a small amount of flash normally visible at the finished joint. As stated in Section 7.4.2.2, the parts should be designed to include a means of alignment. If this is not possible, the fixture can be designed to provide the locating features necessary to keep the parts aligned with respect to each other. Typically, hermetic seals are easier to achieve with amorphous rather than semicrystalline materials. If a hermetic seal is required, it is important that the mating surfaces be as close to perfectly flat and parallel to each other as possible. The butt joint with an energy director is well suited for amorphous resins, because they are capable of molten flow and gradual solidification. However, it is not the best design for semicrystalline resins. With semicrystalline resins, the material displaced from the energy director usually solidifies before it can flow across the joint to form a seal, which causes a reduction in overall strength and makes hermetic seals difficult to achieve. However, sometimes there are certain limitations imposed by the design or size of the part that make it necessary to use an energy director on semicrystalline parts. In situations where an energy director must be used with a semicrystalline resin, the energy director should be larger and have a steeper angle to give it a sharper point (apex). This design enables it to partially embed in the mating surface during the early stages of the weld, thereby reducing the amount of premature solidification and degradation caused by exposure to the air. The larger, sharper design improves the strength and increases the chances of obtaining a hermetic seal. Experimentation has shown that the larger, sharper energy director design is also superior when working with polycarbonates and acrylics, even though both materials are classified as amorphous materials. 7.4.2.2.2 The Step Joint The step joint is a variation of the energy director joint design. Like the energy director, it meets two of the basic requirements of joint design: it provides (1) a uniform contact area and (2) a small initial contact area. A step joint also provides the third requirement: a means of alignment (Figure 7.3). The strength of a step joint is less than that of the butt joint, because only part of the wall is involved in the welding. The recommended minimum wall thickness is 0.080 in. (2.03 mm) to 0.090 in. (2.29 mm). A step joint may be used when cosmetic appearance of the assembly is important. Use of a step joint can eliminate flash on the exterior and produce a strong joint, as material from the energy director will typically flow into the clearance gap between the tongue and the step. The energy director is dimensionally identical to the one used on the butt joint. The height and width of the tongue are each one-third of the wall thickness (T = W/3). The width of the groove is 0.002 in. (0.05 mm) to 0.004 in. (0.10 mm) greater than that of the tongue to ensure that no interference occurs (G = T + 0.002 to 0.004 in.). The depth of the groove should be 0.005 in. (0.13 mm) to 0.010 in. (0.25 mm) greater than the height of the tongue, leaving a slight gap between the finished parts (D = T + 0.005 to 0.010 in.). This design is done for cosmetic purposes, so that it will not be obvious if the surfaces are not perfectly flat or the parts are not perfectly parallel. 7.4.2.2.3 The Tongue-and-Groove Joint The tongue-and-groove joint is another variation of the energy director (Figure 7.4). Like the step joint, it provides the three requirements of a joint design (1) a uniform contact area, (2) a small initial contact area, and (3) a means of alignment. It also prevents internal and external flash, as there are flash traps on both sides of the interface.

CRC_DK8307_CH007.indd 306

10/27/2008 2:28:05 PM

Properties of Materials

307

W

Slip fit

W/8

T

W/3 B

H

A C W/3 G

D

W FIGURE 7.3 Step joint.

FIGURE 7.4 Tongue-and-groove joint.

The tongue-and-groove joint is used primarily for applications where self-location and flash prevention are important. It is an excellent joint design with applications calling for low-pressure hermetic seals. The main disadvantage of the tongue-and-groove joint is that less weld strength is possible, because less area is affected by the joint. The minimum wall thickness recommended for use with the tongue-and-groove joint is 0.120 in. (3.05 mm) to 0.125 in. (3.12 mm). Again, the energy director is dimensionally identical to the one used in butt joint. The height and width of the tongue are both one-third of the thickness of the wall. Clearance should be maintained on each side of the tongue to avoid interference and provide space for the molten material. Therefore, the groove should be 0.004 in. (0.10 mm) to 0.008 in. (0.20 mm) wider than the tongue. The depth of the groove should be 0.005 in. (0.13 mm) to 0.010 in. (0.25 mm) less than the height of the tongue. As with the step joint, a slight gap designed into the finished part assembly proves advantageous for cosmetic reasons. 7.4.2.2.4 The Shear Joint The shear joint is used when a strong hermetic seal is needed, especially with semicrystalline resins. A certain amount of interference is designed into the part for a shear joint (Figure 7.5). Welding is accomplished by first melting the contacting surfaces. As the melting parts telescope together, they continue to melt with a controlled interference along the vertical walls. A flash trap, which is an area used to contain the material displaced from the weld, may be used. The smearing action of the two melt surfaces at the weld interface eliminates leaks and voids, as well as exposure to air, premature solidification, and possible oxidative degradation. The smearing action produces a strong structural weld. Rigid sidewall support is very important with shear joint welding to prevent part deflection during welding. The walls of the fixtured part need to be supported up to the joint interface by the fixture, which should closely conform to the shape of the part.

CRC_DK8307_CH007.indd 307

10/27/2008 2:28:05 PM

Ultrasonics: Data, Equations, and Their Practical Uses

308

Depth of weld

Minimum lead-in 0.020 in. (0.5 mm) Interference

Fixture

Before weld

After weld

FIGURE 7.5 Shear joint.

Horn

Parts

1/4 in. (6 mm) or less

Near field

Greater than 1/4 in. (6 mm)

Far field

FIGURE 7.6 Near-field and far-field welding.

In addition, to make it easier to remove the part from the fixture, the fixture itself should be split so that it can be opened and closed. A shear joint meets the three requirements of joint design. The lead-in provides a means of alignment and self-location of the parts to be welded. Properly designed and molded parts ensure a uniform contact area. The small initial contact area between the parts occurs at the base of the lead-in. 7.4.3

Tools for Ultrasonic Welding

Ultrasonic welding is done by bringing an ultrasonic horn into contact with one of the materials to be welded. The face of the horn must be parallel with the interfaces to be welded. Figure 7.6 illustrates near field and far field welding. Near field welding refers to

CRC_DK8307_CH007.indd 308

10/27/2008 2:28:06 PM

Properties of Materials

309

having a spacing of 1/4 in. (6 mm) or less between the horn and the joint. Far field welding refers to having distances of >1/4 in. between the horn and the weld. It is always better to weld near field. Far field welding requires higher than normal amplitudes, longer weld times, and higher air pressures to achieve a weld comparable to a near field weld. Generally speaking, it is best to use far field welding only for amorphous resins, which transmit energy better than semicrystalline resins. Having the joint interface and the horn area of contact each on single, parallel planes is important to good welding. The energy travels uniformly between the source and the joint. When the energy has to travel different distances through the specimen, the weld joint becomes very inconsistent, making either weak structural bonds or overwelded bonds. Other structural conditions that affect the welding process include sharp corners, holes or voids, and appendages. Sharp corners localize stress. Plastic parts, when subjected to ultrasonic energy, may fracture or melt in these high-stress areas. It is good to use a generous radius in all corners and edges. Holes or voids interfere with the transmission of ultrasonic energy. Little or no welding will be achieved beneath these areas. Eliminate all sharp angles, bends, and holes, where possible. Appendages, tabs, or other protrusions molded onto plastic parts also focus stress when subjected to vibratory energy and have a tendency to fall off (or degate). These problems may be minimized by adding a generous radius to the areas where the appendages join the main part, applying a light force to the appendage(s) to dampen the flexure, making the appendages thicker, or using a higher frequency (40 kHz rather than 20 kHz), if possible. Diaphragming may be a problem in welding thin sections of flat, circular parts, as they may flex under the influence of ultrasonic energy. As the part flexes up and down, it absorbs ultrasound to the extent that heat is formed and causes melting or produces a hole in the part. Diaphragming will often occur in the center of a part or at the gate area. Making those sections thicker may prevent diaphragming. 7.4.4

Staking

Ultrasonic staking allows a thermoplastic material to be attached to a nonthermoplastic material or to a metal using thermoplastic stakes, which are formed into a locked position ultrasonically. It is the process of melting and reforming a stud to mechanically lock a material in place. It provides an alternative to welding when (1) the two parts to be joined are made of dissimilar materials that cannot be welded (e.g., metal and plastic) or (2) simple mechanical retention of one part relative to another is adequate (i.e., molecular bonding is not necessary). The advantages of staking include short cycle time, tight assemblies with no tendencies to spring back, the ability to perform multiple stakes with one horn, good process control and repeatability, simplicity of design, and elimination of consumables, such as screws or adhesives. Staking is most commonly used to attach metal to plastic. A hole in a metal part is designed to receive a stud or boss, which is molded into the plastic part. The stud should be designed with a generous radius at its base to prevent fracturing. A vibrating horn with a contoured tip contacts the stud and creates localized, frictional heat. As the stud melts, light pressure from the horn reforms the head of the stud to the configuration of the horn tip (Figure 7.7). When the horn stops vibrating, the plastic material solidifies and the metal and plastic parts are fastened together.

CRC_DK8307_CH007.indd 309

10/27/2008 2:28:07 PM

Ultrasonics: Data, Equations, and Their Practical Uses

310

Staking cavity

Dissimilar material Plastic Before

After

FIGURE 7.7 Staking.

General guidelines for staking applications include 1. Use of a high-amplitude horn with a small contact area to localize heat and increase the rapidity of the melt. 2. Light initial contact force with controlled horn descent velocity to concentrate the ultrasonic energy at the limited horn and stud contact area. 3. Pretriggering the ultrasonic energy to create an out-of-phase relationship, preventing horn and stud coupling. 4. A slow actuator down speed to prevent stud fracture while allowing plastic material to flow into the horn cavity. 5. A heavier hold force during the hold time to give the stud optimum strength to retain the attached material. The integrity of an ultrasonically staked assembly depends upon the geometric relationship between the stud and the horn cavity, and the ultrasonic parameters used when forming the stud. Proper stake design produces optimum stud strength and appearance with minimum flash. The design depends on the application and physical size of the stud or studs being staked. The principle of staking, however, is always the same—the initial contact area between horn and stud must be kept to a minimum, concentrating the energy to produce a rapid, yet controlled melt. 7.4.4.1

Hydraulic Speed Control

The Dukane Corporation offers a hydraulic speed control for their press/thruster systems. It is especially useful in staking applications, as it regulates the velocity at which the horn descends once it contacts the stud. The horn’s downstroke slows to match the stud’s melt rate, so that intimate contact between the horn and the stud is maintained, giving the stud a finished appearance when the weld cycle is completed. Downstroke speed can be rapid prior to stud contact. Then horn descent can be precisely controlled during the actual staking operation. 7.4.5

Five Basic Staking Designs

The five basic staking designs include (1) the standard rosette profile stake, (2) the dome stake, (3) the hollow stake, (4) the knurled stake, and (5) the flush stake.

CRC_DK8307_CH007.indd 310

10/27/2008 2:28:07 PM

Properties of Materials 7.4.5.1

311

The Standard Rosette Profile Stake

The standard rosette profile stake satisfies most requirements. The finished head is twice the diameter of the original stud. It is designed to stake studs with flat heads and is recommended for studs 1/16 in. (1.6 mm) outside diameter (OD) or larger (Figure 7.8). This stake is ideal for staking nonabrasive rigid and nonrigid thermoplastics. 7.4.5.2

The Dome Stake

The dome stake is typically used for studs of <0.116 in. (1.6 mm) OD or with multiple studs where horn alignment can be a problem (Figure 7.9). It is also recommended for glass-filled resins where horn wear is possible, because the horn cavity can be redressed more easily than standard staking cavities using the inverted staking cone. The stud end of a dome stake should be pointed to provide a small initial contact area prior to staking. Horn and stud alignment is not as critical as with the standard rosette profile stake. 7.4.5.3

The Hollow Stake

The hollow stake is used on studs that are 5/32 in. (4 mm) or larger in diameter (Figure 7.10). The hollow stake offers advantages in molding, because it prevents surface sinks and internal voids. Hollow staking reduces ultrasonic cycle time, because less material is being melted and reformed. Staking a hollow stud produces a large, strong head. In applications where disassembly for repair is a primary requirement, the formed stud head can be removed for access, and reassembly can be accomplished by driving a self-tapping screw into the hole in the hollow stud.

Staking cavity Dissimilar material Plastic Before

After

FIGURE 7.8 Standard rosette profile stake.

Staking cavity

Dissimilar material Plastic Before

After

FIGURE 7.9 Dorne stake.

CRC_DK8307_CH007.indd 311

10/27/2008 2:28:07 PM

Ultrasonics: Data, Equations, and Their Practical Uses

312

Staking cavity

Dissimilar material Plastic Before

After

FIGURE 7.10 Hollow stake.

Staking cavity Dissimilar material Plastic Before

After

FIGURE 7.11 Knurled stake.

Horn

Before

After

FIGURE 7.12 Flush stake.

7.4.5.4

The Knurled Stake

The knurled stake (Figure 7.11) is used for simplicity and a rapid assembly rate. Multiple stakes may be made without concern for precise alignment or stud diameter. The knurled stake may be used with all thermoplastics where appearance is not critical. 7.4.5.5

The Flush Stake

The flush stake is used when a raised stud head is not permitted above the surface of the attached part. The tapered stud design used for dome staking is recommended. The hole in the part to be attached is countersunk so that the volume of the melted stud fills the area, locking the attached part in place. Figure 7.12 shows a flush stake.

CRC_DK8307_CH007.indd 312

10/27/2008 2:28:08 PM

Properties of Materials

7.5

313

Stud Welding

An alternative to staking is ultrasonic stud welding. This process can be used to join plastic parts of similar material at single or multiple localized attachment points (Figure 7.13). The technique is useful in applications that do not require a continuous weld. Stud welding can be used when resin selection, size, or part complexity prevent the use of other techniques. A variation of the shear joint is used in stud welding. Generally, a stud is driven into a hole, with welding occurring along the circumference of the stud.

7.6

Insertion

Insertion is the assembly process of embedding a metal component in a thermoplastic part (Figure 7.14). A hole is premolded into the thermoplastic part slightly smaller than the OD of the insert it is to receive. As ultrasonic energy is applied to the insert, and frictional heat

Horn

Before

After

FIGURE 7.13 Stud welding. Spot welding tip

1.5T 1.5T

3T

T = Top layer of thermoplastic FIGURE 7.14 Hollow stake.

CRC_DK8307_CH007.indd 313

10/27/2008 2:28:08 PM

Ultrasonics: Data, Equations, and Their Practical Uses

314

is generated due to the insert vibrating against the plastic, the plastic melts, permitting the insert to be driven into place. The insert is surrounded by molten plastic, which flows around the knurls, flutes, and undercuts on the OD of the insert. Total process time is usually <1 s. Ultrasonic insertion combines the high-performance strength of conventional moldedin insert with the advantages of postmolded installation. Advantages include short cycle times, no induced stress in the plastic around the metal insert, elimination of possible mold damage and down time should inserts fall into the mold, and reduced molding cycle times. Insertion also allows multiple inserts to be driven simultaneously and is ideal for automated, high-production operations. Insertion is a very repeatable and controlled process. General guidelines for insertion applications include 1. 2. 3. 4. 5.

Low amplitude to reduce horn stress Medium to high pressure to prevent cold pressing the inserts Slow down speed to allow the thermoplastic to soften Pretriggering the ultrasonic energy to prevent a stall condition Horn face should be 3–4 times the diameter of the insert when possible to prevent horn/insert coupling

Ultrasonic insertion can be performed in two ways as follows: 1. The horn can contact the insert, driving it into the plastic part. 2. The horn can contact the plastic part, driving it over the insert. The method used is determined by the particular requirements of an application. However, the advantages of having the horn contact the plastic and driving it over the insert are reduced horn wear and less noise during the assembly process. If the horn must contact the metal insert, it is advised that a hardened steel horn be used, due to the high wear of metal-to-metal horn/insert contact. Horns made of titanium may also be used. Although titanium is not as wear-resistant as hardened steel, it has a higher tensile strength, which makes it capable of handling more stress. Common applications for ultrasonic insertion include threaded bore inserts, eyeglass hinges, machine screws, threaded rods, decorative trims, electrical contacts, and terminal connectors.

7.7

Swaging and Forming

Swaging is a method of assembling two materials without creating a molecular bond. The material that is swaged is always thermoplastic. The swage captures the second material by melting and reforming a ridge of plastic (usually the outside wall) to lock the material in place (Figure 7.15). The part that is captured is typically a dissimilar material, such as glass. Forming is the process of physically changing the shape of a plastic part. The advantages of using swaging include a tight finished assembly, fast cycle times, and the elimination of fasteners or adhesives. Both swaging and forming require special tooling and consideration of the properties of the materials involved.

CRC_DK8307_CH007.indd 314

10/27/2008 2:28:09 PM

Properties of Materials

315 Force

Clamp

Ultrasonic horn

Flexure bar

Ultrsonic motion

Anvil Bond area FIGURE 7.15 Swaging.

In swaging, the horn face dictates how the plastic melts and flows, as well as the final swaged shape. The swage can be continuous or segmented. General guidelines for swaging applications include 1. High initial trigger pressure with controlled velocity to begin cold-forming the plastic 2. Pretriggering of the ultrasonic energy, so that the vibrating horn immediately begins melting the material on contact 3. Using a heavier hold pressure to help overcome material memory

7.8

Spot Welding

Ultrasonic welding joins two like thermoplastic components at localized points with no preformed hole or energy director (Figure 7.16). It produces a strong weld and lends itself to large parts, sheets of extruded or cast thermoplastic, and parts with complicated geometry or hard-to-reach surfaces. Spot welding is often used on vacuum-formed parts such as blister (clamshell) packaging. Most thermoplastics can be spot-welded. In spot welding, a specially designed spot-welding tip melts through the top thermoplastic layer. The weld occurs at the interface between the two sheets. The bottom layer of a spot-welding joint has a smooth appearance. The top layer has a raised ring around the joint. It is a fast assembly process, requiring no extra fasteners, and generally, no special fixturing is needed.

CRC_DK8307_CH007.indd 315

10/27/2008 2:28:09 PM

Ultrasonics: Data, Equations, and Their Practical Uses

316

Hard facing

FIGURE 7.16 Ultrasonic horn for wheel type continuous strip weldings.

The basic guidelines for spot welding include 1. Rigid support directly under the spot-weld area to prevent marking 2. Medium to high amplitude to ensure adequate material penetration 3. Low pressure so that adequate melt can be made at the point interface At times, spot welding is accomplished using a handheld transducer, called a convert-a-probe.

7.9

Degating

Degating is an ultrasonic assembly technique used in separating injection molded parts from their runner systems. By applying ultrasonic energy to the runner in an out-of-phase manner, the parts are melted off at the gate. Degating works best with rigid thermoplastics such as ABS styrene, or acrylics. The advantages that ultrasonic degating offers are speed of operation (typical cycle time is <1 s), low stress on parts, and a clean break at the part surface. There are two major guidelines associated with degating 1. The gate area should be small (0.060 in.) (1.5 mm). 2. Horn contact should be as close to the gate as possible.

7.10

Scan Welding

Scan welding is the continuous, high-speed ultrasonic welding of flat parts that are conveyed beneath a stationary ultrasonic horn. This process is suitable for rigid thermoplastic parts that have at least one flat surface for horn contact. Some fabric or film applications are also suitable for this process. Both large and small thermoplastic parts may be scan-welded. The joint designs that should be used for rigid thermoplastics are self-locating designs such as tongue and groove, step, and pin and socket.

CRC_DK8307_CH007.indd 316

10/27/2008 2:28:09 PM

Properties of Materials

7.11

317

Bonding and Slitting

Ultrasonic assembly techniques are used in several applications in the textile, apparel, and nonwoven industries. Two of the most common are ultrasonic bonding and ultrasonic slitting. 7.11.1

Ultrasonic Bonding

Ultrasonic bonding assembles two or more layers of nonwoven materials by passing them between a vibrating horn and a rotary drum (often referred to as an anvil). The rotary drum is usually made out of steel and has a pattern of raised areas machined into it. The high-frequency mechanical motion of the vibrating horn and the compressive force between the horn and the rotary drum create frictional heat at the point where the horn contacts the material(s), bonding the material(s) together in the pattern of the rotary drum. Bonding takes place only at the contact points between the horn and the material(s). This gives the materials a high degree of softness, breathability, and absorption. These properties are especially critical for hospital gowns, sterile garments, diapers, and other applications used in medical industry and clean room environments. Ultrasonic bonding uses much less energy than thermal bonding, which uses heated rotary drums to bond material(s) together. As with other ultrasonic assembly techniques, ultrasonic bonding requires no consumables, adhesives, or mechanical fasteners. 7.11.2

Ultrasonic Slitting

When thermoplastic material is slit ultrasonically, the edges of the slit are also sealed. Sealing the edges of a woven fabric is beneficial, because yarns are prevented from unraveling and the smooth, beveled edges prevent buildup of the roll material. Two or more layers of woven or nonwoven materials can be slit and melted together during ultrasonic slitting. Speed of operation can be a significant advantage with ultrasonic slitting. The shape of the cutting wheel (anvil) determines the width of the ultrasonic seal and the speed at which it can be formed. There are three methods of ultrasonic slitting and sealing: continuous, plunge, and traversing. The method used depends on the application requirements and the material manufacturing process(es) involved. In the continuous method, the horn and anvil remain in a fixed location and the material is fed through the gap between them. In the plunge method, the material remains in a fixed location and is periodically contacted by the horn. In the traversing method, the material remains in a fixed location and the horn moves over it.

7.12

Other Tool Designs and Application Methods

Other methods of applying ultrasonic energy to bonding materials include fixed-strap bonding and continuous rolling contact bonding in which a shear is transmitted to the weld area.

CRC_DK8307_CH007.indd 317

10/27/2008 2:28:09 PM

Ultrasonics: Data, Equations, and Their Practical Uses

318 7.12.1

Fixed-Strap Bonding

In the fixed-strap method (Figures 7.15 and 7.16) the ends to be bonded are overlapped on an anvil. A short burst of ultrasonic energy is applied to the overlapped juncture in a shearing motion as pressure increases simultaneously from zero to a maximum. Welding occurs as a result of energy being absorbed by the frictional effect of the ultrasound. The joint is cooled rapidly under pressure. This is a very rapid process. 7.12.2

Continuous Rolling Contact

In continuous rolling contact bonding, the pressure for bonding is applied through a wheel-shaped enlargement on the end of a hollow horn (Figure 7.18). Pressure is applied by lever action along the axis of the horn. The horn rotates as the material to be bonded moves between it and a moving anvil. In both the fixed strap method and the continuous rolling method the tool tip should be hard-faced to minimize wear. The horn of the rolling contact bonding is half-wave in length. The enlarged attachment should be machined into the horn contact with a fillet radius sufficient to hold the stresses below the fatigue limit.

7.13

Metals Properties for Choice of Horns [1–3,6]

Ultrasonic horns and transmission lines are made of materials that meet the requirements of the project to which they are to be applied. These include temperatures, environmental conditions, and assembly requirements. Typical applications can be fulfilled by aluminum 6061-T6, aluminum 7075-T6, titanium 6 Al 4V, stainless steel 304, stainless steel 410, and stainless steel 416. Molybdenum has been used for treating materials at very high temperatures (such as molten glass) in a nonoxidizing nitrogen environment. Tantalum has held up well in treating acidic materials at high temperatures and pressures. Tantalum is very expensive for ordinary use on typical daily projects. Some of the factors to be considered in choosing a material to be used in applying ultrasonic energy are wear, corrosion, ability to withstand chemical attack and reaction, and the dynamic stability of the material used. Transmission lines are usually of uniform cross-section. Their design follows the principles given in Chapter 2 for horns. They are used to transfer energy from a horn of a material that may not sustain the environment to be worked, such as acidic material, hot material, or materials subjected to extremely high pressure in a pressurized cell. One example is the application of ultrasound to molten glass through a rod such as a special molybdenum alloy. The rod is protected by a neutral atmosphere, such as nitrogen, while the rod itself protects the driving horn, which may be of titanium alloy or stainless steel, from the heat. A second example is the application of ultrasound to an acidic bath at elevated temperature and high pressure through a tantalum rod. 7.13.1

Horn Materials

Aluminum 6061 and aluminum 7075 are among the earliest materials used in producing ultrasonic horns for power applications. The advantages of aluminum for these purposes are good machinability, low power losses, good mechanical properties, good corrosion

CRC_DK8307_CH007.indd 318

10/27/2008 2:28:10 PM

Properties of Materials

319

resistance, low density, high strength-to-weight ratio, and fracture toughness. Strength can be enhanced during the fabrication process by alloying, heat treatment, cold working, or a combination of all three (ASM Metals Hdbk). Disadvantages of using aluminum alloys for power applications include inability to join extensions by silver soldering to the horn to meet needs inappropriate for the alloy to contact. These inappropriate needs include excess heat and chemical attack under ultrasonic agitation. Methods for manufacturing aluminum alloys affect significantly the material properties. These methods vary considerably. Aluminum alloys are most commonly strengthened by cold working. Other methods include solution heat treatment and precipitation hardening. The most commonly used temper for aluminum is T6. The T6 temper provides good mechanical properties, enhanced machinability, and corrosion resistance. The T7 temper increases the resistance to stress-corrosion cracking. Titanium has become one of the most widely used metals for ultrasonic horns at the present time. There are two allotropic crystal forms of titanium: α-titanium (up to 882°C [1620°F]) and β-titanium. The α-titanium has a hexagonal close-packed (hcp) lattice. The β-titanium is body-centered cubic (bcc) [1]. Aluminum is the most important substitutional additive in binary α-stabilized titanium alloys, because it adds to the ductility and light weight of titanium. Pure titanium is highly ductile. It can be rolled at room temperature without significant cracking to >90% reduction in thickness. Titanium alloys should not be used in environments of chlorides or methyl alcohol. Chlorides and methyl alcohol will cause stress-corrosion cracking producing failure at stress levels much lower than commonly occurs with the particular alloy. This type of failure occurs most often with alpha-phase titanium alloys including Ti-5Al-2.5Sn, Ti7Al-12-Zr, and Ti-5Al-5Sn-5Zr, at ∼370°C (700°F). This type of failure is called hot salt stress-corrosion cracking (HSSCC). The more resistant titanium alloys to HSSCC include Ti-4Al-3Mo-IV, Ti-2.5Al-11Sn-5Zr-1Mo-0.2Si, and Ti-2Al-4Mo-4Ar [7]. 7.13.2

High-Strength Low-Alloy Steels

High-strength, low-alloy steels are included because they offer characteristics that may be useful to an ultrasonics user. One may have to make a special order for this type of material, but there are advantages to be gained by doing this. These materials have good formability, fatigue strength, weldability, and notch toughness. Their good resistance to abrasion and atmospheric corrosion are additional attractive features for design considerations. 7.13.3

Ductile (Nodular) Iron

Ductile iron is a versatile engineering material that can fulfill requirements of highstrength, toughness, and wear resistance—qualities that tend to make good horns and transmission lines. They are noteworthy for their high ductility and hardenability. However, cast parts often contain flotation and chunky graphite, which are detrimental to the high-stress operations to which they would be subjected in use as high-intensity ultrasonic horns. 7.13.4

Hard-Facing Wear Surfaces

For wear resistance, one may want to use a hard-facing material fused to an abrasive surface. Continuous welding surfaces for plastics may be prepared by welding a

CRC_DK8307_CH007.indd 319

10/27/2008 2:28:10 PM

Ultrasonics: Data, Equations, and Their Practical Uses

320

hard-facing surface to a much less expensive horn material or vibratory element. At times, a standard horn material will deteriorate from wear significantly under continuous exposure to the work. This is especially significant when the wear surface includes a design to be transferred to the workpiece. After forming the basic horn or special tool design, the hardfacing alloy can be welded or fused to the sensitive area. The mass of the added material must be considered in the design for the system to work effectively. Principal alloying materials for hard-facing include chromium, molybdenum, nickel, and cobalt. Tungsten carbide is one of the hardest materials known. 7.13.5

Stainless Steels (ASM Hdbk)

Stainless steels offer many good characteristics for use in ultrasonic horns. They exhibit high strength and stiffness, and excellent wear- and corrosion-resistance. They may be used over a wide range of temperatures [8]. Stainless steels are grouped in series such as 200, 300, and 400. The 200 series contains nickel and chromium as alloying elements. Nickel enhances the corrosion resistance of austenitic steels, giving them the best corrosion resistance of all stainless steels. This is especially true when these steels have been annealed to dissolve chromium carbides, then rapidly quenched to retain carbon in solution. These austenitic steels are easily welded. The 300 series of austenitic stainless steels share many of the characteristics of the 200 series. They contain more nickel than the 200 series steels. The nickel content in austenitic stainless steels may vary between 7% and 20%. The 400 series include ferritic steel alloys containing low carbon and high chromium. The amount of chromium ranges from 14% to 18% in Type 430 to a high of 23 to 27% in Type 446. Higher levels of chromium increase the resistance to corrosion and scaling, but reduce the notch impact strength. Adding molybdenum to ferritic stainless steels enhances the corrosion resistance. The 400 series also includes martensitic stainless steels. These steels contain chromium as a major alloying element. They can be heat-treated to a wide range of useful hardness and strength levels. Types 403, 410, 414, and 416 contain up to 0.15% carbon and 11.5–14% chromium. Some types of martensitic steels contain up to 1.2% carbon and 16–18% chromium. These include the 440 series. Martensitic steels show hardnesses to approximately 60 HRC and tensile strengths to ∼1.965 MPa (285 ksi). However, they are not as corrosion-resistant as ferritic or austenitic types. Stainless steels exhibit many characteristics favorable to ultrasonic horns and transmission lines with their good strength, weldability, and corrosion resistance. Perhaps their weakest point is their weight. Stainless steels are nearly three times more dense than aluminum and approximately twice the weight of titanium.

7.14

Important Refractory Metals

Refractory metals are elements that have melting points >2204°C (4000°F) [1]. These include tungsten (W), niobium (Nb), molybdenum (Mo), hafnium (Hf), rhenium (Rd), and tantalum (Ta).

CRC_DK8307_CH007.indd 320

10/27/2008 2:28:10 PM

Properties of Materials

321

Other desirable characteristics of refractory metals include • Resistance to electrochemical corrosion (applies especially to tantalum; tantalum has stress-corrosion-resistance comparable to that of glass) • Excellent high-temperature strength • Excellent resistance to wear and abrasion • Good electrical and heat-conducting properties Tantalum and niobium are favored over the other refractory materials, in that they are easily formed, machined, and joined. Tantalum and niobium suffer an accelerated atmospheric oxidation at temperatures >430°C (800°F). This is better than any of the other refractories, which suffer accelerated oxidation at lower temperature levels.

7.15

Comments and Conclusions

The information presented in this chapter is intended for easy selection of materials to be used in the wide variety of high-intensity applications of ultrasonic energy. Some of the materials listed may have never been used before and may never be considered again. But the reasons for presenting them evolve from some past experiences. For example, in one case there was a need to ultrasonically promote a chemical reaction in an acidic solution (of pH 3) at a high temperature and a high pressure. The requirements were that the material to be treated was placed in a high-pressure cell. The ultrasonic transmission line had to be designed to extend through the bottom of the cell. The line had to withstand the acidic environment. So many of the materials that withstand acids, even at room temperatures, dissolve rapidly after the cavitation of the liquid breaks through the protective coating. The transmission line also had to be long enough that the ultrasonic source was protected from overheating. The solution was to use tantalum for the transmission line. Tantalum was unique for this purpose. It was not a material that was commonly used for transmission lines for horns. The acoustic properties for tantalum were unknown to the designer at the time of this program. He calculated a bar velocity of sound for determining the length using Young’s modulus and density for the material. This was the only information available to him at that time. Fortunately, the calculated value proved to be correct. The material required for the delay line is expensive—$2000 for the single 20 kHz unit. It held up very well for the complete project, making the treatment and cost worthwhile. This example is presented for the purpose of illustrating the reason for giving the choice of materials that are available for use in applying high-intensity ultrasonic energy to specific needs.

References 1. 2. 3. 4.

M. Bauccio, ASM Metals Reference Book, 3rd edition, ASM International, Materials Park, OH, 1993. J.E. Dean, Lange’s Handbook of Chemistry, 15th edition, McGraw-Hill, Inc., New York, 1994. D. Ensminger, Ultrasonics, Marcel Dekker, Inc., New York, pp. 206–213, 1988. Dukane Ultrasonics Corporation, Guide to Ultrasonic Plastics Assembly, Dukane Corporation, St. Charles, IL, 1995.

CRC_DK8307_CH007.indd 321

10/27/2008 2:28:10 PM

322

Ultrasonics: Data, Equations, and Their Practical Uses

5. I.V. Kragelskii, Friction and Wear, Butterworths, Washington, pp. 15–17, 1965. 6. R.C. Weast, Ediotor-in Chief, CRC Handbook of Chemistry and Physics, 60th edition, CRC Press, Inc., Boca Raton, FL, 1985. 7. G.S. Brady and H.R. Clauser, Materials Handbook, 13th edition, McGraw-Hill, Inc., New York, 1991. 8. S.P. Parker, Editor-in-Chief, McGraw-Hill Dictionary of Scientific and Technical Terms, 5th edition, McGraw-Hill, Inc., New York, 1994.

CRC_DK8307_CH007.indd 322

10/27/2008 2:28:10 PM

8 Ultrasonics-Assisted Physical and Chemical Processes Dale Mangaraj, B. Vijayendran, and Dale Ensminger

CONTENTS 8.1 Introduction ........................................................................................................................ 324 8.2 Physical Processes.............................................................................................................. 324 8.3 Chemical Processes ........................................................................................................... 325 8.4 Ultrasonic Effects ............................................................................................................... 325 8.5 Degassing ............................................................................................................................ 325 8.6 Cavitation ............................................................................................................................ 326 8.6.1 Cavitation Phenomena .......................................................................................... 327 8.6.2 Ultrasonic Cleaners ............................................................................................... 328 8.6.2.1 Principles of Ultrasonic Cleaning ......................................................... 328 8.6.3 Ultrasonic Cleaning Fluids................................................................................... 329 8.7 Enzymes and Ultrasonic Cleansers ................................................................................ 332 8.7.1 Enzymes and Cleansers ........................................................................................ 333 8.8 De-inking of Office Waste Paper ..................................................................................... 333 8.9 Homogenization and Emulsification .............................................................................. 335 8.10 Dispersion and Homogenization .................................................................................... 337 8.11 Coagulation, Precipitation, and Filtration ...................................................................... 337 8.12 Atomization ........................................................................................................................ 338 8.13 Preparation of Nanomaterials ..........................................................................................340 8.14 Crystallization .................................................................................................................... 341 8.14.1 Crystallization in Metals ...................................................................................... 341 8.15 Diffusion and Filtration through Membranes ............................................................... 341 8.16 Chemical Effects.................................................................................................................342 8.16.1 Sonochemical Reactions in Aqueous Solutions.................................................344 8.16.1.1 Gases .........................................................................................................344 8.16.1.2 Inorganic Compounds............................................................................344 8.16.1.3 Organic Compounds ..............................................................................345 8.16.1.4 Nonaqueous Sonochemical Reactions ................................................. 347 8.17 Polymerization ................................................................................................................... 350 8.18 Polymer Degradation ........................................................................................................ 353 8.19 Polymer Processing ...........................................................................................................354 8.19.1 Vulcanization..........................................................................................................354 8.20 Devulcanization ................................................................................................................. 356

323

CRC_DK8307_CH008.indd 323

11/14/2008 4:56:15 PM

Ultrasonics: Data, Equations, and Their Practical Uses

324

8.21 Large-Scale Sonochemical Processing ............................................................................ 357 8.22 Summary............................................................................................................................. 360 References .................................................................................................................................... 361

8.1

Introduction

As the previous chapters have revealed, ultrasound has many attributes to offer industry, medicine, and research. This chapter discusses some important applications; in many of these, cavitation plays an important role. Materials undergo physical and chemical processes under external stress, both physical and chemical. Stress is often used to change the nature of materials, to give them shape and size, and to make them useful in different applications. It is therefore essential to have a basic understanding of physical and chemical processes.

8.2

Physical Processes

Most materials exist in three different states: solid, liquid, and gas. In the gaseous state, the molecules of a material move with high speed and occupy any amount of space available. The density of gas is very low at standard atmospheric temperature and pressure but may equal that of light liquids under very high stresses. In the liquid state, molecules retain some of their kinetic motion so as to flow readily. Because of cohesive forces, a liquid does not expand indefinitely like a gas. Relative positions of molecules in both the gaseous and the liquid state are random. In the solid state, molecules or ions have very little freedom to move around their center of mass; they occupy a fixed volume. Materials having fixed configuration are crystalline and materials with random configuration are called amorphous. There are many materials that are semicrystalline: they have both crystalline and amorphous phases, intermingled with each other. Polymers, which are composed of long chain molecules, exhibit a rubbery or leather state, where segments of molecular chains exhibit such hybrid phases. On the other hand, some polymers are crystalline, but behave like liquids. Materials change their physical state under thermal stress. The transition temperature from solid to liquid is known as the melting point, Tm, and the transition temperature from liquid to gas is the boiling point, T b. The temperature at which the glassy materials become leathery is known as the glass transition temperature, Tg. Another important physical process is the mixing of two or more materials. All gases provide a molecularly uniform mixture. Liquids mix only when their intermolecular forces are alike. The degree of miscibility depends on temperature and pressure. Emulsification is a process in which two immiscible liquids form a stable, compatible mixture with the help of a third compound known as an emulsifier. Emulsifiers are chemical compounds that have both nonpolar and polar, or ionic, parts. They stay at the interface and facilitate the coexistence of immiscible liquids that differ in their polarities. Most solids do not mix with each other to provide a homogeneous mass. However, they can be mixed in the liquid, solution, or glassy state under shear stress and can be cooled down to give a homogenous mass. Like mixing, separation is also an important physical process. Typical examples of separation processes include precipitation, filtration, degassing, cleaning, and so on.

CRC_DK8307_CH008.indd 324

11/14/2008 4:56:16 PM

Ultrasonics-Assisted Physical and Chemical Processes

8.3

325

Chemical Processes

Most materials undergo chemical reactions when they are subjected to a stress or they come in contact with appropriate chemicals under suitable physical conditions. Such reactions are activated processes and the rate is accelerated by increase in temperature. The change in the rate of a chemical reaction depends on the activation energy, that is, the amount of energy the reactants need to undergo a reaction on a molecular scale. As an approximation, for most reactions, the rate doubles for every 10°C increase in temperature. Details of different chemical reactions can be obtained from standard textbooks on inorganic and organic chemistry. Materials in general undergo degradation when exposed to heat, light, and chemicals. In other words, they become less functional and finally break down. Metals undergo corrosion, a combined result of oxidation and hydration; ceramics become brittle and crack; and molecular weights of polymers decrease substantially, leading to weakening, crack development, and failure.

8.4

Ultrasonic Effects

As discussed earlier, high-frequency ultrasonic waves are propagated as oscillatory motion through materials: solid, liquid, or gas. The oscillatory effect is attenuated by scattering, absorption, and other mechanisms. The high-intensity oscillations in liquids cause strong bubble formation and collapse (cavitation), producing a large increase in instantaneous local temperature and pressure. Increase in temperature leads to phase changes, acceleration of chemical reactions, and material decomposition. Material decomposition may result in generation of free radicals capable of initiating chemical reactions, including polymerization. High-frequency ultrasonics has therefore found many applications in a variety of chemical and physical processes involving both organic and inorganic materials. Details of these processes have been discussed by Suslik, Ensminger, and Duraswamy [1–3]. The following sections provide brief descriptions of these processes and their uses in both product and process development.

8.5

Degassing

When high-intensity ultrasonic energy is applied to liquids containing dissolved gases, the gases are released into pockets at intensity levels below that at which cavitation of the solution occurs. The bubbles that are formed are not caused by cavitation. They are transient, combining by coalescence, and rise to the surface at a rate dependent upon the sizes of the bubbles and the viscosity of the liquid. When higher intensities are applied with the intention of producing cavitation, degassing occurs first. The rate of removal depends upon the intensity of the ultrasound, the viscosity of the fluid, and the bubble size. If the intensity is too high, the larger bubbles are shattered and move out at a lower rate. This phenomenon is sometimes used to remove gases from liquids to be used later for other purposes.

CRC_DK8307_CH008.indd 325

11/14/2008 4:56:17 PM

Ultrasonics: Data, Equations, and Their Practical Uses

326

Spinosa and Ensminger [4] have studied the use of ultrasonic energy in processing glass. The energy was applied through molybdenum transmission lines protected from oxidation by a nitrogen atmosphere. The viscous modeling indicates that bubble coalescence does occur and that it can contribute to a 15–20% improvement in glass throughput rate, thereby accomplishing an energy reduction in glass throughput rate, thereby accomplishing an energy reduction per pound of melter output. Degassing of carbonated drinks, beer, photographic solutions, and others is carried out by using either ultrasonic cleaning tanks or ultrasonically fitted continuous flow equipment. Degassing of molten metals using ultrasonics has been used to decrease porosity on solidification [5,6]. Ultrasonics has been used to defoam materials during processing. One small whistle aimed into the foam over a tank of oil used in the manufacture of soap reduced the amount of foam in the tank after filling to a satisfactory amount for truck transportation. The whistle operated at 15 kHz.

8.6

Cavitation

Ultrasonic cleaning is especially associated with cavitation. Cavitation is also the key to homogenization and emulsification. It is associated with the dispersion of materials in solvents and fluids and sometimes in the coating of particles and materials. It may or may not be involved in the atomization of liquids and droplet formation depending particularly upon the necessary controls imposed on how the drops are formed. Control of foams may or may not be associated with cavitation depending upon the technique to be used; that is, whether the control is based upon breaking down foam forming above the surface of the liquid or whether it inhibits formation of the foam from inside the bulk. Cavitation refers to the formation or rapid enlargement and collapse of cavities, or bubbles, in a liquid medium being subjected suddenly to low pressures. Cavitation may occur when liquid is forced through certain constrictions or behind a high-speed propeller. In the present context, cavitation is produced by the presence of high-intensity ultrasonic waves in a liquid. When a liquid is subjected to a high-intensity ultrasonic wave, during the rarefaction portion of the cycle when the pressure in the wave is below ambient, gas pockets expand with the impressed field until the pockets collapse violently due to the high stresses developed in the walls. The source of these gas pockets is generally molecules of gas that are very finely dispersed throughout the liquid volume. These may be located at vacant sites of the quasicrystalline structure of the liquid or they may be contained in invisible bubbles of microscopic dimensions [7]. Cavitation is of two types: gaseous cavitation and vaporous cavitation. Gaseous cavitation involves gases dissolved or entrapped in the liquid or existing on surfaces in contact with the liquid. Vaporous cavitation involves gases from the vaporization of the liquid itself. Most liquids contain nuclei about which cavitation bubbles originate. These nuclei may consist of dispersed dust particles, prominences on immersed surfaces, and minute gas bubbles. In fact, unless especially treated, liquids contain dissolved or entrained gas. Various factors influence the onset and intensities of the cavitation bubbles. These factors include the sizes of the nuclei, ambient pressure, amount of dissolved gases, vapor pressure, viscosity, surface tension, and the frequency and duration of the ultrasonic energy.

CRC_DK8307_CH008.indd 326

11/14/2008 4:56:17 PM

Ultrasonics-Assisted Physical and Chemical Processes

327

To be able to produce the effects associated with the expansion and the violent collapse of cavitation bubbles, the bubble must be capable of expanding with the rarefaction part of the cycle of the impressed field and of collapsing before the total pressure reaches its minimum value. That is, the bubble must reach the size where it will collapse catastrophically in less than one-quarter the cycle of the impressed wave. Therefore, generation of intense cavitation depends upon the relationship between the dimensions of the nuclei and the wavelength and the intensity of the sound field. Bubbles larger than a critical radius, Rc, will not expand to an unstable size for catastrophic collapse before the pressure in the wave starts to increase. Frederick [8; 2, p. 67] gives the following relationships for Rc: 3P0 326 1/2 P0 ⬇ f 

Rc 

1


Rc 

1 3  3.9  ⬇ 
R 0  f 

P0  

2 R0

(8.1)

P0 

2 R0

(8.2)

2/3

where Rc is the critical bubble radius (cm), R0 the initial bubble radius, γ the ratio of specific heats of the gas in the bubble, σ the surface tension of the liquid (dyn/cm), P0 the hydrostatic pressure (atm.), ρ the density of liquid (g/cm3), and ω is 2πf where f is frequency. The violence of the collapse of the bubble in a cavitation field depends upon the ratio Rm/R0. The greater the ratio, the more violent the collapse. R m is the maximum bubble radius before collapse. Increasing the ambient pressure does not increase the intensity of the collapse. Equations 8.1 and 8.2 show that the diameter of a resonating bubble must be smaller than the wavelength of the sound in the liquid for the bubble to grow to collapsible size at the particular frequency. In fact, the wavelength of the ultrasonic field usually is approximately two orders of magnitude larger than the critical diameter of the resonating bubble. Only the bubbles that are smaller than critical size are capable of rupture and subsequent collapse before the end of the pressure cycle. This is why production of observable cavitation is increasingly difficult as frequency increases. For example, at 1.0 MHz, the wavelength in water at 20°C is on the order of 0.15 cm. The maximum critical diameter of a resonant bubble under these conditions is ≤15 μm. From Equation 8.1, letting P0 = 16 atm in water and f = 20,000 Hz, for P0 >> 2σ/R0, Rc is ∼0.0336 mm, and λ/Rc is ∼22,050. When P0 is very small compared with 2σ/R0, Rc = 0.652 mm and λ/Rc is 1137. Under similar conditions, the ratios of λ/Rc at 1.0 MHz is ∼113.7 for the high-pressure case and 598.5 for the low-pressure case. 8.6.1

Cavitation Phenomena

Many remarkable phenomena are associated with cavitation produced by ultrasound. Some of the chemical and physical effects are attributable to the tremendous pressures on the gases within the bubbles and the very high temperatures associated with these pressures. Pressures as high as 5000 atm have been reported, causing instantaneous temperatures of at least 7200°C (13,000°F). Some chemical and physical effects associated with high-intensity cavitation include production of OH and other ions, erosion of metal surfaces, disruption of aggregates, and other effects not producible by any other known means.

CRC_DK8307_CH008.indd 327

11/14/2008 4:56:17 PM

Ultrasonics: Data, Equations, and Their Practical Uses

328

When fresh eggs that have been exposed very briefly to ultrasonically produced cavitation within the shell are broken open immediately following treatment, the contents (including the lining) are so thoroughly homogenized that they pour out like water. When the contents are held within the unbroken shell for a short period of time after treatment, they develop a strong odor of rotten eggs indicating that the contents have not only been homogenized but that they have experienced a molecular change during the treatment, indicated by the strong rotten egg odor, which is indicative of a sulfurous acid content (H2SO3). An egg treated at emulsifying intensities for a few seconds within the shell will cook and be edible when it is opened. 8.6.2

Ultrasonic Cleaners

Perhaps one of the oldest and most useful applications of ultrasonic cavitation is cleaning. Ultrasonic cleaning takes many forms. The common cleaner consists of a tank filled with cleaning fluid into which the contaminated surfaces to be cleaned are fully immersed so that the ultrasonic energy can have adequate access to them. The transducers are attached to the bottom of the tank or to the sides. Small parts may be suspended in a basket. Hard surfaces must not be allowed to ride on the bottom. Parts may be rotated slowly or allowed to tumble to assure complete access by the ultrasonic energy to contaminated surfaces. The tank may be specially designed for specific items such as an elongated trough for window blinds or golf clubs. Materials can be cleaned in a hostile environment by means of clamp-on transducers. Special horns may be designed to reach into interior volumes. A cleaning job is limited only by the imagination of the operator. The compatibility of the cleaning fluid with the equipment used for cleaning and the parts to be cleaned is extremely important. Materials to be cleaned must be compatible in dimensions with the cleaner.

8.6.2.1

Principles of Ultrasonic Cleaning

Ultrasonic cleaning is used in a variety of industries, including medical, dental, electronic, optical, and industrial operations. Cleaning is carried out primarily by cavitation in the cleaning fluid. The cavitation activity not only produces kinetic motion but also brings fresh solvent close to the contaminants where they are either dissolved or dispersed as very fine particles. Water and many other solvents are used as cleaning media. Cleaning agents are selected based on their ability to combine cavitational activity with chemical action. The effectiveness of cleaning depends on the type of stress generated between the contaminant and the cleaning fluid, severity of agitation, increase of attraction between the contaminant and cleaning fluid, gas content of the liquid, the adhesive forces between the contaminant and the liquid, and the potential for promoting desirable chemical reaction at the interface. When a surface containing a contaminant is exposed to cavitation, intensity of stress generated depends on the vapor pressure of the cleaning fluid, the gas content of the liquid, and the adhesive forces between the liquid and the surface. The intensity of the ultrasonic energy must exceed the intensity needed to promote cavitation in the cleaning solvent. In most cases, this is 0.5–0.6 W/cm2. The frequencies used in commercial equipment are 20–60 kHz, with 40 kHz being the most common. The power levels are commonly 200 W per each gallon of tank capacity, regardless of the type of irradiating surface used. Conversion efficiency of electronic generator and transducer determines power available to the cleaning solution.

CRC_DK8307_CH008.indd 328

11/14/2008 4:56:17 PM

Ultrasonics-Assisted Physical and Chemical Processes

329

Ultrasonic cleaning is attributable to many factors Ref. [9], especially those which include cavitation. The main factors include (1) development of stresses between the cleaning fluid and the contaminated surface, (2) agitation and dispersion of contaminant throughout the cleaning fluid, (3) increase of the attractive forces between the contaminant and the cleaning fluid, (4) promotion of chemical reactions at the contaminated surfaces in some cases, and (5) effective penetration of pores and crevices. Development of stresses between the cleaning fluid and the contaminated surface is exemplified in nearly all ultrasonic cleaning processes. As cavitation bubbles form at the surface to be cleaned, every component of the forming bubble is stressed at a high repetition rate. The components of the bubble include the internal liquid surface of the entire bubble plus any portion of the solid surface in direct contact with it. The intensity of the stresses under these conditions is a function of vapor pressure of the liquid, the gas content of the liquid, and the adhesive force between the liquid and the surface. These stresses are sufficiently high under cavitation conditions to erode the solid with time, to break suspended solids, to disperse materials throughout the liquid, and, in connection with the instantaneous temperature produced, to accelerate various chemical reactions. These chemical reactions help accelerate the cleaning operation. Agitation occurs not only in the presence of cavitation but also from intensities that promote flow without cavitation. When a compression wave is generated from a solid surface in a fluid, the compression at the interface directs the fluid away from the surface. As the source surface moves to cause rarefaction, other fluid moves to fill in for the outward flowing fluid, thus providing a continuous movement of fluid away from the source (ultrasonic wind). Agitation provides a scrubbing action that promotes the removal of contaminants. Such contaminants may be loose, solid particles or materials that will dissolve or emulsify in the cleaning fluid. A cleaner should never be overloaded. A very important aspect of ultrasonic cleaning is its ability to draw debris out of pores and crevices. 8.6.3

Ultrasonic Cleaning Fluids

Choice of cleaning solvent depends on the chemical compatibility of fluids with materials to be cleaned and the container, as well as on the effectiveness of the cleaning media at removing the contaminant from the surface. The stresses caused by cavitational bubbles are controlled by the vapor pressure of the liquid and volatile impurities present. As vapor pressure increases, surface tension decreases and the maximum stress associated with cavitation decreases. Hence, the cleaning media need to be carefully selected to provide desirable cavitational stress. The intensity of the cavitational stress depends on the ratio of the critical bubble size to the minimum bubble size. High-intensity cavitation stresses can erode plated and coated surfaces. It is important that erosion is minimized or eliminated by the choice of high-vapor-pressure solvents or by blending good solvents with high-vapor-pressure liquids. Addition of a small amount of methyl or ethyl alcohol to water has been used to avoid the destructive effects of ultrasonic cavitation [2, pp. 426–429; 10]. Solubility of a contaminant material in a solvent largely depends on the intensity of their intermolecular interaction. For most nonpolar/organic solvents and organic contaminants such as oil, grease, and paint, the solubility parameter—which is the square root of cohesive energy density (energy of vaporization per unit volume of the solvent)—is a good criterion for evaluating solvent capacity. A solvent will dissolve contaminant, if the difference between the solubility parameter of the solvent and that of the contaminant is

CRC_DK8307_CH008.indd 329

11/14/2008 4:56:17 PM

330

Ultrasonics: Data, Equations, and Their Practical Uses

small [2, pp. 434–435; 11], very often of the order 1–2 units, (Cal/cc)1/2. In case of polymeric contaminants, this difference has to be even smaller. In general, polar contaminants are dissolved by polar solvents and vice versa. Alcohols, ketones, esters, amines, nitriles, etc. are polar solvents. Hydrocarbons and silicon fluids are examples of nonpolar solvents. Ionic materials such as salts are dissolved in solvents with high dielectric constants. Water, which hydrates most ions and has a high dielectric constant, is a good solvent for many ionic solids. Dilute acids are often used to dissolve metal oxides formed due to oxidative degradation and corrosion of metallic substrates. Addition of wetting agents, surfactants, and detergents help in dislodging the contaminants from the substrate and are used both in aqueous and nonaqueous cleaning processes [12]. Selection of solvents should also take into account harmful side effects and environmental compatibility. It is possible that some chemical reaction might take place under acoustic cavitation in certain solvents, resulting in hydrogen embrittlement or dissolution of the container material. Such solvents should be avoided, or if it is necessary to use them, careful planning of materials, seals, and the like should be made before using them. Solvents with high vapor pressure may fume due to temperature rise during ultrasonic application. Thus one has to be very careful in selecting a solvent or solvent system for a particular job. There are many fluids used for ultrasonic cleaning. The manufacturers of the ultrasonic cleaners may offer or recommend certain proprietary trade-name formulations of their own to use with specific classes of soils. We say “may offer,” because they are just as often reluctant to suggest cleaning fluids. They “do not promote or endorse particular products or products from any particular vendor. The selection of a chemistry must be based on an analysis of the substrate and the soil and, finally, by actual laboratory or use environment testing of the product on real case parts. Today’s chemistries are very specialized and offer a wide variety of options which cannot be addressed in generalized recommendations.” [13] Commonly used cleaning agents used in ultrasonic cleaning are listed in Table 8.1 [2, pp. 426–430]. Solubility parameters of organic solvents (δ) are given in (Cal/cc)1/2. Industrial ultrasonic cleaners are standalone units, using stainless steel vessels of 5–150 L capacity for tanks. They are used routinely in the semiconductor industry. Although some cleaners operate at a lower frequency (25 kHz), most cleaners operate at a higher frequency. Temperature-controlled devices are used to maintain constant temperature during the cleaning process. Computer-controlled robotic devices are used for automatic material handling. Additional cleaning functions such as presoaking or vapor rinsing are added to the cleaner to make the cleaning process cost-effective. Small cleaners used in laboratory, jewelry shops, and small part cleaning facilities and ultrasonic cleaning devices (such as sonic toothbrushes) come in one package, containing both the container and the electronics. They operate at 50–60 kHz with a power input of approximately 25–150 W. The effectiveness of a cleaning process is evaluated by measuring the power density inside the cleaner. However, the normal propagation of high-intensity waves is disrupted by the presence of cavitation bubbles, making the measurement by means of pressure-sensitive probes impossible. Calorimetric measurement of heat generated during cleaning can be a good measure of cleaning efficiency, provided that all the energy is absorbed in the cleaning process. The more popular method to evaluate ultrasonic cleaners is to measure sonochemical activity, erosion effects, and probe reactions. Measurement of the iodine released from a solution of potassium iodide in carbon tetrachloride is one of the commonly used methods used. Ultrasonically initiated cavitation releases chlorine from the carbon tetrachloride and iodine from the potassium iodide. The chlorine takes the place of the released iodine in the KI formula and the removed iodine remains free in the solution.

CRC_DK8307_CH008.indd 330

11/14/2008 4:56:17 PM

Ultrasonics-Assisted Physical and Chemical Processes

331

TABLE 8.1 Commonly Used Solvents for Ultrasonic Cleaning Solventsa

Base Materials

HCl with inhibitor

Ferrous alloys

H2SO4 with inhibitor

Ferrous alloys

HNO3 with inhibitor

Stainless steel, some aluminum Iron, steel, brass, zinc, aluminum

Phosphoric acid with inhibitor Water Detergent and water Soap and water Trichloroethylene Tetrachloroethylene Acetone Benzene Xylene Chlorothene

Toluene Freonsb Tricholorotrifluoroethene (Freon 113)

Any material not damaged by water All metals and many other materials All metals and many other materials All metals polar organic, δ = 9.5 All metals polar organic, δ = 9.5 All metals All metals aromatic, δ = 9.2 All metals δ = 8.9 Most materials (bearing materials meters, circuit boards, electronic components, etc.) All metals aromatic, δ = 9.1 Most materials (circuit boards, etc.)

Tetrachlorodifluorethane (Freon 114)

Most materials (circuit boards, etc.)

1,1,1-Trichloroethane

Electrical and electronic assemblies

Contaminants

Remarks

Oxides, heat-treatment scale Oxides, heat-treatment scale Oxides, heat-treatment scale Light scale and oxides, some drawing compounds oil, grease Loose soils, watersoluble soils Oils, greases, loose soils, etc. Oils, greases, loose soils, etc. Oil, grease, buffing compounds, etc. Oil, grease, buffing compounds, paints Oils, some plastic base cements (polar) Oil, grease, adhesive paint Oil, grease (slightly polar) Grease, oils, dust, lint, flux, oxides, pigments, inks

Plastic containers preferred

Oil, grease, adhesive paint Fluxes, loose soils, oils, other organic compounds

Stainless steel containers

Fluxes, loose soils, oils, other organic compounds Flux, dust, etc.

