Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems: State-of-the-Art, Perspectives and Applications

Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems Jan Awrejcewicz Modeling, Simulation and...

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Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems

Jan Awrejcewicz

Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems State-of-the-Art, Perspectives and Applications

Prof. Dr. Jan Awrejcewicz, Phd, DSc. Technical University Lodz Department of Automatics and Biomechanics 1/15 Stefanowski St. 90-924 Lodz Poland [email protected]

ISBN 978-1-4020-8777-6

e-ISBN 978-1-4020-8778-3

Library of Congress Control Number: 2008939501 c 2009 Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Preface

This volume contains the invited papers presented at the 9th International Conference “Dynamical Systems – Theory and Applications” held in Ł´od´z, Poland, December 17–20, 2007 dealing with nonlinear dynamical systems. The conference gathered a numerous group of scientists and engineers, who deal with widely understood problems of dynamics met also in engineering and daily life. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automatics and Biomechanics of the Technical University of Ł´od´z. The patronage over the conference has been taken by the following scientific institutions: Mechanics and Machine Dynamics Committees of the Polish Academy of Sciences, Polish Society of Theoretical and Applied Mechanics, Polish Association for Computational Mechanics, and Technical Committee of Nonlinear Oscillations of IFToMM. The financial support has been given by the Department of Education at the Ł´od´z City Hall, Ministry of National Education and the Polish Association for Computational Mechanics. We welcomed nearly 100 persons from 13 countries all over the world. They decided to share the results of their research and many years of experience in a discipline of