Plastic containers preferred 316 stainless steel and plastic are acceptable materials 316 stainless steel is an acceptable container Stainless steel containers Stainless steel containers Stainless steel containers Stainless steel containers Stainless steel Stainless steel containers Stainless steel containers Stainless steel containers Stainless steel containers

Stainless steel, aluminum, or plastic containers can be used; easily distilled and recycled Stainless steel, aluminum, or plastic can be used; can be used to 150°F but expensive Stainless steel containers

The following are proprietary trade-name formulations (by sonicpro.com) 983 Sonic Booster AC-40

Water conditioner, other cleaning agents Electrical equipment, electronic soft metals

Grease

Medium-duty cleaning performance Ideal for cleaning delicate items and components and painted surfaces. Mildly alkaline; nonbutyl polymeric; environmentally safe, nontoxic, biodegradable, and super-concentrated for economical use. (continued)

CRC_DK8307_CH008.indd 331

11/14/2008 4:56:18 PM

Ultrasonics: Data, Equations, and Their Practical Uses

332 TABLE 8.1

(Continued)

Attack blind cleaning powder BP-70

10-60 LN-42 SC-80

PVC, fabrics, glass, metal, and painted surfaces Alkaline nonbutyl degreaser, safe on most surfaces except highly polished aluminum Molds Porcelain, glass, plastic, and metals Water-damaged metal tools and parts, aluminum and other metals; iron, steel, aluminum, brass, copper, plastic, and painted surfaces

Formulated for use in blind cleaning equipment Grease, oil, carbon deposits, ink pigment (printing press rollers), on industrial soils Carbon residue Smoke and fire residue

Injection mold cleaner Alkaline cleaner/degreaser

Rust, heat scale, tarnish, and oxides

Nonhazardous, biodegradable, and has no harmful vapors. Made from nontoxic organic acids, detergents, and penetrants. Apply OB-300 after cleaning with SC-800 to prevent flash rusting. High pressure; clamp-on transducers

Liquid CO2

a

Wetting agents are recommended for most solvents used in ultrasonic cleaners. DuPont trademark. Source: Ensminger, D., Ultrasonics, Marcel Dekker, New York, 1988: pp. 418, 419.

b

The free iodine produces a blue color in a starch solution. The intensity of the blue color is measured by a colorimetric method. It has been found that the rate of chlorine release is linear with time and appears to be proportional to the intensity of cavitation, which is proportional to power density in the cleaning bath. Carbon tetrabromide may be used in place of carbon tetrachloride, particularly if the measurement has to be done above room temperature [2, pp. 431–438]. Another method used to measure cleaning effectiveness is to measure erosion. This method uses the hanging of aluminum foil in the cleaning bath and measuring the weight loss as a function of time or measuring the time required to make holes in the foil. It is necessary to make certain that the foils have the same metallurgical history from piece to piece and from tank to tank for this method to be effective. Pressure-sensitive probes are used to measure the noise intensity above the threshold to evaluate the effectiveness of a cleaning tank. Other methods of measuring cleaning effectiveness have been discussed by Ensminger [14].

8.7

Enzymes and Ultrasonic Cleansers

Enzymes are biological catalysts. Many of the chemical changes in living organisms are catalyzed by enzymes. They involve the formation of a complex between the enzyme and the substrate, which changes the conformation of the substrate molecule and expedites the reaction by putting a strain on the bond to be broken. This step, called activation, greatly reduces the amount of energy required to cause the reaction. The reaction then

CRC_DK8307_CH008.indd 332

11/14/2008 4:56:18 PM

Ultrasonics-Assisted Physical and Chemical Processes

333

proceeds and the enzyme is regenerated, becoming available to complex with another molecule of the substrate. The reuse of the enzyme makes it effective in minute quantities. Enzymes are widely used in the food industry. “Certain enzyme-catalyzed reactions in food do not occur in the intact tissue because of the physical separation of the enzyme and the substrate. Mechanical damage to the tissue can bring the enzyme and the substrate into contact with each other and permit such a reaction to occur” [15]. The most important factor affecting enzyme activity is temperature. Each enzyme has an optimum temperature which, with a few exceptions, is within the range 35–40°C (95–104°F). At low temperatures, enzyme activity is low and increases with increasing temperatures, approximately doubling with every 10°C increase in temperature up to the optimum temperature. If the temperature increases above the optimum temperature, denaturation occurs at an increasing rate until finally inactivation is complete. Another factor affecting enzyme activity is pH, being in the vicinity of pH7 for all but a few enzymes. For some, the pH range is very narrow; for others, it is relatively broad. If the pH range for activity is narrow, the activity must be buffered to continue. Enzyme and substrate concentrations are also factors in reaction rates. 8.7.1

Enzymes and Cleansers

Because of their activity with respect to biological materials, enzymes are finding use in ultrasonic cleansers used with surgical equipment. Some individuals question this practice because of the thermal effect on the activity of the enzyme. If the enzyme is to be effective, the manufacturer of the cleanser must remind the user of the safe temperature range and its environment to protect the enzyme from inactivation. Ultrasound applied at intensities below cavitation level accelerates the activity by bringing the enzyme into close contact with the biological debris and dispersing it throughout the volume of the cleanser. After removing the enzymatic fluids, a second ultrasonic wash using a standard rinse at a higher temperature can be performed to assure that sterilization is complete. The enzyme to be used must be carefully chosen with respect to the material with which it is to react.

8.8

De-inking of Office Waste Paper

A new area of interest of ultrasonic application is de-inking, for recovering cellulose fibers from waste papers from copiers, printers, and fax machines. Toner is typically removed from office waste paper in a secondary fiber mill by pulping the waste paper and applying mechanical/thermal energy to dislodge the fused toner from the paper surface and separating it by floatation, washing, and screening [16]. However, this process does not work well, because the toner is fused to the fiber by thermoplastic binders such as styrene-butyl acrylate copolymers, polyester resins, and so on. Further, this process gives toner particles ranging from 40µm to 100 μm diameter, making it difficult to remove them by a floatation process. Table 8.2 lists several prominent manufacturers of ultrasonic cleaners and cleaning equipment and the corresponding cleaner types. Ultrasonic generators such as liquid whistles and piezoelectric transducers have been used in the de-inking process and have been very successful in reducing the size of toner particles in the slurry, without the use of chemicals [17–19]. Norman et al. found that the

CRC_DK8307_CH008.indd 333

11/14/2008 4:56:18 PM

334

Ultrasonics: Data, Equations, and Their Practical Uses

TABLE 8.2 Ultrasonic Cleaner Manufacturers Manufacturer

Typical Cleaner Types

Blackstone-Ney Ultrasonics Jamestown, NY Phone: 800-766-6606 FAX: 716-665-2480

A variety of industrial and professional cleaning equipment, including console cleaners, benchtop cleaning tanks, multistage systems, and more to serve a wide range of industries and markets.

Branson Ultrasonics Corporation Danbury, CT

Full line of precision batch cleaning systems for aqueous solvent cleaning, and provides ultrasonic degreasers and ultrasonics parts cleaners; frequencies cover from 20 kHz to 100 kHz includes an applications laboratory to develop processes to meet the customer’s needs.

Coletene/Whaledent (USA) http://www.coltenewhaledent.com

Biosonic Ultrasonic cleaning systems (dental industries).

Crest Ultrasonics Corporation Trenton, NJ Phone: 800-992-7378

Serves a broad range of industries and markets with ultrasonic cleaning equipment and machinery, including aqueous cleaning systems, ultrasonic cleaning tanks, and ultrasonic parts washers.

Dentronix Cuyahoga Falls, OH Phone: 800-523-5944 and Ivyland, PA Phone: 615-800-4220 FAX: 215-364-8607

Specializes in infection controls: manufactures ultrasonic cleaning machines and heat sterilizers.

Elma Ultrasonic Technology http://www.elma-ultrasonic.com

Ultrasonic cleaning equipment for special cleaning tasks.

Greco Brothers, Inc. (USA) http://www.sonicor.com

Manual and automated ultrasonic cleaning systems with aqueousbased solutions or solvents.

Jensen Aqueous Cleaning [JENFAB] Berlin, CT Phone: 860-828-6516

Automated and semiautomated cleaning equipment and machines, ultrasonic parts washers and ultrasonic degreasers, and aqueous cleaning systems supplied to a range of industries and markets.

Lewis Ultrasonics Kiel, WI Phone: 800-545-0661

Complete line of cleaning systems.

S. Morantz, Inc. Philadelphia, PA Phone: 800-695-4522

Full line of ultrasonic cleaning machines featuring True Digital technology. Applications include fire restoration, window-blind cleaning, parts cleaning, and golf-club cleaning. Tanks also built to customer specifications.

Omegasonics Simi Valley, CA Phone: 800-669-8227

Ultrasonic parts washers, ultrasonic blind cleaning equipment, ultrasonic degreasers, and aqueous cleaning systems. Serve industrial, automotive, and commercial markets with ultrasonic equipment.

Pressure Products Company, Inc. Charleston, WV Phone: 800-624-9043

Ultrasonic cleaners for stencils up to 29 in. × 29 in. Economical and environmentally friendly.

RAMCO Equipment Corporation Hillside, NJ Phone: 908-687-6700

Committed to health, safety, and the environment. Ultrasonic cleaning equipment and machinery, aqueous cleaning systems, and ultrasonic cleaning tanks to meet a range of needs.

Smart Sonic Corp. (USA)

Stencil cleaner. Removes solder paster from fine-pitch stencils (RMA no-clean, or water-soluble).

Sonicor Instrument Corporation (USA) http://www.sonicor.com

Ultrasonic cleaning.

Weber Ultrasonics Clarkston, MI (USA) Phone: 248-620-5142

Serving the finishing industries: Industrial parts cleaners and welders offers SONIC—digital generator technology for cleaning, welding, or special applications.

CRC_DK8307_CH008.indd 334

11/14/2008 4:56:18 PM

Ultrasonics-Assisted Physical and Chemical Processes

335

ultrasonic method significantly decreased the particle sizes in xerographic ink paper slurry and that different ultrasonic frequencies affect particles of different size ranges [20]. Scott and Gerber have used piezoelectric transducers in batch processes and liquid whistles in continuous processes for de-inking xerographic waste paper and have successfully reduced particle size for easy removal by the floatation process. They have demonstrated that the performance is strongly affected by treatment pH, number of treatment cycles, and pulp consistency [21]. Ramasubramanian and Madansetty have developed a novel technique, Acoustic Coaxing Induced Microcavitation (ACIM), for removing xerographic ink particles from waste paper [22]. They postulated that the cavitation threshold of hydrated paper is much higher compared to that of hydrophobic xerographic ink. Hence the ink regions can be coaxed acoustically to induce cavitation sites, preferably at the ink–paper interface. Microcavitational implosions generated at high-frequency ultrasonic waves can chisel away the ink paper joints and set the ink particles free without affecting the fiber mat underneath. The authors used ceramic transducers in pulsed mode (short repeated tone modes) at the resonance frequency. In this mode, the transducer operates at low acoustic power, yet develops high peak pressures, producing microcavitation. Focusing the acoustic field led to gain in acoustic intensity and progressive waves and using short tone bursts, precluded standing wave formation. Further, the use of highfrequency fields helps in generating nucleation sites on all particle surfaces. This is called acoustic coaxing effect. The use of low-duty factor acoustics in sonification allows time between pulses for the focal region to relax, equilibrate, restore, and regenerate, thereby reenacting cavitation at every pulse. As de-inking occurs, a white spot is generated in the printed region and the surrounding area remains unaffected. The experimental setup used in this study consists of a transducer, 40 mm in diameter with a focal length of 61.5 mm. The depth of focus is about 1.5 mm and the sample is kept at the focal plane close to the transducer. The entire assembly is immersed in a bath of distilled water. Tone bursts generated by a function generator are fed to the amplifier that drives the focused cavitation generator. The de-inked spot is ∼4 mm in diameter. The residence time for de-inking threshold increases with increasing pressure amplitude and duty factor. Longer pulses at a constant acoustic dose do a good cleaning job. Excessively long pulses, however, often lodge the separated toner particles back onto the paper. The optimal pulse parameters are: pulse width 10 μs, 1 kHz pulse repetition rate (PRR), and acoustic pressure amplitude of 45 atm peak negative, for spot de-inking.

8.9

Homogenization and Emulsification

To emulsify means to make or convert two or more liquid substances into an emulsion. An emulsion is a system of globules or particles of one liquid finely and uniformly dispersed in another. To homogenize means to make or render a mixture homogeneous. It refers to breaking up fat globules and dispersing them uniformly throughout a liquid. In other words, homogenizing a mixture is essentially the same as emulsifying it. Ultrasonic energy provides an excellent means of producing emulsions for laboratory studies. The equipment is rugged and easy to use. The general laboratory device consists of a transducer driving a resonant horn usually made of a titanium alloy such as Ti6Al4V. It can range in size from a small, lightweight unit consuming a very small amount of

CRC_DK8307_CH008.indd 335

11/14/2008 4:56:18 PM

336

Ultrasonics: Data, Equations, and Their Practical Uses

power to large units requiring kilowatts of energy. In most cases, one unit can be used for various studies. A variety of horn tip designs exist, from which one may be selected to fit a particular need. A unit may be made small and light enough to allow free manipulation by hand. This is a quality that is being widely used in operations, such as emulsifying the lens of the eye before replacing it with a new artificial lens. Ultrasonics makes it possible to remove the entire old lens before introducing the new one, without damaging the eye. In the early 1960s, ultrasonic emission from liquid whistles was successfully used for emulsification of a variety of products in water with or without the use of surfactants. The whistle works by impinging a high-pressure, flat jet of liquid on the edge of a thin blade, which is supported at the nodal points and located in a thin cavity [2, pp. 167–170; 23]. The blade vibrates in resonance due to the unstable condition created by the jet and the intense vibration causes localized cavitation. The energy released by the collapse of bubbles accelerates and propels the particles into the media, thereby facilitating the dispersion and emulsification. A typical liquid whistle system handles from 300 gph to 350 gph. In a typical commercial application for preparing resin-bonded solid lubricants consisting of molybdenum disulfide, graphite, and epoxy resin, the production rate by a liquid whistle operated by a 3 hp motor was two-and-half times greater than that of a 12 hp colloid mill [24]. Liquid whistles have been used for the manufacture of waxes and polishes, DDT, margarine emulsions, lotions, and antibiotic dispersions, as well as emulsions of mineral and essential oils. Another method of emulsification is the use of a horn coupled to a transducer so that it can oscillate in a longitudinal mode to produce cavitation when immersed in a liquid containing all of the ingredients. Intensity of cavitation is controlled by the power delivered to the horn that is carefully selected for the specific process. Details of the apparatus, experimental conditions, and the results are given by Weinstein et al. [25] and Hislop [26] and Last [27]. Parabolically focusing bowl types of piezoelectric transducers have been used to concentrate longitudinal ultrasonic waves internally. The liquid mixture is passed through the focus of the parabola to provide good emulsification. Although good emulsions can be prepared by ultrasonics, their stability is enhanced by using surface active agents. Sulfite liquor is used as an emulsifier for DDT emulsification [28] and stearic acid in dispersing mercury in paraffin oil. Mercury stearate formed in the emulsification process acts as a surfactant and stabilizes the dispersion. Similarly, a good dispersion of mercury is accomplished only after aerating the water, possibly by air oxidation of mercury, forming an Hg ion on the surface. The addition of potassium chloride removes the Hg++ by complex formation and destroys the emulsion. Ultrasonic emulsification is being carried out at the present time using powerful piezoelectric and magnetostrictive devices. Ultrasonics has been used to make slurries of coal powder in oil in order to burn it in oil burning plants. Ultrasonic agitation of a slurry containing 40% coal powder and a small amount of water provides a stable dispersion that does not settle during storage or transportation. Heinz [29] has examined the mechanisms of emulsification, both theoretically and experimentally, and has concluded that the efficiency of emulsification is largely controlled by the interfacial tension and viscosities of the liquids. Deformation of larger droplets are caused by the velocity gradients formed by shearing stresses caused by acoustic streaming and droplets burst when the shearing stresses are greater than the restoring forces associated with interfacial tension. The process follows the Taylor relationship η 0G = 8Γ(P + 1)/b(19P + 16)

CRC_DK8307_CH008.indd 336

(8.3)

11/14/2008 4:56:18 PM

Ultrasonics-Assisted Physical and Chemical Processes

337

where η 0 is the viscosity of the continuous medium, G the velocity gradient, P the ratio of the viscosity of the disperse medium to that of the continuous medium, Γ the interfacial tension, and b the radius of the undistorted drop. Efficiencies of emulsification processes have been studied by measuring particle size distribution. It has been shown that the efficiency is also proportional to the intensity of the sound used above the threshold intensity required for the onset of emulsification [30].

8.10

Dispersion and Homogenization

Intense ultrasound has been successfully used in breaking agglomerates and dispersing a variety of solids in liquid media [31–35]. The method requires the use of an ultrasonic horn with its irradiating tip extending into the volume of the mixture. Industrial applications include dye pigment preparations, dispersions of china clay, mica, titanium dioxide, and others used in the rubber and paper industry, and dispersion of magnetic oxides in the manufacture of audio, video, computer tapes and disks. Use of ultrasonic dispersion provides coatings with fewer imperfections and less information dropouts. This will spur greater use of the technique as the emphasis on higher information density grows. Hard-faced ultrasonic horns can be used to minimize wear and tear in dispersing abrasive particles. Ultrasonics has been successfully used in dispersing solid catalysts in liquid epoxy resins without premature cure and in the preparation of mineral slurries [36]. Other potential applications include particle size reduction, removal of paints and surface coats, particle wetting, and slime removal. Precipitation of ultrafine coal and metal dust and separation of sulfur and ash from coal are possible applications. Defibrillation of natural fibers such as flax, hemp, kenaf, and others is an important step in their application as polymer biocomposites. Senapati and Krishnaswamy have successfully defibrillated flax fibers using an ultrasonic water bath operated at 20 kHz and maintaining a flax to water ratio of 1–400 [37]. The radiating surface sometimes has to be designed for the application. For example, a horn with a flanged radiating surface had to be designed to delaminate jig mica. The mica is a heavy mineral that settles rapidly to the tank bottom. The flange throws the mica particles into suspension where the ultrasound can encompass them and peel the layers apart [38].

8.11

Coagulation, Precipitation, and Filtration

Ultrasonics has also been successfully used for coagulation and precipitation. Both the mechanisms and the effectiveness of these functions depend upon the media, the intensity of the sound wave, the frequency, the means of irradiating the materials, the reactions that occur, the materials to be precipitated, and the density (referring to the number of particles per unit of volume) of particles to be coagulated or precipitated. The materials to be coagulated may range from soft and fibrous such as paper pulps to crystalline solids such as metal powders. In some cases, the objective is to remove already existing particles from a volume of material such as ultrafine coal dust from the air or water. In other cases,

CRC_DK8307_CH008.indd 337

11/14/2008 4:56:19 PM

Ultrasonics: Data, Equations, and Their Practical Uses

338

the objective maybe to remove materials as they are produced by means of ultrasonic irradiation, such as extremely small crystals of TiO2 from reactants as they are formed. Continuous vibration brings fibrous particles such as paper pulp to impinge against one another and become entangled to form larger particles, thereby causing precipitation. Precipitation of ultrafine coal and metal dust and separation of sulfur from coal ash has been carried out using ultrasonics. The ultrasonic process has been used to coagulate solids from sewage discharge and to increase the amount of water removed in drying by simultaneously applying ultrasound and electrophoresis to plates during compacting the debris [39]. Ultrasonics has also been used to improve filtration rate as well as to enhance filter life. The filter element is vibrated sometimes by direct contact with the horn. At other times, the vibrations are applied to the fluid to irradiate in the direction of the flow by placing the horn in the vicinity of the filter [40,41]. Filters made from sintered brass, stainless steel, sandstone, and other materials have been used, with pore sizes from few to over 100 μm. Continuous vibration slows the rate of deposition on the filter surface, resulting in less need for cleaning.

8.12

Atomization

Ultrasonic atomization is accomplished by at least three different means: (1) by placing the liquid in a focusing transducer, a bowl-shaped or elongated half cylinder with an operating frequency of 0.4–2.0 MHz with the focal region at or near the surface of the liquid; (2) by passing the liquid over the surface of a horn vibrating ultrasonically in a direction normal to the surface [42]; and (3) by injecting the liquid into the active zone of a high-intensity stem-jet whistle (see Figure 6.8). The production of aerosols is attributed to at least two mechanisms: (1) production of capillary waves on the surface of the liquid and (2) cavitation. Theories based upon these mechanisms are applicable only within certain limits. The more general case is more complicated, involving various dynamic factors and material properties. To a degree, the particle sizes are functions of excitation frequency. If the conditions are correct for optimum atomization, the frequency dependence applies over a wide range of frequencies. In general, cavitation is not a major cause of atomization in industrial systems. It may be a major factor in high-frequency focused nebulizers that are used for atomizing medicines in inhalation therapy. However, in horn-type systems involving atomization from a surface, cavitation can have a deleterious effect on the rate at which mists can be formed. An equation proposed by Lang for the number–mean diameter particle diameter, d, of the droplets in the aerosol is [43] d = 0.34 c

(8.4)

where λc is the wavelength of capillary waves (cm) and, according to Rayleigh [44] is given by  8T 

c   2   f 

CRC_DK8307_CH008.indd 338

1/ 3

(8.5)

11/14/2008 4:56:19 PM

Ultrasonics-Assisted Physical and Chemical Processes

339

where T is surface tension, ρ the density of the liquid, and f the ultrasonic frequency (Hz). This equation applies to very low atomization rates with materials such as very low viscosity oils, very low flow rates, and fairly uniform film thickness. Also the film thickness is large compared with the amplitude of vibration of the atomizing surface. Ensminger discusses additional equations for various conditions of atomization [2, pp. 467–474]. In any feed system, as feed rate increases, atomization will pass through three stages: 1. Low flow rate, at which atomization is attributable entirely to ultrasonic forces. Within this stage, the particles will range in size about a mean diameter that might be approximated by the method of Peskin and Raco, average size increasing with flow rate. 2. Intermediate flow rate, at which atomization is attributable to both ultrasonic forces and fluid dynamic forces. The particle sizes caused by the fluid dynamics are much larger than those produced by ultrasonic forces. 3. High flow rate, at which atomization is primarily a fluid dynamic phenomenon. In general, the efficiency of ultrasonic atomization—that is, ultrasonic energy required to atomize a unit volume of liquid—increases as the frequency decreases. The advantages of ultrasonic atomization include narrow droplet size distribution and the ability to atomize fairly high-viscosity liquids. Further the droplets can be projected in a unique pattern. Ultrasonic atomization has found a variety of applications, including medical inhalants, atomized fuel for efficient combustion, atomization of industrial paints for electrostatic spraying, dispersing cleaners in large tanks, drying fabrics, and producing ultrafine metals and ceramics. Metal powders have been produced by atomizing molten metal [45] and cooling rapidly. Uniform crystals of very small dimensions have been produced by atomizing supersaturated solutions followed by rapid cooling. Medical nebulizers work at frequencies of 1–3 MHz and produce droplets of 1–5 μm. One advantage of the medical devices working within this frequency range is that they produce a mist, without larger particles, that can enter the lungs directly. Absence of any other gas during inhalation makes the technique suitable for anesthetic devices. Ultrasonic humidifiers, which produce atomized water vapor, are commercially available for use in home and offices. Fuel atomization helps efficient combustion. Charpenet [46] found that when an acoustic field, operating at 5–22 kHz, is applied to a steel burner, using both pulverized coal and fuel oil, the flame temperature immediately rose by ∼80°C and the exhaust carbon dioxide increased by 0.8%. Atomization of diesel fuel also improves efficiency, eliminates combustion delay, and enables the engines to run more economically at high loads with low smoke and quieter operation. Use of ultrasonic atomization in liquid fuel burners, such as in home heating, allows the use of large-diameter nozzles, which are self-cleaning under ultrasound and are more fuel-efficient than conventional burners. However, ultrasonic atomizers have not found commercial application, because of the added cost of ultrasonic equipment and recent improvements in conventional burners. It has found some application in special-purpose electric power generation where the rate of dispersion is very low [47]. A small ultrasonic atomizer was designed and built at Battelle Columbus Laboratories for use in portable thermoelectric generators. This system uses a single transistor oscillator with feedback control of the frequency.

CRC_DK8307_CH008.indd 339

11/14/2008 4:56:19 PM

Ultrasonics: Data, Equations, and Their Practical Uses

340

The unit requires 2.5 W from a 12 V battery or a thermoelectric generator without the load but jumps to 5 W when atomizing fuel. The driving frequency is controlled by feedback from the transducer, a pair of piezoelectric crystals. Fuel is fed by gravity and the thermal output is ∼20,000 Btu/h. The types of fuels were gasoline, kerosene, fuel oil, and alcohol. Ultrasonic atomization also helps to improve the sensitivity of flame photometry [48]. Placing high-power ultrasonic horns in direct contact with fabric helps to evaporate a much higher percentage of moisture than mechanical squeezing and leaves a small amount to be dried by heat. The efficiency is higher than for nonabsorbing materials.

8.13

Preparation of Nanomaterials

Nanoparticles of metals and ceramics have found important applications in polymer composites, catalysis, and in the production of amorphous metals. Nanoparticles are a few nanometers in diameter and have very large specific surface. Naturally occurring nanomaterials such as nanoclay, when incorporated into resins such as nylon, polypropylene, and the like produce composites with high strength and impact resistance, at four to five percent loading. In addition, the composites are flame-retardant and have low permeability [49]. Hence a substantial amount of work is being carried out on processes for producing nanofillers. Efficiency of solid catalysts is greatly enhanced as the particle size is reduced to nanoscale. Recently, nanosize metal particles have found important application in magnetic recording media as well in the manufacture of permanent magnets. However, production of nanoparticles on a large scale is a difficult process. Ultrasonics has found important application in the synthesis of nanostructure materials [50]. Ultrasonic cavitation plays an important role in this synthetic procedure. As mentioned earlier, extreme high temperatures and pressures are reached during the implosion of the cavitational bubbles. However, at the end of each cycle, the high temperature is quenched in ∼100 ns with cooling rates of the order of 1010 degrees per second. When a metallic or ceramic solution is subjected to such rapid cooling, amorphous nanoparticles are formed since the small particles do not find enough time to agglomerate or crystallize. Suslik initially used an ultrasonic procedure to synthesize amorphous nanoparticles of iron, by subjecting an alkane solution of iron pentacarbonyl to ultrasonic irradiation [1, pp. 410,411; 51]. His apparatus consisted of a solid titanium rod attached to a piezoelectric ceramic material and powered by a 20 kHz, 500 V power supply. Supported catalysts were prepared by adding an oxide support material, such as finely divided silica, to the carbonyl solution and colloids were prepared by adding a polymeric ligand such as polyvinyl pyrrolidone. However, nanoparticles of iron produced in this process agglomerate readily. Boeing [52] has patented a process for forming amorphous metals, using ultrasonic irradiation of metal carbonyl solution in n-heptane or n-decane and by extracting the nanoparticles from the hydrocarbon solvent by adding polar solvents of high vapor pressure such as ethoxyethyl alcohol, followed by an in situ coating process based on the addition of a polymer such as polyvinyl pyrollidone, acrylics or urethane, or related polymer precursors. The thinly coated nanoparticles do not agglomerate and maintain their miniature size. As mentioned earlier, the nanosize iron particles are being used in magnetic recording media. Quest Integrated has also used this method for synthesizing magnetic nanoparticles [53].

CRC_DK8307_CH008.indd 340

11/14/2008 4:56:19 PM

Ultrasonics-Assisted Physical and Chemical Processes

8.14

341

Crystallization

Ultrasonic energy produces smaller crystals in supersaturated liquids. Crystals of controlled sizes are produced in the pharmaceutical industry by applying high-intensity ultrasonic energy at the bottom of vertical columns in conjunction with fluidized beds. The small-size crystals tend to move to the top, while the larger crystals settle toward the bottom. These oversized crystals are ultrasonically shattered and rise in the columns. Uniform size crystals are collected at a certain height in the column [54]. Lower ultrasonic frequencies, 20–30 kHz, are used for crystallization from melts. Cavitation produces fi ner grain structure [1, p. 129], resulting in improvement of ductility and impact toughness, and specific elongation, in the case of metals. Two categories of crystallization are recognized: (1) crystallization from a solution and (2) crystallization from a melt [2, pp. 410,411; 55]. Crystallization from solution involves the formation and growth of solid crystals of a solute in a supersaturated liquid solution. Crystallization from a melt occurs as the temperature of a liquid is lowered below its freezing point and solid crystals of the substance are allowed to form. 8.14.1

Crystallization in Metals

Applying ultrasonic energy to solidifying metals produces beneficial effects such as homogeneous, distribution of nonmetallic inclusions, reduced gas content, uniform alloying, alloying of normally immiscible substances (such as tin, zinc, and aluminum) and uniform grain refinement. Apparently cavitation and sonic stirring can produce a number of crystal centers far exceeding those forming under solidification without ultrasonic agitation. Sufficient energy cannot be introduced to break metal crystals down into smaller sizes after they have begun to solidify. The major problem is to determine a means of continuously coupling energy into the melt safely and on a suitable production scale. Several methods have been proposed but none has resulted in an evolution from laboratory scale. Coupling elements between the source and the melt must be either highly resistant to melting and eroding into the melt or some means must be found for continuously energizing and feeding a material intended to become at least a part of the melt. The entire volume of melt needs to be exposed until near solidification. Thus, in the metallurgical area, very desirable effects have been seen to be produced ultrasonically, but there are almost formidable problems in applying ultrasonics to large production quantities. Benefits have been realized where treatment of small volumes is involved, such as in arc-welding aluminum [56,57].

8.15

Diffusion and Filtration through Membranes

Ultrasound accelerates diffusion of ions through membranes and cell walls. This effect has been attributed to the disruption of the boundary layer at the interface. Ultrasonic irradiation increases the rate of diffusion of water and sugar through sugar beet membrane. The increased transport continues even after the irradiation is stopped. Substantial increase in the flow rate of crude oil through porous sand stone and that of water through stainless steel filters, under 20 kHz ultrasound has been observed by Fairbanks and Chen [58]. The rate of diffusion of salt through a cellophane membrane has been increased up to 100% by applying ultrasound in the direction of diffusion and is greater when the irradiation is in the direction of gravitational force [59,60]. The effect was not due to temperature

CRC_DK8307_CH008.indd 341

11/14/2008 4:56:19 PM

Ultrasonics: Data, Equations, and Their Practical Uses

342

rise nor due to membrane breakdown. A 69% increase in the diffusion of potassium oxalate through parchment membrane was observed when ultrasonic irradiation was applied in the direction of diffusion [61]. The increase was smaller in both cases when the irradiation was in the opposite direction.

8.16

Chemical Effects

Because ultrasonic frequency varies roughly from 20 kHz to 10 MHz and sound velocity in water at 20°C is ∼1500 m/s, the wavelengths are of the order of 7.5–0.015 cm, and as such are incapable of coupling with small molecules. Hence their effect occurs largely through physical mechanisms, depending on the nature of the system. Suslick has discussed the mechanism and influence of various factors on sonochemical reactions [1, p. 142]. The most important effect of ultrasonic irradiation is cavitation, which takes place in three consecutive steps: nucleation, bubble growth, and implosion. It is generally accepted that as the gas entrapped in small angle crevices of particulate contaminants is subjected to negative acoustic pressure, the bubble volume grows ending in bubble collapse or fragmentation, releasing small free bubbles into solution. Bubble collapse generates microcavities, which serve as nucleation sites for the next cycle of bubble growth and collapse. In homogeneous liquids, there can be either stable cavitation, leading to bubble oscillation, or transient cavitation, in which small bubbles undergo large excursions in volume and then terminate in violent collapse. In practice both stable and transient cavitation takes place simultaneously. Two types of effects have been proposed for implosive collapse of the bubbles; namely, hot spot pyrolysis and electrical discharge. The former hypothesis is largely accepted and has been modeled. The simplest model of bubble collapse, which assumes zero heat transport (adiabatic) and absence of condensable vapor pressure, estimates transient temperature buildup up to 10,000 K and pressure buildup up to 10,000 atm. More sophisticated hydrodynamic models predict temperature and pressure build up at approximately 5000 K and 1000 atm, with residence times of <100 ns. The electrical discharge hypothesis has been thoroughly rebutted. As a bubble constitutes both liquid and vapor phase, the next important question is whether the reaction takes place in the liquid phase or the gaseous phase. Suslick’s work on sonochemical ligand substitutions of metal carbonyls has shown that although the reaction largely occurs in the gas phase, there is a vapor pressure–independent component suggesting some additional reactions taking place in the liquid phase, presumably in the thin liquid shell surrounding the collapsing cavity. Analysis of sonochemical kinetic data, such as the frequency factor and activation energy, in sonochemical ligand substitution reactions mentioned earlier, suggests that in the gas phase, the effective temperature is ∼5200 K and in the liquid phase, it is ∼1900 K. The liquid reaction zone is estimated to be ∼200 nm thick, having a lifetime of less than 2 µs. In short, cavitation plays a very important role in all types of sonochemical reactions through generation of transient high temperature and pressure around collapsing bubbles. Sonochemical reactions are strongly affected by a number of external factors such as acoustic frequency, acoustic intensity, temperature, pressure, choice of solvent, and ambient gas. The frequency of a sound field has relatively little effect on sonochemical reactions, as the reacting atoms or molecules do not couple with the field. Changing frequency, therefore, changes only the resonant size of the cavitation event. At very high frequency, cavitation ceases, leading to no sonochemical reaction. Acoustic intensity,

CRC_DK8307_CH008.indd 342

11/14/2008 4:56:20 PM

Ultrasonics-Assisted Physical and Chemical Processes

343

however, exerts substantial influence on the rates of sonochemical reactions. Below a threshold value, the amplitude of the sound field is too small to induce nucleation or bubble growth. Above the cavitational threshold, the increase in intensity increases the size of the liquid zone undergoing sonochemical reaction, thereby increasing the rate of reaction. Similarly, as the acoustic pressure increases, the range of bubble sizes that undergo transient cavitation increases, leading to enhanced rate of reaction. However, at very high intensities, the cavitation near the radiating surface becomes so intense that it produces a shroud of bubbles that decreases the rate of penetration of sound into the liquid. Further, the bubble growth may become so rapid that the size of bubbles may grow beyond the size range of transient cavitation without the occurrence of implosive collapse. Increase of temperature generally slows down the rates of sonochemical reactions. In other words, sonochemical reactions have negative activation energy. Temperature also influences the solvent effect. The greater the solvent vapor pressure within the bubble, the less effective the bubble collapse. Hence, highly volatile solvents do not promote sonochemical reactions. The effect of external pressure is somewhat complex. Increase in externally imposed pressure suppresses both bubble nucleation and growth as well as the occurrence of implosion. Hence an increase of external pressure slows down the sonochemical reaction rate. Another important factor is the thermal properties of ambient gas. Polytropic ratio, Cp/Cv, of the ambient gas controls the amount of heat released during cavitation. A small difference in this ratio can lead to a great difference in the temperature reached during cavitation. The type of solvent used in the reaction also influences sonochemical reaction rate. The difference in vapor pressure within the bubble and the susceptibility of the liquids to undergo secondary reaction, such as the production of H and OH ions from water, influences the main reaction. Use of robust solvents with low vapor pressure can overcome such problems. Other liquid properties, such as surface tension and viscosity ratio, also influence the yield in sonochemical reaction, due to their influence on bubble formation. In short, acoustic intensity, temperature, ambient gas, and the type of solvent strongly influence the reaction rate of sonochemical reactions. Solvents should be carefully selected to exercise control on the rates and reaction pathways. Because of large differences in sonochemical reactions in aqueous and organic solvents, they have been discussed separately. The important parameters that influence cavitation and sonochemical reactions are given in Table 8.3. TABLE 8.3 The Effects of Extrinsic Variables on Sonochemistry Extrinsic Variable

Physical Property

Effect

Acoustic frequency Acoustic intensity Bulk temperature

Period of collapse Reaction zone size Liquid vapor pressure Thermal activation Total applied pressure Gas solubility Polytropic ratio Thermal conductivity Chemical reactivity Vapor pressure Surface tension Viscosity Chemical reactivity

Resonant bubble size Cavitation events per volume Bubble content, intensity of collapse Enhanced secondary reaction rates Intensity of collapse Bubble content Intensity of collapse Intensity of collapse Primary or secondary sonochemistry Intensity of collapse Transient cavitation threshold Transient cavitation threshold Primary or secondary sonochemistry

Static pressure Ambient gas

Choice of liquid

Source: Suslick, K.S., Ultrasound, VCH Publishers, New York, 1988, 142.

CRC_DK8307_CH008.indd 343

11/14/2008 4:56:20 PM

Ultrasonics: Data, Equations, and Their Practical Uses

344 8.16.1

Sonochemical Reactions in Aqueous Solutions

8.16.1.1

Gases

When water is subjected to cavitation by high-energy ultrasonics, hydrogen and hydroxyl radicals are produced, which subsequently combine to form hydrogen peroxide and hydrogen H2O + e (ultrasonic energy) = H+ + OH− 2OH = H2O2

(8.6)

2H = H2

The formation of these radicals and other high-energy species has been established by Reisz and coworkers by spin-trapping with nitrone and nitroso compounds and studying their electron spin resonance and subsequent scavenging studies by ethanol and sodium formate [62]. These radicals are formed under a variety of reaction conditions, including those used in ultrasonically assisted dental cleaning (25 kHz) and ultrasonic imaging (10 μs pulses of 1 MHz frequency). This refutes the notion that ultrasonic cavitation does not occur under medical diagnostic conditions. In the presence of dissolved oxygen, part of the peroxide is formed by the combination of these radicals with oxygen radicals formed by sonolysis of molecular oxygen. Substantial work using oxygen and hydrogen isotopes has been carried out to establish the reaction mechanism [63]. It has also been found that ozone present in aqueous solution is rapidly destroyed producing molecular and atomic oxygen. The oxygen atom produced in the first step of the reaction attacks a fresh ozone molecule to produce two oxygen molecules [64]. The principal products of sonication of water containing nitrogen are nitrous acid (HNO2) and nitric acid (HNO3) with smaller amounts of ammonia and nitrous oxide [65]. Sonolysis of an aqueous solution containing hydrogen and carbon monoxide produces formaldehyde [66]. Sonolysis of an aqueous solution containing nitrogen, carbon monoxide, and methane produces a variety of amino acids at low yield. Table 8.4 [67] gives products from a number of gases subjected to ultrasonics. 8.16.1.2

Inorganic Compounds

Substantial work was done in sonolysis of aqueous solutions prior to 1980, including sonochemical reactions of solutions containing iodide, bromide and sulfide ions, reaction TABLE 8.4 Aqueous Sonochemistry of Gases Substrate Present HD H2 + N2 H2 + CO 14 N2 + 15N2 N2 N + (CO, CH4, or HCHO) 18 O2 + 16OH2 O3 N2O

Principal Products H2D2 NH3 HCHO 14,15 N2 HNO2, HNO3, NH2OH, N2O, NH3 Amino acids 16,18 O2, 16O2 O2 N2, O2

Source: Weissler, A.J., Cooper, H.W. et al., J. Am. Chem. Soc., 72, 1769, 1950.

CRC_DK8307_CH008.indd 344

11/14/2008 4:56:20 PM

Ultrasonics-Assisted Physical and Chemical Processes

345

TABLE 8.5 Aqueous Sonochemistry of Inorganic Compounds Substrate Present −

−

Br , Cl Ce4+ [Co(NH3)N3]2+ Fe2+ [Fe(III)(C2O4)3]3− H2S XXI− [MnO4]− [NO3]− OsO4 [PO3]2− TI+

Principal Products Br2, Cl2 Ce3+ Co2+ + N3 Fe3+ Fe2+ H2 + S8 I3− MnO2 [NO2]− OsO2 [PO4]2− TI3+

Source: Mead, E.L., Sutherland, R.G. et al., Can. J. Chem., 53, 2394, 1975.

of nitrogen giving nitric acid, reaction of mercuric chloride by aqueous oxalate solution, hydrolysis of metallic acetates, saponification of fat, and rearrangement of benzazide (CcH5CON3) to nitrogen and phenyl isocyanate [68]. Sonolysis of aqueous solutions containing inorganic compounds often results in oxidation. Corrosion of iron is greatly enhanced by sonication. Attempts to characterize sonochemical reaction to provide characteristic g values (number of radicals or oxidation products formed for a unit of ultrasonic energy) has not been successful due to complications arising from the effects of dissolved gas, concentration at nucleation sites, differing behavior of solvent media, and hydrophobicity of added solutes, and so on [69]. The g value of sonochemical oxidation of ferrous compounds is greatly enhanced by the addition of a small amount of alcohol [1, pp. 141,146]. Table 8.5 [70] provides a list of inorganic oxidation–reduction reactions by ultrasonics along with major products.

8.16.1.3

Organic Compounds

Sonochemistry of organic compounds in water has been studied both in solution and in suspension. The higher vapor pressure of water compared to that of most organic liquids, leads to water being preferentially present in the cavitating bubbles and complicating the reaction products due to interference of its own sonolysis reaction. Table 8.6 [71,72] gives products from a number of organic compounds subjected to ultrasonics in aqueous solution. Most of the degradation reactions involve secondary reactions due to hydrogen and hydroxyl radicals, the primary decomposition products of water. Sonolysis of aqueous suspensions of thioethers gives sulfonic acids and sulfoxides. Sonolysis of aldehydes gives caboxylic acids. And sonolysis of carboxylic acids gives carbon dioxide and hydrocarbons. These reactions indicate the manifestation of oxidation and reduction characteristic of free radical induced reactions. Carbon tetrachloride in aqueous solution undergoes sonolysis to produce chlorine, carbon monoxide, and hydrochloric acid [1, p. 146; 73,74]. In the presence of potassium iodide, chlorine reacts with iodide ion to form I3, which produces characteristic yellow color (blue in the presence of starch) that can be used to study the progress of the reaction

CRC_DK8307_CH008.indd 345

11/14/2008 4:56:20 PM

Ultrasonics: Data, Equations, and Their Practical Uses

346 TABLE 8.6

Aqueous Sonochemistry of Organic Compounds Substrate CCl4 CH3I R2CHCl Cl3CCH(OH)2 C6H5Br Maleic acid + Br2 CS2 (C4H9)2S [R2NC(=O)S]− + R′Cl RCHO [HCO2]− C5H5N C6H5OH C6H5CO2H C6H11OH RCO2H ClH2COO2H RCO2R′ RCH2NH3 (CH2NH2)2 Thymine Uracil Various amino acids Cysteine

Principal Products Cl2, CO2, HCl, C2Cl6, HOCl CH4, I2, CH3OH, HI, C2H6 R2CHOH, HCl HCl Br−, C2H2 Fumaric acid S, H2S (C4H9)2SO, polymer R2NC(=O)SR′ CO, CH4, C2H4, C2H4O2, RCO2H CO2 HCN, H2C2, H2C4 C6H4(OH)2 C6H4(OH)(CO2H) C2H2 CO, CH4 H2, CO, CH4, Cl− RCO2H, R′OH H2, CH4, NH3, RCHO, RCH2OH NH3 Hydoxylated products Hydroxylated products H2, CO, NH3, RNH2, HCHO Cystine

Source: Mason, T.J., Lorimar, J.P. et al., Tetrahedron Letters, 23, 1133, 1982; Mason, T.J., Lorimar, J.P. et al., Tetrahedron Letters, 24, 5393, 1964.

by colorimetry. This reaction has been used as a dosimeter for ultrasonic reaction. However, the reproducibility of this reaction is poor because of low solubility of CCl4 in water and high volatility. Sonochemical decomposition of nucleic acids in aqueous solution has been studied [1, p. 147]. As mentioned earlier, hydrogen and hydroxyl radicals produced by sonolysis of water participate in this reaction. The susceptibility of sonochemical reaction follows the order thiamine, uracil, cytosine, guanine, adenine. Pulsed ultrasound has been found to be more effective than continuous irradiation. In a study by Henglein et al. on the scavenging efficiency of different organic liquids on the production of hydrogen peroxide in water, it was found that solubility of the scavenger liquid, not its vapor pressure, produces significant difference in peroxide formation [75]. In other words, trapping of the OH radical is occurring selectively at the interface between the cavitation bubble and the liquid where hydrophobic liquids concentrate. The rate of solvolysis reactions of esters and halocarbons are accelerated by sonication, possibly due to disruption of the solvent structure [71,72]. The rate of solvolysis decreases with increase in temperature and increasing ethanol content. Elpiner and coworkers [76] have used ultrasound to study the isomerization of maleic acid to fumaric acid in presence of a trace amount of water and have shown that rapid isomerization occurs by the catalysis of either bromine or alkyl bromide.

CRC_DK8307_CH008.indd 346

11/14/2008 4:56:20 PM

Ultrasonics-Assisted Physical and Chemical Processes

347

TABLE 8.7 Nonaqueous Sonochemistry of Organic Compounds Substrate Alkanes CH3CN CH3CN + O2 CH3COCOCH3 RCHO + O2 CHCl3 CCl4 C6H6 C6H5CH3 1,3-C6H4(CH3)2 1,3,5-C6H3(CH3)3 C6H5N2+ X C6H5CHCH2 H2CC(CH3)(CO2H) H2CC(CH3)(CO2CH3) H2CCH(CONH2) Many polymers

Principal Products H2, CH4, C2H2I-alkenes H2, CH4, N2 CO, CO2, H2O, N2 (H3C)2CO RCO2H, RCO3H, CO2, CO, CH4 HCl, CCl4, CH2Cl2, C2Cl5H, C2Cl4, C2Cl6, C2HCl3 Cl2, C2Cl6 H2, CH4, C2H2, C6H5CH3, (C6H5)2 H2, CH4, C6H6, C6H4(CH3)2, [C6H5(CH2)]2 H2, CH4, C6H5CH3, C6H3(CH)3, [(H3C)C6H4(CH2)]2 H2, CH4, C5H6, C6Hn(CH3)6−n, [(H3C)2C6H3(CH2)]2 N2, [C6H5]X Polymerization Polymerization Polymerization Polymerization Depolymerization

Source: Prudhomme, R.O., Bull. Soc. Chim. Biol., 39, 425, 1957.

8.16.1.4

Nonaqueous Sonochemical Reactions

8.16.1.4.1 Simple Organic Compounds It was previously believed that sonochemical reactions occur in liquids of high tensile strength and high dielectric constant, such as water. Recent studies have shown that such reactions occur in organic liquids without the presence of water, provided that the temperature is low enough to maintain a low vapor pressure. The latter is important to produce cavitation. The general reaction is bond homolysis and production of free radicals and subsequent secondary reactions [75,77]. Table 8.7 [78] illustrates some of the typical organic chemicals and their sonolysis products [71,72]. In a single class of solvent, the rate of radical production increases with decrease in solvent vapor pressure due to increase in the efficiency of cavitational collapse and increase in peak temperature. Low ambient temperature reduces solvent vapor pressure and facilitates cavitation and bond homolysis. Factors such as bond strength and solvent viscosity should be taken into consideration when solvents of different classes are compared [76]. Sonolysis of alkanes is very similar to pyrolysis and follows the Rice radical-chain mechanism [79]. Hydrogen, methane, and alkenes are the principal products, along with other alkanes, and alkynes. When aromatic hydrocarbons are subjected to sonolysis, they produce deeply colored tars (∼20%) along with hydrogen, methane, substituted benzene, and biphenyls [80]. Substituted benzene products such as toluene and xylene give similar products along with many of their alkyl derivatives. When halocarbons such as carbon tetrachloride and its substituted products are subjected to sonolysis, initially chlorine radicals are formed, which combine to give chlorine gas and other substitution products. Sonolysis of chloroform gives HCl and a whole range of halocarbons including CCl4, CH2, Cl2, and so on, following the carbene pathway mechanism [81].

CRC_DK8307_CH008.indd 347

11/14/2008 4:56:20 PM

348

Ultrasonics: Data, Equations, and Their Practical Uses

Ultrasonic oxidation of aldehydes in the presence of oxygen gives principally carboxylic acids along with some peracids [79]. The rate of ultrasonic reaction is a few times greater than the ordinary oxidation rate. It has been found that sonolysis of aceronitrile gives nitrogen, methane, and hydrogen through the cleavage of C–C bonds [88]. It has also been observed that the initiation of explosion of nitroglycerine and tetranitromethane can be increased severalfold using ultrasonics. A variety of chemical reactions have been studied, using ultrasonics. Reactions that involve free radicals, radical ion intermediates, and single electron transfer (SET) are enhanced by ultrasonics. Thompson and Doriswamy [82] have listed a large number of sonochemical organic reactions including the Diels Alder cyclization reaction, oxidation of indane, hydrolysis of nitrophenyl ester, ketalization of acetophenone, reduction of methoxy-aminosilane, synthesis of di-substituted hydantoins, epoxidation of fatty esters and aryl alkanes, Michael addition of nitroalkanes, oxidation of 2-octanol, Diels Alder cycloaddition of quinone, Ullman coupling of nitrobenzene, reduction of α, β unsaturated carbonyl compounds, and Fridel Krafts reaction involving nucleophilic substitution. Other synthetic reactions such as esterification of several carboxylic acids, aldehydes from olefins using rhodium catalysts, and hydroxylation of phenolic compounds have also been studied [83]. Ultrasonic reactions have been used for the removal of harmful contaminants from water, such as chlorofluorohydrocarbons CFC 11, CFC 13, HCFC 225ca, 225cb, HFC 134a, sodium hypochlorite, and pentachloro phenate [84]. Kinetic analysis of sonochemical reactions has been carried out using electron spin resonance and spin trapping of volatile and nonvolatile solutes [85]. Three zones in the reaction site are (1) the gaseous region of the cavitation bubble containing both permanent gas and reactant vapor, (2) the gas/liquid transition region containing less volatile components and surfactant, if any, and (3) the bulk liquid region. In the gaseous region, volatile chemicals, both solutes and solvents, undergo pyrolysis following the Rice Hertzfield mechanism [80]. Approximately 10% of these radicals escape to the bulk liquid region and undergo different reactions. Super oxide ions (O2−) are formed in argonsaturated aqueous solutions. Reaction with nonvolatile liquids occurs primarily in the transition zone and in the bulk liquid phase. Whereas at low concentration of the nonvolatile solute, reactions occur with radicals generated in Zone 1, at high concentrations the solute undergoes pyrolysis. In general, both types of reaction take place simultaneously as observed in the sonolysis of nonvolatile solutes such as thiamine, organic esters (acetates and propionates), dipeptides, sugars, nucleosides, and nucleotides. Sonolysis of dimethyl formamide, methyl formamide, actamide, toluene, n-acohols, n-alkanes, cyclohexane, tetrahydofuran, etc. have been studied. It has been observed that reaction rates are not enhanced unless the reactants are volatile enough to enter the transition zone. Hart et al. investigated sonolysis of water under argon and acetylene atmospheres. They found products containing two and eight carbon atoms similar to the products obtained from pyrolysis of acetylene. Sonolysis of ozone, carbon dioxide, and nitrous oxide have been studied [86,87]. Sonolysis of carbon tetrachloride and trichloroethylene produces chains of two and four carbon atoms and that of methanol produces methyl and hydroxymethyl radicals [1, p. 152]. Kinetic modeling of these reactions have been carried out by Suslick et al. [88]. 8.16.1.4.2 Organometallic Compounds Suslik and coworkers have made in-depth studies of ligand substitution and pyrolysis of iron carbonyls [89]. Whereas thermal decomposition of iron carbonyl, Fe(CO)5, gives fi nely divided pyrophoric iron powder, ultaviolet irradiation gives Fe2(CO)9 and

CRC_DK8307_CH008.indd 348

11/14/2008 4:56:21 PM

Ultrasonics-Assisted Physical and Chemical Processes

349

multiphoton IR photolysis in the gas phase gives isolated iron atoms. Sonolysis of ferrous carbonyls, however, gives the whole series of iron carbonyls from Fe(CO)3 to Fe3(CO)12, following a first-order rate. The log of the first-order rate coefficient is linear with the solvent vapor pressure and confirms that the disassociation process is activated by local heating, produced by cavitational collapse. The ratio of different products can be varied almost 100-fold by changing vapor pressure of the solvent. Production of Fe3(CO)12 is strongly favored by increasing solvent volatility. Ligand substitution also takes place in other metal carbonyls and the rates of sonochemical ligand substitution vary following their relative volatilities, as expected from the effect of relative volatilities on cavitational collapse. Sonochemical treatment of manganese carbonyls, Mn 2(CO)10, produces a variety of substituted products and follows thermal rather than photochemical mechanisms of CO loss [90,91]. Similar studies have also been carried with cobalt carbonyls [92] and organo-tin compounds [93]. Mechanisms of decomposition reactions have been studied by spin-trapping with nitrosodurene and involves alkyl-tin bond cleavage. Further details of ultrasonically assisted ligand substitution reactions can be found in Suslick’s book [1]. 8.16.1.4.3 Sonocatalysts The systematic ligand substitution observed in sonolysis of metal carbonyls contribute to their ability to act as catalysts. Potential advantages include use of low reaction temperature required to handle thermally sensitive substrates, enhanced selectivity, ability to generate high-energy species, possibility of mimicking reaction conditions used in a bomb on a microscopic scale, and ease of scale-up operation. Isomerization of a 1-pentene to isopentene and trans 2-pentene has been accomplished by using a number of metal carbonyls and sonication [94]. Rate enhancement is approximately 105 times over the thermal rate. Relative order in sonochemical activity of different metal carbonyls follows the same order as photocatalysis, except ruthenium carbonyl, which shows greater activity as sonocatalyst and gives different trans/cis ratio in the products. A lot of terminal alkenes act as substrates. Alkenes without beta hydrogen are not suitable to serve as substrates. The rate of isomerization significantly decreases with increase in steric hindrance. Sonication of alkenes in the presence of molybdenum carbonyl, Mo(CO)6, gives 1-enols and epoxides. A mechanism involving initial allylic C–H bond cleavage caused by cavitational collapse followed by subsequent auto-oxidation and epoxidation has been suggested. Kinetic studies and radical entrapment work supports this mechanism [95]. It may be noted that whereas the catalyst is activated in alkene isomerizaion reaction by ultrasound application, substrate is activated in the oxidation reaction. Since ultrasound has the ability to form fine emulsions and increase the interfacial area, it greatly enhances efficiency of phase transfer catalysts (PTC). This is evident in the synthesis of dibenzyl sulfide from benzyl chloride (organic phase) and sodium sulfide (aqueous phase). Conversion of benzyl chloride was enhanced 1.5 times by ultrasound alone, 5.9 times in the presence of a PTC, tetrabutyl ammonium bromide, and 6.5 times in the presence of both PTC and ultrasound [96]. The catalyst is essential to accelerate the ion exchange reaction between the sparingly soluble sodium sulfide and the organic liquid. Application of ultrasound also enhances the reaction by aiding the mass transfer at the solid–liquid boundary. Madison et al. found that the conversion of ethylene glycol ditosylate to tritosyl triaza cyclononane in 5 hours increased from 6.1% to 73% by using sonication and the PTC tributyl ammonium hydroxide [108,109]. Other PTC-activated reactions include the Cannizaro reaction, β elimination reactions, synthesis of nitriles from alkyl bromides and potassium cyanide, dealkylation and cyclodealkylation of ethyl cyanoacetate [97,98].

CRC_DK8307_CH008.indd 349

11/14/2008 4:56:21 PM

Ultrasonics: Data, Equations, and Their Practical Uses

350

8.17

Polymerization

Because ultrasonic irradiation often generates free radicals, it has been used for the polymerization of vinyl and acrylic monomers in bulk, solution, and emulsions. Earlier, work on ultrasound-activated polymerization has been reviewed by Urban and Salazer Rojas [99]. The rate of free radical polymerization and copolymerization in bulk and solution is accelerated by ultrasonics, and in some cases, polymerization occurs without an initiator. In emulsion polymerization, it helps by creating fine emulsions and in suspension polymerization by preventing agglomeration. As molecules are fractured due to shockwaves created by cavitation, both small and large radicals are produced that are capable of initiating polymerization in the absence of free radical catalysts and of forming block and graft copolymers by reaction with neighboring macroradicals and monomers. Earlier work on polymerization of substituted benzenes in nonaqueous media by Kruus et al. gave darkly colored polymeric tars [100,101]. Subsequently, Kruus has examined polymerization of methyl methacrylate and styrene in organic solvents [102,103]. Polymethyl methacrylate polymers of relatively low average molecular weight (∼400,000) was obtained. The kinetics of polymerization indicated that initiation occurs by radicals formed due to thermolysis of the media from localized hot spots generated by cavitational collapse. The rate was proportional to the square root of ultrasonic intensity. A reaction mechanism was developed that produced rate constants comparable to literature values. Miyata et al. [104] studied ultrasonically assisted bulk polymerization of styrene and copolymerization of styrene with acrylonitrile. They concluded that application of ultrasonics does not change the mechanism. It only accelerates the free radical coupling reaction in the termination process. Lozinski studied the effect of ultrasonics on the polymerization of styrene and methyl methacrylate and their copolymerization with acrylonitrile and vinyl acetate [105]. O’Driscol and Sridharan have used ultrasonics to prepare block copolymers in a stirred tank reactor. In their process, a polymer was subjected to ultrasonic irradiation in the presence of a monomer. Polymer radicals formed initiate polymerization of the monomer forming block copolymers [106]. Kojima, Koda, and Nomura [107] studied the effect of ultrasonic frequency on the bulk polymerization of styrene and Ooi and Bigg studied ultrasonic initiation of polystyrene latex [108]. Fujikura and Kakiuchi obtained block copolymers of styrene and methyl methacrylate by ultrasonic irradiation of polystrene in the presence of methyl methacrylate [109]. Kawase and Kakurai prepared block copolymers of styrene and vinyl acetate and that of styrene with ethylene oxide, using a similar process [110]. Koda et al. [111,112] (Mangaraj, D., Krisnaswamy, P., and Jones, G., private communication) have studied ultrasonic polymerization of vinyl pyrrolidone, without the use of any initiator. Polymerization occurred at 20 and 40 kHz, but not at 540 kHz. At low frequencies, polymer yield increased with time, but became zero when power was turned off, showing that sonication is essential for initiation and propagation steps in ultrasonic polymerization. They also examined the copolymerization of styrene sulfonate and vinyl pyrrolidone, at 20, 40, and 540 kHz. The rate increased with acoustic intensity at 20 and 40 kHz, but was not affected by intensity at 540 kHz. At 40 kHz, the rate was twice that at 20 kHz for the same intensity. Kruus et al. studied copolymerization of acrylic acid and maleic anhydride using ultrasonic irradiation [113]. The sequence length in copolymer structure changed from three to two and the molecular weight decreased, giving a narrower molecular weight distribution compared to free radical polymerization.

CRC_DK8307_CH008.indd 350

11/14/2008 4:56:21 PM

Ultrasonics-Assisted Physical and Chemical Processes

351

Chou and Stoffer have done extensive study of ultrasonics assisted emulsion polymerization and copolymerization of methyl methacrylate, acrylamide, styrene and maleic anhydride, and emulsion polymerization of methyl methacrylate in the presence of sodium lauryl sulfate [114]. They found that purging with argon gas helps in producing stable cavitation. When a substantial amount of argon gas flows through the reaction tube, the cavitation noise is soft. Polymerization occurred giving yields from 20% to 61% (molecular weight from 2.1 million to 3.5 million) within a sonication time of 30–35 min. When argon purging was stopped, only a small amount of black material was formed but no polymerizaion occurred. Based on this finding, the authors suggest that there are two types of cavitation: transient and resonant. Transient cavitation occurs in the absence of purging gas and bubbles form and collapse in the immediate vicinity of the ultrasonic probe. The noise of this type of cavitation is loud and harsh and the bubbles contain largely the vapor of the liquid, and the pressure within the bubbles, close to the vapor pressure of the liquid is usually higher than the applied acoustic pressure. This prevents implosion and cavitation. When transient cavitation takes place, it produces high local temperature, often exceeding 1000 K, resulting in pyrolysis and tar formation. On the other hand, when gas is introduced near the horn, the ultrasound breaks down the large bubbles into microbubbles and disperses them throughout the solution. The gas inside the bubbles is the surging gas and the liquid vapor and the pressure inside the bubbles is low. As a result, a small acoustic pressure can cause cavitation. The audible noise from this type of resonant cavitation is hissing and soft. The local temperature attained during resonant cavitation is low and appropriate for a steady polymerization reaction. The authors also found that free radical scavengers such as hydroquinone inhibit this reaction, confirming that the reaction is a free radical polymerization, similar to thermal emulsion polymerization. Further, it confirms previous results that sonication of organic monomers produces free radicals, capable of undergoing polymerization. They also showed that several recipes containing monomers and surfactants but no initiators, underwent no polymerization on heating to higher temperature, but polymerized when subjected to ultrasonic irradiation. For example, percentage conversion of methyl methacrylate varied between 21% and 61% and the average molecular weight varied from 2.1 million to 3.5 million for 30–35 min sonication at an initial temperature of 5°C and a final temperature of 35°C. These results indicate that ultrasonically initiated emulsion polymerization takes place at relatively low temperature and the reaction is initiated by radicals generated during adiabatic bubble collapse in the cavitation process. Radicals also are generated by shockwaves and shear stress and induce polymerization, particularly graft polymerization. Free radicals produced have sufficient longevity to migrate into the bulk of the liquid monomer and initiate polymerization and copolymerization. The authors also observed that the monomer in a water mixture does not undergo polymerization, even with an argon purge, unless a soap (sodium lauryl sulfate) is added to the system. This indicates lower cavitational efficiency of organic liquids and that radicals formed by sonolysis of monomers combine readily in the presence of argon gas. Radicals generated by sonolysis in the presence of soaps, such as sodium lauryl sulfate, probably have longer life—long enough to initiate polymerization. This was confirmed by radical trapping, using a scavenger (bromoform) and by the analysis of sonolysis product with GC/MS. It was concluded that sonolysis degrades the soap into hydrocarbon radicals and sodium sulfonyl radicals initiate polymerization in the emulsion system. Decomposition of C–O bond takes place either by thermolysis, due to the high-temperaturegenerated bubble implosion or mechanical degradation caused by shock waves and high

CRC_DK8307_CH008.indd 351

11/14/2008 4:56:21 PM

352

Ultrasonics: Data, Equations, and Their Practical Uses

shear. In addition, it was observed that the rate of this polymerization is higher than free-radical-initiated emulsion polymerization of MMA in the presence of the same soap, possibly due to better isolation of growing radicals and greater cavitation efficiency. Average molecular weight of polymethyl methacrylate produced in this study varied from 2.5 million to 3.5 million g/mole and yield was up to 70%. Kinetics deviated from the Smith Ewart theory of thermal emulsion polymerization. The rate was proportional to 0.98 power of acoustic intensity and 0.08 power of the surfactant (instead of 0.5 in the Smith Ewart theory). The rate also varied to the 0.086 power of gas flow rate. However, at surfactant concentration greater than 0.14 M, the rate varied to the 0.58 power of the surfactant concentration, indicating a change in mechanism with increasing concentration of the soap. Gas-purged ultrasonic emulsion polymerization rates at ambient temperature were found to be much greater than conventional emulsion polymerization at a higher temperature. This means that many industrial emulsion polymerization processes can be carried out at a low temperature, using gas-purged ultrasonic assistance, thereby offering substantial savings in energy and cost. Bahattab and Stoffer [115] have studied the kinetics of thermal and ultrasonic copolymerization of styrene with butyl acrylate, using a 20 kHz titanium horn and ammonium persulfate as initiator. Dodecyl-sodium persulfate was used as surfactant and gas was bubbled into the mixture before adding initiator. The acoustic intensity was ∼10 W/cm3. The chemical composition of the polymers were determined using NMR. The reactivity ratios were calculated by comparing –CH2 and –OCH3 absorption bands. They found that ultrasonic-induced polymerization at low temperature has 5–10 min of induction time. However, at 70ºC, the polymerization did not show any induction time. They showed that ultrasonic energy is capable of causing faster polymerization than thermal polymerization. One potential application for the use of ultrasonically induced polymerization is in the removal of trace levels of monomers that are typically present in latex polymers. These latex polymers are used in the manufacture of house paints, carpet adhesives, window sealants, and other products in the building and construction industry. The presence of low levels of monomers in such products leads to emission of volatile organic compounds (VOCs), causing indoor air pollution and problems generally associated with “sick building syndrome.” Okamura, Hayashi et al. have patented a process for grafting vinyl monomers to cellulose by ultrasonic irradiation in the presence of cerium and copper salts [116]. Mangaraj, Jones, and Krishnaswamy have successfully grafted vinyl acetate and metharylatotrimethyl-silane to deactivate flax fibers, using ultrasonics [126]. They carried out graft polymerization of vinyl acetate in the presence of hydrogen peroxide in ceric nitrate catalyst in aqueous metida and that of methracrylato-silane in the presence of benzoyl peroxide in methyl isobutyl ketone (MIBK). The monomer catalyst solutions were purged with nitrogen for half an hour and polymerization was carried out in an ultrasonic bath at 60°C. A weight gain of 8–10% was obtained in 25–30 min. Romanov and Lazar found that when a solution of polypropylene in styrene was subjected to ultrasonic irradiation a block copolymer was formed, whereas the same solution treated with ozone produced a graft polymer [117]. Shiota and coworkers found that ultrasonic irradiation increases the rate of condensation polymerization of terepthaloyl-chloride and ethylene diamine and the molecular weight of the resulting polyamide increased substantially [118]. Hastings and Bolten found that a sharper distribution of molecular weight was obtained when styrene was polymerized by ultrasonic irradiation [119].

CRC_DK8307_CH008.indd 352

11/14/2008 4:56:21 PM

Ultrasonics-Assisted Physical and Chemical Processes

8.18

353

Polymer Degradation

Degradation of synthetic and natural polymers has been studied extensively. A number of workers showed that when starch, gum arabic, gelatin, and polystyrene are subjected to ultrasonic irradiation, they undergo degradation resulting in lower molecular weight [120,121]. Earlier work on ultrasonic degradation has been reviewed by Breck et al. [122]. Rate of ultrasonic degradation depends upon the molecular architecture of the polymer as well as on the molecular weight. Susceptibility to degradation decreases with decrease in molecular weight, and polymers with a certain critical molecular weight do not undergo degradation. The rate of degradation depends on ultrasound intensity and duration of irradiation, chemical structure of the polymer, and solution concentration. This indicates that cavitation plays the predominant role in polymer degradation. This relationship was established by studying the effect of pressure on ultrasonic depolymerization of polystyrene solution in benzene at constant temperature and 500 kHz frequency [123]. When pressure was applied using nitrogen gas from a cylinder, a small decrease took place in the rate of degradation at 450 psi compared to that at 1 atm. However, when the same pressure was exerted using a mercury column, the rate of depolymerization decreased in accordance with the effect of pressure on cavitation. At 450 psi, very little depolymerization took place. Further depolymerization occurred in the presence of dissolved gas and did not occur when the polymer solution was degassed. The nature of the dissolved gas was found to be unimportant, indicating that cleavage does not occur inside the collapsing bubble by thermal processes. Ultrasonic frequency does not have an appreciable effect on depolymerization, because cavitation is independent of frequency as long as the wavelength of the impressed field exceeds the critical bubble diameter sufficiently, as mentioned earlier. Langton and Vaughn found that ultrasonic energy at 20 and 250 kHz are equally effective in depolymerizing dextrin [124,125]. They described depolymerization rate as first order with respect to degree of polymerization given by the following equation: DPt/dt = −k(Pt − P∞)

(8.7)

where Pt and P∞ are the degree of polymerization at time t and at final limiting time. In short, depolymerization is caused by cavitation itself or by shockwaves and high shear fields generated by cavitation. Models have been developed to explain the exact nature of the processes that lead to bond cleavage. It has been suggested that acoustic microstreaming is capable of cleaving large molecules, whereas relatively smaller molecules are cleaved by the frictional force generated by relative movement of the solvent and polymer molecules [126]. When degraded, polymers produce macroradicals, which exist for a short time. Soon they combine disproportionately to give stable molecules, in the same way that macroradicals react during the termination step of free radical polymerization. Henglein used labeled iodine isotopes to trap these radicals, and found that in the absence of oxygen two atoms of iodine are incorporated to each C–C bond broken [127]. This indicates that macro radicals formed under sonolysis undergo disproportionate reaction, forming stable molecules with terminal double bond. Studies on spin trapping have confirmed this observation [128,129]. Head and Lauter studied the ultrasonic depolymerization of several natural gums and came to the conclusion that whereas linear polymers depolymerize at comparable

CRC_DK8307_CH008.indd 353

11/14/2008 4:56:22 PM

Ultrasonics: Data, Equations, and Their Practical Uses

354

rates, nonlinear polymers and complex polymers do not depolymerize under similar conditions [130]. Kossler and Novobilsky studied ultrasonic degradation of polychloroprene in the presence of air and found that the peroxide bonds formed in an earlier stage of aging leads to cross-link formation [131]. Addition of gases retards depolymerization in the order of air, nitrogen, oxygen, and carbon dioxide. Also, the number of units of bonds broken per unit time decreases with increase in concentration of the polymer. Newton and Kissel fractured Tobacco Mosaic virus protein by using ultrasonic irradiation [132]. The polymeric protein was fragmented to give monomeric units. The breakage occurred at a distance of 175–205 mµ from one end, indicating that a weak bond is present at that location in the molecule. At higher intensities, even the monomers were fragmented following a zero-order reaction rate and the virus lost 95% of its activity. Degradation of RNA from yeast has been successfully carried out at frequencies of 2.4 and 4.0 MHz. The degradation of RNA has been attributed to the formation of hydrogen peroxide and nitrous acid due to cavitation and the rate of degradation decreases at higher concentration. Ultrasonic degradation of polysiloxane of broad molecular weight distribution leads to narrow molecular weight distribution [133] and that of agar-agar solution in methyl sulfoxide and water decreases rapidly until the viscosity reaches a critical value [134]. Air contact produces extensive degradation, nitrogen produces less, and argon the least degradation. Temperature did not have any effect on degradation. After 100 min of ultrasonic degradation, pH decreased appreciably, indicating the formation of agar-agar acid, possibly by the reaction of hydroxyl radicals from water sonolysis, with the macroradicals of agar-agar. Degradation is more extensive at higher concentrations. At a viscosity of 2 g/L, agaragar exists in gel form and the viscosity decrease is reversible. Zukal’ko et al. found that whereas polyisobutene (PIB) of 80,000 molecular weight undergoes ultrasonic degradation, PIB having a molecular weight of 20,000 is not affected by ultrasonics [135]. Ultrasonic depolymerization of polyvinyl alcohol in aqueous solution has been studied by Pirkonen and coworkers [136].

8.19

Polymer Processing

Because ultrasonic irradiation of polymers can produce heat, reduce viscosity, and improve wetting of fillers and fibers by resin, it can be used to improve flow, activating the curing process of rubber compounds and composites. At the same time, under different processing conditions, ultrasonic irradiation can destroy chemical bonds. It can therefore be used to devulcanize rubber by destroying the cross-links—a process important to rubber recycling. The following section provides a brief discussion of the work done on ultrasonic assisted vulcanization and devulcanization of rubber compounds, processing of composites, and compatibilization of blends. 8.19.1

Vulcanization

In 1985, Mangaraj and Senapati patented a process for vulcanizing a natural rubber compound using ultrasonic irradiation [137,138]. It was postulated that rubber specimens subjected to ultrasonic irradiation would be heated more uniformly than when heated

CRC_DK8307_CH008.indd 354

11/14/2008 4:56:22 PM

Ultrasonics-Assisted Physical and Chemical Processes

355

by conventional methods, applying heat from the outside. The heat would be generated by the hysteresis loss in ultrasonic wave propagation. Rigid bodies, such as metals and ceramics, generally transmit ultrasonic waves with little loss, as do low-viscosity liquids, but viscoelastic materials like rubber and plastics will absorb a part of their energy by hysteresis loss, exemplified by a lag between stress and strain. The magnitude of this loss depends upon frequency. At a low frequency, the sound waves are highly elastic. As a result, the hysteresis loop is negligibly small and there is little absorption. As frequency increases, absorption increases. Substantial absorption takes place at megahertz frequencies. Transfer of energy from sound to heat is very inefficient at very high frequency. As a result, maximum energy absorption takes place at a medium-frequency range. For moderate frequency, the absorption coefficient is given by α = 4/3(ω2/ ρcη)

(8.8)

where η is the shear viscosity of the medium, c the sound velocity, ρ the density, and ω the frequency. The rate of temperature rise (δT/δt) for one-dimensional heat generation and conduction is given by the equation δT/δt = k/ρC(δ2T/δc2 + 2αIoe−2dx/k)

(8.9)

where T is temperature, t time, k thermal conductivity, Io initial intensity of the sound wave, and dx the distance traveled in the x-direction [139]. Mangaraj, Senapati et al. [149,150] measured α by measuring the instantaneous rate of temperature rise in two nearby locations in rubber slabs and found that α for natural rubber (NR) at 20 kHz is ∼1.3 Np/in. for natural rubber and 2.5 Np/in. for styrene-butadiene rubber (SBR). Ultrasonic intensity decreases with distance from the source. If the ultrasonic applicator is placed on the top of a rubber slab, the intensity decreases from the top to the bottom and the intensity gradient will be different for different rubbers. The authors also observed considerable degradation of unvulcanized rubber compounds when subjected to high-frequency ultrasonics, which was attributed to cavitation. They found that cavitation can be largely suppressed by applying back pressure. They designed an ultrasonic vulcanization equipment in which a 2 in. diameter circular horn was placed in contact with a rubber slab between the two platens of a compression mold and a suitable pressure was exerted while the slab was heated simultaneously with ultrasonics. Cavitation was totally suppressed by exerting back pressure of 600–1000 psi. It was found that the cure time in ultrasonic vulcanization is only half that of thermal vulcanization. Properties are superior in the ultrasonic process, as seen by lower compression set. Four standard rubber compounds based on NR, SBR, nitrile rubber (NBR), and ethylene propylene-diene rubber (EPDM) formulations containing 75 parts per hundred rubber (phr) carbon black filler, as per The Vanderbilt’s Rubber Hand Book, were mixed and cured by ultrasonic and thermal processes. The cure conditions along with hardness, compression set, and swell ratio in toluene are given in the reference. The results showed little difference between the two processes, possibly because of the efficient transfer by carbon black and continued vulcanization after the power was turned off. However, it showed that efficient vulcanization of different rubber compounds can be carried out by ultrasonic processes and will be more efficient in nonblack compounds [140,141]. Mangaraj and Senapati also found that adhesive strength of steel cord to natural rubber after ultrasonic vulcanization is much greater than thermal vulcanization. This was attributed to better degassing and microfriction at the interface. However, adhesive

CRC_DK8307_CH008.indd 355

11/14/2008 4:56:22 PM

Ultrasonics: Data, Equations, and Their Practical Uses

356

strength of natural rubber to polyester cord in ultrasonic cure was inferior, possibly due to ultrasonic degradation of fibers [140,141]. Mangaraj cured glass reinforced polyurethane composites using both ultrasonic and thermal methods and found improved strength and modulus by the ultrasonic method. Ring and Twesme have carried out a substantial amount of work on curing of carbon fiber reinforced epoxy and bis-maleimide composites [142]. They found improved wetting of fibers by resin due to viscosity decrease as well as microstreaming of resin at fiber interfaces. They also found improved dispersion of resin and curative and decrease in volatile content. Resin advance time was retarded and carbon fiber prepegging with both resins were significantly decreased without any damaging effects. Tack level of both resins improved significantly by ultrasonic application. Use of ultrasonics in polyetherether-ketone carbon fiber composites showed improved fiber wetting and flow below melting point. Ultrasonics assisted significantly in the consolidation and curing of composite panels with reduced cycle time. Fujikura Electric of Japan developed an ultrasonic process for continuous vulcanization (CV) of polyethylene-insulated cables, using a 400 kHz/310 W barium titanate cylindrical transducer of 80 mm diameter [143]. They coupled the ultrasonic transducer to a conventional CV tube. The process not only provided uniform curing, but also increased manufacturing speed by 50%. They compared the degree of cross-linking (gel fraction of thermal- and ultrasonic-assisted cure at three locations) for steam cure and ultrasonically assisted steam-cured specimens. Gel fraction is higher for the latter process. Further, the number of microvoids generated during ultrasonic cure was much smaller and were of smaller diameter than those obtained during the steam cure process. Microvoids shorten the service life of cables by generating water trees [144].

8.20

Devulcanization

As mentioned earlier and initially observed by Mangaraj and Senapati, chemical bonds in vulcanized rubber are broken and devulcanization takes place when vulcanized rubber samples are subjected to ultrasonic cavitation [140]. A substantial amount of work has been done by Isayev and coworkers on ultrasonic devulcanization of crumb rubber produced from tire waste [141]. Initially, it was suggested that the dominant mechanism of devulcanization is ultrasonic cavitation and bubble collapse, similar to that in polymer solution. The simplified model treated rubber as pure elastic solid containing noninteracting gas-filled spherical cavities, at pressure equal to ambient pressure. As ultrasonics is applied, the bubbles pulsate at an amplitude, depending largely on the ratio between acoustic pressure and ambient pressure. Under certain conditions, the cavities collapse, generating high temperature and pressure that rupture chemical bonds. The bonds break in order of their relative strength. The weaker polysufidic bonds break in larger proportion compared to relatively stronger mono and disulfide links, ultimately followed by the breakage of C–C bonds. The rupture is confined to a spherical shell of certain thickness surrounding the collapsing bubble. The bond breakage was found to increase with the increase of the amplitude of the ultrasonic horn and with decrease in clearance between the die and the horn [145,146]. The experimental data on cross-link density and gel fraction, however, did not show the strong dependence on processing parameters as predicted by the previous model [147,148].

CRC_DK8307_CH008.indd 356

11/14/2008 4:56:22 PM

Ultrasonics-Assisted Physical and Chemical Processes

357

Variations of cross-link density and gel fraction along the die of the ultrasonic reactor were inconsistent with the prediction, as the collapse theory did not take into account the high ambient pressure generated in the extruder. In other words, the experimental data clearly showed that cavitation and bubble collapse play a less important role in ultrasonic devulcanization. The model was modified and it was proposed that the bonds break due to high cyclic strain imposed by pulsating bubbles, as voids initiate fracture under fatigue stress [149,150]. Further, rubber is not purely elastic, as assumed previously. It is largely viscoelastic, a behavior that suppresses cavitation. Using the Rouse model of viscoelasticity, it was shown that when ambient pressure is sufficiently lower than ultrasonic pressure, the cavities expand in decompression phase, about 20 times higher than the equilibrium radius followed by rebound. Numerical simulations indicated the presence of an unstable regime of acoustic cavitation associated with irregular oscillations of high amplitude. The high level of strain imposed in pulsation leads to the rupture of rubber network. Detailed calculations indicated that ultrasonic pressure in rubber increases with an increase in ambient pressure and remains much higher than ambient pressure up to high levels of compression. These effects are more pronounced at lower clearances and rubber consumes more ultrasonic power with increase in ambient pressure and decrease in gap clearance. Therefore, high-amplitude strains are possible, even at high ambient pressures, leading to breakdown of the rubber network, particularly at lower clearances and higher amplitude. These predictions explained the experimental results better than the bubble collapse theory. Specific acoustic energy consumed per unit mass of rubber is the ratio of ultrasonic power to the mass flow rate. Decrease in cross-link density for various residence times (as defined by flow rate) follows the same pattern. In other words, the residual crosslink density for any flow rate is smaller for higher amplitude and smaller clearance, and it decreases asymptotically with increase in specific energy consumed by rubber [161]. Isayev and Hong have studied ultrasonic devulcanization of unfilled SBR in static and continuous conditions [151]. Isayev has shown that polymer blends can be made compatible by placing the ultrasonic horn in close proximity to the die in a mixing and pelletizing extruder [152]. They have established that the macroradicals formed by ultrasonic degradation of the two polymers combine to produce block copolymers that act as compatibilizer. Ultrasonic vibration is also being used during extrusion to reduce head pressure, die lip buildup, and melt fracture, and to improve surface quality of the extrudate.

8.21

Large-Scale Sonochemical Processing

Mason has reviewed the large-scale application of ultrasonics in chemical processing [153]. Since 1970, general awareness of the potential for ultrasonic processing has increased significantly and has attracted the attention of many chemical industries. It has been realized that ultrasonic cavitation, which is key to most chemical processing, can be as harmful as hydrodynamic cavitation, which causes surface damage of equipment. However, the harmful effects of ultrasonic cavitation can be controlled by using pressure, as shown earlier in ultrasonic vulcanization, and by focusing the acoustic field within the reaction medium. Additionally, it has been demonstrated that efficiency of an ultrasonic process is not related only to acoustic intensity, but to an optimum choice of ultrasonic parameters. As a result, ultrasonics has been used in a variety of industries from food preparation to health care.

CRC_DK8307_CH008.indd 357

11/14/2008 4:56:22 PM

358

Ultrasonics: Data, Equations, and Their Practical Uses

Different types of ultrasonic transducers—namely, the liquid whistle, different types of piezoelectric, and magnetostrictive transducers—have been used. The liquid whistle has been used in the food industry, for preparation of soups, sauces, gravies, and ketchup; in the cosmetic industry, for preparation of skin cream; in textiles, to improve the quality of dyeing of fabrics; and in the plastic industry, for dispersing clay and thickening agents. Terfenol-D has made available greatly improved magnetostrictive transducers. Terfenol-D is an alloy of terbium, dysprosium, and iron and can be produced in various forms such as rods, laminates, and tubes to facilitate horn design. Having very high magnetostriction properties, Terfenol-D transducers can transfer more electrical to mechanical power than other commonly used magnetostrictive materials. Similar improvements in piezoelectric transducers—particularly, production of transducers in the form of flexible sheets—provides many advantages, including better acoustic transmission into aqueous systems [154,155]. As mentioned earlier, ultrasound has found large-scale application in cleaning and decontamination. It has been effective in removing biological contaminants such as algae, fungus, bacteria, and so on, which adhere strongly to the surface of containers such as large food crates. Use of power ultrasonics has been very effective in biocide-assisted decontamination of water, because it breaks down bacterial clumps attached to difficultto-reach internal surfaces and it enhances the permeation of biocides through bacterial cell walls [1, p. 110]. A combination of ultrasonics and electromagnetic radiation, known as Sonoxide, has been used to clean cooling water towers, particularly to break lime scale and to kill algae. Ultrasound is also being used in treating sewage sludge, particularly to accelerate the anaerobic digestion and the rate of fermentation, producing greater volume of biogas [2, pp. 462–466]. Chemical decontamination of water through volatilization and free radical initiation oxidation of pollutants is being carried out on a large scale [1, pp. 462–466]. In offshore drilling operations, ultrasonics is being used to deagglomerate the cleaning mud, so that it does not accumulate around the platform legs [156]. Power ultrasound is also being used to extract pharmaceuticals, flavors, and colorants from vegetable and plant sources [157]. It has also been used to improve the rate of penetration of the dyes into leather from an aqueous solution [158]. Ultrasonics has found substantial application in food processing [159]. Ultrasonic effects such as strong shear forces, particle fragmentation, increased heat, and mass transfer and nucleation of crystallites have been used in a variety of processes, including extraction, filtration, drying, and cooking. Mechanical effects of ultrasound help in greater penetration of solvents into cellular material, resulting in greater mass transfer. Faster disruption of biological cells also helps the extraction process. Emulsification of oils is greatly enhanced when ultrasonic liquid whistles are used. Sonication increases interfacial area, thereby providing better homogenization and enhanced reaction rate. Alcohols produced from hydrolysis of wool wax by an ultrasonic process are cleaner and less colored compared to those obtained by conventional processes. It has been found that ultrasonics can reduce cooking time, particularly that of grains such as rice, by breaking the surface shells of grains. Cavitational microbubbles have been used to provide seeding in saturated solutions and to produce crystals of uniform and designated size, which is very important to the pharmaceutical industry [160]. The jet effect associated with cavitational collapse has been successfully used to reduce the thickness of the diffusion layer around electrodes, resulting in enhanced mass transfer and reduction in electrode fouling, thereby increasing yield and current efficiency in electrolysis [161]. Ultrasound welding of plastics and metals is used extensively in the plastics industry. Assembly operations previously done by adhesives, solvents, and mechanical fasteners

CRC_DK8307_CH008.indd 358

11/14/2008 4:56:23 PM

Ultrasonics-Assisted Physical and Chemical Processes

359

are being replaced by ultrasonic welding. Representative applications include assembly of automotive lenses, dashboards, air ducts, trims, household appliances, TV cases, cassettes, textiles, and microelectronics. Ultrasonic welding has the inherent advantage of high speed, energy efficiency, absence of combustibles, and above all, adaptability to automation [162]. Ultrasonic welding is especially suitable for thermoplastics with low mechanical damping. Waves travel to the joint area without losing energy and produce heat mainly at the line of contact. Welding is carried out at low temperatures, causing less material degradation. Welding is fast and the weld line is strong, because of better wetting at the interface due to microfriction between the joining surfaces. The mating surfaces must be formed to produce good welding conditions. When the surfaces are flat with considerable mating area, the mating surface of the part next to the horn is prepared with an energy director. This director, being much smaller than the mating surfaces, absorbs the energy first and spreads the effect over the area to be bonded. It is triangular in cross section, with the base angles of either 45° or 60°, depending upon the type of material to be welded. It is on the order of 0.005 in. high. It melts fi rst and spreads the melt across the interface, fusing the two parts together. When parts are too large for a single horn, two or more horns are used and amplitude is kept constant in spite of changing load. Automatic frequency control is used for free operation. Other controls currently used include on-line weld quality inspection and exposure correction to assure equal energy consumption in each weld. Table 8.8 provides weldability ratings of various thermoplastics, using ultrasonic power. The following paragraphs explain the terms staking, swaging, inserting, near field welding, and far field welding [2, pp. 462–466]. Staking is a method of attaching plastics and metals (or other materials) by mechanically locking or enclosing the materials in plastic. A small protrusion is extended through a hole in the second material by a distance that is sufficient to form a suitable lock when staking is completed. The dimensions of the protrusions (stud) and the mating hole form a slip fit. The end of the ultrasonic horn is contoured to the geometric design of the completed stake. The staking operation includes applying high ultrasonic amplitude under low clamping pressure to cause the plastic to flow and form a locking head at temperatures that are typically below the melting point of the plastic. This process provides tight joints because there is no material spring-back as with cold heading and it causes only minimal degradation of crystalline materials because of the lower-than-melting temperatures attained. Inserting refers to inserting metal parts in plastic parts. A hole slightly smaller than the metal insert is drilled or molded into the plastic part and the metal is driven into the hole ultrasonically. During insertion (which usually requires less than 1 s) material encapsulates the metal piece and fills flutes, undercuts threads, and so on. Near field welding refers to the condition in which the mating surfaces are within a short distance (1/4 in. at 20 kHz) of the welding tip, such as is the case in lap-welding thin sheets. Far field welding refers to the condition in which the bond line is at a distance from the welding tip (greater than 1/4 in. at 20 kHz). Continuous welding of thermoplastic textiles, such as nonwoven fabrics, is carried out by feeding the fabric between an ultrasonic horn and a rotating anvil engraved with a stitch pattern. Currently, ultrasonic welding is finding major application in fabric laminating and quilting. In the future, it will be extensively used in nonwoven fabric production, particularly in the production of underwear and baby diapers. Ultrasonic welding of metals is practiced in applications where conventional heat welding, thermal or electrical, is not suitable [163]. Because the process does not require electrical resistance to generate heat, it is used for joining high-conductivity metals such

CRC_DK8307_CH008.indd 359

11/14/2008 4:56:23 PM

Ultrasonics: Data, Equations, and Their Practical Uses

360 TABLE 8.8 Thermoplastic Welding by Ultrasound Polymer Amorphous Polymers ABS ABS/Polycarbonate ABS/PVC Acrylic Acrylic multipolymer-XT Acrylic/PVC Acrylic, impact modified Cellulosics (CA, CAB, CAP) Modified phenylene oxide Polyarylate Polycarbonate Polyetherimide Polymer Polystyrene Polystyrene Styrene-malic anhydride PVC, rigid PVC, flexible Sulfone polymers Crystalline Polymers Acetal copolymer Acetal homopolymer Fluoropolymers Nylon Polyester (PBT) Polyester (PET) Polyetheretherketone Polyethylene (LDPE, HDPE) Polyethylene (UHMW) Polymethylpentene Polyphenylene sulfide Polypropylene

Spot Welding

Staking Swaging

Inserting

Field of Welding (Near)

Field of Welding (Far)

E G G G G G F P E F G G

E G G F G G F G E F F G

E G F G G F P E E G G E

E G G G G G F P E G G E

G F F F F F P — G F F G

F F E F P F

F F E G — F

G G E E — G

E G E P P G

E P G P — F

F F — F F F G G — G F E

F F — F F F G F — F P E

G G — G G G E G — E G G

G G — G G G E P — F G F

F F P F F F G P — P F P

Source: Suslick, K.S., Ultrasound, VCH Publishers, New York, 1988, pp. 110,111. E, Excellent; G, Good; F, Fair; P, Poor.

as copper and aluminum. Ultrasonic scrubbing breaks up and disperses surface oxides and other contaminants thereby assuring better weld quality. Because welding is done at relatively lower temperatures, the weld line does not suffer from embrittlement due to recrystallization and formation of intermetallic compounds. Ultrasonic metal welding has found large-scale applications in microelectronics, particularly in welding semiconductor leads to chips. Automatic machines capable of making as many as 10 welds per second are available.

8.22

Summary

In summary, ultrasonics can play a major role in many physical, chemical, and material processes, including acceleration of chemical reactions, enhancement of catalytic efficiency, initiation of novel reactions based on sonolysis of the media—particularly aqueous

CRC_DK8307_CH008.indd 360

11/14/2008 4:56:23 PM

Ultrasonics-Assisted Physical and Chemical Processes

361

media polymerization (free radical polymerization, copolymerization, block and graft polymerization), depolymerization and degradation, vulcanization and devulcanization, and finally in compatibilization of polymer blends. Ultrasonics can enhance a variety of physical processes such as dispersion, emulsification, degassing, coagulation, diffusion through membranes, decontamination, cleaning, de-inking of waste newspapers and laser prints, as well as in welding and processing. They can be used for producing nanoparticles for potential use in emerging nanotechnology markets and for catalyzing chemical and biological reactions. As some of the applications such as ultrasonic cleaning and welding are flourishing, others with great potential—such as ultrasonically assisted polymerization, vulcanization and devulcanization, filtration and separation, drilling, and so on—have not been exploited on a large scale. Although a substantial knowledge base is currently available for using ultrasonics for various applications, more fundamental and applied work is necessary to employ its potential more widely.

References 1. Suslick, K.S., Ultrasound, VCH Publishers, New York 1988, pp. 110, 111, 129, 141, 142, 146, 147, 152. 2. Ensminger, D., Ultrasonics, Marcel Dekker, New York, 1988, pp. 66, 67, 410, 411, 426–438, 462–466, 467–474. 3. Duraswamy, L.K. and Thompson, L.H., Indus. and Engg. Chem. Res., 38(4), 1999, p. 1215. 4. Spinosa, E. and Ensminger, D., Ceram. Eng. Sci. Proce., pp. 410–425, March–April 1986. 5. Kapustina, O.A., Physical Principles of Ultrasonic Technology, Rosenberg, Plenum Press, NY, 1973, 1, p. 379. 6. Brown, B. and Goodman, J.E., High Intensity Ultrasonics, Iliffe Books, 1965, p. 210. 7. Hueter, T.F. and Bolt, R.H., Sonics, Wiley, New York, 1955, pp. 228–232. 8. Frederick, J.R., Ultrasonic Engineering, Wiley, New York, 1965. 9. Agranat, B.A., Bashirikov, V.L. et al., Ultrasonics International 1973 Conference Proceedings, Plenum Press, New York, pp. 247, 515. 10. Ensminger, D., “Improved Process and Apparatus for Cleaning Xerographic Plates”. U.S. Patent No. 3, 483, 037, December 9, 1968; Canadian Patent 899, 825, December 10, 1968. 11. Pohlman, R. and Werden, B., Ultrasonics, 10, 1972, p. 156. 12. Hildebrand, J.H. and Scott, R.L., Solubility of Nonelectrolytes, Reinhold, London, 1950, pp. 270, 356. 13. Fuchs, J., Private communication on selection of cleaning fluids, Blackstone–Ney Ultrasonics, Inc, February 25, 2005. 14. Ensminger, D., “Cavitation Measurement Techniques in General”, 1..E.E. Panel Discussion, New York Hilton Hotel, March 21, 1966. 15. Campbell, A.M. et al., “The Experimental Study of Foods”, 2nd ed., Houghton Mifflin Company, Boson, MA, 1979, pp. 31–33. 16. Cool, M.C. and Silveri, M.A., TAPPI J., 70(11), 1987, p. 75. 17. Turai, L.L. and Teng, C.H., TAPPI J., 61(2), 1978, p. 31. 18. Turai, L.L. and Teng, C.H. TAPPI J., 62(1), 1979, p. 45. 19. Turai, L.L., Proceedings Pulping conference 1982, p. 387. 20. Norman, J.C., Sell, N.J. et al., TAPPI J., 77(3), 1994, p. 151. 21. Scott, W.E. and Gerber, P., TAPPI J., 78(12), 1995, p. l25. 22. Ramasubramanian, M.K. and Madansetty, S.L., 2000, TAPPI Symposium, Proceedings, p. 21. 23. Crawford, A.E., Research, 6, 1953, p. l06. 24. Anonymous, Soap and Chemical Specialties, 39(10), 1963, p. l75. 25. Weinstein, J.N., Yoshikami, S. et al., Science, 193, 1977, p. 489. 26. Hislop, T.W., Ultrasonics, 8, 1970, p. 88. 27. Last, A.J., Ultrasonics, 7, 1969, p. 13.

CRC_DK8307_CH008.indd 361

11/14/2008 4:56:23 PM

Ultrasonics: Data, Equations, and Their Practical Uses

362 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78.

Senapati, N. and Krishnaswamy, P., Private communication. Heinz, J.O., Am. Inst. Of Cehm. Engineers, J.I., 1955, p. 289. Wang, H.C., Yao Hsueh Pao, 12(15), 1965, p. 346. Marboe, H.C. and Weyl, W.A., J. Appl. Phys., 21, 1950, p. 937. Taylor, G.I., Proc. R., Society, London serries, A, 138, 1932, p. 41. Rajagopal, E.S., Indian Academy Scie. Sec. A, 49, 1959, p. 333. Nedjuhii, S.A., Soc. Phys. Acoust., 9, 1963, p. 99. Nosov, V.A., Ultrasonics in Chemical Industry, Consultants Bureau, New York, 2, 1965, pp. 43, 164. Hodge, W.T. and St. Clair, T.L., SAMPE Quart., 14, 1983. Krishnaswamy, P., Senapati, N. et al., Battelle Memorial Inst. 1988. Kunz, E.C. and Ensrm’nger, D., “Method and Apparatus for Treating Mica”, U.S. Patent 2, 798, 673, July 9, 1957. Bjorno, L., Gramn, S. et al., Ultrasonics, 16, 1978, p. 103. Semmelink, A., Ultrasonics, 16, 1978, pp. 7–10. Fairbanks, H.V., Physical Principles of Ultrasonic Technology, Guildford, Surrey, UK, 1973, pp. 11–15. Pohlman, R., Heisler, K. et al., Ultrasonics, 12, 1974, p. 11. Lang, R.J., J. Acoust. Soc. Am., 34(1), 1962, p. 648. Lord Rayleigh, The Theory of Sound, 1st Am. ed, Vol 2, Dover, New York, 1945. Popov, V.F., Goncharenko, G.K. et al., Khim. Technol., 8(2), 1965, p. 331. Charpanet, M.L., Riv. Combust., 18(12), 1964, p. 522. Ensminger, D., “Resonant Vibratory Apparatus, ” US Patent 3, 400, 892, September 10, 1968. Kirsten, W.J. and Bertilson, G.O.B., Analy. Chem., 38(4), 1966, p. 648. Mapleston, P., Modern Plastics, Nov. 2000, p. 72. Suslick, K.S., The Chemistry of Ultrasound, Year Book of Science & The Future, Encyclopedia of Britannica Inc., 1994, p. 138. Suslick, K.S. et al., 353 Nature, Oct. 3, 1991, p. 414. Miller, R.J., Rawlings, D.C. and Olli, L.K., US Patent 5, 520, 717, May 28, 1996. Middler, M., US Patent 3, 510, 266, May 6, 1970. Abrams, O.V. and Teumin, L.L., Physical Principles of Ultrasonic Technology, Edited by Rosenberg, L.D., Plenum Press, NY, 1973, 2, p. 145. Woodford, R. and Morrison, J.C., J. Pharm. Pharmacol., 21(9), 1969, p. 602. Tamas, G. and Tarnoczy, T., Magyar Fisz. Foliorat., 3, 1955, p. 543. Tarnoczy, T., Magyar fisz. Folioirat., 4, 1956, p. 67. Fairbanks, H.V. and Chen W.I., Ultrasonics, 7(30), 1969, p. 543. Baumgarte, F., Arztliche Forschung., 3(21), 1949, p. 525. Frenze, H., Hinsberg, K. et al., Exp. Med., 96, 1935, p. 811. Suslick, K.S. and Hammerton, D.A., IEEE Trans. Ultrason. Ferroelec., Conf. 33, 1986, p. 33. Riesz, P., Berdahl, D. et al., Environ. Health Perspect., 64, 1985, p. 233. Anbar, M. and Pecht, L., Phys. Chem., 68, 1964, p. 352. Henglein, A., Ultrasonics, 25, 1987, p. 6. Fischer, C.H., Hart, E.J. et al., J. Phys. Chem., 90, 1986, p. 1954. Hart, E.J. and Henglein, A.J., J. Phys. Chem., 90, 1986, p. 3961. Weissler, A.J., Cooper, H.W. et al., J. Am. Chem. Soc., 72, 1950, p. l769. Margulis, M.A., Dudenko, U. et al., Zh. Fiz. Khim., 58, 1984, p. 1402. Hart, E.J. and Henglein, A., J. Phys. Chem., 90, 1986, p. 5992. Mead, E.L., Sutherland, R.G. et al., Can. J. Chem., 53, 1975, p. 2394. Mason, T.J., Lorimar, J.P. et al., Tetrahedron Letters, 23, 1982, p. ll33. Mason, T.J., Lorimar, J.P. et al., Tetrahedron Letters, 24, 1964, p. 5393. Sehagal, C.M., Sutherland, R.G. et al., IEEE Trans. Sonics Ultrason., SU30, 1983, p. 2920. Sehagal, C.M., Sutherland, R.G. et al., J. Phys. Chem., 84, 1980, p. 2920. Henglein, A., Kormann, C. et al., Int. J. Radiat. Stud. Phys. Chem. Med., 48, 1985, p. 251. Elpiner, I.E., Sokolkaya, A.V. et al., Nature, London, 208, 1965, p. 945. McKee, J.R., Christman, C.L. et al., Biochemistry, 16, 1977, p. 4651. Prudhomme, R.O. et al., Bull. Soc. Chim. Biol., 39, 1957, p. 425.

CRC_DK8307_CH008.indd 362

11/14/2008 4:56:23 PM

Ultrasonics-Assisted Physical and Chemical Processes

363

79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106.

Henglein, A. and Fischer, C.H., Bur, Bunsenges. Phys. Chem., 88, 1984, p. 1196. Sehagal, C., Yu, T. et al., J. Phys. Chem., 96, 1982, p. 2982. Suslick, K.S., Gawienowski, J.W. et al., J. Phys. Chem, 87, 1983, p. 2299. Thompson, L.H. and Doraiswamy, L.K., Ind. Engg. Chem. Res., 38, 1999, p. 1215. Starcheskii, V.I., Mokryi, E.N. et al., Zh. Fiz. Kim., 58, 1984, p. 1397. Vasilina, T.V., Zh. Fiz. Kim., 58, 1984, p. 1926. Rosenthal, L., Mossoba, M.M. et al., J. Mag. Res., 45, 1961m, p. 359. Hart, E.J., and Henglein, A., J. Phys. Chem., 90, 1986b, p. 5992. Hart, E.J., and Henglein, A., J. Phys. Chem., 90, 1986a, p. 3061. Suslick, K.S. and Hammerman, D.A., IEEE Trans, Ultrason., UFFC 33(2), 1986, p. 143. Suslick, K.S., Schubert, P.F. et al., J. Am. Chem. Soc., 103, 1981, p. 7324. Suslick, K.S. and Schubert, P.F., J. Am. Chem. Soc., 105, 1983, p. 6042. Rehorek, D. and Jansen, E.G., J. Organoraet. Chem., 268, 1984, p. 135. Suslick, K.S., Goodalale, J.W. et al., J. Am. Chem. Soc., 105, 1983, p. 2499. Francony, A. and Petrie, R.C., Ultrason. Sonochem., 3, 1996, p. 877. Lorimer, J.P., Mason, T.J., et al., Ultrasonics, 29(4), 1991, p. 338. Hagensen, L.C., Duraiswamy, L.K. et al., Chem. Eng. Sci., 53(1), 1998, p. 131. Hagensen, L.C., Naik, S.D. et al., Chem. Eng. Sci., 49, 1994, p. 4787. Madison, S.D., Kowek, J.H. et al., US Patent 5,326,61, July 1994. Lorimar, J.P. and Mason, T.I., J. Sonochemistry, Part I, Chem. Soc. Rev., 16, 1987, p. 239. Urban, M.W. and Saiazar Rosas, E.M., Macromolecules, 21, 1988, p. 372. Kruus, P.O., Ultrasonics, 25, 1985, p. 20. Diedrich, G.K., Kruus, P. et al., Can. J. Chem., 50, 1972, p. 1743. Kruus, P., Ultrasonics, 21, 1983, p. 201. Kruus, P. and Patrabody, T.J., J. Phys. Chem., 89, 1985, p. 3379. Miyata, T. and Nakashio, F., J. Chem. Eng. Japan, 8(6), 1975, p. 463, 468. Lozinski, V.I., J. Poly. Sci., Part 1, Polym. Chem., 1(5), 1973, p. 1111. O’Driscol, K.F. and Shridharan, A.U., ACS Meeting, Philadelphia, PA, 1975, Polymer Preprints, 16(1), p. 358. 107. Kojima, Y., Koda, S. et al., Ultrason. Sonochem., 8(2), 2001, p. 75. 108. Ooi, S.K. and Biggs, S., Ultrason. Sonochem., 7(3), 2000, p. 125. 109. Fujikura, A.H., Kakiuchi, H. et al., Kobunshi Robunshu, 33(4), 1976, p. 183. 110. Kawase, S. and Kakurai, T., Kobunshi Robunshu, 35(3), 1978, p. 185. 111. Koda, S., Suzuki, A. and Norman, H., Polym. J., 27(11), 1995, p. 1144. 112. Koda, S., Amano, T. et al., Ultrason. Sonochem., 95, 1996, p. 3391. 113. Kruus, P., Lorimar, J.P., Mason, T.J. et al., Ultrasonics, 29(4), 1991, p. 338. 114. Chou, H.C. and Stoffer, J.O., J. Appl. Polym. Sci., 72(99), 1999, p. 797. 115. Bahattab, M.R. and Stoffer, J.O., Poly. Mat. Sci. Eng., 85(76), 2001, p. 810. 116. Okamura, S.W., Hayashi, K. et al., Japanese Patent 16,155, July 26, 1955. 117. Romanov, A. and Lazar, M., Plaste Kautschuch, 10(8), 1963. 118. Shiota, T., Goto, Y. et al., Kobunshi Kagaku, 22(239), 1965, p. 186. 119. Hastings, G.W. and Bolten, R.F., E. Soc. Chem. Industry, Monograph, No. 20, 1966, p. 274. 120. Flosdurf, E. and Chambers, L.A., J. Am. Chem. Soc., 55, 1933, p. 3057. 121. Schmid, G. and Romnel, O., Z. Phys. Chem. A, 185, 1939, p. 92. 122. Breck, H.W.W. and Jellineck, H.H.G., J. Poly. Sci., 21, 1956, p. 535. 123. Basedow, A.M. and Ebert, K.A., Adv. Poly. Sci., 22, 1977, p. 83. 124. Langton, N.H. and Vaughan, P., Brit. J. Appl. Phys., vol 13, 1962, p. 478. 125. Langton, N.H. and Vaughan, Brit. J. Appl. Phys., vol 14, 1963, p. 563. 126. Henglein, A., Macromol. Chem., 15(188), 1956, pp. 18, 37. 127. Levinson, M.S. and Nefedov, V.P., Izv. Sibirsk OTD. AJkad. Nauk SSR. Ser. Biol. Med. Nauk., 1, 1964, p. 145. 128. Tabata, M. and Sohma, J.J., Eur. Polym., 16, 1980, p. 589. 129. Tabata, M., Miazawa, T. et al., J. Chem. Phys. Lett., 73, 1980, p. 173. 130. Head, W.F. and Lauter, W.M., J. Am. Pharm. Assoc., 46(2), 1957, p. 617.

CRC_DK8307_CH008.indd 363

11/14/2008 4:56:24 PM

Ultrasonics: Data, Equations, and Their Practical Uses

364 131. 132. 133. 134. 135. 136. 137. 138. 139.

140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174.

Kossler, I. and Novobilsky, I., Collection of Czch. Chem. Commn., 28, 1963, p. 578. Newton, N. and Kissel, J.W., Arch. Biochem. Bio. Phs., 47(2), 1953, p. 424. Show, M.T. and Rodriguez, F., J. Appl. Polym. Sci., 11, 1967, p. 991. Witekawa, S., Roeziniki Chem., 36, 1962, p. 693. Zukal’ko, P.P., Kichkin, G. et al., Khim. Technl. Topl. Masel., 14(6), 1969, p. 46. Pirkonen, P., Heikkinen, J. et al., Ultrason. Fonochem., 8(3), 2001, p. 259. Mangaraj, D. and Senapati, N., US Patent 4,599,771, Oct. 22, 1985. Mangaraj, D. and Senapati, N., Int. Conf. On Rubber and Rubber-like Materials, Jamsepur, India, 1986, Proceedings. Bhowmik, A. and Mangaraj, D., Emerging Methods of Rubber Vulcanization In Rubber Products Manufacturing Technology, Ed. Bhowmik, A.K., Hall, M.M., Benarey, H.W., Marcel Dekker, NY, 1994, p. 378. Mangaraj, D. and Senapati, N., Battelle Memorial Institute, Columbus, OH, Multiclient Report, 1988. Mangaraj, D. and Kiss, C., Proc. Conf. On Dielectric Phenomena, Pocono, PA, 1981. Ring, T.B. and Twesme, E.N., Energy & Minerals Research Co., 1984. Mori, E. and Ishiki, S., Fumikura Technol. Ev., 1979, p. 41. Ujani, K., Jap. Plas. Age, May/June 1978, p. 33, July/Aug p. 190. Isayev, A.I., U.S. Patent 5,253,413, 1993. Isayev, A.I. and Chen, J., U.S. Patent 284,625, 1994. Tukachinsky, A., Chen, J. and Isayev, A.I., Rubber Chem. Technol., 58, 1995, p. 267. Tukachinsky, A., Chen, J. and Isayev, A.I., Rubber Chem. Technol., 69, 1996, p. 92. Yushanov, S.P., Isayev, A.I. et al., Paper #59, ACS Rubber Division Meeting, 1997. Levin, V.U., Kim, S.H. et al., Paper #60, ACS Rubber Division Meeting, 1997. Hong, C.K. and Isayev, A.L., Rubber Chem. Technol., 75, 2002, p. 133. Isayev, A.I. and Hong, C.J., Processing of ANTEC, SPE, San Francisco, 2002, p. 1334. Mason, T.J., Ultrason. Sonochem., 8(1), 2001, p. 145. Mason, T.J., Panniwynyk, L. and Lorimar, J., Ultrason. Sonochem., 3, 1996, p. 253. Phull, S.S. and Mason, T.J., Advances in Sonochemistry, Ed., Mason, T.J., V5, JAI Press, Stanford, CA, 1999, p. 175. Cordemans, E. and Hunnecar, B., World Patent 98/01394, 1998 (Water Treatment, Sonoxide). Petrier, C. and Fancony, A., Ultrason. Sonochem., 4, 1997, p. 295. Tiehm, A., Nickel, K. et al., Water Scie. & Technol., 35, 1997, p. 121. Avern, N. and Copercini, P.A., World Oil, 1997, p. 75 (drilling). Vinatora, M., Thoma, M. et al., Advance in Sonochemistry, vol. 5, JAI Press, Stanford, CA, 1999, p. 209. Xie, M.P., Ding, J.F. et al., J. Am. Leather Chem. Assoc., 94, 1999, p. 146. (Die Penetration). Price, C., Pharm. Technol. Euro., October 1997. Mason, T.J., Lorimar, J.P. et al., Ultrasonics, 28, 1990, p. 333. Hart, E.J., Henglein, A.J., J. Phys. Chem., 89, 1985, p. 4342. Sokal’skaya, J., Gen. Chem., USSR, 48, 1978, p. 1289. Polotzki. I.G., Gen. Chem., USSR, 17, 1947, p. 649. Weissler, A.J., J. Acoust. Soc. Am., 32, 1960, p. 283. Weissler, A.J, J. Am. Chem. Soc., 81, 1959, p. 1057. Weissler, A.J., J. Acoust. Soc. Am., 25, 1953, p. 65. Hoffman, M.R., Hua, I. et al., Ultrason. Sonochem., 3, 1996, p. 163. Coate, R.B. and Towels, J.T., US Patent 5,466,367, Nov. 1995. Misk, V. and Riesz, P., Ultrason. Sonochem., 3, 1996a, p. 25. Suslick, K.S. (Ed), High Energy Processes in Oraganometallic Chemistry Symposium Series, ACS, Washington, DC, 1987. Price, G., Ultrason. Sonochem., 3, 1996, S 229.

CRC_DK8307_CH008.indd 364

11/14/2008 4:56:24 PM

9 Advances in Generation and Detection of Ultrasound in the Field of Nondestructive Testing/Evaluation Allan F. Pardini, Gerald J. Posakony, and Theodore T. Taylor

CONTENTS 9.1 Modern Equipment for Ultrasonic Nondestructive Testing........................................ 366 9.1.1 Portable Field Equipment ..................................................................................... 367 9.1.1.1 Handheld Thickness Measurement Equipment................................. 367 9.1.1.2 Handheld Flaw Detection Equipment ................................................. 368 9.1.2 Stationary Field Equipment.................................................................................. 369 9.1.3 Laboratory Systems ............................................................................................... 372 9.1.4 Transducers............................................................................................................. 372 9.1.5 Current List of Selected Ultrasonic Testing Nondestructive Testing Manufacturers .......................................................................................... 374 9.2 Electromagnetic Acoustic Transducers .......................................................................... 375 9.2.1 Overview ................................................................................................................. 375 9.2.2 Introduction ............................................................................................................ 375 9.2.3 Terminology—Specific to the Text ...................................................................... 376 9.2.4 Summary of Electromagnetic Acoustic Transducer Principles ...................... 376 9.2.5 Typical Applications .............................................................................................. 380 9.2.5.1 Velocity Measurements Using Electromagnetic Acoustic Transducers ............................................................................. 380 9.2.5.2 Time Measurement Technique ..............................................................380 9.2.5.3 Ultrasonic Measurement of Texture .....................................................380 9.2.5.4 Ultrasonic Measurement of Stress ........................................................384 9.2.5.5 Measurement Technique ........................................................................ 385 9.2.6 Using Electromagnetic Acoustic Transducers with Composite Materials .................................................................................... 385 9.2.7 Limitations .............................................................................................................. 386 9.3 Phased Arrays .................................................................................................................... 386 9.3.1 Basic Theory ........................................................................................................... 386 9.3.2 One-Dimensional Continuous-Wave Phased Array......................................... 387 9.3.3 The Broadband Emission Phased Array ............................................................ 387 9.3.4 The Broadband Pulse Excited Phased Array ..................................................... 389 9.3.5 Time Delays for Steering and Focusing.............................................................. 390 9.3.6 Angular Response of Elements............................................................................ 390

365

CRC_DK8307_CH009.indd 365

11/14/2008 10:52:24 AM

Ultrasonics: Data, Equations, and Their Practical Uses

366

9.3.7 Beam Apodization ................................................................................................. 391 9.3.8 Reciprocity .............................................................................................................. 391 9.3.9 Summary for the One-Dimensional Case .......................................................... 391 9.4 The Two-Dimensional Case ............................................................................................. 392 9.4.1 Broadband Excitation ............................................................................................ 392 9.4.2 Finite Pulse Length ................................................................................................ 392 9.4.3 Time Delays for Steering and Focusing.............................................................. 393 9.4.4 Angular Response of Elements............................................................................ 395 9.4.5 Reciprocity .............................................................................................................. 395 9.4.6 Summary for the Two-Dimensional Case .......................................................... 395 9.5 Phased Arrays in Industry ............................................................................................... 395 9.5.1 Introduction ............................................................................................................ 395 9.5.2 Potential Applications ........................................................................................... 397 9.5.3 Advantages ............................................................................................................. 397 9.6 Laser Ultrasound ............................................................................................................... 397 9.6.1 Introduction ............................................................................................................ 397 9.6.2 Laser Generation .................................................................................................... 399 9.6.2.1 Thermoelastic Regime ............................................................................ 399 9.6.2.2 Plasma Regime ........................................................................................ 399 9.6.2.3 Laser Detection........................................................................................ 401 References ....................................................................................................................................404

9.1

Modern Equipment for Ultrasonic Nondestructive Testing

The state-of-the-art in modern day ultrasonic nondestructive testing (NDT) instrumentation has come a long way since the early developmental systems of the 1940s. Based on rapid growth in the computer industry and the great strides made in analog-to-digital electronics, modern ultrasonic systems have evolved into very sophisticated inspection tools. Advances have come in all facets of industry. The food industry uses advanced ultrasonic NDT to examine manufacturing processes [1]. The rail industry uses air-coupled ultrasound to examine railroad tracks [2]. Researchers at the national laboratories are using very high-frequency systems to examine frog eggs using a technique known as acoustic microscopy. Most of these applications use the latest commercially available equipment that can be configured to solve complex inspection problems. This section of the book provides examples of modern equipment that can readily be procured and configured for use in these and other demanding ultrasonic NDT applications.* This section of Chapter 9 groups modern ultrasonic testing (UT) equipment into the following categories: Portable field equipment Stationary field equipment Laboratory systems Transducers * Reference to a company or product does not imply approval or recommendation from the Pacific Northwest National Laboratory or the U.S. Department of Energy to the exclusion of others that may be suitable.

CRC_DK8307_CH009.indd 366

11/14/2008 10:52:25 AM

Advances in Generation and Detection of Ultrasound 9.1.1

367

Portable Field Equipment

One of the most common types of ultrasonic NDT instrumentation is the portable field equipment. The intent of this type of equipment is to provide the NDT operator with a complete inspection system that can be easily moved from job to job. This category includes all of the handheld thickness and flaw detection instruments. 9.1.1.1

Handheld Thickness Measurement Equipment

Probably the most widely used ultrasonic equipment in the field is the handheld thickness measuring instrument. The instrument consists of the pulser and receiver packaged together with a digital readout of thickness values. The quality of the instrument is usually based on the accuracy of the readout values. A high-quality thickness measurement instrument can have a measurement accuracy of ±0.0001 in. An example of this type of equipment is shown in Figure 9.1 [3]. The entire ultrasonic system is packaged in a small, extremely compact, handheld unit. Some of the more advanced units provide graphical A-scan displays for verification of true remaining wall thickness. Advanced units also have the capability to distinguish paint and coating thicknesses and measure through certain types of insulation. There are many manufactures of this type of instrument. Figure 9.2 [4] provides an additional examples of a handheld thickness unit.

FIGURE 9.1 Handheld digital thickness measuring equipment. (Courtesy of R/D Tech Inc.)

CRC_DK8307_CH009.indd 367

11/14/2008 10:52:26 AM

Ultrasonics: Data, Equations, and Their Practical Uses

368

FIGURE 9.2 Additional handheld thickness measuring unit. (Courtesy of GE Sensing & Inspection Technologies.).

FIGURE 9.3 Handheld flaw detector. (Courtesy of GE Sensing & Inspection Technologies.)

9.1.1.2

Handheld Flaw Detection Equipment

Another type of commonly used ultrasonic equipment is the portable flaw detector. An example is shown in Figure 9.3. Ultrasonic flaw detectors are used to characterize defects in many different types of materials including metals and composites. These compact units have relatively large liquid crystal displays (LCDs) built in so that the inspector can easily see the display even in direct sunlight. Advanced models have multicolor displays with waveforms that dynamically change color to alert the inspector when conditions warrant attention.

CRC_DK8307_CH009.indd 368

11/14/2008 10:52:26 AM

Advances in Generation and Detection of Ultrasound

369

FIGURE 9.4 Graphic display and keyboard. (Courtesy of R/D Tech. Inc.)

Many of these units have advanced signal processing capabilities and the ability to store large amounts of data. An additional feature allows the inspector to quickly download the data to a personal computer (PC) for postprocessing and report preparation. Systems such as the one shown in Figure 9.4 provide the operator with a fully functional UT system capable of A-, B-, and C-scan views as well as digital readout of thickness. With a fully programmable configuration, the NDT operator can set parameters and acceptance criteria and download final reports directly to a portable printer, alleviating the need to go back to the office. Reports can be delivered to the client immediately and issues can be discussed on the spot. 9.1.2

Stationary Field Equipment

Stationary field equipment encompasses small tabletop systems to large-scale systems used in production facilities and equipment centers. These systems incorporate modular electronics, but the intent is to provide a system that remains in place to acquire and analyze data over extended periods of time. These systems can be simple and provide minimal scanning capability as shown in Figure 9.5 [5], or quite complex and coupled to robotic inspection equipment in a factory floor setting, as shown in Figure 9.6. Modularized components such as the pulser and receivers reside in a ruggedized rack-mountable chassis that can be arranged to form the complete UT inspection system. The basic system components can be procured, interconnected, and rack-mounted for quick installation on the factory floor. Figure 9.7 [6] shows the ease of mounting modular units into a stationary field-deployable unit.

CRC_DK8307_CH009.indd 369

11/14/2008 10:52:27 AM

370

Ultrasonics: Data, Equations, and Their Practical Uses

FIGURE 9.5 Simple X–Y scanning system. (Courtesy of QMI Inc.)

FIGURE 9.6 Examples of ultrasonic testing used in manufacturing systems. (Courtesy of GE Sensing & Inspection Technologies.)

Many new ultrasonic systems are directly integrated with the PC and the UT application software resides on the PC. It is straightforward to add additional signal analysis equipment cards and have the flexibility of using different pulsers that provide spike, squarewave, or tone-burst pulses to meet inspection requirements. The PC has a peripheral

CRC_DK8307_CH009.indd 370

11/14/2008 10:52:27 AM

Advances in Generation and Detection of Ultrasound

371

FIGURE 9.7 Rack-mounted ultrasonic testing system. (Courtesy of Pacific Northwest National Laboratory.)

FIGURE 9.8 Ultrasonic testing pulser/receiver with 100 MHz analog-to-digital converter board and digital signal processing for PCI bus. (Courtesy of US Ultratek Inc.)

component interconnect/industry standard architecture (PCI/ISA) backplane that allows for multiple configurations. Plug-in cards such as the one shown in Figure 9.8 [7] that interface directly to the PC are a favorite for system integrators. They contain the entire UT system, including pulser, receiver, analog-to-digital converter (A/D), and digital

CRC_DK8307_CH009.indd 371

11/14/2008 10:52:28 AM

Ultrasonics: Data, Equations, and Their Practical Uses

372

signal processing (DSP) capability. It allows them to configure the UT system for their specific application using commercial off-the-shelf (COTS) equipment. This is an extremely efficient way to build inspection systems for industrial processes. 9.1.3

Laboratory Systems

The use of ultrasonic NDT systems in the laboratory is worldwide. Laboratory systems tend to be more complex and involve various pieces of equipment put together to provide optimum performance in a very controlled setting. The premise is to get the best data possible so that the limits of technology can be stretched to the next level. Research laboratories continue to procure the very latest in ultrasonic technology; when the latest is not good enough, they develop their own. Many of the ultrasonic manufacturers today got their start as spin-offs from major laboratories. These same manufacturers have developed their own research and development departments to continue to be competitive in today’s market. Laboratory UT systems are usually configured to provide some specific data that can be used to explain some specific scientific phenomena. Depending on which type of laboratory is developing the UT system, the focus of the development can shift from basic science to focused applications. University laboratories tend to develop UT systems that answer fundamental questions about material properties and characteristics. These systems tend to be made up of highly specialized pieces of equipment that, when combined, form the complete UT system. System components may include high-resolution, large-bandwidth oscilloscopes; special tone-burst pulsers or waveform generators; and exotic transducer configurations. Federally funded national laboratories tend to develop UT systems that are focused on a specific application or problem. Although made of specialized pieces of equipment, they are packaged such that the system can be readily (not necessarily easily) moved to attack a specific problem. These systems can be integrated with robotic and process line equipment and be demonstrated both in the laboratory and in actual field trials. Industrial laboratories tend to focus on specific models or types of UT equipment to provide a more sellable, profit-making device. These systems are production-type pieces of equipment that contain all the rudimentary components packaged in sleek, easily configured systems. They can be portable or installed directly as permanent fixtures in a factory installation. Figure 9.9 [6] shows examples of systems that were developed by the different types of laboratories. Frequently ultrasonic systems are developed to solve unique nondestructive inspection problems, but more often than not, the developed technology can be used in unconventional ways. An example of this development is the acoustic inspection device (AID) shown in Figure 9.10, which utilizes simple ultrasonic principals to solve an unconventional type of problem. The AID measures the time of flight through an unknown material such as oil in a drum and can provide information about the contents of the drum rapidly and unintrusively. It is used in homeland security, drug interdiction, fraud monitoring, and international border patrol and treaty verification. 9.1.4

Transducers

At the heart of the UT system is the transducer, which makes ultrasonic inspection possible. The UT transducer takes electronic pulses at various frequencies and converts them to mechanical vibrations within the part being inspected and vice versa. There are companies whose sole existence is to design, fabricate, and deliver different types of UT transducers. There are delay-line, immersion, contact, high-temperature, air-coupled, angle-beam, and

CRC_DK8307_CH009.indd 372

11/14/2008 10:52:30 AM

Advances in Generation and Detection of Ultrasound

373

National lab University Commercial FIGURE 9.9 Examples of ultrasonic systems developed in different types of laboratories. (Courtesy of Pacific Northwest National Laboratory and Force Technology.)

FIGURE 9.10 Acoustic inspection device utilized for drug interdiction. (Courtesy of Pacific Northwest National Laboratory.)

dual-element transducers, to name a few. They are made of many different materials and are fabricated in a variety of proprietary ways. Figure 9.11 [8] illustrates how sound can be transmitted into an inspected part and shows the various components of a typical UT transducer. At the core of this type of transducer is the piezoelectric material. Since the piezoelectric effect was first discovered in the late 1800s, researchers have worked hard to find materials that exhibit this effect. Transducers utilized today, as shown in Figure 9.12 [9], incorporate many of the attributes that were discovered from the various materials exhibiting the piezoelectric effect. Of course, there are other methods of transmitting sound in an inspection part. These other methods will be covered in Sections 9.5 and 9.6.

CRC_DK8307_CH009.indd 373

11/14/2008 10:52:30 AM

Ultrasonics: Data, Equations, and Their Practical Uses

374 Case Epoxy potting Backing material Electrodes Piezoelectric element

Coaxial cable connector Signal wire Ground wire Wear plate

FIGURE 9.11 Piezoelectric contact transducer arrangement. (Courtesy of the Center for NDE at Iowa State University.)

FIGURE 9.12 Transducers used for ultrasonic testing inspections. (Courtesy of NDT Systems, Inc.)

9.1.5

Current List of Selected Ultrasonic Testing Nondestructive Testing Manufacturers

The list of manufacturers of ultrasonic test equipment is considerable, so to provide the readers with a starting point to procure equipment that can assist them in their inspection needs, this book lists a few representative examples. GE Inspection Technologies 50 Industrial Park Rd. Lewistown, PA 17044 (717) 242-0327; Fax: (717) 242-2606 www.geinspectiontechnologies.com

CRC_DK8307_CH009.indd 374

11/14/2008 10:52:32 AM

Advances in Generation and Detection of Ultrasound

375

NDT Systems, Inc. 17811 Georgetown Lane Huntington Beach, CA 92647 (714) 893-2438; Fax: (714) 897-3840 www.ndtsystems.com R/D Tech 505 Boul. du Parc-Technologique Québec, G1P 2J7, Canada (418) 872-1155; Fax: (418) 872-5431 www.rd-tech.com Sigma Transducers, Inc. 8919 W. Grandridge Blvd., Ste. B Kennewick, WA 99336 (509) 783-9497; Fax: (509) 783-9655 www.sigmatx.com

9.2 9.2.1

Electromagnetic Acoustic Transducers Overview

Over the past decade, there have been many advances in the field of NDT and nondestructive evaluation (NDE). Advances have been made in radiography, eddy currents, and infrared; and various algorithms and techniques have been developed to quantify characterization of flaws. This text is designed to address specific advances that have been made in the generation and detection of ultrasonic energy. The basic equations for wave propagation are covered in many available technical books [10,11]. This text is intended to provide a synopsis of specific topics. 9.2.2

Introduction

Electromagnetic acoustic transducers (EMATs) are noncontact, noncouplant transducers capable of developing a variety of ultrasonic waves in electrically conducting materials. EMAT designs specify the frequency, wavelength, and surface displacement direction [12]. These variables are required to isolate specific wave modes. In addition to the conventional longitudinal and vertical shear waves that can be generated with conventional piezoelectric transducers, EMATs can generate a particularly advantageous type of plate wave known as a horizontally polarized shear wave (SH) that propagates parallel to the surface. This type of plate wave is independent of the plate thickness. In contrast to conventional piezoelectric transducers that must use highly viscous couplant or epoxies, this conversion can accommodate scanning using SH waves. The principle of EMATs is based on an understanding that EMATs are related to principles of electromagnetic technology and Lorentz force interaction between an induced

CRC_DK8307_CH009.indd 375

11/14/2008 10:52:33 AM

Ultrasonics: Data, Equations, and Their Practical Uses

376

eddy current in a conductive plate and an applied magnetic field. When an EMAT coil is placed near the surface of an electrically conducting material and driven by a specific ultrasonic frequency, eddy currents are induced in the material. If a static magnetic field is also present, these eddy currents will produce a Lorentz force. The fundamental arrangement of the EMAT transducer consists of explicitly designed patterns of electric conductors (e.g., coils). These patterns may be wire round coils or they may be fabricated on printed circuits as sheet coils or as meander-line coils. Their design, and the associated ultrasonic pulse, determines the frequency and type of wave that is introduced in the conducting material. The pulsed electromagnetic and Lorentz force mechanisms introduce an ultrasonic pulse in the conducting material. Various texts and technical articles [13,14] provide an excellent review of the fundamental principles and application of EMATs for the technology. 9.2.3

Terminology—Specific to the Text [15]

EMAT. An electromagnetic device for converting electrical energy into acoustic energy in the presence of a magnetic field. Lorentz force. Forces applied to electric currents when placed in a magnetic field. Lorentz forces are perpendicular to the direction of both the magnetic field and the current direction. Magnetostrictive forces. Forces arising from magnetic domain wall movements within a magnetic material during magnetization. Meander coil. An EMAT coil consisting of periodic, nonintersecting, winding, and usually evenly spaced conductors. Pancake coil (spiral). An EMAT coil consisting of spirally wound, usually evenly spaced conductors. Bulk wave. An ultrasonic wave, either longitudinal or shear mode, used in NDT to interrogate the volume of a material. 9.2.4

Summary of Electromagnetic Acoustic Transducer Principles

EMATs operate on a different principle than the more conventional piezoelectric transducers [16]. EMATs do not require coupling between the transducer and the test material, but rely on the induction of magnetic and Lorentz forces into electrical conducting materials. When a current carrying wire coil is placed near the surface of a conducting material and is driven at the desired ultrasonic frequency, eddy currents are inducted into the material. If a static magnetic field is also present, Lorentz forces will react to these eddy currents. The forces are defined by the following equation: F J B

(9.1)

where the Lorentz force (F) is a force per unit volume, J is the induced dynamic current density, and B the static magnetic induction. The sketches shown in Figure 9.13 describe the Lorentz force vectors related to the direction of the magnetic fields. The magnetic field can be produced by permanent magnets or by electromagnets (which may be pulsed). EMATs can be designed to provide either narrow or broadband operation. They can generate or receive the ultrasonic waves. Operational frequencies typically range from 0.1 to 10 MHz.

CRC_DK8307_CH009.indd 376

11/14/2008 10:52:33 AM

Advances in Generation and Detection of Ultrasound

377

S

N

S B B

Lorentz force

N Lorentz force

i FIGURE 9.13 Lorentz force direction as related to the direction of the magnetic field.

Spiral coil

Elongated

Meander FIGURE 9.14 Basic coil configurations.

In nonmagnetic materials, such as aluminum and brass, the Lorentz force acts on the lattice of the material. If the electromagnetic field is generated in a conductor via current (i) in an eddy current-type coil, the net force on the lattice is zero, because the forces on the electrons and ions are equal and opposite. However, if a static magnetic field is present, a net Lorentz force is transmitted to the lattice and results in the generation of an ultrasonic wave. The Lorentz force is the only force driving the ultrasonic wave in nonmagnetic electrically conducting materials. In ferromagnetic materials, the presence of a magnetic field causes a change in the shape of the magnetic domains that are present in the material. If an electric current-carrying conductor is placed near the surface of the material (in the presence of a magnetic field), magnetic domains in the ferromagnetic structure change as a function of the strength of the electric current in the coil and the strength of the magnetic field adding to the Lorentz force. This magnetic field/electric-field interaction results in a magnetic induction coupling that generates the ultrasonic waves in the material. The ultrasonic wave mode and the direction of propagation depend on the direction of the magnetic field and the configuration of the EMAT coils. EMATs are most effective at generating ultrasonic waves in ferromagnetic materials. Figure 9.14 shows three types of coils. EMAT coils are designed in a variety of shapes. In general, the spiral coils are designed to produce straight ultrasonic beams. Meander coils are more generally used to generate

CRC_DK8307_CH009.indd 377

11/14/2008 10:52:33 AM

Ultrasonics: Data, Equations, and Their Practical Uses

378

specific waves or angle beams. The frequency of the EMAT is determined by the line spacing in the spiral or meander coils. Spiral coils are generally driven by pulse-spike excitation, and meander coils are generally driven by a sinusoidal tone-burst excitation. The application of EMATs is dependent on the configuration of the magnets and the EMAT coil patterns. Some coil configurations include: • • • • •

Spiral coil EMAT—radially polarized shear waves normal to surface Meander coil EMAT—L or SV waves at selected angles Normal field EMAT—plane-polarized shear waves normal to surface Tangential field EMAT—polarized longitudinal waves normal to surface Periodic permanent magnet EMAT—obliquely propagated SH waves or guided SH modes in plates

For bulk wave Lorentz force EMATs operating such that the ultrasound wavelength is much longer than the electromagnetic skin depth and ignoring diffraction losses, an estimation of the received voltage across the terminals of the EMAT coil is given by:  (G/D)  VEMAT  I EMAT N 2B2 AExp    ZL ,S 

(9.2)

where VEMAT is the received voltage across the terminals of the EMAT coil, IEMAT the current pulse used to drive the EMAT coil, N the number of turns per unit length for

S N + + ++

S N +

(a)

+

+

+

(b) S

N

N + +++

S

(c)

+ + N

S + +

(d) N S N S N S N S N S S N S N S N S N S N

(e) FIGURE 9.15 Cross-section views of some electromagnetic acoustic transducers.

CRC_DK8307_CH009.indd 378

11/14/2008 10:52:34 AM

Advances in Generation and Detection of Ultrasound

379

the EMAT coil, B the static magnetic field, A the active area of the EMAT, G the effective gap between the EMAT coil and the surface of the part, D the diameter of the coil, and α is a geometry-dependent constant. ZL,S is the acoustic impedance of the part, either longitudinal or shear. Figure 9.15 shows permanent magnet/coil configurations. Assuming a noiseless amplifier and ideal matching, the received noise level (referred to the input) is given by: Vnoise  ( 4KTREMAT )

1/ 2

(9.3)

where K is Boltzman’s constant times e, the electronic charge (MKS units), T is temperature in degrees kelvin, β is the bandwidth of the amplifier in hertz, and REMAT the resistance of the EMAT coil. The resistance of an EMAT coil subdivided into N turns can be shown to be REMAT  R0 ( NW )2

(9.4)

where R0 is the resistance of a single turn coil of the same geometry as the EMAT coil and W is the width of the coil such that NW gives the total number of turns in the coil. The current used to drive the EMAT in transmission is related to the power by 2 P0  I EMAT REMAT

(9.5)

A first order approximation for the received signal to noise can be obtained by combining these equations: VEMAT (P0 )1/ 2 B2 AExp(G/D)  Vnoise W 2ZL ,S ( 4KT)1/ 2 R0

(9.6)

Although there are many assumptions in deriving this expression, it is useful to examine each of the terms in regards to optimizing the EMAT received signal-to-noise ratio. Maximizing the peak power delivered to the EMAT coil maximizes the received signal-tonoise ratio. Today, inexpensive, high-pulse power transistors are available allowing 10 kW and higher tone burst pulses to be applied to the EMAT coil if desired. These transistors have simplified drive requirements and are capable of operating in excess of 10 MHz, making 20 MHz operation of bulk wave EMATs practical. Ultimately, the peak power that can be applied to the EMAT coil is limited by average power dissipation in the coil for repetitive pulsing. Methods to cool the coil can be employed, allowing additional peak power or increased repetition rates to be used. Integrated circuit preamplifiers that offer wide bandwidth and very low noise have also become available. Typically, with these preamplifiers, the majority of the random noise at the output is the amplified Johnson noise of the EMAT coils resistance. These preamplifiers typically contribute < 10% of the total noise at the output, approaching the ideal noiseless amplifier. For bulk wave EMATs, the reduction in sensitivity with increasing distance between the EMAT and the part surface is scaled by the diameter of the coil (or wire-to-wire spacing, for meander coils). Using a larger-diameter coil reduces the sensitivity to lift-off. However, the beam properties of the transducer are also related to the coil diameter, so that in practice, there is often little that can be done to reduce the lift-off sensitivity. Maintaining the coil as close to the part as possible minimizes the losses due to lift-off.

CRC_DK8307_CH009.indd 379

11/14/2008 10:52:34 AM

Ultrasonics: Data, Equations, and Their Practical Uses

380

The signal-to-noise ratio is inversely proportional to the square root of the bandwidth (β). By employing optimized bandwidth and roll-off characteristics, further improvement in the received signal-to-noise ratio can be obtained. In some cases, DSP can be employed to provide this filtering, allowing the ultimate in versatile filtering. Digital signal averaging can also be employed to increase the signal-to-noise ratio. For random noise, the signalto-noise ratio is improved by the square root of the number of waveforms averaged together. Because the received signal-to-noise ratio is proportional to the square of the static magnetic field, it is especially beneficial to maximize B. A large improvement in permanent magnet’s field strength was provided with the introduction of neodymium boron iron magnets. The field strength provided by these magnets continues to improve with the introduction of ever-improved magnetic materials. With the availability of finite element electromagnetic modeling tools, it has become practical to optimize the design of the magnets used to generate the magnetic field for EMATs. 9.2.5 9.2.5.1

Typical Applications Velocity Measurements Using Electromagnetic Acoustic Transducers

In the inference of material properties from precise velocity or attenuation measurements, the use of EMATs can eliminate errors associated with couplant variation, particularly in contact measurements. Differential velocity is measured using a T1 − T2—R fixed array of EMAT transducers at 0, 45, and 90° or 0 and 90° relative rotational directions depending on device configuration. 9.2.5.2

Time Measurement Technique

One method that may be used for precise velocity measurements using the EMAT configuration shown in Figure 9.16 is described here. Using EMAT transducers as shown in Figure 9.16, measure the time difference between transmitting search units T1 and T2 using the experimental setup also shown in Figure 9.16. The EMATs in the figure are excited electronically in series to eliminate differential phase shift due to lift-off of the EMAT transducer. Determine the slope of the phase of the received signal by linear regression of weighted data points within the signal bandwidth and a weighted y-intercept. The accuracy obtained with this method can exceed one part in one hundred thousand (1:100,000). 9.2.5.3

Ultrasonic Measurement of Texture

Directed ultrasonic velocity measurements predict formability (texture) by taking advantage of the effects of directionality (anisotropy) that exists in the worked sheet (induced by the rolling process). One consequence of directionality is a change in mechanical properties with direction. For example, the yield strength and ductility may change with the orientation at which a laboratory tensile specimen is cut from a sheet. Generally, minimum and maximum values of these quantities occur at 0, in the vicinity of 45 and at 90° with respect to the rolling direction (see Figure 9.17). Any formation of ears in drawing operations (twofold and fourfold) will also generally take place along these axes. When forming sheet metal, practical consequences of directionality include such phenomena as excess wrinkling, puckering, ear formation, local thinning, or actual rupture.

CRC_DK8307_CH009.indd 380

11/14/2008 10:52:34 AM

Advances in Generation and Detection of Ultrasound

381

I1

V

RT

1

LT1

RT2

Electromagnetic acoustic transducer T1

RT2

Electromagnetic acoustic transducer T2

1 0.8 0.6

Amplitude

0.4 0.2 0 − 0.2

0

20

40

60

80

100

120

Time

− 0.4 − 0.6 − 0.8 −1 FIGURE 9.16 Precise velocity measurement with electromagnetic acoustic transducers—experimental setup.

Likely axes of minimum and maximum ductility 90° 45°

0°

FIGURE 9.17 Directionality properties of a rolled sheet.

At best, these can cause individual pieces to be scrapped. A more serious consequence is the downtime required to correct the manufacturing process. A number of specialized laboratory mechanical tests have been developed to identify the severity of directionality [17]. Included are measurements of plastic strain ratios in tensile tests, limiting drawing ratio measurements, cupping tests, and so on.

CRC_DK8307_CH009.indd 381

11/14/2008 10:52:34 AM

Ultrasonics: Data, Equations, and Their Practical Uses

382

Of particular interest here is the plastic strain ratio, defi ned as r

ew et

(9.7)

where ew is the strain ratio in the width direction and et is the strain in the thickness direction of a tensile coupon loaded in the plastic regime. The plastic strain ratio determines the relative tendency of deformation to occur in the plane of the sheet (ew) as opposed to through the thickness (et). In general, r will vary with the angle at which the tensile coupon is cut with respect to the rolling direction of the sheet. Directions with large values of r will generally correspond to directions of ear formation when a cup is deep drawn, as sketched in Figure 9.18. The RD indicates the rolling direction with respect to the angles that are measured. The upper set of curves shows the variation of r with angle. The lower sketches represent the resulting cup contour. Two commonly used figures of merit are the average plastic strain ratio or normal anisotropy, defined as r

r(0°)
2r( 45°)
r(90°) 4

(9.8)

and the planar anisotropy, defined as ∆r 

r(0°)  2r( 45°)
r(90°) 2

(9.9)

Formability of a drawing quality sheet depends largely on two factors: drawability (capability to be drawn from the flange area of the blank into the die cavity) and stretchability

2

r1

2

2

r1

r1

r = 1.4 r = 1.0 r = 0.7 0

45

90

0

45

RD

RD

Good drawability

Fair drawability

90

0

45

90

RD

Poor drawability

FIGURE 9.18 Relationship of plastic strain ratio to drawability.

CRC_DK8307_CH009.indd 382

11/14/2008 10:52:35 AM

Advances in Generation and Detection of Ultrasound

383

Rolled metal

Directionality (anisotropy)

Mechanical fibering (fiber texture)

Preferred orientation (crystallographic texture)

FIGURE 9.19 Causes of directionality.

(capability to be stretched under biaxial tension to the contours of the punch). Drawability is related primarily to plastic anisotropy, and the average plastic strain ratio, r, is a common measure of its value. This is schematically illustrated in Figure 9.18. The planar anisotropy, ∆r, is thought to be a measure of the tendency to form ears. As will be discussed shortly, directionality is sensed in ultrasonic velocity measurements by taking advantage of another one of its consequences, the dependence of elastic properties on direction. These are determined nondestructively from the elastic wave speeds. Figure 9.19 illustrates the causes for the existence of directionality (anisotropy) in the processed sheet. There are two kinds of anisotropy. One is caused by the alignment of the nonmetallic inclusions existing in the ingot, called mechanical fibering or fiber texture. The other is due to the alignment of the grains (crystals), called preferred orientation or crystallographic texture. The effects of preferred orientation have more profound implications in deep drawing operations, and it is this property that is sensed by ultrasonic measurements. Figure 9.19 demonstrates how preferred orientation is developed through the effects of the rolling operations on the grains of the unprocessed sheet. In response to the force imposed in working the metal, extensive plastic deformation must take place. At a microscopic level, this may be thought of as a result of dislocation motion along planes of low resistance. Two interrelated phenomena result—an elongation of the grains that could be observed visibly, and a change in the crystallographic orientation of the grains. The latter is believed to be the primary cause of directionality of properties associated with deep drawing. It can be sensed by x-ray diffraction or by ultrasonic wave speed measurements. Current ultrasonic methods are based on the relationship between ultrasonic phase velocities and the low-order orientation distribution coefficients (ODCs), a consequence of the application of the Voigt averaging method to infer aggregate elastic properties. Thus, such techniques provide only a partial description of texture. It is well known from scattering theories that although phase velocity of acoustic waves is controlled primarily by averages of single-grain elastic constant fluctuations, redirection of their energy by scattering at the grain boundaries are controlled primarily by two-point averages of these fluctuations. Because these two-point averages depend on higher order ODCs, a more complete and accurate description of texture can be obtained from a measurement of scattered acoustic energy.

CRC_DK8307_CH009.indd 383

11/14/2008 10:52:35 AM

Ultrasonics: Data, Equations, and Their Practical Uses

384

Perhaps the most exciting development in nondestructive determination of texture is presented in a article by Ahmed and Taylor [18]. The article presents the theoretical basis for computing the expected wave vector and backscattered power in a macroscopically orthotropic polycrystalline material with cubic grains, then shows how one can proceed to invert the scattering data (attenuation and backscattered signals) for the twelve ODCs. The article then applies the theory to determine very accurately three lower order ODCs—L-wave and two S-waves propagating only in the normal direction in steel with rolling symmetry. The article demonstrates that it is possible to recover several of the higher order ODCs. Once the theoretical description has been implemented [19], it should be possible to accurately describe the anisotropic nature of metals and use the information for engineering and inspection applications. The correlations between elastic and plastic anisotropy have received extensive experimental study in the steel sheet industry. Experimental results have demonstrated the correlation of the average Young’s modulus [20]: E

E(0°)
2E( 45°)
E(90°) 4

(9.10)

and its anisotropy E(0°)  2E( 45°)
E(90°) (9.11) 2 with the corresponding plastic anisotropy parameters –r and Δr as defined in earlier equations. Here E(θ) is defined as Young’s modulus of a coupon cut at an angle θ with respect to rolling direction. These successful laboratory studies led to a commercial instrument presently used extensively in the steel industry [21]. In summary, there is a quantitative correlation between the directionality of properties of a rolled metal sheet, such as plastic strain ratio or elastic modulus, and the underlying texture. Ultrasound can be used to characterize texture nondestructively and rapidly. Unlike the conventional x-ray techniques, which have limited surface penetration, velocity measurements can assess the drawability and can be of great advantage to the enhancement of forming operations. ∆E 

9.2.5.4

Ultrasonic Measurement of Stress [17]

For sheet and plate specimens experiencing applied or residual stress, the principle stress σa and σb may be inferred from orthogonal velocity measurements. The following equation relates ultrasonic velocities to the principle stresses experienced in sheet or plate. 2 pVavg [V (°)  V (°
90°)]  a  b

(9.12)

Vavg is the average shear velocity. It is understood that velocity difference [V(φ°) − V(φ° + 90°)] will be maximized when the ultrasonic propagation directions are aligned with principal stress axes. The magnitude of this difference, along with the density and mean velocity, can be used to predict the principal stress difference. It is particularly noteworthy that no acousto-elastic constants or other nonlinear properties of the material are needed for stress prediction, which distinguishes this approach from other ultrasonic stress measurement techniques. The nonlinear material characteristics have been suppressed by the process of taking the velocity difference.

CRC_DK8307_CH009.indd 384

11/14/2008 10:52:35 AM

Advances in Generation and Detection of Ultrasound

385

Stress 0, 90

Ferrous Nonferrous

0, 90

Pulsed magnets Permanent magnets

SH0

T1

T2

R

Precise fixed distance

FIGURE 9.20 Differential velocity for stress.

T1

T2

R

Precise fixed distance

FIGURE 9.21 Illustration of electromagnetic acoustic transducers on aluminum foil, which in turn is bonded to a composite material.

9.2.5.5

Measurement Technique

Differential velocity is measured using a T1 − T2—R fixed array of EMAT transducers at 0 and 90° relative rotational directions depending on the device. This configuration is shown in Figure 9.20. The website http://www.ndt-ed.org provides an experimental setup for using differential velocity to measure stress that includes: EMAT driver frequency. 450–600 kHz (nomi-verview_stress.gifnal), sampling period 100 ns Time measurement accuracy. Resolution 0.05 Ns Accuracy required for <2 ksi stress measurements. Variance 2.47 Ns 9.2.6

Using Electromagnetic Acoustic Transducers with Composite Materials

An EMAT requires no couplant and can be noncontacting in the generation and reception of ultrasound. Measurements using EMAT probes can therefore be done with a high degree of reproducibility. In a composite NDE study performed at the Center for Nondestructive Testing (CNDE) and described at the Internet site*, EMATs have been applied to a number of composites, including poorly conducting graphite/epoxy composites and nonconducting glass/epoxy and ceramic matrix composites. To generate sound waves via the Lorentz force mechanism, the surface of the composite must be conducting. An aluminum tape (0.003 in. aluminum foil with adhesive layer) is applied to the composite surface, as shown in Figure 9.21, to achieve this. Ultrasound is generated in the aluminum * The website http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/ EquipmentTrans/emats. Contains many useful articles on EMATs.

CRC_DK8307_CH009.indd 385

11/14/2008 10:52:36 AM

Ultrasonics: Data, Equations, and Their Practical Uses

386

foil and the adhesive bond between the metal and the composite allows the propagation of sound waves into the bulk of the composite. Two types of EMAT probes have been used in the study: SH wave probes originally designed to study rolled plates in a stress and texture project in CNDE and EMATs that generate normal incidence shear waves. Using the SH wave probes, the mechanical anisotropy of composite laminates have been investigated in a configuration somewhat akin to the acousto-ultrasonic technique. Directivity of the received EMAT signals “show excellent correlation with fiber” directions in the laminate. The SH wave probes, although not intended for generating bulk waves, have sufficient fringing field to be used in a transmission measurement through a full inch of graphite epoxy laminate. EMAT-generated plate modes have also been used in the detection of skin-core separation of a honeycomb sandwich structure (a rudder skin). A large rise in amplitude (signal-to-noise ratio of 4:5) is detected over the defect, as expected from damping considerations. Some polymer composites contain a metallized layer in the form of foil or mesh (for electro magnetic interference (EMI) and lightning protection purposes). These composites do not require the help of the aluminum tape in using EMATs. In a graphite epoxy panel containing a 0/90 layup of graphite fiber and a copper mesh, azimuthal scans using a pair of EMATs show the combined effects of the fiber tows at 0 and 90° and the direction of the copper wires at ±35°. In addition, it has been found that an EMAT can also be applied directly to a graphite epoxy panel that contains a top ply of nickel-plated graphite fibers. 9.2.7

Limitations

The principal limitation of EMATs is their very low efficiency. Insertion losses can be 40 dB or more than their piezoelectric counterparts. Furthermore, they can be used only on electrically conducting or ferromagnetic materials. EMATs require specialty instrumentation: application-specific coil designs, high current transmitters, and high gain receivers. The distance between the EMAT coil and the material (lift-off) significantly influences the total system efficiency.

9.3

Phased Arrays

Ultrasonic phased arrays provide technology for generating and receiving ultrasound that enables the angle of the sound in material to be steered and focused dynamically by controlling the sound field from individual piezoelectric elements. Beam steering and focusing permits the user to electronically select beam angles and sound characteristics that are optimized for specific part geometries or inspection problems. Phased arrays offer significant technical advantages for weld inspection compared to conventional ultrasonic inspection techniques [22]. 9.3.1

Basic Theory

This section presents several first order theoretical expressions that describe the pressure distribution of phased arrays in the far-field. The basic theory includes steered and focused beams using continuous wave and pulsed broad-band transducer arrays. The goal of this section is to outline practical methods for the calculation of the pressure field as a function of angle from a pulsed transducer array that is steered to various angles,

CRC_DK8307_CH009.indd 386

11/14/2008 10:52:37 AM

Advances in Generation and Detection of Ultrasound

387

and to thereby calculate the expected system response under such operation. The calculations described in this section consider one-dimensional (1D) and two-dimensional (2D) cases. The method of calculations described in this section may be used as the starting point for the development of software useful for describing and comparing the expected response of simple phased arrays. The material contained in this chapter is not intended to cover all possible aspects of the problem, but rather a limited subset. Multilayer media are not considered. Neither are samples of limited thickness or anisotropic acoustic velocities or transducers with shoes and oblique angles of incidence for the injection of sound. The results are intended for use in the far field of the transducer and are applicable for describing the near field. Extension to account for some of the possibilities mentioned earlier may be attempted in the future. 9.3.2

One-Dimensional Continuous-Wave Phased Array

Equation 9.13 provides an expression for the pressure field of an acoustic array for the case where the acoustic wave is continuous and the far field is considered. The function T(θ) is the normalized pressure at some angle θ and at any distance in the far field from the transducer array. Hence, the equation describes the relative pressure on an arc centered at the transducer. This equation describes the unfocused pressure field of a phased array. Equation 9.13 is valid only for the condition of a continuous wave. The equation is not valid for the conditions of (1) broadband emission of frequency and (2) emission in the form of a short pulse rather than continuous wave (CW). These conditions will be covered next. T (, ) 

sin( N) sin  sin() 

(9.13)

where N is the number of elements; γ = (πd(sin θs − sin θ)/λ); δ = (πa sin θ/λ); d is the element-to-element spacing; a the element width; θ the angle as measured from the normal to the array; θs the angle to which the beam is being steered, as measured from the normal to the array; and λ the acoustic wavelength of the transducer in the medium of interest. Note that in Equation 9.13, the second term, which represents a single element diffraction pattern, does not contain the steering angle. Steering at any particular angle depends on the availability of energy at that angle with which to steer. The second term of Equation 9.13 provides the answer as to energy availability. For a/λ very small, the sinc function is very broad and the beam can be steered over a broad angular range. For a/λ of unity, the width of the single element diffraction pattern is ∼1 rad and the beam can be steered to roughly ±30°. Note that Equation 9.13 is symmetric about the θs = 0 axis. But when steering occurs, the symmetry is broken, and pressure distributions will not be symmetric about the steering direction. Neither term is symmetric about the steering direction except for θs = 0. Because of the sinc function (which is symmetric about and peaked at θ = 0), local peaks nearer to normal to the array will be enhanced. Peaks on the other side of the steering angle will be diminished. In particular, grating lobes of equal order but on opposite sides of the steering angle can be quite different in amplitude, even in the CW case. 9.3.3

The Broadband Emission Phased Array

There are several ways to describe the pressure distribution for broadband emission of a phased array. The method described here is one that is rather straightforward. The broadband pulse of the array elements is described as a distribution of frequencies given by g( f )df over a frequency range fmin to fmax. The range of frequencies is chosen depending upon a specific application and may, as an example, be determined by the

CRC_DK8307_CH009.indd 387

11/14/2008 10:52:37 AM

Ultrasonics: Data, Equations, and Their Practical Uses

388

−3 dB pressure response of a transducer. The resultant pressure distribution, under the preceding conditions, may be described by P(θ), where P() 

∫ T(, f )g( f )df ∫ g( f )df

(9.14)

and where the integral is from fmin to fmax. The integral may or may not be easily evaluated. The integral in the numerator can be approximated by a summation if the frequency range is divided to reasonably small bands: P()  ∑ Ti (, f i )  gi ( f i )∆f

(9.15)

i

Equation 9.15 can also be cast in a form where the independent variable is the wavelength rather than the frequency, and the choice of which form to use depends on convenience: P()  ∑ Ti (, i )  gi ( i )∆

(9.16)

i

For computational purposes, it is suggested that we adopt and use Equation 9.15 or Equation 9.16 to account for the broadband nature of the pulse. As a realistic example, suppose the transducer frequency is nominally 3.0 MHz, but the 6 dB bandwidth extends from 2 to 4 MHz and the frequency components are normally distributed. We obtain  ( f  f 0 )2   ( f  3.0)2  g( f )  exp   exp    ∆f 2 4    

(9.17)

where the frequencies are in megahertz. The integral of g( f ) from 2 to 4 MHz yields ∼1.845. If we divide the interval from 2 to 4 MHz into 20 intervals of width 0.1 MHz, then Equation 9.15 becomes P() 

1  ( f i  3.0)2  Ti (, f i )exp  ∑  0.1 1.845 i 4  

(9.18)

Using Equation 9.18, we obtain

P() 

20

1 ∑ 1.845 i1



{

N[d(sin  s  sin )(3.0
0.1i)] c  d(sin  s  sin )(3.0
0.1i)  sin   c 

sin

}

sin[(a sin )(3.0
0.1i)/c]  (3.0
0.1i  3.0)2  exp   4 (a sin )(3.0
0.1i)/c  

(9.19)

for the pressure at angle θ with the beam steered to angle θs.

CRC_DK8307_CH009.indd 388

11/14/2008 10:52:37 AM

Advances in Generation and Detection of Ultrasound 9.3.4

389

The Broadband Pulse Excited Phased Array

The pulse duration used to excite a phased array has major influence on the grating lobes. When the excitation pulse is very long, similar to a continuous wave excitation, phased arrays form grating lobes which can be a severe limitation when phased arrays are used for practical inspection problems. When the excitation pulse is very short, grating lobes are diminished. Let the pulse length be given by Λ  S p

(9.20)

where λ p is the peak transducer wavelength and S the number of wavelengths in the pulse and is not necessarily an integer—a typical value might be 3 or so. The expression for the grating equation is based on a CW beam. The formation of the grating interference pattern relies on the availability of energy for which coherent summation leads to pressure peaks. Pressure peaks form when delays between various parts of the beam are integral numbers of wavelengths from the location where coherent addition occurs. However, if there are only a few cycles of signal emitted, then coherent summation can occur for the emission from just a few elements, rather than the entire array. The condition for the grating lobe is d(sin  s  sin )  m

(9.21)

 m   g ( i )  sin1  i
sin  s    d

(9.22)

from which

where θg(λ i) is the grating lobe angle for wavelength λ i and m is an integer that we can probably restrict to being less than or equal to 2. Equation 9.21 implies that the coherent sound coming from adjacent elements are delayed by one acoustic wavelength for the first grating lobe and two wavelengths for the second lobe. If the sound is CW, then coherent summation from all N channels will occur at the angular position θ. However, if the pulse length is Λ, then only Λ/λ channels will add coherently at θ. The other channels cannot provide energy at the appropriate time. Hence, the pressure amplitude for the first grating lobe will be reduced from that calculated by Equation 9.13 by a factor of Λ/λN. For the second grating lobe, the pressure reduction factor is Λ/2λN. We can account for the pressure diminution in Equation 9.15 or 9.16 in an approximate manner. We will consider Equation 9.16 as an example. The summation is over the wavelengths emitted by the transducer and different wavelengths will have fi rst and second grating lobes at different angles. There is also an angular width associated with each grating lobe. The steps to follow are: (1) determine the angles at which each wavelength λ i in the summation of Equation 9.15 reaches the first and second grating lobes (using Equation 9.22); (2) for that angle and a range of angles about it, reduce T by the factor S/N, with the angular range being the width of the grating lobe; and (3) continue with the summation. The grating lobe angular width is given by ∆( i ) 

CRC_DK8307_CH009.indd 389

i Nd cos 

(9.23)

11/14/2008 10:52:37 AM

Ultrasonics: Data, Equations, and Their Practical Uses

390

The algorithm for calculating pressure at some angle might then look like this: 1. For each wavelength, calculate the positions of the first and second grating lobes (four angles altogether for each wavelength) and then save these. 2. Proceed with the summation, which is over wavelength (see Equation 9.21) and provides a result at a particular angle. When an angle is chosen at which to exercise the calculation, check to see whether this angle is one that matches one (or more) of the grating lobe angles to within    g ( i )

i Nd cos 

3. For those elements of the summation that match the criterion, reduce the calculated contribution by S/N (or S/2N for a second grating lobe). 4. Complete the summation and try another angle. Adherence to an algorithm such as this or a similar one should account for the pulsed nature of the inspection method. 9.3.5

Time Delays for Steering and Focusing

The calculation of the required element-to-element delays for beam steering to a particular angle or focusing is relatively straightforward and based on simple geometric considerations. Beam steering can be achieved by sequentially exciting individual elements with an interval time delay Δτ. Firing begins at one end of the array and ends at the other end, with a total time span of (N − 1)Δτ. This delay is linear across the elements. It is not hard to figure out which end gets fired first. If focusing occurs in addition to angular steering, another delay must be added to the linear delay that is spherical across the elements. A general expression for the total delay for each element is given by 1/ 2 1/ 2 2 2       (n  N )d   Nd  2(n  N )d F  2Nd
∆ n   1
 sin  s   1
 sin  s     c  F F F  F          

(9.24)

where the elements are numbered from 0 to N − 1, and the zeroth element is opposite the steering angle and obtains the least delay. F is the distance at which the array is being focused as measured from the array center, d the element-to-element spacing, n the element number, c the acoustic velocity of the material under test, and θs the steering angle. And finally, N

N 1 2

This expression should be used for the calculation of all delays and is valid for all steering angles less than or equal to 90°. 9.3.6

Angular Response of Elements

The sin δ/δ term in Equation 9.13 holds in the case that each element can be modeled as a piston in a rigid baffle. Cross-coupling of elements can in many cases reduce the angular response to widths much less than predicted by the sin δ/δ term, further reducing the

CRC_DK8307_CH009.indd 390

11/14/2008 10:52:37 AM

Advances in Generation and Detection of Ultrasound

391

capability for beam steering. If care is taken in the construction of the phased array, then the array can sometimes be modeled as pistons within soft baffles, not rigid baffles. For this case, a more accurate representation of the angular beam width of a single element is given by (cos θ)(sin δ/δ). Hence a further improvement to theoretical calculations would be to include this term in Equation 9.13. 9.3.7

Beam Apodization

Beam apodization is a method for enhancing the directivity of an acoustic beam by increasing the amplitude of elements central in the array while diminishing the amplitudes of elements near the array edges. For instance, a simple cosine apodization seems to be somewhat beneficial. The main idea is to improve the decibels separation between the main lobe and the first side lobe (not the grating lobe). The effect is rather minimal, even for a CW beam, and is likely to be even smaller for a pulsed beam with some bandwidth. Some software packages (RDTech) allow apodization on the receive signal but not the send signal. Given the expected limited enhancement gained by such a procedure, we are probably better served by neglecting its incorporation. 9.3.8

Reciprocity

Sections 9.3.2 to 9.3.8 describe the transmitted pulse but not the receive pulse. In the CW case, reciprocity holds and the system response (point spread response) is given by V ()  P()R()  P 2 ()

(9.25)

where R(θ) is the receive response. For a pulsed broadband system, reciprocity does not exactly hold. The problem is due mostly to pulse stretching at large angles. For instance, for the first grating lobe, the length of the pulse is not S wavelengths, but rather N + S wavelengths. A target located at this angle will receive and reflect energy for a much longer time than a target at the steering angle. Time shifting and summing of signals to observe a target at the beam steering direction may not exclude a contribution from a target at a different angle, if the radiation from the target has a long duration. Hence, the receive response is not as sensitive to pulse length as the transmit response. This problem is difficult to treat analytically. We shall neglect it for the moment and proceed as if reciprocity actually holds. 9.3.9

Summary for the One-Dimensional Case

The system response is given by Equation 9.25. The most important and useful expression is provided in Equation 9.15 or 9.16, which gives the pressure distribution in the far field for a steered beam. The exact nature of the angular distribution of pressure depends on the number of elements in the array, the operating frequency, the bandwidth, the pulse length, and the width of the array elements. Common array design attempts to eliminate the problem of grating lobes by ensuring that the ratio d/λ <0.5, but it is likely that this condition cannot always be met, particularly for broadband excitation. In the case of pulsed arrays, short pulse lengths can be used to mitigate the influence of the grating lobes. The effect of bandwidth is basically to smear out certain features. We can account for the major effects of both pulse length and bandwidth. Although it is certainly possible to achieve element widths that are small compared to the acoustic wavelength, most NDE inspections are operated at the highest possible frequency to obtain spatial resolution. It is likely that conditions where d/λ < 0.5 will not always be obtained.

CRC_DK8307_CH009.indd 391

11/14/2008 10:52:38 AM

Ultrasonics: Data, Equations, and Their Practical Uses

392

In such cases, it is imperative that attempted beam steering angles be limited. Because the expressions for the pressure amplitude have been normalized, it is not evident from them when the condition where very little energy is being steered is reached.

9.4

The Two-Dimensional Case

For the case of 2D arrays, the problem is quite similar. For a CW beam of single frequency, T (, ) 

sin N sin M sin  sin  sin  sin   

(9.26)

where N is the number of elements in one dimension (say horizontal, or x); M the number of elements in the other direction (say vertical, or y); a and b are the transducer element width and height, respectively; d and h the transducer element-to-element spacing in the horizontal and vertical directions, respectively; γ = (πd(sin θs − sin θ)/λ); β = (πh sin φs − sin φ)/λ); δ = (πasin θ)/λ); ε = (πb sin φ/λ); θ and φ the azimuth and elevation angles, respectively; and the subscript s denotes the steered angles. It is seen that the problem is decoupled in the two directions and one simply has two 1D expressions. Hence, all of the development presented for the 1D case can be applied for the 2D case. 9.4.1

Broadband Excitation

We can account for broadband excitation in a manner similar to that employed previously. The pressure amplitude is now a function of the two angles rather than one: P(, )  ∑ Ti (, , f i ) gi ( f i )∆f

(9.27)

P(, )  ∑ Ti (, , i ) gi ( i )∆

(9.28)

i

or, equivalently, i

9.4.2

Finite Pulse Length

The correction for pulse length is more difficult for the 2D case compared to the 1D case. We will again focus our attention on the grating lobes and restrict consideration to the first and second order lobes. These lobes will occur along lines that are parallel to the transducer width and height directions. For a circular array and an unsteered beam, the lobes would describe a circular arc at some angle. Grating lobes occur where sin  g ( i )  sin  s

m i d

or sin g ( i )  sin s

CRC_DK8307_CH009.indd 392

m i h

11/14/2008 10:52:38 AM

Advances in Generation and Detection of Ultrasound

393

Because both angles are involved, lobes may be strongest at particular combinations of θ and φ (i.e., θg and φg), but they can have considerable amplitude if one of the angles is not at the grating lobe angle while the other is at the grating lobe angle. However, because the problem separates into the product of two distributions, we can treat each angle independently in corrections for the pulse length. The correction is the same as described earlier. The algorithm for calculating pressure at some pair of angles might then follow these lines: • For each wavelength, calculate the positions of the first and second grating lobes for each angle pair (eight angles altogether for each wavelength) and then save these. • Proceed with the summation, which is over wavelength (see Equation 9.27) and provides a result at a particular pair of angles θ and φ. When a pair of angles is chosen at which to exercise the calculation, check to see whether either or both of these angles is one that matches one (or more) of the grating lobe angles to within:    g ( i )

i Nd cos 

 g ( i )

i Mh cos

For those elements of the summation that match the criterion, reduce the calculated contribution by S/N or S/M, where S is the pulse length in wavelengths. For the second grating, the reductions are given by S/2N and S/2M. Now the reduction affects only the term containing the particular grating lobe angle (either θ or φ). Only in the particular case where the angles θ and φ match both grating angles will both of the grating terms of Equation 9.26 be reduced. Complete the summation and try another angle. We see that the algorithm for this correction is similar to the one for the 1D case. 9.4.3

Time Delays for Steering and Focusing

There is no readily extendable version of Equation 9.24 to handle the 2D case. Let F be the distance from the center of the array to the point of focus and let Fi,j be the distance from the (i, j)th element to the focal point. Then ∆i , j 

F  Fi , j c

or 1/ 2 2  2    yj   F xi  ∆i , j  1   u0  
 v0  
cos 2 0  
0 c F F      

(9.29)

where u0 = sin θ 0 cos φ 0, v0 = sin θ 0 sin φ 0, θ 0 and φ 0 are the steering angles, and τ0 is some constant that is added so that negative time delays never occur. It is desired to express Equation 9.28 in a form involving the indices of the elements. But a unified single form is not possible, and we have to take separate cases.

CRC_DK8307_CH009.indd 393

11/14/2008 10:52:38 AM

Ultrasonics: Data, Equations, and Their Practical Uses

394

Case 1. Both N and M are odd. Let the central element have index (0, 0) and all other elements have indices (i, j) ranging from (1, 1) to [(N − 1)/2, (M − 1)/2] on the positive side and from (−1, −1) to [−(N − 1)/2, −(M − 1)/2] on the negative side. Then xi = id for all i yj = jh

for all j

Case 2. Both N and M are even. There is no central element. Let the elements have indices ranging from (1, 1) to (N/2, M/2) on the positive side and from (−1, −1) to (−N/2, −M/2) on the negative side. Then 1  xi   i
 d for i  0  2 1  xi   i   d for i  0  2 Likewise 1  y i   j
 h for j  0  2 1  y i   j   h for j  0  2 Case 3. N is odd but M is even. There again is no central element. Let i have the range 0, ±1, ±2, … , ±(N − 1)/2 and j have the range ±1, ±2, … , ±M/2. Then xi  id for all i 1  y j   j
 h for j  0  2 1  y i   j   h for j  0  2 Case 4. N is even but M is odd. There is no central element. Let i have the range ±1, ±2, … , ±N/2 and j have the range 0, ±1, ±2, … , ±(M − 1)/2. Then y j  jh for all j 1  xi   i
 d for i  0  2 1  xi   i   d for i  0  2 These provide the time delays for the transmission of a steered focused beam. In receiving a signal from the region of focus, the same time delays are used for the elements.

CRC_DK8307_CH009.indd 394

11/14/2008 10:52:39 AM

Advances in Generation and Detection of Ultrasound 9.4.4

395

Angular Response of Elements

For realistic phased arrays that admit to some cross-coupling among the elements, evidence in the literature suggests that the CW expressions be modified by inclusion of a cosine term involving the angles. For Equation 9.26, both the term involving the elevation angle and the term involving the azimuthal angle should include a cosine term [23,24]. 9.4.5

Reciprocity

If reciprocity holds, then the point spread response will be given by V(θ, φ) = P2(θ, φ) where P(θ, φ) is given in Equation 9.27 or 9.28. 9.4.6

Summary for the Two-Dimensional Case

The 2D case is similar to the 1D case and contains the same problems with regard to steering and beam forming. Calculation software for describing the beam pressure profile can be expected to be somewhat more complicated, due to the extra dimension.

9.5 9.5.1

Phased Arrays in Industry Introduction

The use of phased arrays in ultrasonic medical diagnostics of soft tissue in the human body preceded their use in industry. The technology has recently been developed for highspeed inspections and specialized applications in industry. Generally, long linear arrays with many individually addressable elements are used in industrial applications, but curved and special shapes can be adapted to meet specific inspection requirements. In this text, the discussion will address the long linear arrays. Arrays are made up of many tiny individual elements. These individual elements are acoustically isolated, and an array may contain 32, 64, 128, 256, or 512 elements. The length of an array containing 512 elements can be no longer than 100 mm (4 in.) with a width of 12 mm (∼0.5 in.) and can be used for both beam steering and electronic sweeping across the length of the array. Shorter arrays are used primarily for beam steering, but the array can be mechanically scanned to cover a section providing both zero and angle of the material being inspected [22]. The advantage of the linear array is that it has the ability to operate in either zero or angle beam modes by changing the electronic triggering protocol. Electronic beam steering and focusing has many advantages over single-element or paintbrush-type transducers. As the elements in the array are individually electronically addressable, they can be triggered to produce a number of different longitudinal and shear beams (0, 15, 30, 45, 60, and 70°), thus eliminating the need for a number of conventional transducers needed to provide equivalent comparative comprehensive inspections [25,26]. Figure 9.22 describes the sound beam pressure pattern that would be generated by pulsing either a single element (Huygens principle) (position A) or multiple (e.g., four or more) elements. The directivity of the beam is dependent on the frequency and number of elements being electronically pulsed.

CRC_DK8307_CH009.indd 395

11/14/2008 10:52:39 AM

Ultrasonics: Data, Equations, and Their Practical Uses

396

Individual array elements

+

+



Position A

Position B

FIGURE 9.22 Beam patterns as a function of frequency and number of elements pulsed.

Position C

Position D

FIGURE 9.23 Electronic switching to achieve beam steering or focusing.

Beam pressure patterns (θ) are governed by the equations provided in Section 9.3. As seen in Figure 9.22, if a single element is pulsed, the beam pattern is nondirectional. However, if multiple elements are pulsed simultaneously, the square pattern of elements becomes directional, but is dependent on the frequency and number of elements that are selected. Using multiple elements, the beam generated can duplicate that generated from a conventional single zero or angle beam element transducer. Although each individual element produces a beam pattern, as seen in Figure 9.22 (position A), the phase of the beam emanating from the individual element controls the directivity of the beam pattern. As illustrated in Figure 9.22, a typical 16-element array can be electronically excited to sweep the length of the array, providing a high-speed inspection swath covering the length of the array. As previously indicated, each of the elements of the array is able to be individually electronically excited (addressed) and thus elements can be addressed to provide a selected angle-beam pattern, or to focus the sound beam, or both. In Figure 9.23, a delay line is used to electronically control the sequence of the excitation pulse signal to the individual elements in the array. The inverse switching sequence is used in the reception mode. The general equation for the phase-steered angle beam is provided in the previous section of this chapter. Following Huygens’ principle, the phase from each element in the array generates a pattern similar to the one shown in Figure 9.22—position A. However, when multiple elements are pulsed, the integration of the phase from different elements

CRC_DK8307_CH009.indd 396

11/14/2008 10:52:39 AM

Advances in Generation and Detection of Ultrasound

397

forms a beam pattern. If the selected elements are pulsed simultaneously, they form a zero beam. If they are pulsed with time-delayed excitation, the time-delayed pulse sequence controls the directivity of the beam. Elements selected can form zero or angle beams that can be swept along the length of the array to provide a paintbrush-like swath for highspeed inspection. In other applications, a multiple element array is positioned at one position and the angle may be changed electronically to provide inspection data for a more detailed analysis. It is not necessary to use a long array. 9.5.2

Potential Applications

For the ultrasonic inspection of materials, the applications for phased arrays are as numerous as they are for conventional single or multiple individual transducers. Some specific applications include: • Real-time, online inspection of flat plates as well as round and square bars using multiple beams for full coverage • Online inspection of railroad wheel rims • Inspection of axles on railroad cars • Nuclear reactor pressure vessel emergency cooling nozzles [27] • In-service inspection of steam turbines • Girth weld inspection of pipelines 9.5.3

Advantages

The control and flexibility of the beam patterns that can be developed with a phased array provides one of its great advantages. Zero, multiple-angle, fixed, and dynamically focused beams can be scanned the length or portions of the array. Other advantages include: • Reduction or elimination of mechanical moving parts • Computer-based calculation of geometric data by CAD or other packages and creation of lookup tables for controlling the beam steering during inspections • Versatility in selecting scanning modes • Time and cost reduction • Software capture of procedure and inspection protocol • Full-coverage inspection with dynamic, high-speed zero, angle and focusing capability at the pulse repetition rate (PRF) of the electronic system • A-scan, B-scan, and C-scan coverage options [28–31]

9.6 9.6.1

Laser Ultrasound Introduction

The technology associated with the use of lasers to generate ultrasound was researched in the early 1980s. Since that time, it has matured and found many applications in NDT and NDE of metals and composite materials, using a variety of techniques in both pulse-echo

CRC_DK8307_CH009.indd 397

11/14/2008 10:52:40 AM

398

Ultrasonics: Data, Equations, and Their Practical Uses

and through-transmission modes. Ultrasonic wave modes that can be generated with lasers include longitudinal, shear, and surface or Rayleigh waves. From a fundamental point of view, the wave propagation mode principles are the same as those generated with the conventional ultrasonic transducers but the generation and reception procedures are different. There are many excellent articles written on this subject* [32]. This text is intended to provide only a brief description of the generation and detection of laser ultrasound. There are advantages in the use of lasers over the more conventional techniques used for NDE of materials, including: • Laser-based ultrasound is a broadband, noncontact method that requires no couplant. • The generation and reception laser is at a distance from the test object. • The technology provides an effective and highly reproducible means for nondestructively inspecting solids for conditions such as flaws, laminations, cracks, and wall thickness changes. • The technology is effective for evaluating material conditions and material properties (grain structure, texture, hardness, porosity). • It can provide for rapid area scanning and imaging. • Frequencies from kilohertz to gigahertz can be generated. • It can be used to remotely investigate complex geometries that are difficult to reach with conventional transducers. • Being noncontact, laser ultrasound can be used to inspect fragile, thin film and delicate samples. • It can be used for inspecting irregular shapes that would be difficult to inspect with conventional techniques. • It can be used for examination of samples at extreme temperatures such as 1800°C, as well as other hostile environments. Applications for laser ultrasound include: • • • • • • •

Detection of anomalies (inclusions, flaws, cracks) in materials Determining the wall thicknesses (e.g., metals and glass) Determining elastic constants at temperatures to 1800°C Inspections involving noncontact, long standoff distance requirements Noncontact grain and texture measurements in thin sheet metals Real-time inspection of railroad rail Inspection of aircraft airframe composites without surface preparation and without detachment of parts during maintenance • Measurement of wall thickness of steel tubing at speeds of 12 ft/min • Detection of lap seam corrosion in aluminum airframe structures The main disadvantage for laser ultrasound is the cost associated with operator education and capital equipment.

* The website http://www.ndt.net/cgi-bin/htsearch contains numerous reference articles.

CRC_DK8307_CH009.indd 398

11/14/2008 10:52:40 AM

Advances in Generation and Detection of Ultrasound 9.6.2

399

Laser Generation

Ultrasonic laser technology is generally described in terms of three principal regimes, namely, thermoelastic, plasma/ablative, and constrained surface. 9.6.2.1

Thermoelastic Regime

When ultrasound is generated in the thermoelastic regime, power densities are typically >107 W cm2. In this generation regime, the amplitudes of the compression, shear, and surface (Rayleigh) waves increase linearly with increasing power density. Generation of ultrasound in the thermoelastic regime occurs through localized thermal expansion due to local absorption of a laser pulse [33]. The nature of the ultrasonic wave that forms depends upon the distribution and size of the laser beam, the thermal properties of the material, and whether the surface of the material is constrained. Marc Dubois et al. developed a 1D mathematical model that can be used to describe development of ultrasonic waves by thermal expansion [34]. The model assumes that local temperature elevation is due only to optical absorption. The model is presented in the following description. Assume that an infinite plate of a given thickness is made of an orthotropic material. The x-axis is oriented in the direction of thickness and is assumed to be one of the principal axes of the material. A laser pulse strikes the top of the plate at time equal t = 0. The mechanical displacement field for the thickness (x direction) can be described mathematically in the following equation:  2u( x , t) 1  2u( x , t) T ( x , t)   2 2 v t x x

(9.30)

where v is the longitudinal velocity in the x direction, χ an apparent thermal expansion coefficient taking into account ( χ = 3λ + 2µα/(λ + 2µ) in the particular case of isotropic material, λ and µ Lame coefficients and α its linear thermal expansion coefficient), and T(x, t) the temperature elevation field imposed by the penetration of the light inside the material. This field is defined in the following way: T ( x , t) 

I 0 C p

ex H (t)

(9.31)

where β is the optical absorption coefficient, I0 the energy per surface unit absorbed by the sample, ρ the density, Cp the specific heat, and H(t) the integral over time from 0 to t of the normalized temporal shape f(t) of the laser pulse. When the model is compared with more elaborate 3D formulations, the results and agreement are very good. 9.6.2.2

Plasma Regime [35]

At higher power densities, thermoelastic generation is supplemented by the ablation of electrons and ions, forming a plasma. As the plasma expands away from the surface of the sample, a momentum pulse is transmitted into the solid, thus producing the compression wave pulse. This method of producing ultrasound also produces a small ablation pit, ∼5 µm deep. This damage may make laser-generated ultrasound in the plasma regime impractical for some applications. The source of ultrasound in the plasma regime has a stress distribution that may be simply modeled as the transient response of an elastic half-space due to the application of

CRC_DK8307_CH009.indd 399

11/14/2008 10:52:40 AM

Ultrasonics: Data, Equations, and Their Practical Uses

400

a normal point force at the surface with arbitrary time dependence. Its time dependence consists of the recoil force of evaporating surface material and a longer time scale decreasing force, due to the pressure exerted over an increasing circular area by the expanding plasma. When ultrasound is generated in the constrained surface source regime, which occurs for power densities in both the thermoelastic and plasma regimes, the amplitudes of the compression, shear and surface (Rayleigh) waves are enhanced: It 

(1  R)2 Ii (1  R)2
4R sin 2 ()

(9.32)

where It is the intensity of the transmitted light, R is the reflectivity of the mirrors (if any), and φ = (2πnL / λi) cosθ is the phase difference due to the mirror separation, in which n is the index of refraction of the cavity, λ I the incident optical wavelength, and θ the angle of incidence of the light on the mirror and the cavity length or mirror separation. Pulsed lasers generate broadband signals. Their exact frequency response is dependent on the rise time of the laser pulse, the bandwidth of the system, and the property of the material being examined. The rise time of the laser pulse can range from nanoseconds to picoseconds and can generate ultrasonic signals from a few kilohertz up to several gigahertz to establish the center frequency of operation. Figure 9.24 shows the components of a typical pulsed laser system, including the generation and detection lasers, Fabry–Perot interferometer, optical components, and detector. This generation of the ultrasonic waves is typically by a pulsed YAG (neodymium-doped yttrium aluminium garnet) laser. Depending on the application, the pulse duration, wavelength (frequency) and energy levels will vary. The sketch shows a laser system such as might be used for pulse-echo detection of flaws or specimen thicknesses, stress or specimen anomalies. The typical beam diameters from a YAG laser system for the application shown in the figure might be in the order of 30 nm. The pulsed-generation laser beam is reflected from a polarized beam-splitter mirror and strikes the surface of the material. This configuration generates a zero-beam ultrasonic wave that travels into the material and reflects signals from flaws, material thicknesses, or material anomalies. The beam from the detection, or reference, laser reflects from the second beam-splitter mirror, through the first beam splitter onto the surface of the material and provides the basis for comparing phase differences associated with any interferometric surface displacement changes that may be present. As the wave propagates into the material, echoes from an inclusion, flaw, and the back surface are reflected to the surface.

Surface displacement

Reference laser

Laser

Confocal Fabry−Perot interferometer

Flaw

Detector Optical lens

Beam splitter mirrors

FIGURE 9.24 Schematic of components of laser generation and reception.

CRC_DK8307_CH009.indd 400

11/14/2008 10:52:41 AM

Advances in Generation and Detection of Ultrasound

401

The microscopic (a few tenths of a nanometer) mechanical displacement signals at the surface create an interferometric pattern that can be detected through the interferometer. The Fabry–Perot interferometer provides the means for analyzing these surface phase changes and produces an output for the detector. The detector converts the signals into an echo pattern typical of conventional ultrasonic signal patterns. The signals can then be displayed both on an oscilloscope and digitized for further use. In conjunction with a mechanical scanning bridge, the signals can be converted into images of the anomalies. For example, to generate a pulse with a center frequency of 5 MHz would typically involve the use of a Q-switched pulsed YAG laser with pulse duration of 20 ns, a wavelength of 1064 nm, and an energy level of ∼200 mJ. Figure 9.24 also shows a technique for a point source generation of an ultrasonic wave. For angle beam inspections or evaluation of material properties, a shear (or longitudinal) wave is generated. Such a technique is shown in Figure 9.25, where a laser beam reflects from a mirror and progresses through optical lenses. The cylindrical and spherical lenses produce a beam that, when passing through the diffraction grating, establishes a phase relationship at the surface that generates the angle-beam in the material. The angle of propagation in the material is controlled by the phase of the optical beam incident at the surface. The angle beam can be used for weld examination or flaw detection. The laser optics for the generation of an ultrasonic wave need not be initiated with the laser beam impinging directly on the surface. A fiber optic cable may be more appropriate, such as for the examination of complex geometries or for ID inspections. The fiber optic may be several feet in length and still perform an effective inspection. Figure 9.26 shows a sketch of the concept for the use of fiber optic cable in laser generation of the ultrasonic beam. In Rayleigh wave examination, the laser and detection optics (or alternate transducer configuration) are physically separated (as shown in Figure 9.26) providing a means for evaluating other parameters, such as materials properties [36,37]. 9.6.2.3

Laser Detection

The most commonly used laser detectors fall into two categories. One category is interferometric, in which Fabry–Perot, Michelson, time delay, vibrometers, and the like are used.

Laser Cylindrical lens

Spherical lens

Diffraction grating

Detector

FIGURE 9.25 Sketch of angle beam or surface wave generation with lasers.

CRC_DK8307_CH009.indd 401

11/14/2008 10:52:41 AM

Ultrasonics: Data, Equations, and Their Practical Uses

402 Laser

Circular lens Optical fiber

Cylindrical lens Ultrasonic wave

Detector

Test object FIGURE 9.26 Long fiber optic being used to generate the ultrasonic beam.

Laser

T e s t o b j e c t

Surface displacement

Optical interferometer

Lens

Polarized beam splitter

FIGURE 9.27 Components of an interferometer laser-optic detection system.

A second category of laser detection is amplitude-variation detection, in which detectors such as optical knife-edge technology are employed. Not all detectors are laser-based. EMAT technology can be used for noncontact examinations, and conventional contact piezoelectric elements may be used where contact can be made with the part. Selection of the detection technique is dependent on the application. 9.6.2.3.1 Interferometric Surface displacement patterns from anomalies result in angstrom changes that define the condition. Figure 9.27 shows the elements required for interferometric laser detection. In this simplified sketch, the beam from the laser is reflected from a mirror through the polarized beam splitter. A portion of the beam is directed to the part under examination and a portion is used as a reference beam to the interferometer. When the laser beam strikes, it generates a wave that reflects from the surface and transmits a wave into the specimen. A lens captures the microscopic optical patterns that occur at the surface. When processed with the reference beam from the beam splitter, the phase interference pattern can be demodulated into a conventional time domain ultrasonic waveform or can be produced into an image. Consider the parameters associated with an examination of an aluminum plate, with the laser excitation of 1064 nm wavelength radiation and a power level of ∼3 J/cm2. In this example, this method would simultaneously generate broadband surface and longitudinal,

CRC_DK8307_CH009.indd 402

11/14/2008 10:52:41 AM

Advances in Generation and Detection of Ultrasound Laser

403

Circular lens Optical fiber

Cylindrical lens

Oscilloscope

Amplifier EMAT receiver

Ultrasonic wave FIGURE 9.28 Laser fiber-optic generation with an electromagnetic acoustic transducer receiver.

Laser

Circular lens Optical fiber Oscilloscope

Amplifier Cylindrical lens Piezoelectric receiver

Ultrasonic wave FIGURE 9.29 Laser fiber-optical generation and piezoelectric receiver.

Rayleigh, and shear waves in the material. This inspection would be within the thermoelastic regime. 9.6.2.3.2 Amplitude-Variation Detection In Figure 9.28, the noncontact EMAT sensor mounted 0.02 in. (0.5 mm) above the surface would detect the in-plane ultrasonic waves. As the ultrasonic waves pass through the magnetic field of the EMAT, the incident waves caused by the surface movement induce eddy currents that are converted to ultrasonic signals. The orientation of the magnetic field and the geometry of the EMAT are critical to the modes of ultrasonic waves that can be detected (see Section 9.2). There are other techniques for detecting the surface displacements of the ultrasonic wave modes. In Figure 9.29, a conventional piezoelectric receiver is used. However, this is a contact technique and couplant is required. The wave modes from Rayleigh or angle-beam waves are detected, amplified, and displayed as with other techniques. Capacitive transducers have also been used to detect the ultrasonic surface displacement signals.

CRC_DK8307_CH009.indd 403

11/14/2008 10:52:42 AM

404

Ultrasonics: Data, Equations, and Their Practical Uses

References 1. J Benedito, JA Carcel, R Gonzalez, and A Mulet, 2002. Application of low intensity ultrasonics to cheese manufacturing processes, Ultrasonics, 40, 19–23. 2. S Kenderian, B Boro Djordjevic, and RE Green Jr., 2002. Laser-based and air-coupled ultrasound as noncontact and remote techniques for testing of railroad tracks, Mater. Eval., 60(1), 65–70. 3. R/D Tech Inc. 4. GE Sensing & Inspection Technologies. 5. QMI Inc. 6. Pacific Northwest National Laboratory and Force Technology. 7. US Ultratek Inc. 8. Center for NDE at Iowa State University. 9. NDT Systems, Inc. 10. M Hirao and H Ogi, 2003. EMATs for Science and Industry—Non-contacting Ultrasonic Measurement, Kluwer Academic Publishers, Boston, MA, ISBN 1-4020-7494-8. 11. GA Alers, G Huebschen, BW Maxfield, W Salzburger, RB Thompson, and A Wildbrand, 2000. Nondestructive Testing Handbook, Second Edition, Vol. 7, Ultrasonic Testing (ASNT), 326–340 (2000, 13). 12. RB Thompson, 1990. Physical principles of measurements with EMAT transducers, Chapter 3 in Ultrasonic Measurement Methods (RN Thurston and AD Pierce, eds), Academic Press Inc., Boston. 13. S Hanada and T Sugiura, 2003. Flaw identification by angle beam electromagnetic acoustic transducers, Department of Mechanical Engineering, Keio University, Yokohama, Japan. 14. AS Birks, RE Green Jr., P McIntire, eds, 1991. Nondestructive Testing Handbook Second Edition, Vol. 7, Ultrasonic Testing, American Society for Nondestructive Testing, Columbus, OH, ISBN 0-931403-049. 15. ASTM E1774, Standard Guide for Electromagnetic Acoustic Transducers (EMATs). 16. D MacLauchlan, S Clark, B Cox, T Doyle, B Grimmett, J Hancock, K Hour, and C Rutherford, Recent advancements in the application of EMATs to NDE, Proceedings of 16th WCNDT 2004 – World Conference on NDT, Aug 30–Sep 3, Montreal, Canada, 2004. 17. PR Mould and TE Johnson, 1973. Sheet Metal Ind., 50, 328–348. 18. S Ahmed and TT Taylor. Using scattering theory to determine orientation distribution coefficients of polycrystals with preferred grain orientations, Ultrasonics. Proceedings Indian Society for Non-Destructive Testing, Kolkata Chapter, National Seminar on NDT (NDE-2005), Kolkata, India, December 2–4, 2005. 19. RB Thompson, JF Smith, SS Lee, and GC Johnson, 1989. Met. Trans., 20A, 2431–2447. 20. G Davies, DJ Goodwill, and J Kallend, 1972. Met. Trans., 3, 1627–1631. 21. CA Stickles and PR Mould, 1970. Met. Trans., 1, 1303–1312. 22. PA Meyer and JW Anderson, Ultrasonic Testing Using Phased Arrays, Krautkramer, Branson, Lewiston, PA (company literature.) 23. S-C Wooh and Y Shi, 1998, Influence of phased array element size on beam steering behavior, Ultrasonics, 36, 737–749. 24. S-C Wooh and Y Shi, 1999, Optimum beam steering of linear phased arrays, Ultrasonics, 29, 245–265. 25. DK Lemon and GJ Posakony, 1980. Linear array technology in NDE applications, Mater. Eval., Research Supplement, 38, 34–37. 26. JM Smith, 1978. Practical linear array imaging for nondestructive testing applications, Proc. Eighth Int. Symp. on Acoustic Imaging, May 29–June 2, Key Biscayne, FL. 27. GJ Posakony, 1978. Acoustic imaging—A review of current techniques for utilizing ultrasonic linear arrays for producing images of flaws in solids, Elastic Waves and Nondestructive Testing of Materials, ASME, New York.

CRC_DK8307_CH009.indd 404

11/14/2008 10:52:42 AM

Advances in Generation and Detection of Ultrasound

405

28. FL Becker, VL Crow, TJ Davis, SR Doctor, BP Hildebrand, DK Lemon, and GJ Posakony, 1979. Development of an ultrasonic imaging system for the inspection of nuclear reactor pressure vessels, NP-1229, Electric Power Research Institute, Palo Alto, CA. 29. H. M. Wilcox, 1975. Ultrasonic cross-sectional imaging system, U.S. Patent No. 3,881,466, U.S. Patent Office, Washington, D.C. 30. S Mahaut, S Chatillon, R Raillon-Picot, and P Calmon, 2004. Simulation and application of dynamic inspection modes using ultrasonic phased arrays, Review of Quantitative Nondestructive Evaluation, Vol. 23 (DO Thompson and DE Chimenti, eds), American Institute of Physics, New York, London. 31. H Wustenberg, A Erhard, and G Schenk, 1999. Some characteristic parameters of ultrasonic array probes and equipment, NDT.net, 4(4), BAM, Berlin. 32. G Birnbaum and GS White, 1984. Laser techniques in NDE, Research Techniques in Nondestructive Testing, Vol. 7 (RS Sharpe, ed.), Academic Press, New York, 259–265. 33. RM White. Generation of elastic waves by transient surface heating, J. Appl. Phys., 34, 35–59. 34. DA Hutchins, 1988. Ultrasonic generation by pulsed lasers, Physical Acoustics, Vol. 18 (WP Mason and RN Thurston, eds), Academic Press, New York, 21–123. 35. RJ Dewhurst and Q Shan, 1999. Optical remote measurement of ultrasound, Meas. Sci. Technol., 10(11), 129–168. 36. AS Murfin, RA Soden, D Hetrick and RJ Dewhurst, 2000. Laser-ultrasound detection systems: A comparative study with Rayleigh waves, Meas. Sci. Technol., 11, 1208–1219. 37. CB Scruby, 1993. Evaluation of Materials and Structures by Quantitative Ultrasound (JD Achenbach, ed.), Springer, New York, 223–237.

CRC_DK8307_CH009.indd 405

11/14/2008 10:52:42 AM

CRC_DK8307_CH009.indd 406

11/14/2008 10:52:42 AM

10 Medical Ultrasound: Therapeutic and Diagnostic Imaging Foster B. Stulen

CONTENTS 10.1 Introduction.......................................................................................................................408 10.1.1 Contact Ultrasound Applications .....................................................................409 10.1.2 Wound Healing ...................................................................................................409 10.1.3 Lithotripsy............................................................................................................409 10.1.4 Liposuction ..........................................................................................................409 10.1.5 High-Intensity Focused Ultrasound ................................................................409 10.1.6 Diagnostic Ultrasound ....................................................................................... 410 10.2 Therapeutic Ultrasound .................................................................................................. 410 10.2.1 Contact Ultrasonic Surgical Systems ............................................................... 410 10.2.2 Phacoemulsification ........................................................................................... 414 10.2.3 Cavitating Ultrasonic Surgical Aspiration ..................................................... 415 10.2.4 Ultrasonic-Assisted Liposuction ...................................................................... 417 10.2.5 Lithotripsy ........................................................................................................... 417 10.2.6 High-Intensity Focused Ultrasound ................................................................ 419 10.2.6.1 General Principles of High-Intensity Focused Ultrasound ......... 419 10.2.6.2 High-Intensity Focused Ultrasound Cancer Treatment .............. 420 10.2.6.3 High-Intensity Focused Ultrasound Atrial Fibrillation Treatment ............................................................................................ 421 10.2.6.4 High-Intensity Focused Ultrasound Wrinkle Reduction.............423 10.2.6.5 High-Intensity Focused Ultrasound Fat Removal ........................423 10.2.6.6 High-Intensity Focused Ultrasound Acute Puncture and Wound Closure........................................................................... 424 10.2.7 Ultrasonic Healing ............................................................................................. 424 10.2.7.1 Ultrasonic Bone Healing ................................................................... 424 10.2.7.2 Ultrasonic Wound Healing ...............................................................425 10.2.7.3 Ultrasonic Wound Debridement ...................................................... 426 10.2.8 Sonophoresis ....................................................................................................... 427 10.2.9 In Vivo Ultrasonic Welding................................................................................ 427 10.2.10 Ultrasonic Thrombolysis.................................................................................... 429 10.3 Diagnostic Ultrasound Imaging ....................................................................................430 10.3.1 Introduction .........................................................................................................430 10.3.2 Ultrasonic Properties of Tissue......................................................................... 431 10.3.3 Modes ................................................................................................................... 432 10.3.4 Doppler Imaging................................................................................................. 432 407

CRC_DK8307_CH010.indd 407

10/27/2008 11:36:26 AM

Ultrasonics: Data, Equations, and Their Practical Uses

408

10.3.5 Harmonic Imaging .............................................................................................434 10.3.6 Transducers, Beams, and Arrays ...................................................................... 436 10.3.7 Ultrasonic Exposimetry ..................................................................................... 439 10.3.8 Advancement in Diagnostic Imaging ..............................................................440 10.3.8.1 Contrast Agents .................................................................................. 441 10.3.8.2 Sonoelastography ............................................................................... 441 10.3.8.3 Intravascular Ultrasound .................................................................442 10.4 Summary ...........................................................................................................................443 References ....................................................................................................................................444

10.1

Introduction

Ultrasound is applied in virtually all areas of medicine, due to the number of desirable physical effects produced by ultrasound, the flexibility in the design of transducers, and the delivery of either high or low power at high or low frequencies. The applications of ultrasound are shown by frequency in Figure 10.1. The spectrum is divided into two categories: therapeutic and diagnostic. Therapeutic describes applications that make a physical change to a patient to improve the patient’s health. Therapeutic ultrasound can be based on direct contact with the tissue or with ultrasonic energy radiated to the targeted tissue. Diagnostic ultrasound is used to display an image of the internal structures of the body for diagnosis. The term ultrasound is used here in the broadest sense, meaning the generation, propagation and delivery of ultrasound. The specific term ultrasonics is used to distinguish

Medical ultrasound

Therapeutic

Diagnostic

Surgery • Soft tissue and vessel coagulation • Soft tissue fragmentation and aspiration • Liposuction

Transdermal drug delivery

Orthopedics and arthroscopy • Bone cutting • Bone cement softening • Suture welding

Opthalmology Phacoemuslification Thrombolysis

20 MHz higher

1 MHz

20 kHz

High-intensity focused ultrasound • A-Fib treatment • Liver cancer • Prostate cancer • Cosmetic surgery • Vessel sealing Wound healing • Recalcitrant wounds • Nonunion fracture • Wound debridement

Lithotripsy • Extracorporeal shockwave lithotripsy • Percutaneous

Medical imaging • General imaging • M-mode • Intravascular • Guidance • Ablation • Nonlinear mechanisms

Enhanced imaging and drug/gene delivery with microspheres

Sonoelastography

FIGURE 10.1 Medical device spectrum.

CRC_DK8307_CH010.indd 408

10/27/2008 11:36:28 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

409

applications in which energy is applied to a material to produce a desired change of state. Clearly the noun ultrasonics is derived from the adjective ultrasonic, which applies to energy, as in the phrase ultrasonic energy. Ultrasound is more specifically employed in applications in which a structure is being interrogated or imaged to obtain information about its state. Although the levels in diagnostic ultrasound are much lower compared with ultrasonic energy applications, energy is still being delivered. This difference between the terms ultrasonics and ultrasound is mirrored in the terms therapeutic and diagnostic. Although this distinction between therapeutic and diagnostic ultrasound was well defined in the past, this is no longer the case. High-intensity focused ultrasound, commonly referred to as HIFU, is a combination of therapeutic and diagnostic ultrasound, typically operating in the range of 1–10 MHz. Sonoelastography is also under development as a diagnostic modality that uses both low mechanical vibrations <10 kHz and highfrequency imaging. 10.1.1

Contact Ultrasound Applications

These applications rely on the direct interaction of a vibrating instrument with the target tissue. The word contact is used to differentiate applications that rely on the propagation of ultrasonic energy from a transducer to the target tissue. Applications include: ultrasonic surgical systems, phacoemulsifiers, cavitating ultrasonic surgical aspirations (CUSAs)®, liposuction, in vivo welding of sutures, transdermal drug delivery, and bone cutting. 10.1.2

Wound Healing

Ultrasound is used to promote the healing of tissues after the normal healing process has failed to close the wound. Radiating ultrasound in the low megahertz frequencies is used for bone healing, and ultrasound in the tens of kilohertz is used to generate a mist of saline to heal open wounds. Direct contact instruments are used for wound debridement, which is the clearing of necrosed tissue in a wound. 10.1.3

Lithotripsy

Extracorporeal shockwave lithotripsy (ESWL) was the first medical application of focused intense energy. A shockwave pulse is generated externally and focused on the kidney stones to fragment them. Percutaneous ultrasonic lithotripsy is a needle-like device that contacts the stone directly. 10.1.4

Liposuction

Adipose tissue, otherwise known as fat, is removed by liposuction. The instruments used in liposuction are large hollow tubes that simply suck out the highly friable fat cells located between the skin and the muscular walls of the abdomen. Smaller versions are used in cosmetic procedures and plastic surgery. Ultrasonic-assisted liposuction (UAL) imparts ultrasonic energy to the tube. So as the suction tube is advanced in the fatty layers, the cells are disrupted and tissue is removed more easily with potentially better hemostasis. 10.1.5

High-Intensity Focused Ultrasound

HIFU is based on propagating and focusing ultrasound to a lesion. At the focus, the intense field heats the tissue by its absorption of the ultrasonic energy. This is the therapeutic

CRC_DK8307_CH010.indd 409

10/27/2008 11:36:28 AM

Ultrasonics: Data, Equations, and Their Practical Uses

410

action. Diagnostic ultrasonic imaging is an integral part of many HIFU systems. The diagnostic ultrasound is used both to target the treatment site and track the volume of ablation. Other imaging modalities, such as Magnetic Resonance Imaging (MRI), are also used. 10.1.6

Diagnostic Ultrasound

Ultrasound is commonly used to image underlying tissues to make diagnoses. The ultrasound is of relatively high frequency, typically in the range of 2–20 MHz. The high frequencies allow the ultrasound to propagate in a relatively narrow beam. The focus of the ultrasonic beam is swept through the volume of interest with electronic beam-forming algorithms. The higher the frequency, the better the spatial resolution. However the higher the frequency, the higher the attenuation, and depth of an ultrasonic scan is limited. Recently microspheres have been introduced into the blood in conjunction with diagnostic imaging. They act as contrast agents to enhance images or as vesicles to deliver contained drugs when the imaging beam disrupts them. This latter application represents another example of the blurring between therapeutic and diagnostic ultrasound. This chapter presents many of these applications and discusses the underlying ultrasonic mechanisms on which they rely. Actual devices are shown in figures as representative examples only. Each device has any number of competing devices and technologies, and the examples given herein are not intended as endorsements of the product or its manufacturer. Also, the use of such images in any book will always date the work. Some products may fade from the market, and undoubtedly new products and their applications will be introduced.

10.2 10.2.1

Therapeutic Ultrasound Contact Ultrasonic Surgical Systems

The first attempts at an ultrasonic surgical system literally added ultrasonic energy to a cold steel scalpel. In 1955, Balamuth and Kuris [1] filed a patent on just such an implementation, as shown in Figure 10.2 [2], which was taken from the patent. Given the sharpness of a scalpel blade, the ultrasonic energy significantly increased the speed of cutting. With the speed, the blade was in contact with any tissue for only a brief period of time. Therefore little power—that is, heat—was delivered into any volume of tissue. Hemostasis was poor, because temperature must be elevated to coagulate blood. The additional speed and ease of cut would naturally be perceived as advantages; however, they limited the tactile feedback that a surgeon relies on. The lack of both tactile feedback and hemostasis are likely reasons this approach was not adopted. With the vantage point of several decades of advancement, speed of cut and hemostasis are clearly a tradeoff in blade design that depends on the application. The benefits of ultrasonics in surgery were proven in the early 1990s, when a new approach was developed [3]. These ultrasonic surgery systems have proved revolutionary in general surgery and more so in laparoscopy. The blades used today look nothing like a scalpel at all, but are relatively thin dull rods that vibrate in a longitudinal mode. The size of the blade, roughness of the surface, and sharpness of edges are designed to produce the cut speed and hemostasis for the targeted tissue. Different surfaces with specific features on the same blade may be presented to the tissue to achieve different effects, such as greater

CRC_DK8307_CH010.indd 410

10/27/2008 11:36:28 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

26 4 28 24 36 22

34

2b

6

10

2

2c 12 14 16 2a

18

20 40

38 42

12a

8

32

28

411

40 62 56

58 64

60

54

68

66

74 70

76

72

FIGURE 10.2 A literal ultrasonic scalpel. (From US Patent No. 3,086,288. With permission.)

cutting speed or greater hemostasis. For example, a sharp side drawn through tissue cuts fast, a blunt tip creates hemostatic otomies (holes through tissue layers), and a board surface provides coagulation without transaction. The systems consist of a generator, handpiece, and instrument. The generator converts electrical line power to electrical energy at the ultrasonic system frequency. The generator is connected to the handpiece that contains the ultrasonic transducer. The transducer uses a stack of piezoelectric elements to convert the electrical energy to mechanical energy. The proximal end of the transducer is free and is within the handpiece housing. The distal end of the transducer extends through the handpiece. Here, any one of a family of instruments is connected. The ultrasonic energy propagates down a waveguide within the instrument to its blade tip. Most systems are designed to generate primarily longitudinal vibrations. There are also systems that use transverse or torsional motions or combinations of all three types of motion. The instrument consists of a waveguide and a blade portion. The waveguide and blade designs can modify the input motion so that the distal tip vibrates in a complex combination of motions to achieve the desired vibrational displacements. One of the more widely accepted systems is the Harmonic™, from Ethicon Endo-Surgery, which is shown in Figure 10.3 [4]. Its family of instruments consists of two basic types: blades and shears. The blades are instruments that have a continuous metal waveguide, which is functionally divided into extender and blade sections. The instrument is sheathed up to the proximal end of the blade; the tip extends past the sheath to expose the active surface of the instrument. The sheath protects the surgeon and patient from the vibrating extender. Tissue is forced against the blade by tissue tension applied by the surgeon. In contrast, the shear instruments include a clamp mechanism to force tissue between the vibrating blade tip and the passive clamp arm. The clamp arm has a polymeric pad, which is in contact with the tissue. When transection occurs and the clamp arm and the blade are in contact, the pad prevents metal-to-metal contact, which could otherwise potentially fracture the tip.

CRC_DK8307_CH010.indd 411

10/27/2008 11:36:28 AM

412

Ultrasonics: Data, Equations, and Their Practical Uses

FIGURE 10.3 Harmonic™ product family. (Courtesy of Ethicon Endo-Surgery.)

The mechanical system is the transducer and instrument together. It operates at a nominal frequency and is a very high-Q system. The power drawn in air, referred to as quiescent power, should be as low as possible, for three reasons. First, the power loss heats the handpiece housing and raises its temperature. Second, losses in the blade/instrument lead to the blade self-heating in air under no load. Finally, the amount of quiescent power reduces the available power that can be delivered to the tissue. Power losses include: the piezoelectric mechanical and dielectric losses, interface losses, losses at connecting points to the handpiece housing and instrument, seals and other mounting features, and internal losses in the metallic components. By placing seals and connection points at displacement nodes, these losses are minimal. Proper preparation and handling of the piezoelectric elements and the electrode shims help to minimize these losses. High-Q materials such as titanium and aluminum are used in the blade and handpiece, so their losses are extremely low. Most of the quiescent power draw is in the piezoelectric stack. A poor stack design and lack of attention to details in assembling can increase the losses by a factor of two or more. Delivered power is what drives coagulation and transection. For clamped coagulation, the delivered power, PD, can be shown to behave as a simple frictional process: PD  vN

(10.1)

where v is the rms velocity, in meters per second; N the normal clamp force, in newtons; and µ the proportionality coefficient. The velocity, v, is the rms value of the excursion along the blade that contacts the tissue. It equals the rms value of displacement times the angular frequency, ω. N is the normal force pressing the tissue against the blade. The coefficient, µ, is normally referred to as the coefficient of friction. In this surgical application, it is referred to as the coefficient of coagulation.

CRC_DK8307_CH010.indd 412

10/27/2008 11:36:29 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

413

Interestingly, the maximum velocity depends on the stress limitation in the material and does not depend on frequency. One can show that 

v E c

(10.2a)

c max E

(10.2b)

and therefore vmax 

where E is the elasticity, in newtons per meter square; c the speed of sound, in meters per second; σ the stress, in newtons per meter square. The maximum stress, σmax, is the endurance limit of the material. One should note that this value is generally lower when driven at ultrasonic frequencies as compared with lower frequency mechanical vibrations. Equation 10.1 can be interpreted at a high level. The delivered power is a function of the tissue, µ; the Harmonic instrument, v; and the surgeon, N. The ultrasonic system controls velocity, the surgeon and instrument design control the normal force, and tissue is represented by the coefficient of coagulation. When ultrasound is applied to the tissue, the blade and tissue heat up to coagulate and seal. As the tissue temperature rises, collagen is first denatured ∼65°C, and then the water vaporizes at 100°C. As pressure and power continue to be applied, the temperature rises >100°C, and the desiccated friable tissue separates. In medical terms, the tissue is transected. As the blade tip heats up, its elasticity lowers, which causes the resonant frequency to decrease. Therefore, the ultrasonic generator is a tracking generator to maintain the system in resonance. A lock range is used to ensure that only the proper longitudinal mode is excited, because modal densities can be relatively high, especially with longer instruments. Many neophytes in ultrasound assume that the decrease is due to thermal expansion. The longer length does create a lower frequency, but this accounts for only a very small fraction of the typical decrease. Laparoscopy, otherwise known as keyhole surgery, involves inflating the abdomen with CO2 to raise the abdominal wall off the viscera. This creates a working space for the surgery. Instruments are introduced through trocars, which are small-diameter tubes that penetrate the abdominal wall. The trocars have features to seal around the instrument to maintain CO2 pressure. Greater skill, experience, and care are required by the laparoscopist, but the patients’ benefits are significant: less pain, less blood loss, quicker return to normal activity, and better cosmesis. Now consider the functionality that an ultrasonic shears provides the laparoscopic surgeon. First, without the ultrasonic energy applied, the instrument tip is basically dull and inherently safe. It can be used as a probe for blunt dissection. The clamping mechanism can be used as a grasper to manipulate tissue. With the ultrasound on, the back of the blade (side opposite the clamp arm) can be used to assist in dissection. The blade tip can be used to create otomies. When tissue is clamped and the ultrasonic energy is on, the tissue is sealed and transected in one step. The adoption of the ultrasonic systems in laparoscopy is continuing to grow and is now being used in many open procedures. The cutting of hard tissues—that is, bone—is dominated by mechanical mechanisms rather than thermal mechanisms used to transect soft tissue. The cut is due to the superposition of the applied static load and ultrasonically induced stresses exceeding a maximum

CRC_DK8307_CH010.indd 413

10/27/2008 11:36:29 AM

Ultrasonics: Data, Equations, and Their Practical Uses

414

critical stress, which is a property of bone. The ultrasonic vibrations against the cut planes generate friction, heat, and temperatures rise. A couple of studies have used linear elastic fracture mechanics and Coulomb friction models to reasonably represent bone cutting [5,6]. The system configuration and design issues described in this section apply to most ultrasonic systems, including many of the systems presented later. This level of description will not be repeated. 10.2.2

Phacoemulsification

The occurrence of cataracts increases with age. As cataracts grow, vision becomes impaired, affecting daily life. At this stage, lens replacement by a surgical procedure is recommended. Of the 3.1 million cataract surgeries performed in the United States in 2003, 2.8 million were done using phacoemulsification. Phacoemulsification is a step in the procedure to remove the cataracts and lens material by emulsifying and aspirating cataracts using an ultrasonic needle. Phaco is derived from the Greek word phakos, meaning “lens-shaped.” Phacoemulsification is an enabling technology that allows the procedure to be performed on an outpatient basis. Phacoemulsification was the invention of Dr. Charles D. Kelman [7], an ophthalmologist practicing in New York City. In 1964, Kelman was sitting in his dentist’s chair; the dentist started to use an ultrasonic dental scaler. Kelman felt the vibration and heard the high-pitched sound when the scaler touched his teeth. At that moment, the proverbial lightbulb went on, and he invented phacoemulsification. After intense research to develop the procedure and instrument, he performed the first procedure on a patient in 1967. By 1969, he had performed 12 cases. By 1985, 16% of all cataract operations were performed using phacoemulsification, and by 1996 the percentage increased to a stellar 97%. Kelman was awarded the National Medal of Technology in 1992 and was inducted into the Inventor’s Hall of Fame. The phacoemulsifier system is similar to the ultrasonic surgical system. It consists of a generator, handpiece, and disposable instrument, which is basically a hollow needle. A commercial unit from Alcon is shown in Figure 10.4 [8]. The handpiece is constructed to introduce irrigation to the needle tip and suction through the needle. Irrigation fluid is introduced through a polymeric sleeve that covers most of the needle. The last 1 mm of the needle extends past the sleeve. The needle has its central cannula, and the handpiece has a central bore that communicates with the needle cannula. Saline is introduced to the sleeve, which flows to the tip to keep it cool and to provide irrigation to the lens capsule. The polymeric sleeve also isolates the vibrating shaft of the needle from adjacent tissues. The center bore of the transducer and needle are connected to suction to aspirate the fluid, cataracts, and natural lens material. The ultrasonic action and the suction-irrigation are basically separate units that are integrated into a single system. In cataract surgery, the cornea is first cut with a small scalpel at the edge of the sclera. The tip of the phacoemulsifier is introduced to the anterior capsule of the eye. It is acti-

FIGURE 10.4 The OZil® torsional phacoemulsifier. (Courtesy of Alcon.) This unit combines longitudinal and torsional motions at the tip on the left to improve performance versus longitudinal motion alone.

CRC_DK8307_CH010.indd 414

10/27/2008 11:36:30 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

415

vated to emulsify the cataracts and the lens. The suction of the unit draws in tissue to force it against the vibrating tip and then removes the fragments. Once the space is prepared, an artificial intraocular lens (IOL) is inserted and positioned. The ultrasonic mechanisms of cataract emulsification continue to be researched and debated, even though the procedure is 40 years old. Research on the mechanisms generally concludes either that the mechanism is direct mechanical action (the so-called jack hammer effect) or that it is cavitation. Review articles, such as by Packer et al. [9], suggest that both mechanisms are present, but more recent research favors cavitation. 10.2.3

Cavitating Ultrasonic Surgical Aspiration

The CUSA was developed by Cavitron in the 1970s. This development evolved shortly after they developed the phacoemulsifier in collaboration with Kelman. The initial product release was in 1980 and was targeted for the removal of brain tumors and other abnormalities of the brain. The CUSA is surgical system that includes irrigation and suction at its tip. A commercial unit is shown in Figure 10.5 [10]. The resonant frequency is typically in the range of 25–40 kHz. It is basically a large phacoemulsifier used to remove tissue, rather than cataracts. The tip of a phacoemulsifier is typically 19 ga and that of the CUSA is typically 12 ga. An irrigation flue covers the vibrating member up to the last 3 or 4 mm. The flue delivers the fluid near the tip and protects the patient and surgeon from the vibrating sides of the cannula. Suction is not drawn through a center lumen running from the tip through the handpiece, as in a phacoemulsifier. Typically, the suction port is connected in a segment between the transducer and the instrument. This attachment is, of course, at a vibrational node. The suction is provided by the suction system in the operating room (OR). The irrigation fluid is either gravity-fed from a saline bag hung from an intravenous (IV) pole or a small peristaltic pump. The CUSA models from the 1980s through the mid-1990s used magnetostrictive stacks of laminated sheets of nickel or permalloy. The magnetostrictive stacks had relatively low Q and therefore high losses. This required the handpiece to be water-cooled. So inlet and outlet tubes were included in the handpiece cable that connected to the front of the console.

FIGURE 10.5 The EXcel™ cavitating ultrasonic surgical aspirator. (Courtesy of Integra LifeSciences.)

CRC_DK8307_CH010.indd 415

10/27/2008 11:36:30 AM

Ultrasonics: Data, Equations, and Their Practical Uses

416

The water was circulated in a close loop with a small reservoir of water mounted to the backside of the console. This was a different and separate fluid path from the irrigation and suction. The magnetostrictive stacks required a biasing voltage to maximize their efficiency, in addition to an alternating voltage at the resonant frequency of the system. Today, the CUSA handpieces use piezoelectric materials, which eliminate the need for a biasing voltage and circulating water. As a result, the size of the generator unit is much smaller in comparison to the original units, because of the reduced complexity. Much of the reduction is due to the continuous trend in electronics to decrease size and increase power. The handpiece and its cable are also significantly smaller, because the cooling fluid tubes are eliminated and high-efficiency piezoelectric materials are used. The CUSA devices are known for tissue selectivity [11]. Tissue may be broadly classified in three categories. The first is soft friable tissue with high fluid content, such as liver parenchyma, fat, or tumors. The third is hard tissue—that is, bone. The second category contains less water and more structural components such as collagen and elastin. It includes vessels, ducts, nerves, ligaments, and other structures. In soft tissue, cavitation effects appear to be the dominant ultrasonic mechanism. Cavitation may occur in the irrigation, the interstitial, or the intracellular fluids. In hard tissue, stress induced failure or fracture is the likely dominate ultrasonic action. Both phacoemulsifers and CUSAs can in fact readily core cortical and cancellous bone. In the second category, neither mechanism is effective in ablating these tissues. Recent numerical simulations by Fong et al. [12] support the role of cavitation in soft tissues. Predictions of the velocity jet of a collapsing bubble are shown to be highest in the region of tissue with low compressibility, as seen in Figure 10.6 [13]. Therefore, the cells are very likely to be disrupted by the jet. The jet velocity for bone does increase from this minimum, but it is still an order of magnitude less than that of friable tissue, such as fat. These numerical results tend to support cavitation as the primary mechanism in phacoemulsification of cataracts. To fail a material by overcoming its stress limit implies that one must be able to exert a force. This is difficult to achieve in soft tissues that offer very low mechanical resistance. So ultrasonic stresses are not generated in soft tissue, the tissue just gives way, which accounts in part for the selectivity for which the CUSA is known. 980

1000

Maximum jet velocity (m/s)

Fat

153 Skin 98

100

83

Cornea

Bone 37 24

Cartilage

Muscle

0.01

0.1

10 1

10

100

K∗

FIGURE 10.6 Cavitation efficiency as a function of tissue type. (Published with permission from Elsevier Limited.)

CRC_DK8307_CH010.indd 416

10/27/2008 11:36:31 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

417

Interestingly, where the CUSA devices have little effect is where the ultrasonic surgical systems have their best performance. The second category, defined previously, includes vessels and ducts. These structures are readily sealed and transected with the clamp, coagulate, and cut instruments, such as the Harmonic™ product family, using heat produced through the frictional mechanism. Broadly speaking, both the phacoemulsifiers and the CUSA devices mobilize targeted soft tissues that are then removed from the body. 10.2.4

Ultrasonic-Assisted Liposuction

Liposuction is the procedure to remove fat from the abdominal wall. It involves injecting a tumescent fluid that contains anesthetics and vaso-restrictors. The anesthetics lessen the pain associated with the procedure, and the vaso-restrictors lessen the bleeding. In the procedure, the surgeon makes a small incision on the periphery of the treatment area to gain access to the fatty layers. This allows the surgeon to sweep the fat layer with a reciprocating motion of the instrument. The basic liposuction instrument is a simple rigid aspirating tube. As the surgeon plunges and sweeps the instrument, friable fat cells are sucked out. This procedure is relatively quick and safe. One can easily picture the aspirating tube being a big CUSA device without the irrigation. A UAL instrument from Mentor is shown in Figure 10.7 [14]. UAL offers several benefits over conventional liposuction. First, it does increase speed, although conventional liposuction is fast. It is less traumatic, because the ultrasonic vibrations help to follow the tissue planes. UAL also provides some hemostasis, although a tumescent fluid is still used. It also lessens clogging of the suction channel. 10.2.5

Lithotripsy

One million Americans, mostly men, are treated annually for kidney stones. Kidney stones can be very painful. Although many stones will clear naturally or can be dissolved with medications, stones that contain calcium do not readily dissolve. One ultrasonic approach is a minimally invasive approach called percutaneous nephrolithotomy (PCNL). Even though

FIGURE 10.7 The LySonix™ ultrasonically assisted liposuction system. (Courtesy of Mentor, Inc., and Misonix.)

CRC_DK8307_CH010.indd 417

10/27/2008 11:36:31 AM

Ultrasonics: Data, Equations, and Their Practical Uses

418

PCNL can be done with minimally invasive surgery, it is still a surgical procedure that requires the placement of a drain tube and a couple of days of hospital stay. ESWL is a routine noninvasive approach that can be performed on an outpatient basis. Manufacturers of ESWL systems consider their technology to be different than either ultrasound or ultrasonics. This differentiation is based on the creation of an intense shockwave directly from the ESWL source, which is substantially different from the typical generation used in most ultrasonic and ultrasound systems. Yet once the shockwave is produced, it is reflected and propagated according to the same physical mechanisms that govern ultrasound, whether of high or low intensity. The disintegration of a kidney stone is also attributed to common ultrasonic mechanisms including cavitation and spalling. So in homage to the manufacturers and their industry, ESWL is included in this chapter on medical ultrasound. The first ESWL units were specially installed free-standing stations that required patients to be partially immersed in a tub of water that was part of the system. Today portable ESWL units are used with C-arm x-ray systems, which are used to image the kidney stone. The patient is positioned comfortably on a table especially designed for use with a C-arm. The ESWL unit is located near the kidney undergoing treatment. The ESWL head/ bladder is then placed against the patient and targeted at the stone. The principle of operation is to create high-intensity transient shockwaves external to the body. The shockwave from the source propagates in all directions. Therefore, the energy must be reflected and focused on the kidney stone. Typically 2000–3000 shocks are given in a treatment to disintegrate one stone. The resulting small sand-like fragments are cleared by urine flow within a day or so. The typical source is an underwater spark gap, known as an electrohydraulic generator. To focus the energy on a stone, an ellipsoid reflector is used. The source is located at one of the two focuses. Through imaging, the kidney stone is positioned at the second focus of the ellipsoid. In this way, almost all of the source energy is captured and focused on the stone. The focusing mechanism of an ellipsoid is easy to understand in terms of an ellipse (see Figure 10.8). A simple ellipse is drawn on paper by stretching a string of fixed length with a pencil between two tacks (the foci). So a property of an ellipse and ellipsoid is that all reflected paths pass through the focus. The pencil is moved around the tacks against the string to create the ellipse. At all times, the length of the string connected to the two tacks is fixed. Therefore, all the energy originating from one focus arrives in phase at the second focus at the kidney stone, because all rays of energy travel the same distance. A second type of source is a piezoelectric source. In this case, many piezoelectric elements are shaped into a spherical bowl. The center of the bowl is the focus and is where the Ellipsoid reflector

Source

Stone

FIGURE 10.8 Ellipsoid, rays and foci.

CRC_DK8307_CH010.indd 418

10/27/2008 11:36:32 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

419 Incident wave impinges on stone Cavitation occurs on proximal surface

Wave propagates through stone

Wave is reflected from distal surface Reflected and incident waves combine to increase stress

Stone spalls

FIGURE 10.9 Disintegration mechanisms of kidney stones.

stone is positioned for treatment. A large amount of charge is stored in the piezoelectric elements. When discharged, the faces of the elements rapidly expand to create a pressure wave. A third type of source is an electromagnetic source. In this case, a metallic membrane is moved rapidly when an impulse drives an electromagnetic coil. Several mechanisms lead to the fracture and disintegration of stone. Rassweiler et al. [15] describe four mechanisms commonly suggested for stone comminution. The primary action on the stone is the fracture of the back surface. The fracture is generated by the stress due to the combination of the reflected wave from the rear side of the stone with the incoming wave on the front side, as depicted in Figure 10.9. The energy is reflected without a sign change, so that the maximum stress occurs near the back face of the stone. So the stone is disintegrated from back to front, as more and more shockwaves are delivered. Typically 2000–3000 shocks are given in a treatment. A second important mechanism of kidney stone removal is erosion caused by cavitation, as depicted in Figure 10.9. Systems currently under investigation use two frequencies: one high and one low. The higher frequency first creates a shockwave to fragment the front surface. This also produces a local cavitation cloud. By then driving with a lower frequency, the cavitation is directed to the back surface and locally implodes to rapidly erode the stone. ESWL is only one tool in the urologist’s armamentarium of treatments for kidney stones. But it has proven to be a safe and attractive noninvasive outpatient procedure. 10.2.6

High-Intensity Focused Ultrasound

10.2.6.1

General Principles of High-Intensity Focused Ultrasound

HIFU is basically the focusing of radiating ultrasonic energy to the targeted area for treatment. The high intensity in HIFU is used to distinguish it from diagnostic imaging systems that use low intensities to avoid any tissue effects. In essence, ESWL may be

CRC_DK8307_CH010.indd 419

10/27/2008 11:36:32 AM

420

Ultrasonics: Data, Equations, and Their Practical Uses

considered to be the original HIFU system. Current HIFU systems are sophisticated and are integrated with diagnostic ultrasound to image the treatment site. This type of system, which provides real-time diagnostics and therapy, is referred to as a see and treat modality. HIFU is being developed for many procedures, including the treatment of cancers, sealing of femoral arteries after catheter procedures, treatment of atrial fibrillation (A-Fib), cosmetic procedures, and trauma injuries. These are all minimally invasive procedures, meaning that they are extracorporeal or natural-orifice (transluminal) approaches. Both the principles and basic devices of HIFU are reviewed in an article by ter Haar and Coussios [16]. HIFU’s main action is to concentrate ultrasonic energy in a small volume of tissue to ablate it thermally. These are relatively high-frequency ultrasound systems, but somewhat lower frequency than diagnostic systems. The high frequencies are beneficial for two reasons. First, the higher frequency produces a smaller focus volume, so treatment can be precise. Focus volume is important, because targeted tissue may be near vital structures. Consider HIFU’s application to prostate cancer, where the nerves on the capsule of the prostate need to be spared to avoid erectile dysfunction. Second, the absorption of ultrasound and, therefore, the amount of heat produced increases with frequency. The transducers are driven at high amplitudes, so that there is nonlinear propagation in the tissue. This is a basic property of propagation in water and naturally occurs as the intensity increases to therapeutic levels. Although many ultrasound mechanisms are leveraged in HIFU, several issues exist. The small focus volume is good for precision, but it means that the focus must be moved throughout the treatment volume, so treatment time is relatively high. As the temperature of the tissue rises in areas outside of the volume treated, the speed of sound changes. This distorts the diagnostic ultrasound imaging and thereby degrades the effective treatment resolution. Optimal dosage planning and mapping and advanced signal processing of the diagnostic images minimizes these otherwise deleterious effects. The existing and potential benefits of HIFU are evident in the current approved procedures and those in the approval process. The promise is to move treatment from surgery in the hospital to noninvasive procedures performed on an outpatient basis. 10.2.6.2

High-Intensity Focused Ultrasound Cancer Treatment

Prostate cancer is diagnosed in 10 million men annually in the United States. It is also the second leading cancer death in men, behind only lung cancer. Prostate cancer in older men is indicated when the prostate specific antigen (PSA) level is over 4 ng/mL, a rapid increase in PSA levels, or by irregularities discovered by a digital rectal exam. Treatment depends on several factors, including the individual’s age and stage of cancer at the time of diagnosis. If a man is in his 70s and is diagnosed at a T1 or a T2 cancer, then he and his physician may consider watchful waiting or hormonal therapy, because the growth of the most prostate cancers is relatively slow, and frankly, he is likely to die of other causes. Younger men with longer life expectancies generally undergo brachytherapy or surgery. Brachytherapy is the implantation of radioactive seeds that slowly kill the prostate tissue and the cancer within. As long as the cancer has not spread beyond the capsule of the prostate, then a patient can consider a nerve-sparing surgical procedure that is performed either open or laparoscopically. HIFU offers a third alternative. Two companies, Focus Surgery and EDAP, have systems on the market to treat prostate cancer in men. Both systems use a phased array to deliver ultrasound to the prostate across the rectal wall transluminally, as shown in Figure 10.10. [17,18].

CRC_DK8307_CH010.indd 420

10/27/2008 11:36:33 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

421

FIGURE 10.10 High-intensity focused ultrasound treatment of prostate cancer. The probe is in the center of the Sonoblate 500® system in the photo on the right. (Courtesy of Focus Surgery, Inc.) The image on the left depicts the probe treating the prostate. (Courtesy of Takai Hospital Supply Co., Ltd.)

The array is used to image the prostate, target the cancer, and deliver the ablative energy to it. HIFU is also under development for liver and breast cancer. 10.2.6.3

High-Intensity Focused Ultrasound Atrial Fibrillation Treatment

A-Fib is an abnormal cardiac rhythm. As the name implies, the left atrium goes into random high-rate contractions called fibrillations. A-Fib is rarely fatal in and of itself, because the symptoms generally drive patients to see their physician early. However, A-Fib can cause clots to become dislodged and to travel to the brain leading to a devastating or fatal stroke. About 120,000 strokes out of an annual number of 700,000 strokes are caused by A-Fib. Also, roughly a third of the patients undergoing open-heart surgery develop A-Fib. A-Fib is caused by electrical events occurring outside the normal electrical pathways that produce normal sinus rhythm, which leads to the fibrillations. Although these errant events can occur anywhere, they typically occur in the pulmonary veins and propagate to the bundle of His. Here they disrupt the normal sinus rhythm, which is normally moderated by this anatomical structure. Several treatments exist. If episodes of A-Fib are relatively infrequent and not severe, the patient is often placed on a chronic regimen of medications. If the events are more severe or resistant to drug therapies, then the patient might undergo a catheter-based procedure. In this treatment, a cardiologist uses a catheter, the tip of which is a monopolar radio frequency (RF) electrode. Under fluoroscopy, the cardiologist maneuvers the tip inside the heart to the periphery of the opening (the ostium) of the pulmonary vein. Then RF energy delivered through the tip creates a small spot of ablation. The cardiologist laboriously repeats this maneuver 10–20 times to create a ring of coagulation surrounding the entry of the vein. The coagulation halts the flow of the electrical signal into the heart and thereby mitigates A-Fib. In the most severe cases, the patient may undergo open-heart surgery for a Cox-Maze procedure. Here a surgeon cuts the heart muscle and sutures it back together. The incision heals with a line of scar tissue. This scar tissue blocks the flow of the errant electrical events. In this procedure, a number of incisions are made. The resulting incisional scars form a fence of protection around normal electrical pathways. Although these treatments all manage to mitigate A-Fib, each has it disadvantages. Patients generally will endure chronic medication, but there are potential side effects, and

CRC_DK8307_CH010.indd 421

10/27/2008 11:36:33 AM

422

Ultrasonics: Data, Equations, and Their Practical Uses Focal/coagulation zone

Pulmonary vein

Myocardium

CO2-filled balloon

Fluid-filled balloon

FIGURE 10.11 Catheter-based ultrasound treatment of atrial fibrillation. (Courtesy of ProRhythm, Inc.)

no one can be sure of the long-term effects. Also, patient compliance is always an issue for self-administered drugs. The catheter-based procedure is basically safe, but still time-consuming, and the basic risks to a patient increase with increased length of any procedure or surgery. In addition, the cardiologist does not know whether the number of spots of ablation is enough to block the errant foci. The Cox-Maze procedure is an open-heart onthe-pump procedure, with all of its inherent risks. Several devices are on the market to coagulate cardiac tissue for the treatment of A-Fib. These include both RF and ultrasound. One HIFU device from a company called ProRhythm is catheter-based, as shown in Figure 10.11 [19]. It has the advantage of being much less invasive than surgery, as is true for any catheter system. But unlike the monopolar RF catheter, it delivers one application of energy in about a minute. Because the monopolar RF catheter needs to be repositioned and activated many times, the procedure can last for more than an hour. The HIFU catheter has a double-wall balloon that is wrapped around a small piezoelectric cylinder. The catheter is positioned and then the balloons are inflated. The inner balloon is filled with saline to couple the ultrasonic energy from the cylinder. The proximal outer balloon is filled with CO2. Thus a gas–liquid interface is created and the differences in acoustic impedances means that the energy will be reflected at the interface. The energy from the cylinder radiates outward to the interface. The energy is then reflected at an angle such that it is focused into an annulus just outside the distal edge of the balloon. The distal end of the balloon is seated in the pulmonary vein opening. The focus annulus is thereby located at the base of the pulmonary vein. The ultrasonic energy is turned on for ∼1 min to create a ring of ablation surrounding the base of the vein. Another company, Epicor, has developed a minimally invasive HIFU procedure. They have created an ultrasonic bracelet that is composed of 12–17 individual ultrasonic HIFU transducers. The bracelet is introduced through a small port and wrapped around the atrium at the base of the pulmonary veins. It is cinched down onto the myocardium. The extra transducers not in contact with the myocardium are noted and are not activated during treatment. The transducers are driven individually to denature the underlying area of myocardium through its entire thickness. After the treatment is finished, all transducers have been activated and a complete band of denatured tissue is achieved to block the errant signals. Clinical trials have demonstrated the safety and efficacy of HIFU treatment of A-Fib. It promises to improve patient outcomes while decreasing procedure time, shifting the site of care, and reducing costs.

CRC_DK8307_CH010.indd 422

10/27/2008 11:36:34 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging 10.2.6.4

423

High-Intensity Focused Ultrasound Wrinkle Reduction

With age come wrinkles, such as crow’s feet. Although minimizing sun exposure is known to defer the onset of wrinkles and other signs of aging, most people will develop wrinkles. HIFU devices are being introduced to the market to remove wrinkles in noninvasive officebased procedures. In this HIFU application, the ultrasonic energy is focused just below the surface of the skin. This means that the HIFU transducer needs an intervening bladder to position the focus at this shallow depth. Sufficient energy is delivered to raise the temperature to denature the underlying collagen. The objective is to raise the temperature in this layer to ∼65°C and no more. The fluid in the bladder keeps the skin cool to avoid potential surface burning. When collagen dentures, it goes from highly ordered cross-linked strands to a tighter, more amorphous structure. This is known as the helix-coil transformation, and causes the collagen to shrink, and thereby to pull the adjoining layers. This action locally flattens the skin to reduce the appearance of wrinkles. This treatment is currently under development, so whether the effects are significant and long-lasting have yet to be demonstrated. It is known from other procedures that are based on collagen shrinkage that the effects may be transitory; the tissues relax. It may be the case that an individual would have periodic treatments depending on the remaining (or renewed) capacity for collagen to shrink. 10.2.6.5

High-Intensity Focused Ultrasound Fat Removal

An intensely focused ultrasound system, shown in Figure 10.12 [20], is now being used to literally melt away fat. An extracorporeal transducer is coupled to skin above the layer of fat to be removed. The transducer focuses the energy at the layer of fat below the skin. The intense energy disrupts the fat cell walls as well as emulsifies the fat. The transducer is swept over the treatment area to disrupt the undesired fat. The normal clearing processes of the body then removes the remnant cell components and the fatty molecules. Ultrasonic fat removal has several potential benefits over conventional UAL. First, no incision or injection of a tumescent is necessary. Second, there is significant reduction of the risk of damaging the abdominal wall and the underlying viscera. Third, though fat

FIGURE 10.12 Noninvasive fat cell disruption using focused ultrasound. The photo on the left is the transducer that delivers the ultrasonic beam, and the drawing on the right depicts ultrasonic energy reaching the subcutaneous fat. (Courtesy of UltraShape.)

CRC_DK8307_CH010.indd 423

10/27/2008 11:36:35 AM

Ultrasonics: Data, Equations, and Their Practical Uses

424

layers are basically avascular, the focused ultrasound should seal off any small bleeders. Finally, because fat cells are friable and readily absorb ultrasound, there is selectivity. That is, the intensity level can be set to a range that will emulsify fat but deposit relatively low heat to intervening and neighboring tissues. A few companies are developing systems for extracorporeal ultrasonic fat removal at the time of this writing. If proven effective with long lasting benefits, then this would truly be a distributive technology. 10.2.6.6

High-Intensity Focused Ultrasound Acute Puncture and Wound Closure

Catheters are generally inserted through the femoral artery located in the inner thigh. Obviously, the last step in these procedures is to close the puncture wound in the femoral artery. In the past, the patient rests in a recovery room bed with a sandbag over the leg to apply compression to the puncture and seal the wound. Recently, suturing devices have been introduced, which are inserted through the puncture and then deploy needles with sutures or anchor-like devices. The sutures are drawn tight, the puncture is closed, and a cinching unit then holds the sutures taut. About 30% of patients today receive these closure devices. A company, Therus, is developing a HIFU system to close a femoral artery punctures with ultrasonic energy. Here an ultrasonic image is taken with the M-mode (“M” for motion) to identify the location of the puncture. This information then targets the HIFU system to that section of femoral wall. Ultrasonic energy is delivered to coagulate the leaking blood. Because the blood is flowing in the artery, there is little risk of accidentally occluding this major vessel. Benefits include the same as achieved with the suture devices. In addition, nothing is left behind in the patient, and the method requires less skill, in principle. Obviously, this same system has applications in trauma. AcousTx, a spin-off of Therus, is developing such a system for the battlefield. The system is used for the detection, localization, and sealing of wounds. It is to be portable and lightweight and can be readily operated by combat personnel with minimal training. 10.2.7

Ultrasonic Healing

Standard treatments for wound healing and bone healing are very effective. However, in some cases bones are slow to heal. Nonunion fractures are defined as fractures lasting longer than six months. Similarly wounds often resist treatments or are challenging, because the environments leading to the wounds are difficult to remove, including pressure sores and diabetic foot ulcers. Ultrasound has been shown not only to speed healing, but more importantly, to heal the nonunion fractures and the recalcitrant wounds. 10.2.7.1

Ultrasonic Bone Healing

In 1892, Wolff [21] observed that bone responds to micromechanical stress induced by acoustic pressure waves. Nearly a century later, in 1983, Dyson and Brooks [22] and Duarte [23] both reported that application of low-intensity ultrasonic energy could stimulate bone growth. By the early 1990s, the use of low-frequency ultrasound had been used on human patients with nonunion fractures. In the mid-1990s, the Exogen Company introduced its commercial product (see Figure 10.13; [24]) for accelerated healing of bone. Ultrasound in the frequency range of 0.5–1.5 MHz at intensity levels of 0.03–1.0 W/cm2 is effective. Various explanations have been given for the stimulation. Yang et al. [25] present data that suggests improved mechanical properties of healing callus by stimulating the synthesis of extracellular matrix proteins in cartilage, accelerating chondrocyte

CRC_DK8307_CH010.indd 424

10/27/2008 11:36:35 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

Nonunion fracture

425

After treatment

Animal model FIGURE 10.13 The Exogen 4000+™ ultrasonic bone healing system. (Courtesy of Smith & Nephew, Inc.)

maturation and endochondral bone formation. Because of the low intensity and pulsed delivery of the energy, ultrasonically induced thermal mechanisms have little, if any, contribution. Therefore the mechanical effects of ultrasound—including the basic pressure fluctuations, microstreaming, and mixing—must play a siginifcant role in bone healing. Cavitation is likely not a significant mechanism, because of the low intensity levels and its more disruptive nature. Experimental data clearly demonstrates that the ultrasonic treatment improves the blood flow to fracture site compared with no treatment. In a study, Doppler ultrasound showed significantly greater flow in the treated side versus the untreated side within 9 days of bilateral factures in a canine model. Clinical studies have demonstrated improved outcomes in humans. In a multicenter prospective study of 96 patients with 97 tibia fractures, two different evaluations estimated an improvement in healing rate by 38 and 48%, respectively. Another study by Cook et al. [26] looked at improvements in healing in smokers using low-intensity ultrasound. Cigarette smoking is known to have a profound effect on bone and bone healing. Delayed healing of nonunion tibia and mandibular fractures is known to occur more frequently in patients who smoke. Cook et al. present data that demonstrates a 41% increase in healing rate for smokers and 26% increase in nonsmokers with tibia factures. Their data also show similar results in radius fractures: a 51% increase in healing for smokers and a 34% improvement in nonsmokers. So although the exact mechanisms of ultrasonic accelerated bone healing are not precisely known, clinical outcomes demonstrate the effectiveness and safety of this application of ultrasound. 10.2.7.2

Ultrasonic Wound Healing

Wound care is a major health care issue worldwide. In the United States, 89 million patients are treated annually for some type of skin condition, including wound care. Recalcitrant skin ulcers account for 6 million of these numbers. Roughly 3 million are pressure sores,

CRC_DK8307_CH010.indd 425

10/27/2008 11:36:35 AM

426

Ultrasonics: Data, Equations, and Their Practical Uses

Depiction of deep penetration of mist and ultrasonic energy to kill bacteria and promote healthy cell growth

FIGURE 10.14 Ultrasonic wound healing with ultrasonic energy and saline mist. (Courtesy of Celleration, Inc.)

1 million are venous stasis ulcers, and 2 million are chronic foot ulcers in diabetic patients. Unfortunately, many of the diabetic foot ulcers lead to amputation. In 2001, more than 85,000 amputations occurred. An airborne ultrasound mist of sterile saline has been shown to accelerate wound healing and close otherwise recalcitrant skin ulcers. A commercial unit is shown in Figure 10.14 [27]. A low-frequency transducer drives at 40 kHz. At the distal end, sterile saline is introduced to the face of an ultrasonic horn. Capillary waves in the liquid on the vibrating surface create a mist. The droplets stream outward from the inertia imparted by the extension of the vibrating surface and acoustic radiation pressure. The mist also acts a coupling agent to improve the transfer of ultrasonic energy to the wound. Both the mist and the ultrasonic energy impinge on the open wound. Their combined action stimulates healthy cell growth while significantly reducing bacteria colonization. In a double blind study [28], diabetic patients with foot ulcers were either treated with the ultrasonic mist system or a sham unit. The sham unit was developed to create the same amount of saline flow and mist, but not using ultrasound. Each wound was treated with 4 min of mist with or without ultrasound. The ultrasound unit was held a distance of 5–15 mm away from the wound, corresponding to an intensity of 0.5 and 0.1 W/cm2, respectively. After treatment, dressings were applied per a standard protocol. The duration of the study was 12 weeks. At the end of the study, 41% of the wounds were closed in the ultrasonic group and 14% in the sham group. Statistical analyses demonstrated that there were no significant confounding effects from other parameters. The mechanisms of wound healing are under active investigation. In a study in a diabetes-induced rat model, the blood supply was observed to increase with the ultrasonic mist treatment [29]. In a laboratory study, colonies of cultured bacteria were smaller with the ultrasonic treatment when compared with colonies grown without treatment [30]. 10.2.7.3

Ultrasonic Wound Debridement

Debridement refers to clearing of necrosed and infected tissue and foreign matter from a wound site. It is typically done with a steel scalpel. In 1996, King and his coworkers used a CUSA-like device for debridement and stated that it provided meticulous wound debridement [31]. Recently, ultrasonic devices that directly contact the wound have been

CRC_DK8307_CH010.indd 426

10/27/2008 11:36:36 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

427

evaluated for debridement [32,33]. The instrument is an ultrasonic blade that is similar in construction to those used in ultrasonic surgical systems. Saline is introduced to the tip to keep it cool, to avoid denaturing of proteins in the healthy tissue, and to provide irrigation of the site. The necrosed or infected tissue is structurally weak and friable. An ultrasonic instrument can readily remove these tissues. The level of ultrasonic energy can be selected so that it removes the friable weakened tissue but is insufficient to affect healthy tissue. It lessens the damage to the healthy tissue and is less traumatic compared with a sharp scalpel, which can easily cut healthy tissue as well. The combined actions of the saline and the direct ultrasonic action may also significantly lessen the bacteria burden at the margins of the wound. 10.2.8

Sonophoresis

The skin or dermis is the largest organ in the body. It keeps body fluids inside and prevents external substances from entering. The dermis is divided into three layers: the outermost layer is the stratum corneum. It is a thin, highly organized layer of bilipid cells of nonviable keratincytes. This layer is the body’s natural barrier, and only a few low-molecularweight drugs are able to penetrate it. Sonophoresis is the ultrasonically assisted transport of molecules across the stratum corneum. It is analogous to electrophoresis, which is also used to drive molecules into the body through the skin. Mitragotri and Kost [34] report on the use of low-frequency sonophoresis and the mechanisms involved. Ultrasound applied to the skin disrupts the bilipid layers of the tightly packed cells. This creates larger pathways between the surface of the skin and its deeper layers. So large molecular weight molecules can diffuse from the surface to the capillary bed of the skin and thereby enter the bloodstream. A simplistic view of this ultrasonic pathway is something like a crack in the mortar in between the bricks of a brick wall. The pathways enable transdermal drug delivery of larger-molecule-weight molecules. Sontra Medical has introduced its first product (see Figure 10.15; [35]) for the administration of analgesics to pediatric patients. A low-frequency and relatively low-power transducer radiates ultrasonic energy from its face into the skin through a liquid packet containing the drug. The biggest potential market for transdermal delivery is needle-free injection of insulin for diabetic patients. Although results to date have shown that the intake of insulin through the skin increases dramatically with sonophoresis, it does not have the required delivery profile to the blood stream to be viable at this time. 10.2.9

In Vivo Ultrasonic Welding

Suture knots are similar to square knots. Surgeons also throw in a few more half hitches for security. Properly tied knots have relatively low profiles, but nonetheless sit above the adjoined tissues. In most surgical procedures, the presence of the knot is not an issue. However, it is an issue in the surgical repair of joints (as opposed to joint replacement). One of the most successful industrial applications of ultrasound is the welding of plastics. Sutures are made of polymers (i.e., plastics), such as polypropylene, nylon, and polyester. Therefore, ultrasound can be used to weld suture materials as an alternative to conventional tying. Monofilament nylon sutures are the most amenable for welding versus multifilament materials, because there is only one melt zone. In the case of multifilaments, each filament has its own melt zone, and therefore the consistency and strength of welded multifilament suture are low. Richmond [36] showed that welded monofilament sutures perform nearly as well as knotted sutures.

CRC_DK8307_CH010.indd 427

10/27/2008 11:36:37 AM

Ultrasonics: Data, Equations, and Their Practical Uses

428

Ultrasonic cavitation Stratum corneum

100 µ

25 µ

Viable epidermis

FIGURE 10.15 SonoPrep® skin permeation system. (Courtesy of Sontra Medical Corporation.) Action of ultrasound and cavitation create 25–100 µm channels.

FIGURE 10.16 Ultrasonic suture welding system. (Courtesy of Axya Corporation.) Note that the welding instrument is the unit on the right, and the small loops are ultrasonically welded sutures.

CRC_DK8307_CH010.indd 428

10/27/2008 11:36:37 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

429

A commercial device from Axya is shown in Figure 10.16 [37]. In the Axya System, the suture is first passed through the two edges to be approximated and joined. The sutures are then pulled through the port and the welder is introduced. The ends of the suture are drawn tight and anchored to the body of the ultrasonic handpiece. The unit is activated to weld the overlapped region of the suture together. The suture ends are then cut adjacent to the weld. A low-profile knot has been created, with strength approaching the native strength of the suture material itself. Although ultrasonically welded sutures have been evaluated in laparoscopic procedures and mitral valve replacements, welded sutures offer greater utility in arthroscopic procedures. These procedures include rotator cuff tears of the shoulder, meniscal tears, cartilage damage, and acute cruciate ligament (ACL) repair. The surgeon prepares the torn surfaces, approximates the prepared edges, and sutures them together. Unlike other surgeries, here the objective is to make low-profile knots, because the repaired tissue has relative motion against other tissues. Standard knots can lead to irritation, pain, and potentially fretting of the moving cartilage surfaces. 10.2.10

Ultrasonic Thrombolysis

Blood clots or thrombus are the cause of peripheral arterial disease, heart disease, ischemic stroke, and blockage of the carotids. In each case, the tissues on the distal side of the thrombus are starved for oxygen. The treatment of clots varies with their location and corresponding risk to the patient. The surgical treatment for clogged coronary arteries is bypass surgery. Often when coronary stenosis involves only one or two vessels, a cardiologist uses stents to force and hold the arteries open. In ischemic stroke, recombinant tissue plasminogen activator (r-tpa) is used to dissolve the clot. It is also given at the onset of heart attack and for the treatment of peripheral arterial disease (PAD). Clot removal by breaking down the clot is known as thrombolysis. Ultrasonically active wires in catheter-based thrombolysis systems have been under development for at least a decade. The systems are all similar, in that ultrasound is delivered into or at the clot with or without the addition of thrombolytic drugs. One approach is to generate the ultrasound externally and propagate the energy down a wire. Either longitudinal or transverse vibrations or a combination of the two are used to excite the wire. The length of the vibrating wire is shielded with a thin polymeric tube, except for the distal end, which interacts with the clot. The other approach is to put the transducing elements on the distal end of the wire, so that the energy is generated and delivered immediately at the clot. The need for a shielding tube is mitigated in this approach, but electrical wires connected to the generator need to be carried along to drive the ultrasonic elements. One device uses transverse motion of a vibrating wire that is indicated for the clearing of thrombus in the peripheral vasculature shunt placed in dialysis patients (see Figure 10.17; [38]). A device from another company is targeted for the treatment of PAD. It uses a series of small piezoelectric cylinders at the distal end of a wire. The cylinders radiate ultrasound radially outward directly into the clot. Two big clinical challenges remain for ultrasonic thrombolysis systems. The first is to break up chronic total occlusion in the coronary arteries. These are old thrombi that become more dense and harder with time. The other challenge is clearing the thrombus in ischemic stroke. The ability of ultrasound to enhance the effectiveness of r-tpa in the treatment of ischemic stroke is well known. Therefore any ultrasonic system that can be easily, quickly, and accurately deployed would be a benefit in the treatment of acute stroke.

CRC_DK8307_CH010.indd 429

10/27/2008 11:36:39 AM

Ultrasonics: Data, Equations, and Their Practical Uses

430

Vibrating wire FIGURE 10.17 Ultrasonic catheter-based thrombolysis system. (Courtesy of OmniSonics™, Inc.)

10.3 10.3.1

Diagnostic Ultrasound Imaging Introduction

The body is nearly an ideal system to image with ultrasound for two primary reasons. First, the speeds of sound in soft tissues are nearly the same as water and differ by only a few percent. For spatial imaging, round-trip speeds therefore can be used to accurately calculate distances, independent of the various tissues that the ultrasound passes through. Second, even though the speeds of sounds are the same, their acoustic impedances are sufficiently different to reflect energy from an interface back to the ultrasonic imaging transducer. Generally, only a portion of the incident energy is reflected. Energy is still transmitted across the interface, so that deeper structures in the body can be imaged as well. Other factors are also advantageous in imaging. Body temperature is regulated, therefore, the normal temperature dependence of the speed of sound is not an issue. Attenuation is frequency-dependent but relatively low. So the ultrasonic energy can propagate deep into the body. Spatial resolution can be traded off against attenuation to select the system frequency. Mechanical mechanisms of ultrasound, such as Doppler shift, cavitation, nonlinear propagation, and microstreaming, can all be utilized to further extend the benefits and applications of an ultrasound imaging system. From the two basic acoustic properties of tissue—speed and attenuation—the capabilities of ultrasound imaging systems are derived. Yet advances in imaging are obtained by further utilization of ultrasonic mechanisms and the continuous increase in computing power and speed. Ultrasonic imaging started with a room full of equipment in the late 1940s. Currently, there is a compact portable 5 MHz system that is the size of a briefcase. This unit is capable of three-dimensional (3D) and Doppler imaging, and has the potential for HIFU applications. Detailed explanation of the physics, system architecture, transducer design, and image reconstruction are beyond the intent of this chapter. This section presents the basics of

CRC_DK8307_CH010.indd 430

10/27/2008 11:36:39 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

431

imaging and summarize the current capabilities of imagers. The reader is referred to Diagnostic Ultrasound Imaging: Inside and Out by Szabo [39] for more in depth description. 10.3.2

Ultrasonic Properties of Tissue

The speeds of sound in soft tissues are nearly the same as water: 1530 m/s, assuming a body temperature of 37°C and a salinity of 4500 ppm. In 1950, Ludwig [40] reported the average speed of sound of soft tissues to be 1540 m/s. This value is still used as the standard in modern imaging systems. The range of values reported by various sources is included in Table 10.1. The acoustic impedances, Z1 and Z2, of two mediums determine the amount of energy reflected and transmitted across their boundary. The reflection coefficient (amplitude) is given by R = (Z2 − Z1)/(Z2 + Z1), and the transmission coefficient (amplitude) is given by T = 2 . Z2/(Z2 + Z1). Clearly, if Z1 and Z2 are equal, the situation is equivalent to propagating in the same medium with R = 0 and T = 1. A list of acoustic impedances is included in Table 10.1. Another acoustic property is attenuation of energy as it propagates through tissue. Attenuation has the form A(z, f )  A oez

(10.3a)

  af b

(10.3b)

The absorption coefficient, α, includes a factor that depends on frequency raised to a power, b. For classic absorption based on viscosity and thermal conduction, the power is 2.0. In tissues as well as other real materials, the values typically range from 1.0 to 2.0 and are determined experimentally. Absorption coefficients are also included in Table 10.1 [41]. Attenuation arises primarily from two mechanisms: absorption and scattering. Absorption is the conversion of ultrasonic energy to heat and scattering is the reflection and transmission of energy in all directions. Scattering depends on the structural components in the tissue and their geometry relative to a wavelength. Both mechanisms are frequency-dependent, so the measured attenuation coefficient implicitly represents a weighted average of both. Attenuation can also be caused by thermal relaxation phenomena associated with translation, rotation, and bond deformations at the molecular level. Sound is then preferentially

TABLE 10.1 Ultrasonic Tissue Properties Material Water 20°C Saline (at 37°) Soft tissue (average) Hard tissue (bone) Blood Fat Heart Muscle Liver

Density (kg/m3)

Speed (m/s)

Impedance (×106 rayl)

Absorption (dB/MHz-cm)

1000 1000 1050 1990 1060 930 1050 1040 1050

1482 1530 1540 3200 1580 1430 1550 1580 1580

1.48 1.53 1.62 6.37 1.67 1.33 1.63 1.64 1.66

0.002 0.002 0.50 3.50 0.14 0.60 0.50 0.60 0.45

Source: Courtesy of Volcano Co.

CRC_DK8307_CH010.indd 431

10/27/2008 11:36:40 AM

Ultrasonics: Data, Equations, and Their Practical Uses

432

absorbed in a relatively narrow band of frequencies. This mechanism also changes the velocity slightly. As absorption is frequency dependent, the speed of sound is frequency dependent and therefore is slightly dispersive. The change in speed in tissues from 0 to 20 MHz represents <10 m/s increase in most tissues. Therefore one may conclude that relaxation mechanisms in tissue are minimal. 10.3.3

Modes

The term mode is used to describe or define the operation and display of the imaging system. There are three common modes: the amplitude mode or A-mode, the brightness mode or B-mode, and the motion mode or M-mode. The A-mode is the most basic. Here the received amplitude is plotted as a function of time. Because the speed of sound is essentially constant, the amplitude is also displayed as a function of distance where time is multiplied by speed of sound, 1540 m/s and divided by 2, because the time is for a round trip. The A-mode is used to better visualize interfaces. The A-mode is commonly used in nondestructive testing and research systems for the study of wave propagation in materials and structures. The B-mode or B-scan is used to display a 2D image. The received amplitude is used to control the brightness of a spot, whose 2D position represents a location in the body. Interfaces between tissues with very different impedances are displayed with high brightness. When the impedances are the same or nearly so, the position appears dark. Small ultrasonic coherent scatterers are located throughout tissue, so that even in the same tissue, the ultrasonic image will generally contain low-intensity speckles. If a dark shadow is observed in an image beyond which there is no speckle, then the ultrasonic energy has been blocked by a structure. If a dark circular shadow exists, then the shadow is likely a liquid-filled cyst that does not have structural components to create speckles. In an M-mode scan, the frequency of the returning signal is measured. Any difference in the received frequency from the transmit frequency is due to a Doppler shift where the sound beam has interacted with a moving medium. For the most part, Doppler shift is used to image blood flow within the heart and arteries. The M-mode data are generally plotted in color over a grayscale B-scan, which allows the clinician to evaluate the blood flow in and out of a structure of interest. There are numerous modes of an imaging system. Most are combinations and overlays of the aforementioned basic three. 10.3.4

Doppler Imaging

In the M-mode, either continuous or pulsed Doppler is used. Both depend on the wellknown Doppler shift that occurs when radiating energy impinges on a moving object, as depicted in Figure 10.18. The frequency of the received signal is shifted based on the relative speeds between the moving object and moving observer: ∆F  Fi ( v/c o )[2 cos()]

(10.4)

where ∆F is the Doppler shifted frequency, Fi the input frequency, co the speed of sound, v the velocity between the target and the source, and θ the angle of object movement to the direction of the ultrasound. The cosine factor is used to obtain the projection of the object’s velocity in the direction of the sound propagation. The velocity is assumed to be much smaller than the speed of

CRC_DK8307_CH010.indd 432

10/27/2008 11:36:40 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

433

Source frequency: fo Speed of sound: Co Speed of source: Vs = 0 Doppler shift: ∆f = (Vs /Co) . fo = 0

∆f = 0 ∆f = +2 −1/2 . (Vs /Co) . fo

45° ∆f = − (Vs /Co) . fo

∆f = +(Vs /Co) . fo

Vs

Doppler shift ∆f = 0

FIGURE 10.18 Doppler shift principles. Sound waves are depicted as radiating from a point source. The source is fixed in the top image.

sound, assumption v << co. This assumption is reasonable, because the nominal speed of sound, co, is 1540 m/s and the speed of blood pumping in the heart is <3 m/s. The ratio of the two velocities is <0.002. The factor of 2 in the equation arises from the fact that in a pulse-echo configuration, there are both an incident path and a reflected path. Continuous wave (CW) Doppler uses both transmit and receive transducers with overlapping focal volumes. A continuous sine wave drives the transmit transducer. The sine wave propagates out into the body based on its beam pattern. The receiving transducer detects the reflected beam from the region of overlap. Quadrature signal processing is used to recover the Doppler frequency. The information can be displayed in an A-mode to the operator. The Doppler frequency can also be used to drive stereo headsets. Depending on the transmit frequency, the Doppler frequencies produced by flowing blood are within the range of human hearing. Experienced trained operators can often hear subtle abnormalities that may not be obvious in the displayed image. CW implemented with a single transmit transducer and a single receiving transducer is very sensitive to flows in their mutual focal volumes, but no explicit location information is achieved with respect to the underlying anatomy. The individual transmit and receive transducers can be implemented with transducers arrays so that their focal volume can be scanned through an area and overlapped with a B-mode image. Pulsed wave Doppler drives the elements in a transducer array with short packets of a sinusoidal wave at the transmit frequency. These packets propagate from the array, are reflected at boundaries, backscattered, and frequency-shifted by flowing blood. The energy returning from a known location is then used to set the grayscale for the location, and the Doppler frequency is extracted. This procedure allows the B-mode image to be produced with velocities displayed in precise registration with the image. From the image,

CRC_DK8307_CH010.indd 433

10/27/2008 11:36:40 AM

Ultrasonics: Data, Equations, and Their Practical Uses

434

FIGURE 10.19 Doppler measured blood flow in the abdominal aorta and the vena cava. (Courtesy of Philips Medical Systems.)

the angle between the beam and the vessel can be used to estimate the cosine term in Equation 10.4. The velocity estimate can then be improved and displayed. Figure 10.19 [42] the blood flow in the abdominal aorta and the vena cava are indicated in the small sector outlined in white that is overlaid on the larger B-Scan. In this particular case, the flows are in opposite directions normally indicated by blue and red. In this B&W image the Doppler signatures are apparent by the “filled in” section of the vessels. The pulsatile nature of blood flow is observed in the Doppler signal itself in the bottom traces of the image. The information in the figure includes the B-Mode, M-Mode and Doppler signal, which is referred to as the triplex mode. 10.3.5

Harmonic Imaging

Models of ultrasound and acoustic wave propagation are developed based on infinitesimal displacements and material properties being constant. As long as these conditions are approached, the linear models are accurate descriptions of the real physical systems. These assumptions are best approximated by propagation in solid material especially metals and least applicable in gases. Water and tissue fall in between these cases. For linear propagation, the density is assumed to be constant. Yet in gases, the density is known to increase with pressure from the ideal gas law. Thus the speed of sound is faster in compression and slower in rarefaction compared with the speed based on the nominal density. The nonlinearity increases with amplitude. The same behavior exists in water and tissues. The peak amplitudes in compression move the fastest and the peaks in rarefaction move the slowest. As amplitude increases, the compress peak skews forward and the rarefaction peak skews backward. A series of single cycles of compression and rarefaction is depicted in Figure 10.20. At a certain amplitude, a vertical drop occurs from compression to rarefaction, representing a shockwave. As amplitude is further increased, the relative height of the drop increases.

CRC_DK8307_CH010.indd 434

10/27/2008 11:36:41 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

435

σ=0 z = 0.0 mm σ = π/4 z = 8.5 mm σ = π/2 z = 16.9 mm σ = 3π/4 f = 10 MHz β = 3.5 (water) Co = 1540 m/s (tissue) λ = 0.154 mm

z = 25.4 mm σ=π z = 33.8 mm

FIGURE 10.20 Increased nonlinearity as wave propagates in tissue. σ is the normalized nonlinearity distance parameter.

At even higher amplitude, the vertical drop extends from maximum compression to maximum rarefaction and the wave appears as a sawtooth curve. In a linear analysis of wave propagation, the fluctuations in pressure, p, about equilibrium are proportional to fluctuations in density, ρ. In the nonlinear analysis, the fluctuations in pressure are expanded in a Taylor series expansion of density fluctuations given by 2

   o     o  p  po  A 
B
   o   o 

(10.5)

where the subscript “o” indicates equilibrium conditions. When the coefficient, B, approaches zero, linear behavior is observed. The coefficient of nonlinearity is defined by   1
B/2A

(10.6)

For soft tissues, the value of β falls between 3 and 7 [43]. A normalized nonlinearity distance, σ, can be expressed as a function of β: 

po 2f z oc o3

(10.7)

Several features of nonlinear propagation can be observed by this expression. The nonlinearity distance increases with β, pressure amplitude, frequency, and distance. In addition, the amplitude saturates when the waveform shocks as can be inferred from Figure 10.20. That is, no additional increase in amplitude is achieved, even when the input pressure is increased at the source. Furthermore, the nonlinearity takes some distance, z, to develop. The nonlinearity grows, even though the source is a simple sinusoid. The growth is depicted in Figure 10.21, which also indicates the effects of attenuation. As the wave propagates, the harmonics experience more attenuation, due to its frequency effects. Therefore, at deeper depths, the wave is primarily a single sine wave at the drive frequency.

CRC_DK8307_CH010.indd 435

10/27/2008 11:36:41 AM

Ultrasonics: Data, Equations, and Their Practical Uses

436

Plane wave generator

Initial pure sine wave

Distortion increases

Full triangular wave

High frequencies absorbed

FIGURE 10.21 Nonlinear growth and attenuation with distance.

In a certain range of depth, ultrasonic components at both the fundamental and the harmonics exist. Obviously, the second harmonic generally has the second-highest energy compared with the fundamental. So for the most part, harmonic imaging is based on the second harmonic. The benefit of higher frequency imaging is improved resolution. The resolution improves by two mechanisms. First, the higher frequency naturally produces higher resolution. Second, the backscattered energy from smaller objects increases, so they are more readily detected. 10.3.6

Transducers, Beams, and Arrays

The understanding of transducers, arrays, beam steering, and focusing is very detailed and deep. This section presents a high-level overview of the involved physics. The reader is referred to other texts for a more comprehensive presentation of the subjects [39,40]. A circular piston radiator in a baffle is a basic model of an ultrasonic transducer radiating into a fluid medium. The face of the piston is assumed to be moving with uniform displacement. A simple relation gives the pressure developed at a position in space from any differential area on the face. The acoustic pressure produced at any position can be determined by integrating the contribution of all differential areas on the face at that position as depicted in Figure 10.22a. The acoustic field is simply a map of the acoustic pressures for all positions in the space of interest. The pressure along the central axis is shown in Figure 10.22b, and the predicted directivity pattern is shown in Figure 10.22c. The angle of the center beam is determined by the size of the aperture, a, relative to the wavelength of sound in the medium. When the wavelength is much larger than the aperture, the piston appears as a point source. When the wavelength is shorter than the aperture, the beam pattern forms. The shorter the wavelength, the narrower the central beam. The most striking features of the beam are the rapid pressure fluctuations in the Fresnel zone (or near field), the single peak pressure in the Fraunhofer zone (far field), and the presence of side lobes. From the beam pattern, a number of parameters are defined and are used to characterize the transducer output. Some of these are indicated in Figure 10.22b.

CRC_DK8307_CH010.indd 436

10/27/2008 11:36:42 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging x

437

Piston radiator

Field point z a y

(a)

Position vector Differential area

Pressure amplitude (atm)

0.12

3 dB

0.08 0.06

a = 0.01 m

0.04 a = 0.0075 m 0.02 0

(b)

Depth of focus

0.1

Fresnel Fraunhofer zone zone 0.02

0.07

0.12

Fresnel number, S = zλ/a2 S = 1 Wavelength, λ = c/f

0.17

0.22

0.27

0.32

z-distance (m)

tio

ec

dir

y-

tion

n

(c)

irec

z-d

FIGURE 10.22 Radiation pattern from a circular source: (a) piston radiator of radius, a, and field point, (b) on-axis response and beam definitions, and (c) logarithmic value of far-field directivity pattern.

Original imagers used a single transducer that was mechanically translated across the skin to produce the images. Now arrays are used to electronically steer and focus beams to provide much more flexibility compared with a solid continuous disk. In the development of pressure field from a piston radiator, the contributions from each differential surface area are summed through integration. Assume that the differential elements are replaced with small but finite radiators, all vibrating in phase. Clearly the same pattern would emerge as the size of the radiators decreases and their density increases. Now consider that the signals driving each radiator can be varied. Suppose that a linear delay is imposed across the radiators, as shown at the top of Figure 10.23. The wave front (constant phase) is angled from the array. The radiated field would still be estimated in the same way, only the delay

CRC_DK8307_CH010.indd 437

10/27/2008 11:36:42 AM

Ultrasonics: Data, Equations, and Their Practical Uses

438

Time Linear delay

Delay

Space Transducers

Delay Delay

Signal generator

Delay Delay Delay Delay

Focused delay

Delay

Wavefronts in fluid

Delay Delay

Signal generator

Delay Delay Delay Delay

FIGURE 10.23 Phased arrays: linear and focused delay schemes.

in each radiator needs to be explicitly included. If the signals are delayed by a circular arc in time, as depicted in the bottom of Figure 10.23, then the wave front is focused. With combinations of linear and circular delays, the beam can be steered off-axis and focused at a desired distance. In addition to delaying, the voltage driving the beam can be used to shape the beam. One advantage is that the amplitudes of the side beams can be lessened, thereby, speckle is reduced and image quality improves. Arrays are described as 1D, 2D, or 1.5D. In 1D arrays, the elements are long and narrow, so that each forms a slice of a beam pattern similar to the one shown in Figure 10.22b. Each element is driven individually with delay and voltage control, and the number of elements range from 32 to 256, in a 1D array. The beam is steered and an image is produced in the azimuth plane. To obtain volumetric data, the beam needs to be steered in both the azimuth and elevation planes. Obviously, a 2D array of elements would accomplish this. However, the number of elements quickly becomes unwieldy. A modest 2D array of 32 × 32 is still a total of 1024 elements. A compromise is the 1.5D array. In this case, the number of elements in an azimuth plane is still high, but only a small number of elements span the elevation. For example, a 5 × 128 element array has only 640 elements, so the span and resolution in the azimuth direction are still high, but the range and resolution are much less in the elevation direction. Composite transducers are formed by cutting a matrix of pillars of a soft piezoelectric ceramic and the spaces are filled with a low-impedance polymer. This construction is referred as a 1–3 composite. The composite has two benefits. First, its effective coupling (i.e., the amount of energy that can be transformed from the electrical to the mechanical domain) is higher than the solid piezoelectric element. Second its acoustic impedance is less, so it is better matched to tissue impedance [44]. A recent development in transducers is the capacitive micromachined ultrasonic transducer (CMUT) [45]. These small devices have a space under vacuum to create a miniature capacitor between two electrode surfaces. The top surface deflects when a voltage is applied or produces a voltage when a deflection occurs (basically a capacitive microphone).

CRC_DK8307_CH010.indd 438

10/27/2008 11:36:43 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

439

Their sensitivity and bandwidth are both high, and their density can be high, because they are micromachined. They can potentially produce very high-resolution images at shallow depths. 10.3.7

Ultrasonic Exposimetry

In the initial sections of this chapter, the use of ultrasound to coagulate and ablate tissue is presented. The goal in diagnostic imaging is to interrogate the body and leave it unchanged. Clearly, systems must limit their intensities and times of exposure to avoid irreversible tissue damage. Two mechanisms lead to tissue damage. The first is inertial cavitation, which is the rapid growth and violent collapse of gas or vapor bubbles within a host liquid. Upon collapse, pressures and temperatures within the bubble reach extremely high values. When these occur near a solid surface, such as an organ or bone, the surface is rapidly eroded. The classical example of the destructive power of cavitation is the erosion of marine propellers by cavitation. Thus, the potential for tissue damage is also high with the onset of cavitation. The second mechanism is elevated temperatures. As tissue temperature increases, irreversible damage occurs for periods of relatively long exposure. As an example, the goal of hyperthermia treatment of cancer is to increase the temperature to 45°C. Without placing sensors in the tissue of interest, these conditions cannot be measured directly. So the ultrasonic imaging industry has developed two indices: the mechanical index (MI) and thermal index (TI). These are indications of when damage is likely. They are not accurate predictors of the effects, but nonetheless do set defined standard safety limits for imaging systems. The MI combines two factors that influence the onset of inertial cavitations. One is the peak rarefaction pressure (derated). This is the driving force for violent collapse. The other is frequency, fc. The cavitation threshold depends on the inverse of the square root of the frequency. So the MI becomes MI 

pr ,3 fc

(10.8)

The pressure, pr,3 is a derated value of the peak rarefaction in water that is used to characterize the system. The derating factor represents a conservative average value for tissue. The Food and Drug Administration (FDA) has recommended limits on the MI. The limit is 1.9 for most soft tissues. One exception is the eye, its limit is only 0.23. The TI is based on the temperature rise in tissue. The bioheat equation is often used to model temperature response to thermal inputs and its dependence on various factors: tc t

dT d  dT   t (T  Tb )
q p
q m  k    bc bm dt dx  dx 

(10.9)

where ρ is the density, c the heat capacity, q̇ the heat absorbed or generated, ṁ the volumetric perfusion flow rate (in cubic meters per kilogram per second), the subscripts denote: t the tissue, b the blood, p the power in, and m the metabolic heating. The power in, q̇ p, is the ultrasonic energy absorbed by the tissue. For a single frequency, this value is given by q p  2I(x, y, z)

CRC_DK8307_CH010.indd 439

(10.10)

10/27/2008 11:36:43 AM

Ultrasonics: Data, Equations, and Their Practical Uses

440 TABLE 10.2

Perfusion Times and Lengths Tissue

Perfusion Time, τ (s)

Perfusion Length, L (mm)

Fat Muscle Liver Heart

4000 2140 98 69

19.5 18.0 3.8 3.2

Source: Szabo, T.L., Diagnostic Ultrasound Imaging: Inside Out, Elsevier Academic Press, Boston, MA, 2004.

The metabolic heating is usually considered small or can be viewed as the heat necessary to keep body temperature at 37°C. Therefore, the metabolic term can be ignored under normal conditions. The perfusion term, m ̇ represents the cooling due to blood flow and must be included in calculation. Nyborg [46] manipulated the bioheat equation to show that the temperature, T, from a small heat source has a steady-state spatial exponential falloff with distance, r, given by T2

C exp ( r/L) r

(10.11a)

where L is a characteristic length and C is a constant that depends on the source and the tissue properties. Nyborg also showed that the after the source is turned off, the temperature falls off as an exponential: T  To exp(t/)

(10.11b)

where τ is a characteristic time. The characteristic time and length are specific for a given tissue. As seen from Equations 10.9 through 10.11, the actual temperature rise and heat required depends on local tissue properties and local perfusion. Characteristic perfusion times and lengths are given in Table 10.2. The TI is the ratio of delivered power to the average power required to raise the tissue temperature 1°C: TI 

W W1° C

(10.12)

The values of W1°C are based on a perfusion length of 10 mm. The values of power are calculated at positions most relevant for the tissue being imaged. There are three common thermal indices: soft tissue (TIS), bone (TIB), and cranium (TIC). In the cases of TIS and TIC, the location is the skin surface directly under the transducer. The TIB index was developed for fetal monitoring, where the bones of the fetus are at the focal depth. Bone has higher absorption than soft tissue and so could potentially reach much higher temperatures compared to the neighboring soft tissues. 10.3.8

Advancement in Diagnostic Imaging

The system capabilities of diagnostic ultrasonic imagers continually improve. However, the most dramatic improvements evolve by understanding and employing the various ultrasound effects on tissue. This section covers a few of these advancements.

CRC_DK8307_CH010.indd 440

10/27/2008 11:36:43 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging 10.3.8.1

441

Contrast Agents

Contrast agents are small spheres of gas such as perfluorocarbons or air. The gas is often encased or captured in an elastic shell or within a surfactant. The criteria for a good contrast agent are that it is highly echogenic, and it has a diameter in the range of 2–6 µm. The first criterion is by definition a must-have for a contrast agent. The second criterion basically sizes the contrast agents to the size of red blood cells, which allows the agent to pass through the small capillaries of the lung and move on into the circulatory system. Contrast agents are introduced by either a bolus injection or through IV infusion. These agents are echogenic, because they grow and shrink in a nonlinear fashion, even in a linear acoustic field. The bubbles are relatively free to expand, but can compress only so much. The effective backscattering cross section may be 100 times greater than that of a rigid sphere, and the nonlinearity creates harmonics of the drive frequency. When insonified, the agents have different behaviors, depending on the pressures and nominal diameters. At low pressures in terms of MI, the bubble behavior is linear with symmetric uniform expansion and contractions. At higher pressures and bubble sizes corresponding to the drive frequency, stable nonlinear bubble vibrations occur. At even higher pressures, the shells can fragment, even though the MI is below FDA-recommended limits. When the shells are shattered, the scattering is typical of simple gas bubbles. Once shattered, the gas bubbles are absorbed within a fraction of a second, and their echoes fade away. The contrast agents can be viewed as small micro vesicles to carry and deliver drugs. The most general concept is to focus the beam of imaging system in the structure of interest and increase the pressure to fragment the shells and release the drugs. An effective level of drug could be delivered locally, while the systemic level of drug is lower, to avoid side effects. Methods for driving the shells or attracting the shells to a preferred site are also under development. By so doing, the concentration of vesicles is increased and a higher dosage delivered when the shells are fragmented. Suggested applications include delivering thrombolytic drugs to thrombi and plaque, and cytotoxic drugs to the weak small vessels that develop by the process of angiogenesis to feed a tumor. The objective of this approach is to starve the tumor to death. 10.3.8.2

Sonoelastography

Sonoelastography is the study of the elastic properties of tissue with induced or imposed mechanical vibrations at frequency <10 Hz. The concept is relatively simple. Stiffer materials show less vibration than softer tissues. The basic concept is to vibrate the tissue, and detect the vibration amplitudes with Doppler (M-mode) and overlay on a B-scan. So rather than use Doppler to detect and display blood flow, sonoelastography is used to detect and display minute tissue movements. Dramatic defi nition of tumor boundaries has been achieved using sonoelastography (see Figure 10.24; [47]). Sonoelastography can be viewed as highly sensitive ultrasonic palpation that produces a high-resolution 2D image. Driving the skin with what amounts to mechanical shakers or speakers can impose mechanical vibrations. This approach is effective in detecting large volumetric changes, such as cirrhosis of the liver. But this approach requires additional equipment and setup beyond the ultrasonic imager. A more eloquent approach is to combine: (1) the mixing of two close frequencies to produce a low-frequency beat frequency, (2) the ensuing radiation force at the low frequencies to vibrate the structures, and (3) Doppler ultrasound to detect and display these vibrations.

CRC_DK8307_CH010.indd 441

10/27/2008 11:36:44 AM

Ultrasonics: Data, Equations, and Their Practical Uses

442

FIGURE 10.24 A real-time elastography image of a right breast with a tumor. The elastographic image on the left shows the tumor as a well-defined sphere. In the conventional B-Scan on the right, the backside of the tumor is marked by the shadowing effect of ultrasound. (Courtesy of Hitachi Medical Systems.)

When two frequencies are mixed, frequencies at their sum and difference are produced. The difference frequency is used to produce the mechanical vibrations of sonoelastography. As an example, a 5.000 MHz signal mixed with a 5.001 MHz signal produces a difference frequency at 1 kHz. Radiation force is given by F  W/c

(10.13)

where W is the time average power and c the speed of sound. Given the nominal speed of 1540 m/s in tissue, the force is 0.65 mN/W. For a diagnostic beam of 100 mW/cm2, the force on a square centimeter target is 65 mN. So even though the radiation force appears relatively low, it still proves effective. The actual value of force depends also on the nature of the interface and its angle relative to the beam. Greenleaf [48] and his coworkers are currently investigating use of the resonance response of a targeted structure. An estimate of the elasticity of an artery is obtained from its low-frequency resonances. The assumption is made that an artery behaves as a cylindrical shell, and its resonances depend on the diameter, density, and elasticity. By assuming values for density and diameter, an estimate of elasticity can be obtained from a single mode. If several modes are identified, then all three parameters can be estimated, in principle. The radiation force or pressure even in a uniform medium has spatial variations. The spatial variations are pressure gradients. In a fluid medium, the gradients drive fluid flow, called microstreaming. Once again, the fluid motion can be detected with Doppler ultrasound, just as for observing blood flow. Nightingale and her coworkers [49] use this mechanism to differentiate between benign fluid-filled cysts and tumors in breast exams. 10.3.8.3

Intravascular Ultrasound

Vulnerable plaque is plaque on the inside of an arterial wall and has various forms and stages of development. Generally, plaque is sequestered by a thin fibrous layer. A plaque rupture can lead to blockage of the arteries and capillaries starving the distal tissues they feed. It can cause a blood clot at the site to create a coronary occlusion and a heart attack. It can break off and lead to blockage of the arteries of the brain to cause ischemic stroke or lodge deeper in the peripheral vascular system.

CRC_DK8307_CH010.indd 442

10/27/2008 11:36:44 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

443

FIGURE 10.25 Image of artery with plaque. (Courtesy of Volcano Co.)

Intravascular ultrasound (IVUS) is a high-frequency ultrasonic transducer at the distal end of a catheter. The frequency of an IVUS system frequency can be especially high, because it is imaging the coronary arteries for vulnerable plaque, stent placement and deployment, and other conditions of the structure. The high frequency allows high spatial resolution to image the layers of the arterial wall: adventitia, media, and intima. An IVUS image is shown in Figure 10.25. The deployment and use of an IVUS catheter is basically the same as other catheterizations. The cardiologist inserts the catheter/guide wire through a needle puncture in the femoral artery of the leg. The catheter is then snaked up through the artery to the aorta into the left or right coronary arteries. Under fluoroscopy, the imaging portion of the IVUS tip is position in the region of interest. The transducer is used in an imaging mode and rotated 360°. The data are used to construct a 2D circumferential scan of the artery. From the image, the cardiologist can ascertain the severity and stability of the plaque and then prescribe appropriate treatments or medications.

10.4

Summary

Medical ultrasound started with diagnostic imaging, and later moved into therapy. Imaging and therapies were located as two different applications, and this distinction remained so until the 1990s. At that time, HIFU and contrast agents were emerging that combined imaging and therapy for see and treat procedures. While ultrasound imagers and ultrasonic therapies are mainstrays in the medical profession today, the see and treat combinations, promise to improve patient outcomes and to shift the site care away from the operating room for treatment of some conditions.

CRC_DK8307_CH010.indd 443

10/27/2008 11:36:44 AM

444

Ultrasonics: Data, Equations, and Their Practical Uses

The use of ultrasound in medicine continues to expand. Much of the future growth will be fueled by advancements in allied technologies, such as signal and image processing, computing power and speed, transducer materials, and further miniaturization of electronics. Not touched upon herein are the areas of ultrasonic micromachined electromechanical systems (MEMS), and ultrasonic motors and actuators. Their combination may allow for more precise targeting, ultrasound-based micro-analyzers and use for manipulation of instruments in MRI-enabled procedures. These technological advances will help expand the capabilities and utility of ultrasound in medicine. However, whole new approaches and applications will be driven by a greater understanding of the interactions of ultrasound energy with healthy and diseased tissues even down to the cellular level.

References 1. Amaral, J.F., and Berci, G., Laparoscopic application of an ultrasonically activated scalpel, Gastrointest. Endosc. Clin. N. Am., 3(2), 381–392, 1993. 2. US Patent No. 3,086,288. 3. Balamuth, L., and Kuris, A., Ultrasonically vibrated cutting knives, U.S. Patent No. 3,086,288, April 23, 1963. 4. Ethicon Endo-Surgery. 5. Lucas, M., MacBeath, A., McColloch, E., and Cardoni, A., A finite element model for ultrasonic cutting, Ultrasonics, 44(suppl.), e503–e509, 2006. 6. MacBeath A., Ultrasonic Bone Cutting, Ph.D. Thesis, University of Glasgow, 2006. 7. Goldstein, J.L., How a jolt and a bolt in a dentist’s chair revolutionized cataract surgery, Nat. Med., 10(10), xix–xx, 2004. 8. Alcon. 9. Packer, M., Fishkind, W., Fine, I.H., and Seibel, B.S., and Hoffman, R.S., The physics of phaco: A review, J. Cataract Refract. Surg., 31, 424–413, 2005. 10. Integra LifeSciences. 11. Fasano, V.A., Zeme, S., Frego, L., and Gunetti, R., Ultrasonic aspiration in the surgical treatment of intracranial tumors, J. Neurosurg. Sci., 25(1), 35–40, 1981. 12. Fong, S.W., Klaseboer, E., Turangan, C.K., Khoo, B.C., and Hung, K.C., Numerical analysis of a gas bubble near bio-materials in an ultrasound field, Ultrasound Med. Biol., 32(6), 925–942, 2006. 13. Elsevier Limited. 14. Mentor, Inc., and Misonix. 15. Rassweiler, J.J., Tailly, G.G., and Chaussy, C., Progress in lithotriptor technology, EAU Update Series, 3(1, spec. iss.), 17–36, 2005. 16. ter Haar, G., and Coussios, C., High intensity focused ultrasound: Physical principles and devices, Int. J. Hyperthermia, 23(2), 89–104, 2007. 17. Focus Surgery, Inc. 18. Takai Hospital Supply Co., Ltd. 19. ProRhythm, Inc. 20. UtraShape. 21. Wolff, J., Das Gesetz der Transformation der Knochen, Berlin August Hirschwald, 1892. 22. Dyson, M., and Brooks, M., Stimulation of bone repair by ultrasound, Ultrasound Med. Biol., Suppl. 2, 61–66, 1983. 23. Duarte, L.R., The stimulation of bone growth by ultrasound, Arch. Orthop. Trauma. Surg., 101(2), 153–159, 1983.

CRC_DK8307_CH010.indd 444

10/27/2008 11:36:45 AM

Medical Ultrasound: Therapeutic and Diagnostic Imaging

445

24. Smith & Nephew, Inc. 25. Yang, K-H., Parvizi, J., Wang, S-J., Lewallen, D.G., Kinnick, R.R., Greenleaf, J.F., and Bolander, M.E., Exposure to low-intensity ultrasound increase, aggrecan gene expression in rat femur fracture model, J. Orthop. Res., 14(5), 802–809, 1996. 26. Cook, S.D., Ryaby, J.R., McCabe, J., Frey, J.J., Heckman, J.D., and Kristiansen, T.K., Acceleration of tibia and distal radius fracture healing in patients who smoke, Clin. Orthop. Relat. Res., 337, 198–207, 1997. 27. Celleration, Inc. 28. Ennis, W.J., Formann, P., Mosen, N., Massey, J., Sonner-Kerr, T., and Meneses, P., Ultrasound therapy for recalcitrant diabetic foot ulcers: Results of a randomized, double-blind, controlled, multicenter study, Ostomy Wound Manage., 1(8), 24–39, 2005. 29. Thawer, H.A., and Houghton, P.E., Effects of ultrasound delivered through a mist of saline to wounds in mice with diabetes mellitus, J. Wound Care, 12(5), 171–176, 2004. 30. Kavros, S.J., Wagner, S.A., Wennberg, P.W., and Cockerill, F.R., The effect of ultrasound mist therapy on common bacterial wound pathogens, presented at Symposium of Advanced Wound Care, 2002. 31. King, W.W.K., Zekri, A., Lee, D.W.H., and Li, A.K.C., Debridement of burn wounds with a surgical ultrasonic aspirator, Burns, 22(4), 307–309, 1996. 32. Tan, J., Abisi, S., Smith, A., and Burnand, K.G., A painless method of ultrasonically assisted debridement of chronic leg ulcers: A pilot study, Eur. J. Vasc. Endovasc. Surg., 33(2), 234–238, 2007. 33. Weir, D., Blakely, M., and Chakavarthy, D., Enhanced wound bed preparation and healing outcomes utilizing low-frequency ultrasound assisted wound treatment (UAWT), poster presentation, 2006 Advanced Wound Care Congress, Colorado Springs, CO, 2006. 34. Mitragotri, S., and Kost, J., Low-frequency sonophoresis: A noninvasive method of drug delivery and diagnostics, Biotechnol. Prog., 16(3), 488–492, 2000. 35. Sontra Medical Corporation. 36. Richmond, J.C., A comparison of ultrasonic suture welding and traditional knotting, Am. J. Sports Med., 29(3), 297–299, 2001. 37. Axya Corporation. 38. OmniSonics™, Inc. 39. Szabo, T.L., Diagnostic Ultrasound Imaging: Inside Out, Elsevier Academic Press, Boston, MA, 2004. 40. Ludwig, G.D., The velocity of sound through tissues and the acoustic impedance of tissues, J. Acoust. Soc. Am., 22, 862–866, 1950. 41. Volcano Co. 42. Philips Medical Systems. 43. Duck, F.A., Physical Properties of Tissue: A Comprehensive Reference Book, Academic Press, London, 1990. 44. Justice, J.H., Owsley, N.L., Yen, J.L., and Kak, A.C., Array Signal Processing, Prentice-Hall, Inc., New Jersey, 1985. 45. Wong, S.H., Wygant, I.O., Yeh, D.T., Zhuang, X., Bayram, B., Kupnik, M., Oralkan, O., Ergun, A.S., Yaralioglu, G.G., and Khuri-Yakub, B.T., Capacitive micromachined ultrasonic transducer arras for integrated diagnostic/therapeutic catheters, AIP Conference Proceedings, 829, 395–399, 2006. 46. Nyborg, W.L., Solutions to the bio-heat transfer equation, Phys. Med. Biol., 33(7), 785–792, 1988. 47. Hitachi Medical Systems. 48. Zhang, X.M., and Greenleaf, J.F., An anisotropic model for frequency analysis of arterial walls with the wave propagation approach, Appl. Phys., 68(9), 953–969, 2007. 49. Soo, M.S., Ghate, S.V., Baker, J.A., Rosen, E.L., Walsh, R., Warwick, B.N., Ramachandran, A.R., and Nightingale, K.R., Streaming detection for evaluation of indeterminate sonographic breast masses: A pilot study, Am. J. Roentgenol., 186(5), 1335–1341, 2006.

CRC_DK8307_CH010.indd 445

10/27/2008 11:36:45 AM

CRC_DK8307_CH010.indd 446

10/27/2008 11:36:45 AM

11 Mechanical Effects of Ultrasonic Energy Dale Ensminger

CONTENTS 11.1 Introduction .......................................................................................................................447 11.2 Atomization .......................................................................................................................448 11.2.1 Atomization by Focusing on a Liquid Surface................................................448 11.2.2 Whistles ................................................................................................................448 11.3 Small Uniform Drops from a Liquid Jet........................................................................ 449 11.4 Degassing Liquids ............................................................................................................ 449 11.5 Drying and Dewatering .................................................................................................. 450 11.5.1 Acoustical Drying ............................................................................................... 450 11.5.2 Electroacoustic Dewatering ............................................................................... 451 11.6 Curing Epoxy .................................................................................................................... 453 11.7 Ultrasonic Tools ................................................................................................................ 453 11.8 Wire and Bar Drawing..................................................................................................... 455 11.9 Ultrasonic Soldering......................................................................................................... 459 11.10 Treatments in Hostile Environments ............................................................................. 459 11.10.1 Degassing Molten Glass.................................................................................... 460 11.10.2 Treating Materials of Low and High pH........................................................ 461 11.11 Compaction of Metal Parts .............................................................................................. 462 11.12 Metallurgical Processes ...................................................................................................464 11.13 Deburring ..........................................................................................................................465 11.14 Accelerated Fatigue Testing............................................................................................. 466 11.15 Welding of Metals and Plastics....................................................................................... 467 11.16 Conclusions ........................................................................................................................ 469 References .................................................................................................................................... 470

11.1

Introduction

Ultrasonic energy is mechanical energy. Its transmission is dependent upon the elastic properties and the densities of the media through which it is propagated. The stresses associated with the propagation of ultrasonic waves are the basic cause of the numerous mechanical effects attributable to applying ultrasonic energy. The stresses may operate directly or may be converted into thermal energy by absorption or into chemical energy by their effects upon the molecular conditions of the materials. Examples of the direct effect of ultrasonic stresses are breaking particles down into smaller particles, emulsification, 447

CRC_DK8307_CH011.indd 447

10/27/2008 2:16:10 PM

Ultrasonics: Data, Equations, and Their Practical Uses

448

degassing of liquids, drying and dewatering of materials, ultrasonic machining, atomization of liquids, and metal forming. Examples of thermal effects of ultrasonic irradiation are ultrasonic welding of polymers and metals. Chemical factors include ultrasonic cleaning involving assisting chemical attack of contaminants, acceleration of chemical reactions by mixing and curing epoxy materials, and the production of new products by accelerating reactions. The objective of this chapter is to review the mechanisms of several ultrasonic operations.

11.2

Atomization

Ultrasonic atomization is accomplished by several means: by focusing high-frequency ultrasonic energy on the surface of a liquid in a bowl-shaped transducer (0.4–10.0 MHz), by ultrasonically vibrating a surface over which the liquid flows (18–100 kHz), or by feeding the fluid into the active zone of a whistle (8–30 kHz). Small droplets of a uniform size may be formed by feeding the fluid at a controlled rate through a small orifice in the tip of a horn vibrating ultrasonically in a longitudinal mode [1, 467–474]. 11.2.1

Atomization by Focusing on a Liquid Surface

Focusing ultrasonic energy from beneath the surface of the liquid in a bowl-shaped transducer operating at frequencies from 0.4 MHz to 10.0 MHz produces a disturbance with characteristics that depend primarily upon the intensity of the ultrasound. Very low intensities produce a small bulge on the surface around the focal region. Higher intensities produce a fountain. Very high intensities produce a fog of small particles, which is the principle of operation of small ultrasonic atomizers for medical inhalants. A fog of medical inhalants is formed of particles small enough to be breathed deep into the lungs with very little loss within the bronchial tubes [1, p. 467]. 11.2.2

Whistles

The use of whistles for coating very fine particles and for breaking down foams was discussed in Chapter 6. When a stream of fluids is introduced into the active zone of a stemjet whistle, the stream is shattered into small particles or droplets. The best position to accomplish good atomization is within a region slightly downstream of the point at which the sound wave is fully formed. The stem-jet whistle offers a unique method of applying a thin coat of polymers to individual fine particles without agglomeration. For example, a stream of particles 1–5 µm in diameter can be fed into the active zone through a port in the reflector. These are introduced slightly downstream from the active zone to prevent interfering with the formation of the sound wave. The coating material is introduced slightly beyond the particle supply through a second port. The resulting fog leaves a coating on each individual particle. The polymer is cured as it coats the particles and no agglomerates appear in the end product. These are modified Hartmann whistles. They operate at frequencies from 10 to 30 kHz for practical reasons. Lower frequencies are also possible (see Figure 6.8). The second application mentioned in Chapter 6 was controlling foam in large vessels. Figure 11.1 shows schematically the method used. In this application, an oil that is used in large quantities in the production of soap is stored in large vessels. It is drained from the vessels into tank trucks and shipped to its final destination. To solve the truckers’

CRC_DK8307_CH011.indd 448

10/27/2008 2:16:11 PM

Mechanical Effects of Ultrasonic Energy

449

Whistle

Exhaust

Foam

Fluid fill

FIGURE 11.1 Controlling foam in large vessels.

complaints of excessive foaming during shipping, one 15 kHz stem-jet whistle is applied to the cavity above the oil in each of the large vessels as it is being filled. This action has proved to be effective in reducing the foaming sufficiently for shipping in the trucks. The whistle is energized by compressed nitrogen. The intensity of the sound wave generated by the whistle is very high (exceeding 145 dB) within the tank, however, the radiated sound is reduced to a safe level outside of the tank by the tank’s structure.

11.3

Small Uniform Drops from a Liquid Jet

Ultrasonic energy applied to small jets of liquid flowing through an axial orifice in a longitudinally vibrating horn can produce a series of small droplets of uniform size. Occasionally, minor satellite droplets are formed between the larger droplets on the axis of the stream. An equation for determining the optimum frequency for controlling the size of droplets formed from liquid jets is f = uj/4.508dj

(11.1)

where uj is the velocity of the jet and dj the diameter of the jet [1, p. 472]. The distance the jet moves during each cycle is given by

= u/f

(11.2)

This process is used in some ink-jet printers.

11.4

Degassing Liquids

When sonic energy is applied to liquids containing gas, various results are obtained, depending upon the intensity, direction of exposure (horizontal, vertical, or at an angle with the horizontal), frequency, and hydrostatic pressure. The application of an acoustic wave

CRC_DK8307_CH011.indd 449

10/27/2008 2:16:11 PM

Ultrasonics: Data, Equations, and Their Practical Uses

450

to a liquid containing gases causes (1) migration of bubbles already present in the fluid, (2) release of dissolved gases into small bubbles at low intensities, (3) release of larger bubbles at moderate intensities, and (4) formation of collapsible cavitation bubbles at high intensities. These conditions may be associated with (1) quiet degassing, (2) linear resonance of gas bubbles, and (3) nonlinear collapse of vapor bubbles. The larger bubbles resulting from (1) and (2) remain after the power is terminated. The smaller bubbles redissolve. When small bubbles coalesce, they form larger bubbles that move toward the top of the vat at a rate dependent upon bubble size and viscosity of the fluid. At very low intensities, the bubbles migrate in a direction depending upon the status of the sound wave (i.e., progressive or standing, vertical or horizontal). When the sound wave is cut off the bubbles rise by buoyancy, being much lighter than the liquid that they displace. Bubbles of radius R rise in a free field, according to Stokes law, at a rate given by v = 2 0gR2/9

(11.3)

where ρ 0 is the density of the liquid and η is the viscosity of the liquid. The bubbles grow in an acoustic field at a rate determined by the difference in the exchange of gas between the inside of the bubbles and the liquid surrounding the bubbles. Letting internal pressure of the bubble be pb and the liquid pressure be p, the difference in pressure is given by pb − p = 2 /R

(11.4)

where σ is the surface tension of the liquid. When pb < p, the gas goes from outside to inside. When pb > p, the direction is outward [2]. An acoustic wave at low intensities traveling through a maze of small bubbles causes the bubbles to coalesce into larger bubbles. Larger bubbles rise at a faster rate than the smaller ones. Very high intensities cause larger bubbles to break up into smaller ones.

11.5 11.5.1

Drying and Dewatering [1, pp. 474–478] Acoustical Drying

Acoustical drying is best used for drying heat-sensitive materials, especially those that degrade in a moisture environment, and chemicals that have a long drying cycle. Production of pharmaceutical materials is one area in which acoustic drying could potentially be a benefit. The cost of the drying process is offset by the protection from degradation of the materials that could be caused by the delays in drying or the heat that might be necessary otherwise. Much drying has been done by air-jet types of whistles emitting intensities exceeding 145 dB at frequencies between 6 and 15 kHz. Operators have to be protected from such intense sound at these frequencies through the use of acoustically prepared chambers into which the materials to be dried are placed. The materials are dried in layers less than 2.5 cm (1 in.) to 5 cm (2 in.) thick. The exotic application of sonic energy to drying is interesting, and it is practical for meeting the general small-scale needs of industry.

CRC_DK8307_CH011.indd 450

10/27/2008 2:16:12 PM

Mechanical Effects of Ultrasonic Energy 11.5.2

451

Electroacoustic Dewatering

Very often, combining two forms of energy produces results on a scale that improves upon the results of either alone. In the case of electroacoustic dewatering, electrical and acoustical methods combine to remove water from a product without the expense of boiling the water from the product. In most cases, removing water from a product is a two-stage process; the two stages are dewatering and drying. Dewatering is illustrated by wringing water from clothes after they have been laundered. Drying is the familiar process that follows the dewatering (wringing) operation. A significant amount of the energy for drying is consumed in supplying the heat of vaporization, which is approximately 1.055 × 106 joules per pound of water evaporated under standard atmospheric conditions. Any process that can remove a large percentage of the water from a product prior to applying heat will reduce the thermal load on the dryer accordingly. If the dewatering process costs less than the drying process, the overall result might be a financial savings in total costs. The effectiveness of the dewatering process depends upon the content of the types of water associated with solid particles. There are four types of water to consider: 1. Bulk or free water, which is water present on or intermingled with the product but not bound chemically or mechanically to it. Some of this water can be separated by conventional equipment. 2. Micropore water, which is water located in micropores and capillaries of the product. Some of this type of water can be removed by filter presses. 3. Colloidally bound water, which is water held by strong surface forces to the particles—particularly those of colloidal size—to which it is attached. This type of water cannot be removed by filters, centrifuges, filter presses, or similar mechanical devices. 4. Chemisorbed water, which is water bound chemically onto the surface layer of an absorbent. Chemisorbed water cannot be released by filters, centrifuges, filter presses, or similar mechanical devices. Systems requiring dewatering fall into two categories: (1) low-moisture-content systems, in which the space between solids (such as particles and fibers) to be dried contains considerable air and (2) high-moisture-content systems, or continuous phase systems, in which all potentially void space is completely filled with moisture and the systems have the characteristics of a liquid or a solid [3,4]. The water in the low-moisture category may be in three forms: free, entrapped, or bound. The free liquid appears in a thin film on the surface of the material to be dried and the material may have the appearance of a dry product. The entrapped water includes micropore water, colloidally bound water, and chemisorbed water. In the high-moisture category, the liquid fills the available volume of the product, therefore showing an appreciably greater amount of water than the first category. It may also appear in the free, entrapped, or bound state. The effective dewatering mechanisms differ for the two conditions. It is important to know the mechanisms associated with each form of energy to be applied to design an optimum combined system. Some very promising techniques for dewatering and materials separation have been developed by combining acoustical and electrical phenomena both with and without vacuum [3–5]. The acoustical mechanisms may be categorized into groups of (1) those that

CRC_DK8307_CH011.indd 451

10/27/2008 2:16:12 PM

452

Ultrasonics: Data, Equations, and Their Practical Uses

play a major direct role in the separation process and (2) those that assist the electrical forces (electrokinetic phenomena) in their role in the separation process. Acoustical mechanisms most closely associated with dewatering in the high-moisture content systems and where the acoustical energy is expected to have a major role in dewatering are radiation forces, and, in some combinations of media, possible localized electrical charge generation appearing on the surfaces of particles through mechanical movement of one of the phases. Cavitation produces free chemical radicals, which may increase the rate at which electrophoresis (see explanation shortly) is able to separate the solid materials from the water. However, cavitation is energy-intensive and interferes with wave propagation, making it difficult to treat a large volume uniformly. Radiation pressure at intensity levels below cavitation can help move particles unidirectionally away from an acoustic source. The same mechanisms are active in low-moisture-content systems, but not necessarily to the same intensities. Other mechanisms may play a part, such as low-temperature boiling off or accelerated evaporation during the rarefaction phase of each cycle and strong turbulence in the gas phase surrounding the particles. Among acoustical phenomena that assist the electrokinetic forces are localized cleaning of electrodes to maintain good electrical contact with the media to be separated, acceleration of moisture transfer and removal through filters or screens, and ultrasonically assisted compaction of solids as moisture is removed to maintain more sustained electrical continuity through a cake and increase the amount of fluid removal. The two electrokinetic principles that are utilized in dewatering are electrophoresis and electro-osmosis. Electrophoresis refers to the movement of solid particles through a relatively stationary liquid under the influence of an electric field. Electro-osmosis refers to the movement of a fluid through a relatively stationary permeable membrane or porous cake under the influence of an electric field. The direction in which the particle moves depends upon the sign of the electrical charge on the particle. The rate at which it moves is a function of the strength of the electrical field. Thus, in a broad sense, electrophoresis is used in a high-moisture system to move particles in any desired direction, particularly away from filter media, to prevent or delay clogging the filter until the bulk of the free water has been removed through the filter. Electro-osmosis is used to move water through a porous cake. By combining ultrasonics and electrokinetic principles, either with or without vacuum, it is possible to obtain synergistic effects. As with any other high-intensity application of ultrasonic energy, the best results are obtained by considering several factors related to the effects on the products, the mechanisms that offer the greatest benefit to the process, and the nature of the product as it is being treated. Reviewing these factors with the various principles of electrokinetics and ultrasonics may reveal when and how to apply ultrasound to the process. Important factors include not only the mechanisms that promote dewatering but also other mechanisms, such as flocculation (particularly of fibrous materials) and degassing, and whether these might have beneficial or deleterious effects on the product. Figure 11.2 shows schematically a plan combining ultrasonic energy with an electric field to assist in dewatering. Vacuum may be applied on the side of the ultrasonic horn and an electric field can be applied across the sludge cake to increase the total amount of water removed from the cake by a significant amount. Probably not all of the acoustic mechanisms have been recognized, but it is possible to identify the following: 1. Greater compaction of the cake through the influence of ultrasonic stresses across the cake. This mechanism maintains electrical continuity across the cake for a greater period of time and increases the amount of water that can be removed by electro-osmosis.

CRC_DK8307_CH011.indd 452

10/27/2008 2:16:12 PM

Mechanical Effects of Ultrasonic Energy

453 Applied force V

Sludge cake Filter Ultrasonic horn

Gr

FIGURE 11.2 Electroacoustic dewatering.

2. Ultrasonic removal of materials that tend to coat the electrode surfaces, thus promoting better electrical contact with the product. 3. Ultrasonic removal of liquid droplets from the bottom of the filter by inertial forces. 4. Ultrasonic assistance in transferring moisture through the filter. Thus, the local activity of the ultrasound, its effect on the cake (which does not require excessively high intensities) and the electric field, the vacuum (not shown), and the applied pressure in operating on the bulk of the cake cooperate to provide a very effective dewatering system.

11.6

Curing Epoxy

Applying ultrasound at 20–50 kHz removes the gas bubbles from epoxies and accelerates the mixing and the curing process. Applying the ultrasound at too high of an intensity may produce an excessive exothermic reaction, resulting in very rapid darkening of the specimen. Applications of this effect include not only the mixing and degassing of the components of an adhesive material but it has also been used to completely encapsulate products such as nondestructive testing (NDT) transducers to make them safe for use underwater (see Figure 11.3).

11.7

Ultrasonic Tools

Applying ultrasound to cutting tools improves the ease and speed of cutting. The motion helps overcome the frictional drag. Applying ultrasound to a rotary drill eases the drilling in materials that are otherwise difficult to machine. A twist drill is driven parallel to the length. The need for changing the cutting tools often due to wear depends upon how hard and tough the tool materials are relative to the work (see Figure 11.4).

CRC_DK8307_CH011.indd 453

10/27/2008 2:16:12 PM

Ultrasonics: Data, Equations, and Their Practical Uses

454

Ultrasonic transducer

Ultrasonic horn

Epoxy mix

FIGURE 11.3 Ultrasonic curing of epoxies.

Transducer

Coupling horn

Workpiece Drill

FIGURE 11.4 Ultrasonically assisted rotary drilling.

Applying ultrasonic energy to conventional lathe tools and grinding operations can be beneficial [6]. Ultrasonically agitating the tool can eliminate the poor surface quality often observed when cutting light metal alloys at low speeds. The effect drops sharply with increasing cutting speed. The amount of improvement is dependent primarily upon vibratory motion of the tool in the direction of main cutting forces, increasing as the ratio of maximum vibratory velocity to cutting speed increases. Grinding with ultrasonic assistance can produce results superior to those obtained with regular grinding [7]. Fatigue properties of the work are not impaired and burning is almost totally absent. Vibration-assisted grinding with heavy cuts produces less chatter

CRC_DK8307_CH011.indd 454

10/27/2008 2:16:12 PM

Mechanical Effects of Ultrasonic Energy

455

and a generally superior surface finish. When properly applied, ultrasonically assisted grinding may use about a third as much power on the spindle as does an identical but unaided grind. The amount of energy necessary depends upon the force applied between the ultrasonic tool and the work. In all applications of ultrasonic energy, whether they are forming operations or any other types, the design of the system must conform to good, basic ultrasonic principles to ensure that the ultrasonic energy is delivered into the proper regions to be treated and in the anticipated most effective manner in order for the ultrasound to be effective. Ultrasonic machining using an abrasive slurry was promoted extensively during the late 1940s and 1950s. It was used primarily to form holes and shaped impressions in ceramic materials. A tool shaped according to the impression to be made was attached to the end of an ultrasonic horn vibrating in a longitudinal mode. An abrasive slurry of boron carbide was washed between the object and the vibrating tool brought into close proximity to the material to be formed. The intense vibrations produced cavitation, which accelerated the particles of the slurry into the work to erode the surface of the work away according to the shape of the tool face. The abrasive slurry was recirculated. Over a period of time, the surfaces of the slurry particles became smooth, reducing their machining effectiveness. A weighted force was applied to the tool system. Deep holes were difficult to machine ultrasonically with a hollow tool for two reasons: (1) the difficulty of delivering abrasive material to the tip of the tool increased with depth and (2) an air pocket would form in the hollow tool preventing deep penetration into the material. A small hole was drilled radially through the tool near the top to prevent entrapped air from forming the cushioning effect, which stopped the progress of the tool into the work. Adding weights to the transducer and tool assembly could not overcome the resistance of the cushion without the release hole. The slurry had to make contact with the cutting face of the tool for effective machining. Ultrasonic drilling was also applied in dentistry, in preparing cavities for filling teeth using aluminum oxide crystals instead of the boron carbide crystals used in the machining of hard ceramics. Patients would often experience pain as the ultrasound impinged on a nerve. New dental tools have replaced the ultrasonic units. Ultrasonics is now used occasionally for removing dental plaque, calculus, food debris, and stain. These operations are done without the slurry. It has also been applied to treating certain gum diseases. Today, laser machining of ceramics has replaced much of the previous ultrasonic abrasive machining, producing much faster and cleaner cuts.

11.8

Wire and Bar Drawing

Ultrasonics has been investigated as a possible means of enhancing many types of forming operations, including wire and bar drawing, tubing drawing, sheet metal rolling, densification of powder-metal parts, and grinding. Ultrasonically vibrating the dies reduces the force required to draw wires and bars when the movement of the dies is parallel to the drawing direction. Small copper wires can be pulled by hand through the vibrating dies. The drawing becomes much more difficult when the die vibration stops. Prolonged exposure of wires to intense ultrasound in the dies before and after forming can produce undesirable effects. Halting drawing with the die still in motion may lead to rapid fatigue failure at a position of maximum vibrational stress (see Figure 11.5).

CRC_DK8307_CH011.indd 455

10/27/2008 2:16:13 PM

Ultrasonics: Data, Equations, and Their Practical Uses

456 Transmission horn Piezoelectric crystals

Retaining block Die

Bar

Pulling block (Schematic) Pulling force

Transducer Node FIGURE 11.5 Ultrasonic bar drawing.

Ultrasonics applied to the die reduces the number of draws necessary to reduce bar diameters. How much to reduce per draw depends upon the hardness, the toughness, the stress applied by the ultrasonic coupling, the shape of the die, and the material of the die. There must be sufficient difference between the materials to prevent the bars from welding to the dies. The die must remain tough and hard enough to maintain rigid shape during the drawing process. Potential advantages of the application of ultrasonic energy to drawing applications include the increase in metal formability, the beneficial influence of ultrasonic energy in forming difficult-to-form materials such as case-hardened steel and tungsten, increased area reduction per pass, better size control, improved inside-diameter surface finish (tubing), greater tool life, and minimized chatter. The drawing force necessary for some forming processes can be reduced by more than 50% by using ultrasound. Two factors believed to improve formability of metals with the application of ultrasonic energy are (1) the softening effect on the crystals (volume effect) and (2) the reduction of frictional forces between the workpiece and the tool. A plate of soft-annealed copper as thick as 3.175 mm (1/8 in.) can be reduced to a thin foil by forcing it between a flat tip of an ultrasonically vibrating tool and an anvil when the spacing between the anvil and the tool is fixed. The drawing force necessary to maintain a continuous feed rate can be applied easily by hand. Different individuals have offered theories as to why ultrasound should be effective in drawing and forming operations. Following are some theories proposed to explain the volume effect relative to the drop in force required to draw materials under the influence of ultrasonics. 1. Thermal-equivalent theory, based upon the internal thermal energy of a solid being present in the form of incoherent atomic vibrations. The combination corresponds to an effectively increased kinetic energy content per atom. This argument suggests that ultrasonic energy introduced into a solid should have the same effect on the behavior of the metal as elevating the temperature of the material by an equivalent amount. Balamuth claimed to show experimental evidence of a drop in drawing force under ultrasonic excitation that exceeds the dynamic force applied [8]. Reduction in apparent static yield stress occurs under the influence of high-intensity ultrasonic energy for strain rates up to 100/s. 2. Plastic deformation from the total energy viewpoint. Acoustic strain energy interacts with materials in several time-dependent ways. Any resonance absorbs energy in the form of higher dynamic strains. Absorption due to internal friction or absorption at lattice defects or grain boundaries raises the total thermal energy

CRC_DK8307_CH011.indd 456

10/27/2008 2:16:13 PM

Mechanical Effects of Ultrasonic Energy

457

of the material. Some acoustic energy is converted to residual strain energy through plastic deformation [9]. Drawing forces can be reduced by 80% at drawing speeds of 30.5 m/min (100 ft/min). It has been claimed that yield strengths of metals, such as zinc and aluminum, are greatly increased by a few minutes application of ultrasonic energy at an acoustic pressure exceeding 2.5 × 107 dyne/cm2 [10]. The ultrasound oscillates and distributes the crystal lattice dislocations, similar to work hardening, but without change in shape. It was further claimed that yield strength in shear is increased 50% by radiation at 4.5 × 107 dyne/cm2 and as much as 150% at 5 × 107 dyne/cm2. Fracture along grain boundaries occurs at stresses exceeding 5.5 × 107 dyne/cm2. Investigations of the metallurgical effects of high-intensity ultrasound on a specimen of carbon steel reveal, via the microstructure of sections cut from the maximum stress zone, that grain boundaries between pearlite and ferrite are less sharp than those in untreated material [11]. Compressive deformation studies, with ultrasonic vibrations superimposed, show that the flow stress of compression can be reduced ultrasonically, but the effectiveness is dependent upon the acoustic impedance, Young’s modulus, melting point, work hardenability, and stacking-fault energy [12]. Temperature rise in the material with increase in intensity also is material-dependent. Utrasonics influences the hardness distribution of the material in accordance with the distribution of the stresses in the field. Work hardening and tensile strength of wires of face-centered cubic metals increases, especially when these materials (copper, gold, silver, and nickel) are treated at 20 kHz at various temperatures [13]. Experiments with copper, spring steel, and other steel alloys have shown that the internal friction can be reduced by ultrasound and considerable reduction of the external friction between the die and the workpiece can be achieved in wire- and tube-drawing operations [14]. The surface conditions and the microhardness of copper wires are similar to those of wires drawn without ultrasound. Multiplication of dislocations in nickel, copper, and aluminum (as well as lithium fluoride and sodium chloride crystals) begin at a certain threshold amplitude of vibrations, and this threshold decreases with increasing temperature [15]. Above the threshold amplitude, the dislocation density increases with the ultrasonic intensity. The dislocation density also increases linearly with time, initially, but afterward tends toward saturation. Saturation level increases with both increasing temperature and increasing amplitude of vibration. The use of ultrasonics during the tempering of cutting blades increases their wear resistance over that of the normal treatment [16]. Above the threshold amplitude, the dislocation density increases with the ultrasonic intensity. The dislocation density also increases linearly with time, initially, but afterward tends toward saturation. Saturation level increases both with increasing temperature and with increasing amplitude of vibration. Specimens of an alloy steel containing 0.87% carbon, 4% chromium, 6.3% tungsten, 2.5% molybdenum, and 1.7% vanadium were subjected to intense ultrasonics in a salt bath at 560° for periods from 15 to 60 min. Comparisons of hardness, impact and bending strength, red hardness, and wear resistance of these specimens with those of specimens similarly heattreated but without ultrasonics showed that (1) hardness increased with a 15 min application of ultrasonics over that of usual tempering, (2) impact strength and bending strength after a 60 min application of ultrasonics corresponded to values obtained after a 120 min normal tempering period, and (3) ultrasonics applied during tempering also increased the red hardness and wear resistance of this alloy. These effects probably are associated both with internal mechanisms and with mechanisms associated with heat transfer.

CRC_DK8307_CH011.indd 457

10/27/2008 2:16:14 PM

458

Ultrasonics: Data, Equations, and Their Practical Uses

The application of ultrasonic energy during a static test causes softening of a metal, but if the insonation is superimposed prior to the static deformation, the sample is usually hardened by an amount that depends upon the intensity of the ultrasound, temperature, and amount of prestrain [17]. Increasing prestrain is accomplished by decreasing hardening effect of ultrasonics until, at a certain level of prestrain, softening occurs. The decrease in hardening may be due to recovery and recrystallization. Recovery occurs for polycrystalline materials if the insonation of the samples is stopped before fracture occurs. Increases in hardening are less when insonation continues to fracture due to recrystallization. Ultrasonics induces the same types of dislocation structure as does alternating stress at low frequencies. Experiments with 18.7 kHz ultrasonic vibration applied to a drawing die for mild-steel wire with the oscillations in the axial direction indicated that the only reduction in force was attributable to the superposition of ultrasonic force on the drawing force rather than to heat and reduction of friction [18]. No reduction in drawing forces was observed for drawing speeds equal to or greater than the peak velocity of the oscillations. Specimens drawn under the influence of ultrasound revealed no changes in surface finish or mechanical properties [14]. Lehfeldt [19] believed that the lowering of the yield point under the influence of ultrasound is a spurious effect rather than a specific change in the material. If wires are drawn through orifices located at the node of a rod vibrating at 20 kHz—the vibration direction being perpendicular to the direction of drawing—the rotational symmetry of the drawing texture is destroyed [20]. Severdenko and Clubovich [21] found that vibration amplitudes of 0.012–0.02 mm at 25 kHz applied to a wire-drawing die for copper wire caused a 50% reduction in drawing force, but also increased the elongation with a corresponding reduction in quality, estimated to be ∼15%. The diameter of standard copper wire was reduced from 1.57 to 1.25 mm. Rather than attaching the transducers to the die, Lorant [22] used a wire-drawing machine, consisting of 11 dies, immersed in a small ultrasonically agitated bath of liquid to draw wires of high-purity copper, aluminum, and nickel to a diameter of 0.01778 mm (0.0007 in.) under the influence of ultrasound. Reduction was 30% per die at 300 m/min. He claimed the following benefits: (1) reduction in wire breakages, (2) increases in the life span of drawing dies, and (3) closer dimensional control. Neither the intensity of the ultrasound nor the temperature of the bath were given. Reductions in bar-drawing forces and improvement in surface finish without affecting the microhardness and microstructure are possible with ultrasound. This approach has been proven in drawing such materials as 6061 aluminum, AISI 4340 alloy steel, and Ti–6Al–4V alloys at a frequency of 15 kHz, with the die driven in the axial direction. At 4.5 kW input to the transducer, drawing force is reduced 15–20% for the aluminum alloys and 4.5–6.5% for the steel alloys. Increasing the ultrasonic intensity increases the magnitude of the effectiveness. With power input of 7 kW, stock of 17.4625 mm (11/16 in.) initial diameter can be reduced in area 15% per pass at 18.3 m/min (60 ft/min) [23]. Reduction of drawing forces of at least 10% can be expected with these and similar processes applied to other metal alloys. The most significant factor to consider is the toughness or rigidity of the dies and forming materials. Applying ultrasonic energy reduces drawing time for 316 stainless steel tubing by 30% [24]. With a fixed-plug mandrel, thin-walled tubing can be drawn to 4.6355 mm (0.1825 in.) OD and a wall-thickness of 0.254 mm (0.001 in.) at tolerances of 0.0254 mm (0.001 in.) and an inside tolerance of 0.0003 mm (12 μin.); 321 stainless steel bellows tubing held to a tolerance of 0.00752 mm (0.0003 in.) on a 0.00762 mm (0.003 in.) wall; Fe–Ni–Co tube with a 0.381 mm (0.015 in.) wall drawn to a minimum inside corner radius of 0.381 mm (0.015 in.) [25].

CRC_DK8307_CH011.indd 458

10/27/2008 2:16:14 PM

Mechanical Effects of Ultrasonic Energy

459

Forming operations using ultrasonic energy also include 7075 aluminum alloy, 601-T6 alloy tubing, 1100 aluminum (impact-extruded) 3003-H14 alloy covers, 304 stainless steel, phosphor bronze, titanium and beryllium copper, Inconel 718, Kovar, brass, and nickelplated steel. Forming operations also include flaring required to form stainless steel tubing connections and production of brass cartridge cases [26].

11.9

Ultrasonic Soldering

One of the earliest commercial applications of high-intensity ultrasonics was to solder difficult-to-solder materials, especially aluminum. Aluminum oxidizes very rapidly, therefore, aluminum materials are naturally protected by an oxide layer that prevents standard methods of soldering. Ultrasonic energy abrades the oxide layer and brings the molten solder into direct contact with the base material (see Figure 11.6). The ultrasonic energy disperses the oxide coatings and causes the solder to penetrate cavities by reducing the apparent surface tension of the solder and by the imposed stresses that promote flow. Soldering baths of various sizes can be obtained that contain built-in heaters and attached transducers [1, p. 458]. Ultrasonic soldering guns are also available. Soldering is accomplished by ultrasonically agitating tips placed against a joint under molten solder. Elements also may be joined by first coating them ultrasonically and then completing the soldering by conventional means.

11.10

Treatments in Hostile Environments

Some problems associated with treating materials with ultrasonic energy in a hostile environment include (1) excessive heat (molten glass), (2) explosive atmospheres (hydrogen), (3) materials of low or high pH, (4) high pressures, and (5) combinations such as a (a) heat and low pH (acidic) and (b) heat, low pH, and pressure in a nonoxidizing atmosphere (nitrogen, argon, etc.).

Handle Ultrasonic tool Solder Aluminum wire

Trigger Aluminum plate FIGURE 11.6 Ultrasonic soldering gun.

CRC_DK8307_CH011.indd 459

10/27/2008 2:16:14 PM

Ultrasonics: Data, Equations, and Their Practical Uses

460 11.10.1

Degassing Molten Glass

Applying ultrasound to molten glass is a challenge. The objective of applying ultrasound is to accelerate the degassing of the glass by causing the smaller bubbles to coalesce into larger bubbles. The first consideration is the choice of material to transmit the energy between the transducer and the molten glass [27]. What temperature must the transmission line sustain? What transmission line can be used to withstand the temperature range? Is there a material with sufficient Young’s modulus at the required temperatures? What atmospheres are necessary to protect the transmission line from oxidizing? What size unit is to be used? Is the proposed setup an experimental laboratory one or a commercial one? What kind of transducers can be used? Will the energy be fed into the melt from above the melt, through the bottom, or at some angle through the sides of the container? All of these questions must be answered before preparing to degas molten glass. A typical temperature for molten glass is 760°C (1400°F). Molybdenum has a melting point of ∼2620°C (4750°F). A few alloys of molybdenum have the elastic properties at temperatures above 760°C (1400°F) to be able to be used for treating molten glass. This material is a good choice for an ultrasonic transmission line to transfer ultrasonic energy from a transducer into the glass. However, molybdenum needs a protective atmosphere against oxidation at 760°C (1400°F) temperature. This protection can be provided by placing a ceramic shield around the molybdenum line and feeding nitrogen into the space between the molybdenum rod and the shield. Three methods of introducing the energy into the melt exist in an experimental system. These are from the top, from the bottom, or from the sides of the melt. In the technique applying ultrasonic energy from the top, the molybdenum transmission line is suspended from a magnetostrictive transducer mounted above an Al2O3 nitrogen shield. The ultrasonic transmission line is lowered into the melt to a position near the bottom of the tank. The bottom of the transmission line is flanged to provide a general upward force on the bubbles. The ceramic shield is lowered to a position where the lower end is only slightly above the fluid glass. Nitrogen is fed from the top of the assembly into the space between the transmission line and the protective shield at a rate to keep the enclosed space filled with nonoxidizing gas. Figure 11.7 is a schematic of an experimental method used to treat a melt from the top. The shield is not shown. Introducing the energy from the bottom is a bit more difficult, because it involves lifting the vat holding the molten glass to a level to provide a space beneath the melt sufficient for the transducers and the coupling systems. Cooling the horns, the transducers, and the electrical transmission lines must be considered to keep the operating system safe. This problem is less of an issue in small laboratory-size setups. Another means of introducing the ultrasonic energy is through the sides near the bottom of the tank. The same principle applies. The ultrasonic energy will cause the smaller bubbles to coalesce and rise at a more rapid rate than those that have not been treated. Radiation pressure will increase the rise rate by a slight amount, but to use this method, there must be an upward radiation by the source. In each of these methods, the increase in rise rate is dependent primarily upon the increased bubble size through coalescence and forced bubble growth through gas seepage while the ultrasound is applied.

CRC_DK8307_CH011.indd 460

10/27/2008 2:16:14 PM

Mechanical Effects of Ultrasonic Energy

461 Water-cooled magnetostrictive transducer

Transmission line

Ultrasonic horn

Top of molten glass

FIGURE 11.7 Ultrasonically assisted degassing of molten glass.

Radiating the volume of glass from the bottom provides an additional advantage in that the radiating forces of the ultrasound exert a force on the bubbles, forcing them to accelerate toward the top of the bath. Control of ultrasonic intensity is always important. Intensities at too high a level will cause the larger bubbles to break up into smaller bubbles, thus defeating the purpose. Very low intensities have insufficient energy to produce worthwhile effects. Tests conducted at Battelle Columbus Laboratories have shown considerable promise for ultrasonic degassing of molten glass. 11.10.2

Treating Materials of Low and High pH

Treating materials of low or high pH at room temperatures requires the need to find compatible materials and to protect the environment from exposure to dangerous chemicals. Sealed systems include a means of conducting the energy into the material to be treated without loss. The horn or transmission line is equipped with a shoulder, which is located at the velocity node. Various means of applying the seal are available. Using suitable gaskets to avoid leakage of fluid vapors, pressure can be applied to seal the entrance at the nodal region. Properly designed seals allow only little restraint on the motion of the transmission line. Some means of applying the pressure include (1) a screw-down nut capable of enveloping the nodal region, (2) a matching part that is forced against the shoulder, pressing it against the rim on the horn and then into the stationary blocking holder from either the bottom or the top, or (3) threads on the horn matching a set of threads in the top of the container that can be used to screw the horn into a locked

CRC_DK8307_CH011.indd 461

10/27/2008 2:16:14 PM

Ultrasonics: Data, Equations, and Their Practical Uses

462

Transducer

Ultrasonic horn Tantalum horn Gasket

Pressure ports

Acid mix Pressure in Gasket

Hot plate

FIGURE 11.8 Treating acids under high pressure and temperature.

position. These suggestions are not a complete list of all methods, but serve to illustrate approaches that can be used. Tantalum resists acids well. It is also very expensive. Its melting point is approximately 2295°C (5420°F). It is also very malleable. These characteristics make tantalum a very useful material for exotic experiments at high temperatures, high pressures, and low or high pH. Figure 11.8 illustrates a setup that has been used on a practical scale. It is used for treating materials under very high pressure. The chemical process produces very fine particles that are dispersed immediately upon formation. This process avoids the formation of larger crystals and of agglomerates.

11.11

Compaction of Metal Parts

There are two methods of ultrasonically compacting powder-metal parts: (1) by locating nonprecompacted materials in a mold located at the free end of an ultrasonically vibrating horn and (2) by locating precompacted materials at the high stress (velocity node) of an ultrasonically vibrating horn [1, 456–462]. Both methods should be done in a noncorrosive atmosphere, such as nitrogen. In the first method, the particles have a relatively free but constrained motion, reflecting a low acoustic impedance (see Figure 11.9). A light pressure is applied to couple the energy into the mold. Energy is absorbed primarily by the relative motion between particles. Materials that can be compacted in this manner generally have low melting points. Some materials that have been compacted in this manner are polymers, tin, lead, copper, silver, and even stainless steel. The mold reaches a red-hot temperature within a few seconds at 1.0 kW and 20 kHz. Even 304 stainless steel specimens 1.27 cm × 5.72 cm × 6.35 cm pressed in low-thermal conductivity dies reach a red-hot temperature within a few seconds at 1.0 kW input to the transducer. Under similar conditions, silver will melt within a few seconds.

CRC_DK8307_CH011.indd 462

10/27/2008 2:16:15 PM

Mechanical Effects of Ultrasonic Energy

463

Transducer

Ultrasonic horn

Powdered metal for molding

Backup plate

FIGURE 11.9 Ultrasonically compacting powdered metal.

The second method is used with specimens that have been precompressed to ∼75% theoretical density or greater. Particle motion within the compact is restricted, corresponding to a high acoustic impedance. For this reason, the specimen is located at a high-impedance position in the vibratory system. This position corresponds to the velocity node and stress antinode of a bar in longitudinal vibration. Ultrasonic compaction is limited to specimens with thicknesses of a fraction of a wavelength in the material of the compact (≈0.03λ). Various loose iron compacts, without presintering or reduction of oxides and using only dead-load clamping pressure, showed improvements in transverse strengths ranging from 7.5 to 14 times the green fiber stress values for the materials. Corresponding values in density were between 8 and 10%. Higher strengths were associated with higher intensities. Massive oxides may be present near the pores and outside surfaces. These oxides may be produced partially during ultrasonic excitation and partially as a result of agglomerations of oxides already present in the compact. In a method illustrated by Figure 11.10 specimens of various alloys precompacted to ∼75% theoretical density or greater may be compacted to 100% of theoretical density with power inputs of 1200–1650 W. The compact must be of the same thickness (≈0.03λ) in the precompacted specimens as in the nonprecompacted specimens. The specimen is located at the vibration node. The pressure is applied (between the shoulders marked F) by a press capable of applying up to 75 tons. This type of compaction has produced some interesting results. Transverse strengths for iron specimens, hydrogen-annealed before green pressing and presintered in a hydrogen atmosphere, showed improved transverse strengths of 1.5–2.5 times greater than those of samples prepared from their as-received state. Increases in strength were obtained, with increases of both intensity and pressure. The increase in strength by doubling pressure was as high as 66%. Only a normal amount of oxides was present. Extrusion of metal powders has shown an improvement in density with the application of ultrasonic energy and some reduction in temperature for equivalent density in hot pressing of metal powders with ultrasonic energy applied [28]. Pokryshev and Marchenko [29] report similar results from applying ultrasound to hot extrusion of iron powder. The temperature was reduced up to 20% when ultrasound was imposed.

CRC_DK8307_CH011.indd 463

10/27/2008 2:16:15 PM

Ultrasonics: Data, Equations, and Their Practical Uses

464

Transducer

F

F

Coupling horn

Compression horn Metal compact

Mold

Compression horn

Coupling horn

F

F

FIGURE 11.10 Ultrasonically compacting precompacted powdered metal.

Application of vibratory energy at lower frequencies has also been effective. Pines and coworkers [30] applied sonic vibrations in the range 5–15 kHz at displacements of 3–6 μm to activating sintering and interparticle diffusion.

11.12

Metallurgical Processes

Metallurgists could realize many useful effects if efficient methods of coupling ultrasonic energy into molten metals were available. Choices of materials for each of the components, containers, molds, transmission lines, and transducers are critical to the design of a system for applying ultrasound to a metallurgical melt. Several beneficial effects have been demonstrated, including degassing, reduction of grain size, and uniform mixing of different metals. The major problems that must be overcome in applying ultrasound to molten metals are related primarily to the excessive heat involved. The difference between the melting point of the ultrasonic transmission line, Tt, and that of the material into which it is working, Tm, must be great enough to minimize erosion of the transmission line into the melt. As the melt temperature increases, the potential difference Tm − Tt decreases for available materials. Differences of near zero to zero must be avoided.

CRC_DK8307_CH011.indd 464

10/27/2008 2:16:15 PM

Mechanical Effects of Ultrasonic Energy

465

Certain alloys of molybdenum have high melting points, but their use requires protection against oxidation. Nitrogen is probably the least-expensive protective atmosphere. Any two materials that must be exposed to each other but must not enter into the metallurgical result must be compatible; they must not dissolve one another. The material transmitting the energy into the melt must be able to resist dissolution into the melt. The mold material must not wear or dissolve into the melt excessively. Similarly, two or more materials that are to be combined must be compatible. The transducers must be carefully chosen and protected. Most of these have Curie points well below the temperature of the melt. They must be well insulated from the heat of the molten metal. The transmission line must be of sufficient length that it can go from being hot at the melt and be cooled near the transducer. Combining vacuum and ultrasonic degassing of molten metals results in considerable improvement over the rate and effectiveness of either method used separately. Reduction of grain size is obtained when ultrasonic or low-frequency vibratory energy is applied to a melt. Uniform and refined grain sizes are produced when the distribution of energy throughout the melt is such that the entire melt is subjected to intensities above a certain threshold level. Refinement is independent of intensity above the threshold level. In molten metals, refinement appears to be most effective in alloys that have a low alloy content and are predominantly solid solutions in structure. Grain refinement is best when the melt is cooled slowly with irradiation continuing until solidification is nearly complete. Abramov [31] lists the beneficial mechanisms of ultrasonics in metallurgy as follows: 1. Cavitation: Cavitation promotes crystallization and grain refinement. 2. Degassing. 3. Microstreaming: Microstreaming disperses crystallization centers and significantly diminishes the thickness of a diffusion layer, thus intensifying processes where diffusion through a boundary layer is a limiting factor, and increases the homogeneity of the ingot. These mechanisms are responsible for eliminating columnar structure, forming fine equiaxed grains, breaking up precipitates of excessive phases, and reducing zonal and dendritic segregation.

11.13

Deburring [1, pp. 455–456]

Ultrasonics is used to debur precision parts where fine deburring is required, if tolerances are not <0.0025 mm (0.0001 in.). Fine burs form cavitation nuclei. Cavitation accelerates the dissolution of small metallic burs. Items to be deburred are placed in a cleaning-type tank that has been prepared with an acid-resistant lining. The tank contains an acidic slurry formulated specifically for the type of metal to be deburred. The unit is turned on for a predetermined time and the bath temperature is set at a predetermined level. The deburring cycle is followed by ultrasonic cleaning in an alkaline bath to neutralize the acid. This step is followed by a clean-water rinse. Ultrasonic deburring has been used on such materials as stainless steels, mild steels, copper, and nickel alloys. It should be possible to debur other types of alloys as well. Its use

CRC_DK8307_CH011.indd 465

10/27/2008 2:16:15 PM

Ultrasonics: Data, Equations, and Their Practical Uses

466

requires careful formulation of the deburring bath and good control of temperature and time to avoid excessive erosion of the part. Poor choice of combined materials, deburring fluid, and temperature could result in complete destruction of the part. Ultrasonic deburring does not remove heavy burs without excessively eroding the part. It should be used on parts where the economics of the operation justify its use, but not on inexpensive, nonprecision parts.

11.14

Accelerated Fatigue Testing

Most failures in dynamic structural elements are traceable to fatigue. Considerable effort is expended in fatigue-life measurements or the development of S–N curves for various materials and structures. The S–N curve relates stress level (S) to number of cycles (N). Standard fatigue testing is done at frequencies ranging from a fraction of a cycle per second to 200 Hz. Developing a complete S–N curve by conventional methods for any given material occupies months of testing. For example, it takes more than 1.6 years to expose a test material to 1010 cycles at the rate of 200 Hz. By comparison at 20 kHz, materials are subject to 1.728 × 109 cycles/day or 5.184 × 1010 cycles in a 30 day period. At 50 kHz, materials are subjected to 4.32 × 109 cycles per day and to 1.296 × 1011 cycles in 30 days. These figures indicate the difference between the time required at 20 Hz at the conventional method and that obtainable at ultrasonic frequencies. Therefore, if fatigue data obtained at ultrasonic frequencies are applicable to low-frequency needs, they are an asset to fatigue testing, and permit testing time to be shortened immensely. Ultrasonic fatigue testing requires very careful preparation. The specimens are halfwave longitudinally vibrating bars. Their diameters are small compared with the length. Preparation includes determining the bar velocity of sound, which means avoiding published values of elastic constants and densities. Specimens are very carefully and accurately machined. They are then carefully polished to remove any burrs and inspected for cracks and other spurious stress concentrators. The density is determined by weighing the specimen prior to the test. The modulus of elasticity is determined by the equation for bar velocity, C

E 

(11.6)

Accuracy in measuring wavelength, frequency, and displacement is essential to a reliable S–N curve at ultrasonic frequencies. Stepped horns are preferable for this purpose. Displacement amplitude is continuously measured by a capacitance type of probe. The stress concentration at the step between the smaller and the larger diameter is determined by the shape of the fillet used and calculated using equations provided by Roark and Young [32]. Measurements are made at constant strain amplitude. Feedback from the transducer controls the frequency of the driver. Feedback from the displacement monitor holds the strain at a constant level. Stress level is determined by the displacement amplitude and calculated by using the elastic constants of the specimen material. Tests are suspended immediately at the time that resonance frequency begins to shift rapidly and unidirectionally—a certain indication that failure is in progress.

CRC_DK8307_CH011.indd 466

10/27/2008 2:16:15 PM

Mechanical Effects of Ultrasonic Energy

467

Power oscillator

Transducer

Test specimen

Displacement feedback

FIGURE 11.11 Fatigue-life test setup.

An appropriate high-frequency fatigue testing facility includes: 1. A suitable electronic power oscillator with feedback control to the electronic oscillator to maintain resonance operation throughout the test 2. An accurate, preferably noncontacting, displacement-measuring device for monitoring displacement at the free end of a resonant test specimen 3. An accurate means of locating the first displacement node from the free end of the specimen and thus determining a quarter-wavelength of sound in the bar 4. An accurate frequency meter for monitoring frequency and providing a basis for determining a count of the number of cycles of stress applied 5. An accurate timer 6. A means of stopping the test and indicating the time that the test was stopped at definite signs of failure 7. A strip chart recorder for continuously monitoring displacement amplitude and time 8. An appropriate means of cooling 9. A thermocouple and associated instrumentation for monitoring specimen temperature (see Figure 11.11)

11.15

Welding of Metals and Plastics

Welding is the largest commercial application of high-intensity ultrasonic energy. Metals were the first materials to which ultrasonic welding was applied. Welds have been made in metals such as aluminum, brass, 304 and 321 stainless steel, copper, zirconium, titanium, niobium, commercial iron, chromel, nickel, molybdenum, titanium, tantalum, gold, and platinum. In many of the applications, the motion of the welding tip is applied to produce a shear stress at the interface. The total energy absorbed at the interface is a function of the shear stress at the junction between the elements to be joined, the frequency, and the total time for which the motion is applied. The shear stress depends upon the force applied and the

CRC_DK8307_CH011.indd 467

10/27/2008 2:16:16 PM

Ultrasonics: Data, Equations, and Their Practical Uses

468 Power oscillator

Horn

Transducer

Force

Polymer layers

Pivot Anvil

FIGURE 11.12 Horizontal shear welder.

F

Resonant bar when unloaded Transducer

Horn

Motion

Polymer layers Anvil FIGURE 11.13 Strap welder.

surface condition, which includes coefficient of friction and roughness. Both coefficient of friction and roughness will decrease with the progress of the weld. The effective thickness of the top specimen is limited by the stiffness of the material of the strip to welded. Figures 11.12 and 11.13 illustrate the method of shear stress at the interface. Figure 11.14 illustrates the spot welder, whose motion is normal to the interface. This method is also applied to materials of different construction in which the ultrasonic energy is transmitted through one of the components to the joints to be welded. Figure 11.12 illustrates a horizontal shear welder. The tip can be made to slide or rotate with the movement of the weld area to form a seam weld. The design of the system places a limit on the pressure that can be applied to the weld area and limits the thickness and type of material that can be welded. This design can be used for welding metals as well as polymers.

CRC_DK8307_CH011.indd 468

10/27/2008 2:16:17 PM

Mechanical Effects of Ultrasonic Energy

469

Transducer

Horn

Tool

Polymer layers Anvil FIGURE 11.14 Normal spot welder.

Figure 11.13 is a diagram of a welder designed to apply greater forces to the weld area. This design has been used to bond straps and similar structures. The vertical bar resonates at the driven frequency before the load is applied, giving a maximum motion to the tip of the bar at the weld contact end. The weld begins on the application of the load. The bar is pressed into the weld area and the pinning force changes the bar to vibrate in a pinned mode, making the end a nodal position. It is held in this position while the weld solidifies and the power is turned off. Figure 11.14 illustrates a typical spot welder, with which the motion of the tool is normal to the surface to be welded. Figure 11.14 could also illustrate a torsional type of welder. In this type, the longitudinal mode transducer could be replaced by a torsional mode unit and a means of applying a force across the weld is added. The tool is hollow. It is placed with axis normal to the weld area. The typical weld occurs as 680 kg (1500 lb.) of pressure and 5400 W of energy are applied, for example, to 0.0127 cm (0.005 in.) thick 3003-H19 aluminum alloy to produce hermetic seals in propellant containers. Welding time is 0.5 s.

11.16

Conclusions

There are many other things that can be accomplished by the use of ultrasonics. An ultrasonic whistle can be used to pollinate self-pollinating plants such as tomatoes. A standing wave over a still body of light liquid can move small droplets of uniform size into the nodal positions of the wave. Sonic and ultrasonic energy can cause dust to agglomerate and thus can remove the dust from a very dusty region. The applications mentioned within this book are those that have been proven to be practical. There are many other practical applications of ultrasonic energy. Some other reasons to use ultrasonic energy are given in Chapter 12.

CRC_DK8307_CH011.indd 469

10/27/2008 2:16:17 PM

470

Ultrasonics: Data, Equations, and Their Practical Uses

Occasionally, some intriguing applications have been reported as having been developed or are still in the experimental stage. Applying ultrasound to the cracking of corn to enhance the removal of ethanol is one interesting application under study. Supposedly, this application could increase the total amount of ethanol removed economically. Whether this is a logical hope depends upon how the alcohol is presently removed.

References 1. D. Ensminger, Ultrasonics, 2nd edition, Marcel Dekker, New York, pp. 456–458, 467–474, 1988. 2. P. R. Hill, J. C. Bamba, and G. R. ter Haar, Ultrasonics, 2nd edition, John Wiley & Sons, Hoboken, NJ, p. 259, 2004. 3. H. S. Muralidhara, N. Senapati, D. Ensminger, and S. P. Chauhan, Sep. Fifth, pp. 351–353, November/December, 1986. 4. H. S. Muralidhara, D. Ensminger, and A. Putnam, Dry. Technol., 3(4), 529–566, 1985. 5. H. S. Muralidhara, N. Senapati, and R. B. Beard, Advances in Solid-Liquid Separation, H. S. Muralidhara (ed.), Battelle Press, Columbus, OH, pp. 335–374, 1986. 6. H. G. Dokmen, Aluminum, 42(9), 559–566, 1966. 7. L. Balamuth, Ultrasonics, 4, 125–130, 1966. 8. L. Balamuth, SAE Paper No. 650762 presented at the National Aeronautic and Space Engineering and Manufacturing Meeting, Los Angeles, October 4–8, 1965. 9. B. Langenecker, IEEE Trans. Son. Ultrason., SU13(1), 1–8, 1966. 10. B. Langenecker, U.S. Patent 3,276,918, October 4, 1966. 11. N. L. Pozen, V. N. Semirog-Orlik, and I. A. Troyan, Fiz. Khim. Mekh. Mate. Akad. Nauk. Ukr. SSR, 5(1), 112–113, 1969. 12. O. Izumi, K. Oyama, and Y. Szuki, Trans. Japan Inst. Met., 7(3), 162–167, 1966. 13. G. Kralik and B. Weiss, Z. Metallkd., 58(7), 471–475, 1967. 14. R. Pohlmann and E. Lehfeldt, Ultrasonics, 4, 178–185, 1966. 15. B. Ya. Pines and I. F. Omel’yanko, Fiz. Met. Metalloved., 28(1), 110–114, 1969. 16. V. N. Chackin and A. L. Skripnichenko, Izv. Aikad. Nauk. Beloruss SSR (Fiz.- Tekh.), 3, 43–46, 1968. 17. E. Schmid, Proc. International Conference on Strength of Metal Alloys 1967, 9798–9804, 1968. 18. C. E. Winsper and D. H. Sansome, J. Inst. Met., 97(9), 274–280, 1969. 19. E. Lehfeldt, VDI, 111(6), 359–363, 1969. 20. G. Kralik, Acta Phys. Austriaca, 20(1–4), 370–375, 1965. 21. V.P. Severdenko and V. V. Clubovich, Dolkl. Akad. Nauk SSSR, 7(2), 95, 1963. 22. M. Lorant, Tooling, 20(12), 51, 1966. 23. N. Maropis and J. C. Clement, ASM Technical Report No. C6-26.3, American Society for Metals, Metals Park, Ohio, OH, 1966. 24. Anon., Steel, 159(20), 38, 1966. 25. Anon, Machinery, 74(9), 88–89, 1968. 26. J. B. Jones, Met. Prog., 93(5), 103–107, 1968. 27. E. Spinosa and D. Ensminger, Ceramic Engineering and Science Proceedings, 7(3–4), 410–425, 1986. 28. E. Lehfeldt and R. Pohlman, Plauseeber. Pulvermetall., 16(4), 263–276, 1968. 29. V. R. Pokryshev and V. I. Marchenko, Poroshk. Metall. Akad. Nauk Ukr. SSR, 2, 30–33, 1969. 30. B. Ya. Pines, I. B. Omal’panenko, and A. F. Sirenko, Pooshk. Metal., 8, 106–111, 1967. 31. O. V. Abramov, Ultrasonics, 25(2), 73–82, 1987. 32. R. J. Roark and W. C. Young, Formulas for Stress and Strain, 5th edition, McGraw-Hill, New York, 1975.

CRC_DK8307_CH011.indd 470

10/27/2008 2:16:17 PM

12 Criteria for Choosing Ultrasonics Dale Ensminger

CONTENTS 12.1 Introduction ...................................................................................................................... 471 12.2 Low-Intensity Applications ............................................................................................ 474 12.2.1 Nondestructive Testing .................................................................................... 474 12.2.2 Medical Diagnosis.............................................................................................. 475 12.3 Commercialization Problems ......................................................................................... 475 12.3.1 Mica Mill.............................................................................................................. 475 12.3.2 Treatment of Perfume ........................................................................................ 476 12.3.3 Measuring Flow of Paper Sludge ..................................................................... 477 12.3.4 Powdered Coffee ................................................................................................ 477 12.3.5 Treatment of Laryngeal Papilloma .................................................................. 477 12.3.6 Effects on Eggs .................................................................................................... 478 12.3.7 Pollinating Tomatoes ......................................................................................... 478 12.3.8 Pest Control ......................................................................................................... 479 12.3.8.1 Insects .................................................................................................. 479 12.3.8.2 Birds .................................................................................................... 479 12.4 Separation of Materials.................................................................................................... 479 12.5 Summary ...........................................................................................................................480 References ....................................................................................................................................480

12.1

Introduction

Ultrasonics is a form of mechanical energy existing in a multitude of waveforms. It is used successfully in a wide range of applications, including almost every area of materials science such as medicine, metallurgy, cleaning, NDT, underwater sound, welding of plastics and metals, atomization of liquids, chemical effects, and cell disruption. There seems to be no end to its potential applications. The primary objective of this present chapter is to explain the criteria necessary to recognize the feasibility of applying ultrasonic energy to any application and to recognize its economic factors. Dr. Bergmann’s preface to his book, which states, “It is rare for a physical phenomenon to have found within a few years so many applications in all branches of science and technology as ultrasonics,” [1] was written prior to the 1940s [2]. The statement, “so many applications in all branches of science and technology as ultrasonics” is still applicable. Improvements in electronics and in the means of generating ultrasonic waves have contributed much to the development of the ultrasonic industry. 471

CRC_DK8307_CH012.indd 471

10/24/2008 1:12:01 PM

472

Ultrasonics: Data, Equations, and Their Practical Uses

Previous chapters illustrated the number of practical uses that have been developed for ultrasonic energy. These may be divided into at least three groups: (1) low-intensity applications using ultrasonic energy that makes no change in the materials exposed to it; (2) moderate-intensity applications, such as those intended only to produce heat or perhaps some associated activity such as relieving pain, and not to make a permanent change in the physical characteristics of a material; and (3) high-intensity applications intended to disrupt, chemically affect, or otherwise change the physical characteristics of the exposed materials. NDT and medical diagnosis are two examples of widespread uses in the first, low-intensity applications. Using ultrasound in therapeutical applications is an example of the second category: medium-intensity applications. The third category, high-intensity applications, includes such uses as liquid atomization, emulsification, welding, and ultrasonic cleaning. There is a fundamental scientific basis for every useful effect behind every successful industrial and medical application of ultrasonics. Ultrasonics is not magic. Each application requires an understanding of related basic principles. There are some questions that one must always answer either subconsciously or consciously in choosing to apply ultrasonics to anything. What do you want to do? Why do you expect ultrasonics to do the job for you? What are the mechanisms involved in bringing about the desired end? How can ultrasonics be applied effectively and to advantage over other methods? In what ways does the use of ultrasonic energy appear to have an economical advantage over an alternative method? How can the ultrasonic energy be applied to the product of interest? This chapter discusses certain factors that need to be addressed in developing any new idea into a commercial practicality and to obtain the maximum benefit from ultrasonics where it is already being successfully applied. Principles that are basic to the successful development of practical applications of ultrasonic energy are discussed with reference to experiences and historical records of the past 75 years. In many cases, these principles are fairly simple and easily applied, while in other cases, the principles are somewhat more complicated. The applications are divided into the various groups to simplify this topic. There is no need to discuss again in any great detail common commercial applications, such as medical, NDT, cleaning, or welding, that have been discussed in previous chapters. Some very good companies have undertaken continuous development in each of these areas. Perhaps the discussions that follow may bring challenges to the reader that may result in industrial uses equivalent to those already attained in NDT, cleaning, welding, and medicine. Principles of the developments in low- and medium-intensity applications of ultrasonics include the following: 1. The acoustical properties of the material being investigated. 2. The versatile means of testing: pulse-echo, through-transmission, and resonance. 3. Coupling ease: piezoelectric transducers, electromagnetic properties, passive (listening to the structural responses to shock waves), and so on. 4. Versatile means of generating and receiving energy. 5. Multiple modes of vibration, leading to complete testing of a material. 6. A ready market. Mechanisms that promote ultrasonic effects at high intensities in materials that are of value to industrial markets include the following: 1. Heat. Heat is formed by absorption of energy from a traveling wave and shear or friction at both internal and external boundaries.

CRC_DK8307_CH012.indd 472

10/24/2008 1:12:02 PM

Criteria for Choosing Ultrasonics

473

2. Stirring. Intense ultrasound will produce violent agitation in a liquid medium of low viscosity and disperse material by resulting in currents of liquid or accelerations imparted to the particles. 3. Cavitation. Many of the effects associated with ultrasonics occur in the presence of cavitation. Cavitation may produce emulsification of otherwise immiscible liquids and accelerate the emulsification of miscible liquids. 4. Chemical effects. Cavitation may produce such chemical reactions as oxidation. Other chemical effects also appear including polymerization, depolymerization, and other useful activities. 5. Mechanical effects. Stresses developed ultrasonically may cause ruptures to occur, bonding between mating parts (as in ultrasonic welding), or stress erosion. 6. Electrolytic effects. When two metals separated on the electrolytic scale by even a small amount are exposed to intense ultrasonic irradiation in water, an accelerated galvanic action may be induced which causes electrolytic corrosion. 7. Diffusion. Ultrasound promotes diffusion through cell walls, into gels, and through porous membranes. 8. Vacuum effects. During the low-pressure phase of each cycle, materials may be drawn from pores, and fluids may be drawn into tiny pores. 9. Cleansing. Ultrasonic high-intensity waves have been proven to be good cleansers. Often ultrasonic energy can remove protective coating, so that a chemical may gain access to the base material, causing effects that cannot be produced otherwise. The acceptance of high-intensity ultrasonic processes in industry has depended upon the following: 1. 2. 3. 4.

Design simplicity—for example, welders and cleaners Unique capabilities offered Utilization of localized reaction zones to ease access to the zone to be treated Operational simplicity

The factors that hinder large-scale applications of high-intensity ultrasonic energy include: 1. Competition from other less-expensive methods of accomplishing the same thing. One good example is the replacement of ultrasonic machining by lasers. Laser machining has limited the usefulness of ultrasonic machining by being faster and cleaner. 2. Scale-up problems—particularly with volumetric effects requiring cavitation. a. Cavitation is energy-consuming. b. Associated energy loss limits the total volume through which cavitation can be effective. c. Total energy required for an effective process is a function of volume to be treated. 3. General lack of information, training, and equipment developed for the proposed application. 4. Interest or ability in marketing.

CRC_DK8307_CH012.indd 473

10/24/2008 1:12:02 PM

Ultrasonics: Data, Equations, and Their Practical Uses

474

Intensity levels and time of application are very important factors in the use of ultrasonics. In the medical profession, the diagnostic levels requiring very low intensities are set by experience. At higher intensities, the effects to be obtained may be due to a combination such as heat and ultrasonic agitation. Ultrasonic energy may be used to treat a tumor at a given temperature. The heat produced is a function of absorption of sound within the tumor, the absorption within the exposed area being slightly higher than that of the surrounding tissues. The ultrasound is focused to concentrate the energy in the tumor without overheating the surrounding tissue. The exposure may be given in brief pulses of high intensity to bring the temperature to a predetermined level and to hold the temperature at that level for a given length of time. This approach has been very successful in treating brain cancers, for example. The temptation to accelerate the reaction by increasing the ultrasonic energy, giving little heed to the time and temperature, is dangerous. The result could be the spread of the disease rather than its destruction. In other areas, the application of high-intensity ultrasonic energy to a diseased tissue may be desirable; however, intense energy may damage the surrounding tissue. Many good effects of ultrasonic energy may not be achieved if there is a lack of sufficient study of the location and conditions surrounding the spot to be irradiated. Crystallization occurs within a metastable range of temperatures. Ultrasonic energy may be added to the crystallization process at a given rate to control the size of the crystals. As shown in Chapter 8, when ultrasonic energy is applied at high intensity for an excessive period of time, the excessive heating causes the crystals to dissolve and crystallization is slowed down or eliminated.

12.2 Low-Intensity Applications 12.2.1

Nondestructive Testing

The use of ultrasonic waves for NDT was suggested by Russian and German scientists as early as 1929 and 1931. They recognized the ability of ultrasonic waves to travel in materials and to be reflected by defects such as cracks as a basis for NDT. Floyd Firestone developed the Reflectoscope in 1942, which opened the way for ultrasonic testing in the United States. The early Reflectoscope used a single transducer and employed pulses of ultrasonic energy transmitted into and reflected from surfaces of defects. It was possible to study the shadow effect of a discontinuity by using two transducers in a through-transmission mode. Early in the 1940s, Norman Branson—experimenting in his garage with ultrasonic testing—became discouraged at the poor reception that his instruments seemed to be getting in industry. However, friends kept him from quitting his developments by telling him there was a lot of interest “out there” in what he was doing. In 1946, he introduced the Audigage and the Vidigage, both thickness-measuring instruments. Today, the Branson companies are among the leaders in the ultrasonics industry, producing ultrasonic cleaners, welders, and NDT equipment. These developments have been aided by the developments in electronics, of the understanding of principles involved in wave transmission through solids, and the improvements in transducer materials. A few questions that may need answers can be expected if the proposed application involves new or unusual situations, such as environment. A need to test extremely hot specimens is an environmental problem. How can you couple to a specimen that is extremely

CRC_DK8307_CH012.indd 474

10/24/2008 1:12:02 PM

Criteria for Choosing Ultrasonics

475

hot? Electromagnetic acoustic transducer (EMAT) techniques, capacitance transducers, and laser techniques have been developed that adequately test hot materials (see Chapter 9). It may be possible, if the test has to be restricted to the piezoelectric technique, to use a transmission line that is long enough to be cooled at a location close to the transducers and that withstands the thermal condition into which it is working. A similar delay line would be required to transmit the received signal to a second transducer, or the echo from the discontinuity may be transmitted back to the transmitting transducer used now as a receiver. The transmission line would have to be free of defects and a reliable means of coupling the ends to the specimen would be necessary, such as a molten material that would wet both mating surfaces. Such a through-transmission method would have very limited use. 12.2.2

Medical Diagnosis

The popularity of ultrasonics in NDT encouraged the use of ultrasound in medical diagnosis. The use of ultrasound is similar to that used in NDT. Pulses transmitted from the surface into the main parts of the body are reflected from discontinuities, scattered, attenuated, and shadowed by conditions within the body (see Chapter 10). Cancerous tumors are stiff, like springs. Normal tissues and benign lesions are softer and more easily compressed. When a transducer is pressed against the breast containing a cancerous tumor, the echoes produce a clear image. Echoes received from pressing over softer, benign tumors cause less clear images. The method utilizing this principle is called elastography [3]. Early studies using ultrasonic energy reflected from a lump in the breast identified cancerous formations correctly more often than biopsies. In research applying elastographic techniques, correct identification is nearly 100%. Until further research has been conducted to establish certainty of identification, there is a general fear of lawsuits if biopsies are not prescribed and cancers are missed. The number of medical applications is growing, as physicians see their need and gain confidence in the technology. The development of new equipment is progressing in several areas. Progress is being made in methods of homogenizing, methods of evaluating internal conditions within a body, and many other areas. Ultrasonic imaging has become a routine practice in hospitals and clinics. These are only a few of the uses of ultrasonics, as discussed in previous chapters. There are many others that have also been discussed, such as homogenization of tissues, atomization of liquids, and degassing. What has made these practical is primarily the ready need for the system, the developments in electronics and miniaturization of the complete processes, and an understanding of the values of the needs. Medical ultrasonics is discussed more thoroughly in Chapter 10.

12.3

Commercialization Problems

A discussion of a few systems may help to understand why certain apparently successful applications have not become commercially practical. 12.3.1

Mica Mill

Mica is a rock of complex aluminum silicates, laminated in thin layers. The laminations can be separated into thin, flexible leaves. Different types of mica can be found in

CRC_DK8307_CH012.indd 475

10/24/2008 1:12:02 PM

Ultrasonics: Data, Equations, and Their Practical Uses

476

different parts of the world. It may be translucent or colored: yellow, brown, green, red, or black. The type of mica found in the Appalachian Mountains is called muscovite, or white mica. Mica is resistant to heat and electricity. Jig mica is mined as small chunks. It is sold as either dry mica or wet ground mica, depending upon the end use. Wet ground mica is the more expensive grade. Jig mica is used in oils as a lubricant, in fireproofing materials, and in some decorative applications such as wallpapers, to provide a shiny luster to the surface. During the 1950s, Dr. Eric C. Kunz, a retired president of a large perfume manufacturing corporation, witnessed the mining and wet grinding of jig mica in North Carolina. He had heard how ultrasonics could break small particles to smaller particles by cavitation. He sponsored a project with Battelle Columbus Laboratories to ultrasonically grind jig mica and as a result, a patent for the process was issued to Kunz and Ensminger [4]. A 10 kW system utilizing ten 1 kW 20 kHz transducers designed to treat mica in suspension, where they could be operated on from all sides, was constructed. Mica is a rock and, without stirring and the ultrasonic agitation, it would settle rapidly, leaving very little surface exposed from which particles could be removed ultrasonically. With this system, the end product was clean and clear, which increased its value as an additive in many products. Why isn’t it available today? Some possible reasons include the following: 1. It was developed at a time when the cost of the ultrasonic equipment was high. The electronics would be much cheaper and housed in a much smaller volume today. 2. The placement of a system would be an engineering task. 3. The cost was too high for the market at the time. This application illustrates the issues that arise when marketing a new idea: 1. There should be a need. There seemed to be one. 2. The cost should be right. It was too expensive. With the development in electronics and in piezoelectric and magnetostrictive devices during more recent years, this objection should be much weaker. 3. There should be a market for the development. Interest in developing one did not exist. There might be a market for such a service today. One could construct a system to do the same thing and lease it out to users or sell it outright to them. 12.3.2

Treatment of Perfume

When oils to be used in perfumes are first extracted from flowers, they are allowed to age under various conditions of exposure. At Dr. Kunz’s suggestion, some oils were treated at frequencies between 1.0 and 5.0 MHz for a few minutes to as much as 16 h, under both cavitating and noncavitating conditions, and with exposures to both air and nitrogen during the treatments. There was a noted change in each sample treated. Odors such as the “weedy” odor, a smell resembling that of newly mown hay, were removed. A patent [5] was obtained by Kunz and Ensminger on the use of ultrasound to change the odors of perfume oils.

CRC_DK8307_CH012.indd 476

10/24/2008 1:12:02 PM

Criteria for Choosing Ultrasonics

477

Why isn’t this treatment being applied today? 1. It was simple and inexpensive. 2. It worked very well. 3. Marketable? No one seemed interested in picking up on it. Again, there should be a market or a demand for the service today. 12.3.3

Measuring Flow of Paper Sludge

Ultrasonics has been used to measure the flow of liquids in pipes. However, measuring the flow of high-density paper sludge causes problems, due to high attenuation by scattering. A means of measuring flow rate in a 7% slurry has been developed. The unit worked on the principle of the difference of travel time between an upstream and a downstream ultrasonic pulse. A prototype model was delivered to the laboratories at Syracuse University, where it was given a final test. Following the test, the unit was delivered to a location in Texas, where it was set aside. No other tests were tried and no commercialization of the method was attempted. What happened? 1. The development was successful, in that the unit worked well. 2. Marketability? Apparently other methods are acceptable. 12.3.4

Powdered Coffee

A company once expressed a desire to incorporate ultrasonic energy as a step in the preparation of instant coffee. The ultrasonic energy would have replaced the mechanical pulverization step in the original procedure. The flavor was removed from the coffee in the original process and returned again as a final step in the preparation for sale to the public. In the suggested procedure, the flavor would remain in the coffee throughout the pulverization process. Adding ultrasonics to the former process would probably affect the flavor and add unnecessary steps to the original procedure. Using only a mechanical procedure for grinding the coffee would lead to a new product to sell, which could be identified by the coarser grains. The suggestion was made that no grinding other than that which was normal be done. The coarser material would be easily discernible from the former instant coffee. The company took the suggestion and produces a product that is widely accepted. The original flavor is protected. 12.3.5

Treatment of Laryngeal Papilloma

Laryngeal papilloma is a disease that attacks very young children and disappears after the children reach adulthood, if they can be kept alive that long. A growth forms on the larynx, making it difficult for the child to speak. The former practice in operating was to pull the growths out, but the growths usually return at least once a year. Dr. Burke, a professor at the Ohio State University Hospital, suggested applying ultrasonic energy during the operation. A small device was designed to apply ultrasonic energy at a frequency between 1.0 and 10.0 MHz from a tiny transducer attached to the end of a long rod. Immediately following removal of the growth, high-intensity ultrasonic energy was applied for a short period of time by contacting the wound with the transducer attached to the rod.

CRC_DK8307_CH012.indd 477

10/24/2008 1:12:02 PM

Ultrasonics: Data, Equations, and Their Practical Uses

478

The results have been very satisfactory, with only two out of more than 100 patients treated having to return for further treatments. Where is it used today? Nowhere, to our knowledge. Dr. Burke has switched to a laser system, which apparently is working well. 12.3.6

Effects on Eggs

This suggestion came from a customer who witnessed a waiter homogenizing a chicken egg using an appliance from which a stem was inserted into the egg at one end and spun at high speed. After treatment, the contents poured out like water. He wondered whether ultrasonic energy would do the same thing without necessitating the breaking of the eggshell. A series of tests using several dozen eggs was conducted using frequencies at 1.0 MHz. After a very brief exposure, the contents of the eggs, including the liner, the yolk, and the white, poured out like water. If the treatment was at too high of an intensity or treatment was for too long, the egg would be partially to fully cooked when it was opened. Several volunteers accepted these treated eggs. One advantage was that the shell had not been broken until the egg was ready to be used. Only one accepted a follow-up offer. Others never asked for another egg. One who accepted them said that when they were opened they gave off an odor like rotten eggs. The one who repeatedly accepted the offered eggs used them for baking immediately on getting them home. Apparently she used them before the rotten egg odor appeared. How could this process be used practically? It would seem that the waiters using the mechanical method could also use the ultrasonic method. If it were used immediately after treatment, the bad odor would be avoided. This might be a good suggested use for the product of this research: to use a miniature ultrasonic unit for preparing a service in a restaurant. Such a unit also could be made for use in homes, for example, in the preparation of breakfast. In both cases the egg would be used immediately after treatment. Such a device would not be expensive today. An attempt was made to make a practical use of the process by applying it to eggs of flies that infected crops. A leaf coated with a large supply of the insect eggs was treated by the output of an ultrasonic whistle operating at 15 kHz. The day following the treatment, live larvae from each of the eggs coated the leaf. The sizes of the eggs compared with the wavelength imposed were large in the case of the chicken eggs, while the sizes of the insect eggs were very small compared with the wavelengths imposed by the whistles. The sizes of the insect eggs left no room for developing cavitation to emulsify the contents. No further tests were conducted along this line. 12.3.7

Pollinating Tomatoes

Tomatoes are difficult for bees to pollinate. The conventional method of pollinating tomatoes industrially is to send a group of workers through the rows of tomatoes at the proper time and to contact the stems containing blossoms that are ready to be pollinated with a vibrator. The ultrasonic method tried was to pollinate the blossoms using an ultrasonic whistle. The results were favorable, with the fruit described as “fuller.” This ended the experiments. Should the ultrasonic treatment of tomatoes be continued today? If nitrogen bottles could be carried by the workers pollinating the tomatoes and the test could be equivalent in cost to the mechanical devices, it might be economically advantageous compared with the mechanical method. The tomatoes obtained by ultrasonic application are fuller in size, therefore, the application of the ultrasound might be economical.

CRC_DK8307_CH012.indd 478

10/24/2008 1:12:02 PM

Criteria for Choosing Ultrasonics 12.3.8

479

Pest Control

This application has been investigated for many years. Possible methods include (1) using lethal intensities, (2) using intensities of frequencies that prove to be annoying to the pests, (3) duplicating the sounds of distress calls, and (4) duplicating the sounds of natural enemies. Some suggestions follow. 12.3.8.1

Insects

How is the energy to be applied? Can it be annoying to the insects and thus drive them away? Can it kill them if the source is brought into close contact with them? The first question relates to low-intensity signals, which could be applied in a general area infested by the insects. A person using this approach would have to consider several things: What frequencies or “tunes” would the insects respond to? How far would such ultrasound reach at such frequencies? How would it penetrate soils and brush to a successful level of intensity? Regarding the third question: yes, ultrasound can be sufficiently loud to kill insects. However, this would have to be done at very close range. It could be just as easily accomplished using a flyswatter. Lethal intensities are impractical for any pest. 12.3.8.2

Birds

This idea has been followed for several decades. One can still buy ultrasonic devices to drive off birds. The devices use several ultrasonic speakers that generate frequencies that irritate the birds and repels them. These frequencies are above the hearing range of humans. Crows and gulls are typical of birds that use ultrasound or sound for assembling together and for distress signals. Gulls or crows from one area may or may not respond to distress or assembly calls from another area, although they will always respond to the calls from birds in their own flock. Can this information be used to drive crows away [6]? Studies show that copying the distress signals of crows from a given region may be effective for a day or two, but the crows will return after they learn that the signals are artificial [6, p. 480, 7].

12.4

Separation of Materials

At times, two or more materials may be separated by means of ultrasonics. This phenomenon has been experienced in ultrasonic cleaning. Ultrasonics may be used to separate the particles in a mixture, to remove coatings or materials combined within or attached to another. When brown sand in water is exposed to cavitation levels of ultrasound, cavitation will remove the outer layer of clay from each particle of sand. The clean sand is more dense than the muddy slurry that it leaves behind, so it settles rapidly to the bottom, making a white deposit. This example is only one of many. The fluid may be other than water. A chemical reaction may be necessary to separate one material from the other. When ultrasonic energy is applied at cavitation levels, the material to be recovered or the material in which it is embedded may chemically react with the solvent, thus producing a material that is more dense than any other in the tank and that will settle out first. Or, the material desired may be separated from a heavier material and appear at the top of the slurry.

CRC_DK8307_CH012.indd 479

10/24/2008 1:12:02 PM

Ultrasonics: Data, Equations, and Their Practical Uses

480

The use of ultrasonic energy to remove dust from cotton has been studied. A sonic whistle radiating into a batch of freshly harvested cotton can raise a cloud of dust that can be removed by a gentle flow of air. This process leaves less dust to be removed by common washing (laundry) practices. The continuous flow of air through the whistle is sufficient for this purpose [8].

12.5

Summary

The use of ultrasonic energy either commercially or for convenience seems to be unlimited. An entrepreneur must have what it takes to put the product on the market. He must understand the potential market size and he must understand what his product can do and why it can do it. He must understand ultrasonics and the market. These facts may be illustrated by two different success stories. The first deals with breaking up clumps of synthetic mica into individual flakes and maintaining the original quality of each flake. Other methods were tried, including ultrasonics, but by trying to accomplish the separation while laying the clumps in the bottom of a beaker of water. They did not work. The only successful one was to apply ultrasonics by the method proposed in this chapter, in Section 12.3.1. This method was able to supply enough of the product to meet the need. A second program required the coating of fine particles (spheres) of ∼0.127 mm (0.005 in.) in diameter with a thin coating of polymer, ∼0.025 mm (0.001 in.) thick. No agglomerates were allowed. These beads were fed through one orifice of a stem-jet whistle operating at 15 kHz and the coating polymer was fed through another orifice. The jets of powder and polymer were aimed into the active zone of the whistle, which dispersed the powder into the atomized vapor of polymer. The material remained in suspension until the polymer had an opportunity to cure. The final product showed every particle being perfectly coated with no agglomerates (see Section 11.2.2). One must know what his product can do and be able to demonstrate it. He must have confidence enough in the usefulness of his product to push it in the commercial marketplace.

References 1. L. Bergmann. Translated by Dr. H. Stafford Hatfield, Ultrasonics, John Wiley and Sons, New York, NY, Authors Preface, p. v, 1938. 2. L. Bergmann, Der Ultraschall, Hirzel Verlag, Stuttgart, 1954. 3. The San Diego Union Tribune, December 2, 2006, pp. A1,2, 2006. 4. E. C. Kunz and D. Ensminger, Method and apparatus for treating mica, U.S. Patent No. 2,798,673. July 9, 1957. 5. E. C. Kunz and D. Ensminger, Preparation of perfume, U.S. Patent No. 3,025,220. March 3, 1962. 6. D. Ensminger, Ultrasonics, Marcel Dekker, New york, p. 7, 1988. 7. R. G. Busnel, J. Giban, Ph. Gramet, H. Frings, M. Frings, and J. Jumber, Paper HA6 presented at the Second ICA (International Congress on Acoustics) Congress, Boston, MA, 1956. 8. D. Ensminger, Joseph G. Montalvo, Jr. and Albert Baril, Jr., Application of ultrasonic forces to remove dust from cotton, Trans. ASME Journal of Engineering for Industry. Paper Delivered at Textile Industries Conference, October 7, 1983, Raleigh, NC, 1983.

CRC_DK8307_CH012.indd 480

10/24/2008 1:12:03 PM

Index A Abampere, 238 Abvolt, 238 Acoustical drying, ultrasonic technology, 450 Acoustic coaxing effect, sonification chemistry, 335 Acoustic impedance, piezoelectric materials, 191 Acoustic wave general wave equation, 14 horns, 20–22 plane wave equation, 12–14 vibratory systems, 2 Acute puncture closure, high-intensity focused ultrasound, 424 Adhesive, defined, 302 Adhesive bond/bonding, defined, 302 Aging, of piezoelectric materials, 232 Agricultural applications, ultrasound technology, 478–479 Amorphous materials, ultrasonic processing, 324 Ampere, 238 Ampere’s law, 238 Amplification factor, wedge-shaped horns, 44 Amplitude mode (A-mode), diagnostic ultrasound imaging, 432 Amplitude-variation detection, laser ultrasound, nondestructive testing, 403 Angular element response, nondestructive testing one-dimensional phased array, 390–391 two-dimensional phased array, 395 Anisotropy, defined, 187 Annular plates, plate wave equation free-free annular plate, 104–106 inside-outside clamps, 101–104 outside-inside clamps, 100–101 uniform thickness, 100–106 Antiferromagnetism, 238 Aqueous solutions, sonochemistry, 344–347 gases, 344 inorganic compounds, 344–345 organic compounds, 345–347 Array technologies, medical ultrasound technology, 436–439

Atomic number, elements properties, 287–291 Atomization mechanical applications, 448–449 ultrasonic applications, 338–340 Atrial fibrillation, high-intensity focused ultrasound, 421–422

B Bar drawing, ultrasonic technology for, 455–459 Bar impedance equation, wide blade-type horns, 120–121 Bar velocity of sound, horn performance and design factors, 56 Beam apodization, nondestructive testing, 391 Beam systems, medical ultrasound technology, 436–439 Bird management, ultrasound technology, 479 Bolt holes, finite element analysis, 159–162 Bond, defined, 301 Bone properties contact ultrasound technology and, 413–414 high-intensity focused ultrasound healing, 424–425 Boundary conditions, finite element analysis, 132, 145 free plate vibrations, 147–148 Bracelet devices, high-intensity focused ultrasound, atrial fibrillation, 423 Brightness mode (B-mode), diagnostic ultrasound imaging, 432 Broadband emission phased array, nondestructive testing, 387–388 Broadband excitation, two-dimensional phased array, 392 Broadband pulse excited phased array, nondestructive testing, 389–390 Bulk modulus of elasticity, acoustic waves, 22 Bulk velocity of sound, horn performance and design factors, 56 Bulk wave, electromagnetic acoustic transducers, nondestructive testing, 376 481

CRC_DK8307_INDEX.indd 481

11/3/2008 12:52:49 PM

482

C Cancer treatments, high-intensity focused ultrasound, 420–421 Capacitance defined, 187, 238 piezoelectric materials, 195 Capacitive micromachined ultrasonic transducer (CMUT), medical applications, 438–439 Cataract surgery, ultrasonic technology for, 414–415 Catenoidal horns, equations for, 50–52 Catheterization procedures, high-intensity focused ultrasound, atrial fibrillation, 422–423 Cavitating ultrasonic surgical aspiration (CUSA), techniques and applications, 415–417 Cavitation electroacoustic dewatering, 451–453 large-scale sonochemical processing, 358–359 medical ultrasound exposimetry, 439–440 ultrasonic lithotripsy, 419 ultrasonic mechanisms, 473 ultrasonic processing, 326–328 chemical effects, 342–343 Characteristic acoustic impedance, vibratory systems, 12 Checkerboard modes, flow-through ultrasonic processor, 153–154 Chemical environments materials design, harsh materials treatment, 127–128 ultrasonic mechanisms, 473 Circle diagram, magnetostrictive transducers, 248 Circular plates, plate wave equation, 97–98 Clamped-hinged bar, equations, 81, 86 Clamped-sliding bar, wave equations, 87 Class one-dimensional techniques, finite element analysis, large horns, 162–167 Cleaning fluids large-scale sonochemical processing, 358 ultrasonic processing, 329–332 Cleansers enzyme-based ultrasonic cleansers, 332–333 ultrasonic processing, 328–329, 473 manufacturers, 334 Closed loop systems magnetostrictive bars, 261–265

CRC_DK8307_INDEX.indd 482

Index magnetostrictive transducer design, uniform, rectangular bar stack, 267–268 Coagulation contact ultrasound technology, 411–413 ultrasonic applications, 337–338 Coil positioning, magnetostrictive bars, 260–261 Colorants, welding materials, 304 Commercialization issues, ultrasound applications, 475–479 Compatibility, welding materials, 302 Composite materials, electromagnetic acoustic transducers, 385–386 Compressibility, acoustic waves, 22 Compressional strength, piezoelectric materials, 193 Conductivity, 238 Conically tapered horns design equations, 44–50 area decrease, 48–50 area increase, 45–48 loss effects, 64–65 Constitutive relations, finite element analysis, 140–142 Contact medical ultrasound basic principles, 409 phacoemulsification, 414–415 surgical systems, 410–414 Continuous-wave (CW) Doppler mode, diagnostic ultrasound imaging, 433–434 Contrast agents, diagnostic ultrasonic imaging, 441 Corrosion, ultrasonic processing, 325 Coulomb unit, 238 Couplers defined, 29 driver-flexural bar coupling, 94 Cross sections, horn equation geometries, 37 Crystal axes, piezoelectric materials, 187–188 Crystallization piezoelectric materials, 196–218 ultrasonic processing, 341 Curie temperature, piezoelectric materials, 188 Current, 238 Cutting tools, ultrasound development of, 453, 457–459 Cylindrical shell and cavity finite element analysis, Donnell’s model, 153–162 horn equations, 122–123

11/3/2008 12:52:50 PM

Index

D Damping calculations, finite element analysis, 138 ultrasonic drivers, 179–180 Debridement procedures, high-intensity focused ultrasound wound healing, 426–427 Deburring operations, ultrasonic technology, 465–466 Degassing of liquids, 449–450 ultrasonic processing, 325–326 molten glass, 460–461 Degating, materials properties, 316 De-inking processes, ultrasonic applications, 333–335 Dental tools, ultrasonic technology for, 455 Design factors finite element analysis, ultrasonic drivers, 168–170 harsh chemical environment, materials treatment, 127–128 high-temperature materials treatment, 127 horn equations, 55–69 loss effects, 61–68 Poisson’s ratio, 56–60 temperature effects, 68–69 stress concentration, 127 ultrasonic horns, 126–128 Devulcanization, ultrasonic polymer processing, 356–357 Dewatering, electroacoustic processes, 451–452 Diagnostic ultrasound basic principles, 430–431 contrast agents, 441 Doppler imaging, 432–434 harmonic imaging, 434–436 imaging modes, 432 intravascular ultrasound, 442–443 overview, 410 sonoelastography, 441–442 tissue properties, 431–432 Die applications, ultrasonic technology for, 455–459 Dielectric constant, piezoelectric materials, 188–189, 192 Dielectric displacement, piezoelectric materials, 189 Dielectric loss factor, piezoelectric materials, 189 Diffusion, ultrasonic processing, 341–342, 473 Dipole, piezoelectric materials, 189 Directionality, electromagnetic acoustic transducers, 380–383

CRC_DK8307_INDEX.indd 483

483 Direct numerical integration, finite element analysis, ultrasonic drivers, 176–179 Dispersion, ultrasonic homogenization and, 337 Displacement nodes finite element analysis, 134–135 large horns, 165–167 transient analysis, 177–179 ultrasonic drivers, 170–173 plate wave equation, 94–95 shell vibrations, finite element analysis, 149–151 wedge-shaped horn equation, 41 Distributed load, horn equations, 72–74 Donnell’s model, finite element analysis cylindrical shell, 153–162 shell vibrations, 148–151 Doppler imaging, diagnostic ultrasound imaging, 432–434 Double cylinder horn equation, 32–36 stepped horn equations, 35–36 Drilling tools, ultrasound development of, 453–455 Driver-flexural bar coupling, 94 Dysprosium, 238

E Eddy currents, magnetostrictive transducer impedance, 249, 251 Efficiency, piezoelectric materials, 189 Egg processing and handling, ultrasound technology, 478 Eigenvalues, finite element analysis, 138 ultrasonic drivers, 175–179 Eigenvectors, finite element analysis, 136–138 Elastic compliance, piezoelectric materials, 189–190 Elasticity acoustic waves, 22–24 piezoelectric materials, 193–194 Electro-osmosis, electroacoustic dewatering, 452–453 Electrical conductivity, piezoelectric materials, 190 Electrical impedance magnetostrictive transducer impedance, 249–250 magnetostrictive transducers, toroid transducers, 253–254 Electrical quality factor (Q ), vibratory systems, 11–12 Electrical resistivity, piezoelectric materials, 190

11/3/2008 12:52:51 PM

Index

484 Electric charge, 238 Electric field, piezoelectric materials, 190 Electric potential, 238 Electroacoustic dewatering, ultrasonic technology, 451–453 Electrodes, piezoelectric materials design criteria, 232 soldering, 231 Electrohydraulic generator, ultrasonic lithotripsy, 418 Electrolytic effects, ultrasound technology, 473 Electromagnetic acoustic transducers (EMATs), nondestructive testing, 375–386 basic principles, 375–380 composite materials, 385–386 limitations, 386 measurement techniques, 385 terminology, 376 time measurement technique, 380 ultrasonic stress measurement, 384–385 ultrasonic texture measurement, 380–384 velocity measurements, 380 Electromechanical coupling coefficient piezoelectric materials, 190–191 stored energy and sensitivity, 195–196 toroidal transducers, 257–258 Electromotive force, 238 Electrophoresis, electroacoustic dewatering, 452–453 Electrostrictive effect, piezoelectric materials, 191 Elements alphabetical order and atomic numbers, 287 angular response, nondestructive testing, 390–391, 395 physical properties, 288–291 Emulsifiers, ultrasonic processing, 324 homogenization and, 335–337 Energy storage, piezoelectric materials, 194–195 Engineering materials, physical properties, 292–299 Enzyme-ultrasonic cleanser interactions, 332–333 Epoxy curing, ultrasound technology, 453 Equivalent circuits magnetostrictive materials, 245–246 magnetostrictive bars, 260–265 magnetostrictive transducers, impedance, 246–250 toroidal transducers, 255–256 Erbium, 238 Exponential tapered horn equations longitudinal geometry, 37–38 lumped mass, 72

CRC_DK8307_INDEX.indd 484

rectangular cross section and constant width, 39–40 velocity of sound corrections, 58–60, 63–68 Exposimetry, ultrasonic, medical applications, 439–440 Extracorporeal shock wave lithotripsy (ESWL), ultrasonic techniques, 418

F Failure analysis, stress concentration, 127 Fatigue testing materials properties, 299–300 ultrasonic technology, 466–467 Fat removal, high-intensity focused ultrasound, 423–424 Ferroelectric crystals, piezoelectric materials, 191 Fillers, welding materials, 303 Filtration, ultrasonic applications, 337–338 membrane diffusion and, 341–342 Fine particle coating, whistle devices for, 280 Finite element, defined, 133 Finite element analysis (FEA), ultrasonic transducers basic principles, 130–135 boundary conditions, 145 commercial software packages, 180–181 conventional design approach, 168–170 cylindrical shell examples, 153–158 damping and power loss, 179–180 Donnell’s model comparisons, 158–162 flat plate example, 151–153 interpolation, 142–145 large horn example, 162–167 linear elastic isotropic properties, 145–146 piezoelectricity, 139–142 plate vibrations, 146–148 pre- and post-processors, 146 resonances and displacements, 170–173 shell vibrations, 148–151 solutions, 136–139 summary and future applications, 181–182 symmetry, 145 theory, 134–136 transient analysis, 173–179 ultrasonic driver, 167–168 Finite pulse length, two-dimensional phased array, 392–393 Fixed-free bar, Rayleigh frequency, 91–92 Flame retardants, welding materials, 303 Flat plate design, finite element analysis, 151–153 Flexible string, transverse wave equation, 14–15

11/3/2008 12:52:51 PM

Index Flexural bar clamped-clamped solution, 81–88 driver-flexural bar coupling, 94 longitudinal mode, 75 stresses, 93 Flexural mode, ultrasonic drivers, fi nite element analysis, 171–173 Flexural rings, vibration principles, 116–117 Flow-through processor, finite element analysis, 155–157 Flux penetration, 238 Foam breakdown, whistle devices for, 280–281 Force, vibratory systems, 2 Forced boundary conditions, fi nite element analysis, 145 Forming operations materials properties, 314–315 ultrasonic technology for, 456–459 Fourier decomposition, finite element analysis, ultrasonic drivers, 176–179 Fraunhofer zone, medical ultrasound technology, 436–437 Free circular plaques, plate wave equation, 98–99 Free-free annual plates, plate wave equation, 104–106 Free-free uniform bar, horn equation, 30–32 Free plate vibrations, finite element analysis, 147–148 Free radical scavengers, ultrasonic processes, polymerization, 351–352 Free-sliding, wave equations, 87 Frequency constant, piezoelectric materials, 191 Frequency parameters, annular plates, 103–106 Fresnel zone, medical ultrasound technology, 436–437 Friction, materials properties, 300–302

G Gadolinium, 238 Galton whistles, pneumatic transducer design, 275 Gases, aqueous sonochemical reactions, 344 General wave equation, acoustic wave, 14 Geometrical forms, 69–94 bar flexural stresses, 93 boundary conditions, 85–88 clamped-clamped bar, 81 distributed loads, 72–74 driver-flexural bar coupling, 94 flexural bar, longitudinal mode, 75 flexural bar solutions, 81–94 inertial moments, 76–78

CRC_DK8307_INDEX.indd 485

485 lumped mass, 71–72 multiple mode systems, 74–94 Rayleigh determination, fundamental frequency, 88–92 thin bar transverse wave equation, 78–81 uniform bar characteristics, 70–71 Geometric boundary conditions, finite element analysis, 145 Gilbert force defined, 238 magnetostrictive transducers, 237 Glass transition temperature, ultrasonic processing, 324 Glue, defined, 301 Grinding operations, ultrasonic tools for, 454–455

H Half-wave horn length horn equations, 39, 42–43 large, cup-shaped horns, 124–126 Half-wave magnetostrictive rods, mass and resistance loading, 265–266 Hamiltonian function, 238 Hamilton-Jacobi equation, 239 Hamilton-Jacobi theory, 239 Hamilton’s equation of motion, 239 Hamilton’s principle, 239 Handheld devices, ultrasonic nondestructive testing flaw detection equipment, 368–369 thickness measurement devices, 367–368 Harmonic™ contact ultrasonic device, 411–412 Harmonic imaging, diagnostic applications, 434–436 Hartmann whistle basic theory, 275–277 defined, 273 mechanical applications, 448–449 modification of, 279–280 power increases, 281 Heat mechanisms, ultrasonics-assisted processing, 472 Henry unit, 239 High-intensity focused ultrasound (HIFU) acute puncture and wound closure, 424 atrial fibrillation, 421–422 basic principles, 409–410 cancer treatments, 420–421 fat removal, 423–424 general principles, 419–420 wrinkle reduction, 423

11/3/2008 12:52:51 PM

Index

486 High-Q ultrasonic driver, shell modes, 153–155 High-temperature stress, materials design, 127 Hollow cylinders piezoelectric materials, 224–225 vibration principles, 117–118 Homogenization, ultrasonic applications dispersion and, 337 emulsification and, 335–337 Hook’s law, finite element analysis, piezoelectric materials, 140–142 Horizontal shear welding, metals and plastics, ultrasonic technology, 467–469 Horn devices design and performance factors, 55–69 loss effects, 61–68 Poisson’s ratio, 56–60 temperature effects, 68–69 large area block-type horns, 121–122 large, cup-shaped horns, 122–126 liquid jet applications, 449 lumped mass, 71–72 solutions, 30–55 basic equation, 20–22 catenoidal horns, 50–52 conically tapered horns, 44–50 double cylinder/stepped horn, 32–36 exponential tapering, 37–40 free-free uniform bar, 30–32 hyperbolic horns, 52–55 noncircular cross-sections, 37 slender wedge-shaped horn, 40–44 ultrasonic applications, 20–22 wide blade-type horns, 119–121 Hostile environments, ultrasound applications in, 459–462 Hygroscopicity, welding materials, 302 Hyperbolic horns, equations for, 52–55 Hysteresis loss, magnetostrictive transducer impedance, 251–252

I Impedance horn equations, mechanical impedance, 39 magnetostrictive transducers, 246–250 circle diagram, 248 electrical impedance and efficiency, 249–250 mechanical impedance, 248–249 motional impedance, 247–248 toroidal transducers, 253–255

CRC_DK8307_INDEX.indd 486

piezoelectric materials, 191 vibratory systems, 10–12 wedge-shaped horn equation, 43–44 Inductance, 239 Industrial phased arrays, nondestructive testing, 395–397 Inorganic compounds, sonochemical reactions, aqueous solutions, 344–345 Insecticides, ultrasound technology, 479 Insertion, materials properties, 313–314 Inside-outside clamps, annular plates, 101–104 Intensity of magnetization, 239 Interferometric detection, laser ultrasound, nondestructive testing, 402–403 Internal damping factors, horn performance losses, 61–68 Interpolation matrix, finite element analysis, 142–145 Intravascular ultrasound, basic principles, 442–443 Isoparametric function, finite element analysis, 142–145 Isotropy, piezoelectric materials, 192

J Jacobian operator, finite element analysis, 144–145 Joint designs, welding materials, 304–308

K Kynar piezo film, 192, 222

L Laboratory systems, ultrasonic nondestructive testing, 372 Lagrangian equation, 239 Lagrangian function, 239 Lamb waves, oscillatory motions, 17–19 Laparoscopy, contact ultrasound technology, 413–414 Laplace operators plate wave equation, 94–96 wave equations, 19–20 Large horns block-type horns, properties and applications, 121–122 cup-shaped horns, properties and applications, 122–126 finite element analysis, 162–167

11/3/2008 12:52:51 PM

Index Large-scale sonochemical processing, ultrasonic transducers, 357–360 Laryngeal papilloma, ultrasound management of, 477–478 Laser detectors, laser ultrasound, nondestructive testing, 401–403 Laser ultrasound, nondestructive testing, 397–403 amplitude-variation detection, 403 interferometric detection, 402–403 plasma regime, 399–401 thermoelastic regime, 399 Lateral coupling factor, piezoelectric materials, 190 Lathe tools, ultrasound development of, 454–455 Laves phases, 239 Linear elastic isotropic materials, finite element analysis, 145–146 ultrasonic drivers, 175–179 Liposuction, ultrasound technology, 409, 417 Liquid surface atomization, ultrasound applications, 448–449 Lithotripsy, ultrasound technology, 409, 417–419 Load effects finite element analysis, 132 horn performance losses, 61–68 Load impedance, toroidal transducers, 255 Longitudinal coupling factor, piezoelectric materials, 190, 222 Longitudinal mode, flexural bar, 75 Longitudinal vibrations, contact ultrasound technology, 411–412 Lorentz force, electromagnetic acoustic transducers, nondestructive testing, 376 Loss mechanisms horn performance, 61–68 spring/mass oscillator, 9–10 Love waves, oscillatory motions, 19 Low-intensity ultrasonic applications, 473–474 Lubricants, welding materials, 303 Lumped mass, horn equation, 71–72

M Machining operations, ultrasonic tools for, 455 Magnetic field strength, 239 Magnetic flux, 239 Magnetic flux density, 239 Magnetic induction, 239 Magnetic pole strength, 239 Magnetostriction defined, 239

CRC_DK8307_INDEX.indd 487

487 eddy currents, 251 electromagnetic acoustic transducers, nondestructive testing, 376 equivalent circuits, 245–246 historical background, 236–237 hysteresis losses, 251–252 materials availability, 271 properties, 241 notation, 244–245 steady-state magnetic relationships, 241–243 strain curves, 241 terminology and nomenclature, 238–241, 244–245 transducers applications, 271 closed-loop, uniform, rectangular bar stack, 267–268 design and construction, 266 impedance, 246–250 circle diagram, 248 electrical impedance and efficiency, 249–250 mechanical impedance, 248–249 motional impedance, 247–248 magnetic properties, 237 magnetostrictive bar, 258–266 multiple-bar type, 268–269 permanent magnet bias, 269–270 silver soldering and brazing, 270 simple uniform stack, 267 toroid transducer, 253–258 Magnetostrictive bars basic properties, 258–260 designs and coil positions, 260–261 equivalent circuits, 261–265 half-wave rods, 265–266 Mass properties, spring/mass oscillator, 9 Matching elements, piezoelectric transducer structures, 230–231 Matching principle, uniform bars, 73–74 Material geometry, acoustic waves, 22–24 Materials properties atomization, 338–340 cavitation, 326–328 ultrasonic cleaning, 328–333 chemical effects, 342–343 chemical processes, 325 coagulation, precipitation and filtration, 337–338 crystallization, 341 degassing, 325–326 design factors, 126–128 devulcanization, 356–357

11/3/2008 12:52:51 PM

488 Materials properties (contd.) dispersion and homogenization, 337–338 electromagnetic acoustic transducers, 385–386 elements, alphabetical order and atomic numbers, 287 engineering materials, 292–299 enzyme-based ultrasonic cleaners, 332–333 homogenization and emulsification, 335–337 magnetostrictive properties, 241–244 commercial availability, 271 membrane diffusion and filtration, 341–342 nanomaterials preparation, 340 physical properties, 286–299, 324 polymer degradation, 352–354 polymerization, 350–352 polymer processing, 354–356 sonochemical reactions aqueous solutions, 344–349 gases, 344 inorganic compounds, 344–345 large-scale processing, 357–361 nonaqueous compounds, 347–349 organic compounds, 345–347 theoretical background, 286 ultimate stress and fatigue limit, 299–300 ultrasonic processing, 325 friction and wear, 300–302 office waste paper de-inking, 333–334 pH levels and, 461–462 staking, 309–313 ultrasonic cleaner manufacturers, 333–335 ultrasonic welding, 308–309 welding and staking, 302–308 vulcanization, 354–356 weights and measures, 286 Materials separation, ultrasound technology, 479–480 Mathematical relationships, piezoelectric materials, 194–195 Matrix equation, ultrasonic drivers, finite element analysis, 169–170 Maximum kinetic energy, Rayleigh frequency, 90–92 Maximum power transfer, toroidal transducers, 258 Meander coil, electromagnetic acoustic transducers, nondestructive testing, 376 Mechanical energy storage, piezoelectric materials, 195 Mechanical impedance horn equations, 39 magnetostrictive transducers, 248–249 toroid transducers, 253

CRC_DK8307_INDEX.indd 488

Index Mechanical index (MI), medical ultrasound exposimetry, 439–440 Mechanical quality factor (Q ) piezoelectric materials, 193 vibratory systems, 11–12 Mechanical ultrasonic technology acoustical drying, 450 atomization, 448–449 deburring, 465–466 degassing liquids, 449–450 electroacoustic dewatering, 451–453 epoxy curing, 453 fatigue testing, 466–467 in hostile environments, 459–462 metallurgical processes, 464–465 metal parts compaction, 462–464 small liquid jets, 449 soldering, 459 stress effects, 473 tools development, 453–455 welding of metals and plastics, 467–469 wire and bar drawing, 455–459 Medical ultrasound bone healing, 424–425 cavitating ultrasonic surgical aspiration, 415–417 contact applications, 409 phacoemulsification, 414–415 surgical systems, 410–414 diagnostic imaging, 410 basic principles, 430–431 contrast agents, 441 Doppler imaging, 432–434 harmonic imaging, 434–436 imaging modes, 432 intravascular ultrasound, 442–443 sonoelastography, 441–442 tissue properties, 431–432 exposimetry, 439–440 future applications, 443–444 healing treatments, 424–427 high-intensity focusing, 409–410 acute puncture and wound closure, 424 atrial fibrillation, 421–422 cancer treatments, 420–421 fat removal, 423–424 general principles, 419–420 wrinkle reduction, 423 laryngeal papilloma, 477–478 liposuction, 409, 417 lithotripsy, 409, 417–419 low-intensity applications, 475 overview of applications, 408–409 sonophoresis, 427

11/3/2008 12:52:51 PM

Index thrombolysis, 429–430 transducers, beams and arrays, 436–439 in vivo surgical welding, 427–429 wound healing, 409, 425–427 Membranes diffusion and filtration, ultrasonic processing and, 341–342 transverse wave equation, 15–16 Mesh density finite element analysis, 131–132 large horns, finite element analysis, 163–167 Metallurgical processes, ultrasonic technology, 464–465 Metals compaction of metal parts, ultrasonic processing, 462–464 crystallization in, 341 formation of, ultrasound technology, 457–459 nonaqueous sonochemical reactions, 348–349 Metastable physics, 240 Mica milling, ultrasound applications, 475–476 Miscibility, ultrasonic processing, 324 Modal frequencies, finite element analysis, cylindrical designs, 159–162 Modal superposition, finite element analysis, 139 Modulus of elasticity, piezoelectric materials, 193–194 Mold release agents, welding materials, 302–303 Moments of inertia, multiple mode systems, 76–78 Mossbauer effect, 240 Motional impedance, magnetostrictive transducers, 247–248 Motion mode (M-mode), diagnostic ultrasound imaging, 432 Multiple-bar transducer design, 268–269 Multiple mode systems geometrical forms, 74–94 longitudinal mode horn, flexural bar, 75 moments of inertia, 76–78 Mutual impedance, toroid transducers, 254

489 Nondestructive testing (NDT) electromagnetic acoustic transducers, 375–386 basic principles, 375–380 composite materials, 385–386 limitations, 386 measurement techniques, 385 terminology, 376 time measurement technique, 380 ultrasonic stress measurement, 384–385 ultrasonic texture measurement, 380–384 velocity measurements, 380 laser ultrasound, 397–403 amplitude-variation detection, 403 interferometric detection, 402–403 plasma regime, 399–401 thermoelastic regime, 399 phased arrays, 386–392 angular element response, 390–391, 395 basic principles, 386–387 beam apodization, 391 broadband emission phased array, 387–388 broadband excitation, 392 broadband pulse excited phased array, 389–390 finite pulse length, 392–393 industrial application, 395–397 one-dimensional continuous-wave phased array, 387–392 reciprocity, 391, 395 steering and focusing time delays, 390, 393–395 two-dimensional case, 392–395 ultrasonic techniques, 474–475 laboratory systems, 372 manufacturers of, 374–375 modern equipment, 366–375 portable field equipment, 367–369 stationary field equipment, 369–372 transducers, 372–374 Nonlinearity, diagnostic harmonic imaging, 435–436

O N Nanomaterials, ultrasonic processing, 340 Neel temperature, 240 Nodal patterns, finite element analysis cylindrical designs, 159–162 flat plate design, 151–153

CRC_DK8307_INDEX.indd 489

Oersted, 240 Office paper de-inking, ultrasonic applications, 333–335 One-dimensional continuous-wave phased array, nondestructive testing, 387–392

11/3/2008 12:52:52 PM

490 Organic compounds, sonochemical reactions aqueous solutions, 345–347 nonaqueous reactions, 347–348 Orientation distribution coefficients (ODCs), electromagnetic acoustic transducers, 383–384 Oscillations ultrasonic processing, 325 vibratory systems pendulum, 3–8 simple and compound pendulums, 4–7 torsional pendulums, 7–8 spring/mass oscillator, 8–10 ultrasonic energy, 2–3 Outside-inside clamps, annular plates, 100–101

P Pancake coil, electromagnetic acoustic transducers, nondestructive testing, 376 Paper sludge flow measurement, ultrasound applications, 477 Paramagnetism, 240 Parametric studies, finite element analysis, 132–133 Particle acceleration distribution, horn equations, 39 Particle displacement distribution, horn equations, 38–39 Pendulum, oscillatory motion, 3–8 simple and compound pendulums, 4–7 torsional pendulums, 7–8 Percutaneous nephrolithotomy (PCNL), ultrasonic techniques, 417–418 Performance factors, horn equations, 55–56, 55–69 loss effects, 61–68 Poisson’s ratio, 56–60 temperature effects, 68–69 Perfume extraction, ultrasound applications, 476–477 Permanent magnet bias, magnetostrictive transducer design, 269–270 Permeability, 240 Permendur, 240 Permittivity, piezoelectric materials, 188, 192 Pest control, ultrasound technology, 479 Phacoemulsification, ultrasonic technology for, 414–415 Phased arrays, nondestructive testing, 386–392 angular element response, 390–391, 395 basic principles, 386–387 beam apodization, 391

CRC_DK8307_INDEX.indd 490

Index broadband emission phased array, 387–388 broadband excitation, 392 broadband pulse excited phased array, 389–390 finite pulse length, 392–393 industrial application, 395–397 one-dimensional continuous-wave phased array, 387–392 reciprocity, 391, 395 steering and focusing time delays, 390, 393–395 two-dimensional case, 392–395 Phonons, horn performance, 68–69 Piezoelectric devices and materials aging, 232 basic properties, 185–186 crystalline properties, 196, 218–220 definitions, 187–193 electrochemical coupling coefficient, stored energy, and sensitivity, 194–196 finite element analysis, 139–142 damping and power losses, 179–180 mathetmatical relationships, 194–196 mechanical data, 193–194 polymer properties, 196, 221–222 properties, 196–222 selection and design guidelines, 196, 222–227 electrode types, 232 transducer assembly, 227–231 transducer design, 231–232 symbols and direction notations, 186–187 thermal data, 194 transducer applications, 232–233 ultrasonic drivers finite element analysis, 168–170 transient analysis, 173–179 ultrasonic lithotripsy, 418–419 Piezoelectric effect defined, 192 fundamental actions, 196, 222 Piezoelectric strain, defined, 192–193 Pin-jet whistle, pneumatic transducer design, 280 Pinned-free bar, wave equations, 86 Pinned-sliding bar, wave equations, 87–88 Planar coupling factor, piezoelectric materials, 190 Planck’s constant, 240 Plane wave equation, acoustic wave, 12–14 Plasma regime, laser ultrasound, nondestructive testing, 399–401 Plastic deformation, ultrasound technology, 456–459

11/3/2008 12:52:52 PM

Index Plasticizers, welding materials, 303 Plate theory, finite element analysis basic equations, 146–147 flat plate design, 151–153 free plate vibrations, 147–148 Plate wave equation annular plates, 100–106 free-free annular plate, 104–106 inside-outside clamps, 101–104 outside-inside clamps, 100–101 basic components, 94–96 fixed center circular plate, 98–99 oscillatory motions, 17–19 polar coordinate solutions, 96 rectangular coordinates, 97 rectangular plates, 106–114 sides simply supported, 109–110 SS-C-SS-C condition, 110–112 simply supported plates, 99–100 uniform circular plate thickness and radius, 97–98 Pneumatic transducers basic properties, 273–274 vibrating blade, 282–283 whistles applications, 280–281 basic properties, 274 common whistles, 274–275 Galton whistles, 275 Hartmann whistle, 275–277, 279–280 pin-jet whistle, 280 stem-jet whistle, 277–279 vortex whistles, 282 Poisson’s ratio acoustic waves, 22–24 horn performance and design factors, 56–60 velocity of sound, 55–56 large-area block-type horns, 121–122 large, cup-shaped horns, 123–126 piezoelectric materials, 194 plate wave equation, 94–95 wide blade-type horns, 121 Polar coordinates, plate wave equation, 94–96 Polymer degradation, ultrasonic processes, 353–354 Polymerization, ultrasonic processes, 350–352 Portable field equipment, ultrasonic nondestructive testing, 367–369 Postprocessors, finite element analysis, 146 Potential efficiency, magnetostrictive transducer impedance, 250 Powdered coffee, ultrasound applications, 477 Power applications, piezoelectric transducer assembly, 227–229

CRC_DK8307_INDEX.indd 491

491 Power loss contact ultrasound technology, 412 finite element analysis, ultrasonic drivers, 179–180 Precipitation, ultrasonic applications, 337–338 Pre-processors, finite element analysis, 146 Prostate cancer, high-intensity focused ultrasound, 420–421 Puncture wounds, high-intensity focused ultrasound closure, 424 Pure radial vibration, rings, 114–116

Q Quality factor (Q ) piezoelectric materials, 193 vibratory systems, 11–12 Quantum, 240 Quantum theory, 240

R Radial coupling factor, piezoelectric materials, 190 Radiation efficiency, toroidal transducers, 256–258 Rayleigh frequency, distributed mass systems, 89–92 Rayleigh surface velocity, acoustic waves, 23–24 Reciprocity, nondestructive testing one-dimensional phased array, 391 two-dimensional phased array, 395 Rectangular coordinates, plate wave equation, 97 Rectangular plates, plate wave equation, 106–114 sides simply supported, 109–110 SS-C-SS-C condition, 110–112 Rectified diffusion, cylindrical shell, finite element analysis, 154–162 Regrind, welding materials, 304 Relative Permittivity, piezoelectric materials, 188–189, 192 Reluctance, 240 Resin grade, welding materials, 304 Resistance, 240 Resonance ultrasonic drivers, fi nite element analysis, 170–173 vibratory systems, 10–12 Ring modes, piezoelectric materials, 224

11/3/2008 12:52:52 PM

492 Rings, vibration principles, 114–117 flexural modes, 116–117 pure radial vibration, 114–116 Round disks, piezoelectric materials, 224

S Samarium, 240 Sandwich transducers, piezoelectric materials, 227–229 Scalpels, contact ultrasound technology, 410–411 Scan welding, materials properties, 316 Schrödinger equation, 240 Second-order differential matrix equation, finite element analysis, 136–138 Sensitivity parameters, piezoelectric materials, 194–195 Shear coupling factor, piezoelectric materials, 190 Shear modulus of elasticity, acoustic waves, 22 Shear velocity of sound, horn performance and design factors, 56 Shell theory, finite element analysis basic equations, 146–147 cylindrical shells, 153–162 shell vibrations, 148–151 Shell vibrations, finite element analysis, 148–151 Silver soldering and brazing, magnetostrictive transducer fabrication, 270 Simple uniform stack, magnetostrictive transducer design, 267 Simply supported circular plates, plate wave equation, 99–100 Slender wedge-shaped horn equations, 40–44 Slotted horns, finite element analysis, 162–167 Software packages, finite element analysis, 180–181 Soldering operations, ultrasound technology, 459 Sonic whistles, design and applications, 280–281 Sonocatalysts, nonaqueous sonochemical reactions, 349 Sonochemistry aqueous solutions gases, 344 inorganic compounds, 344–345 organic compounds, 345–347 nonaqueous reactions, 347–349 organic compounds, 347–348 organometallic compounds, 348–349 sonocatalysts, 349 ultrasonic processing, 342–343

CRC_DK8307_INDEX.indd 492

Index Sonoelastography, diagnostic ultrasonic imaging, 441–442 Sonophoresis, medical applications, 427 Specific heat, piezoelectric materials, 194 Spot welding materials properties, 315–316 metals and plastics, ultrasonic technology, 468–469 Spring/mass oscillator principles of, 8–9 Rayleigh frequency, 89–92 Static equilibrium equation, finite element analysis, 138–139 Stationary field equipment, ultrasonic nondestructive testing, 369–372 Steady-state harmonic analysis, ultrasonic drivers, 178–179 Steady-state magnetic relationships, magnetostrictive materials, 241–244 Steam-jet whistle, mechanical applications, 448 Stem-jet whistle construction of, 278–279 pneumatic transducer design, 277–279 Stepped horn equation, 32–36 Stirring mechanisms, ultrasonics-assisted processing, 473 Stored electrical energy, piezoelectric materials, 195 Strain energy piezoelectric materials, 194 shell modes, finite element analysis, 157–162 Strap welding, metals and plastics, ultrasonic technology, 467–469 Stress concentration flexural bars, 93 materials design, 127 stepped horn equations, 33–35 Stress distribution horn equations, 39 wedge-shaped horn equation, 42 Stress measurement, electromagnetic acoustic transducers, 384–385 Stud welding, materials properties, 313 Superposition problem, finite element analysis, ultrasonic drivers, 175–179 Surgical ultrasound contact systems, 410–414 phacoemulsification, 414–415 suture welding techniques, 427–429 Susceptibility, 240 Swaging, materials properties, 314–315 Symmetry, finite element analysis, 145 Donnell’s model v., 159–162

11/3/2008 12:52:52 PM

Index

T Tapered horns, velocity of sound corrections, 58–60, 63–68 Temperature effects, horn performance, 68–69 Tempering operations, ultrasound development of, 457–459 Tensile strength, piezoelectric materials, 194 Terbium, 240 Terfenol-D, 240 Texture measurement, electromagnetic acoustic transducers, 380–383 Thermal conductivity, piezoelectric materials, 194 Thermal data, piezoelectric materials, 194 Thermal-equivalent theory, drawing and forming operations, 456–459 Thermal index (TI), medical ultrasound exposimetry, 439–440 Thermoelastic effects, horn performance, 69 Thermoplastic regime, laser ultrasound, nondestructive testing, 399 Thickness coupling factor, piezoelectric materials, 191, 223 Thin bars, transverse wave equation, 78–81 Thrombolysis, ultrasound technologies, 429–430 Time constant, piezoelectric materials, 193 Time delays, steering and focusing one-dimensional phased array, 390 two-dimensional phased array, 393–394 Time measurements, electromagnetic acoustic transducers, 380 Tissue properties, diagnostic ultrasound imaging, 431–432 Tomato pollination, ultrasound technology, 478 Tools defined, 29 ultrasound development of, 453–455 Toroid transducers electrical impedance, 253–254 equivalent circuits, 255–256 load impedance, 255 maximum power transfer, 258 mechanical impedance, 253 mutual impedance, 254 radiation efficiency, 256–258 Torque pendulum systems, 4–7 piezoelectric transducer assembly, 228–229 Torsional mode piezoelectric materials, 225–226 ultrasonic drivers, fi nite element analysis, 171–173 Torsional pendulums, oscillatory motion, 7–8

CRC_DK8307_INDEX.indd 493

493 Transducers electromagnetic acoustic transducers, nondestructive testing, 375–386 basic principles, 375–380 composite materials, 385–386 limitations, 386 measurement techniques, 385 terminology, 376 time measurement technique, 380 ultrasonic stress measurement, 384–385 ultrasonic texture measurement, 380–384 velocity measurements, 380 magnetostrictive transducers applications, 271 closed-loop, uniform, rectangular bar stack, 267–268 design and construction, 266 impedance, 246–250 circle diagram, 248 electrical impedance and efficiency, 249–250 mechanical impedance, 248–249 motional impedance, 247–248 magnetic properties, 237 magnetostrictive bar, 258–266 multiple-bar type, 268–269 permanent magnet bias, 269–270 silver soldering and brazing, 270 simple uniform stack, 267 toroid transducer, 253–258 medical ultrasound technology, 436–439 piezoelectric materials applications, 232–233 assembly, 227–229 design criteria, 231–232 structural elements, 229–231 pneumatic transducers basic properties, 273–274 vibrating blade, 282–283 whistles applications, 280–281 basic properties, 274 common whistles, 274–275 Galton whistles, 275 Hartmann whistle, 275–277, 279–280 pin-jet whistle, 280 stem-jet whistle, 277–279 vortex whistles, 282 ultrasonic nondestructive testing, 372–375 current manufacturers list, 374–375 metallurgic processes, 465 Transient analysis, ultrasonic drivers, 173–179 Transverse coupling factor, piezoelectric materials, 190, 223

11/3/2008 12:52:52 PM

Index

494 Transverse wave equation, thin bars, 78–81 Transverse wave equations, acoustic waves, 14–16 Two-dimensional phased arrays, nondestructive testing, 392–395 angular element response, 395 broadband excitation, 392 finite pulse length, 392–393 reciprocity, 395 steering and focusing time delays, 393–394

U Ultimate stress, materials properties, 299–300 Ultrasonics-assisted processing atomization, 338–340 basic principles, 324 cavitation, 326–328 chemical processing, 325 cleaning fluids, 329–332 cleaning processes, 328–329 coagulation, precipitation, and filtration, 337–338 commercialization issues, 475–479 crystallization, 341 degassing, 325–326 devulcanization, 356–357 dispersion and homogenization, 337 enzyme-based cleansers, 332–333 homogenization and emulsification, 335–337 low-intensity applications, 474–475 materials separation, 479–480 membrane diffusion and filtration, 341–342 nanomaterials, 340 office waste paper de-inking, 333–335 oscillatory effect, 325 physical processes, 324 polymer degradation, 353–354 polymerization, 350–352 polymer processing, 354–356 selection criteria, 471–474 sonochemistry, 342–343 aqueous solutions, 344–347 large-scale processing, 357–360 nonaqueous solutions, 347–349 vulcanization, 355–356 Ultrasonic transducers drivers, finite element analysis, 167–180 conventional design, 168–170 damping and power loss, 179–180 resonances and displacements, 170–173 transient analysis, 173–179 finite element analysis

CRC_DK8307_INDEX.indd 494

basic principles, 130–135 boundary conditions, 145 commercial software packages, 180–181 conventional design approach, 168–170 cylindrical shell examples, 153–158 damping and power loss, 179–180 Donnell’s model comparisons, 158–162 flat plate example, 151–153 interpolation, 142–145 large horn example, 162–167 linear elastic isotropic properties, 145–146 piezoelectricity, 139–142 plate vibrations, 146–148 pre- and post-processors, 146 resonances and displacements, 170–173 shell vibrations, 148–151 solutions, 136–139 summary and future applications, 181–182 symmetry, 145 theory, 134–136 transient analysis, 173–179 ultrasonic driver, 167–168 horns, 20–22 defined, 29 impedance, resonance and Q factor, 10–12 materials properties, 300–312 friction and wear, 300–302 geometry and elasticity, 22–24 staking, 309–313 ultrasonic welding, 308–309 welding and staking, 302–308 physical factors, 24–25 piezoelectric materials, 194–195 vibratory systems, 2–3 Uniform bar distributed load matching, 73–74 equal diameters, 70–71 horn performance losses, 62–63 lumped mass, 71–72 magnetostrictive properties, 258–266 designs and coil positions, 260–261 equivalent circuits, 261–265 half-wave rods, 265–266 pendulum torque, 6–7 transverse wave equation, 16

V Vacuum effects, ultrasound technology, 473 Velocity distribution horn equations, 38 wedge-shaped horn equation, 41

11/3/2008 12:52:52 PM

Index Velocity measurements, electromagnetic acoustic transducers, 380 Velocity of sound horn performance and design factors, 55–56 tapered horn corrections, 58–60 material geometry and elasticity, 22–24 Vibrating blade systems, pneumatic transducer design, 282–283 Vibrational modes, piezoelectric transducer design, 231–232 Vibratory systems hollow cylinders, 117–118 impedance, resonance and Q factors, 10–12 pendulum, 3–8 simple and compound pendulums, 4–7 torsional pendulums, 7–8 rings, 114–117 flexural modes, 116–117 pure radial vibration, 114–116 spring/mass oscillator, 8–10 ultrasonic energy, 2–3 Villari effect, 240 magnetostrictive transducer impedance, 248–249 Virtual work principles, finite element analysis, 134 Vortex whistles, pneumatic transducer design, 282 Vulcanization, ultrasonic polymer processing, 354–356

W Wave equations acoustic wave equations, 12–20 general equation, 14 Lamb waves, 17–19 Laplace operators, 19–20 Love waves, 19 plane wave equations, 12–14 plate wave equation, 17–19 transverse wave equation, 14–16 material geometry and elasticity, 22–24 plane wave equation, 12–14 plate wave equation annular plates, 100–106 free-free annular plate, 104–106 inside-outside clamps, 101–104 outside-inside clamps, 100–101

CRC_DK8307_INDEX.indd 495

495 basic components, 94–96 fixed center circular plate, 98–99 oscillatory motions, 17–19 polar coordinate solutions, 96 rectangular coordinates, 97 simply supported plates, 99–100 uniform circular plate thickness and radius, 97–98 Wear, materials properties, 300–302 Wedge-shaped horn equation, 40–44 amplification factor, 44 impedance, 44–45 Weldability, defined, 301 Welding defined, 301 large-scale sonochemical processing, 359–360 materials properties, 302–308 colorants, 304 compatibility, 302 fillers, 303 flame retardants, 303 hygroscopicity, 302 joint design, 304–308 lubricants, 303 model release agents, 302–303 plasticizers, 303 regrind, 304 resin grade, 304 medical ultrasonic technology, in vivo welding, 427–429 metals and plastics, ultrasonic technology, 467–469 Whistles fine particle coating applications, 280 foam breakdown applications, 280–281 mechanical applications using, 448–449 pneumatic transducers applications, 280–281 basic properties, 274 common whistles, 274–275 Galton whistles, 275 Hartmann whistle, 275–277, 279–280 pin-jet whistle, 280 stem-jet whistle, 277–279 vortex whistles, 282 Wide blade-type horns, properties and applications, 119–121 Wire drawing operations, ultrasonic technology for, 455–459 Work coupling surface, large, cup-shaped horns, 124–126

11/3/2008 12:52:53 PM

Index

496 Wound healing, ultrasound technology, 409, 425–427 Wrinkle reduction, high-intensity focused ultrasound, 423

X X-Y scanning system, ultrasonic nondestructive testing, 370–372

CRC_DK8307_INDEX.indd 496

Y Young’s modulus of elasticity acoustic waves, 22 magnetostrictive transducer impedance, 248–249 piezoelectric materials, 196 plate wave equation, 94–95 Ytterbium, 241 Yttrium, 241

11/3/2008 12:52:53 PM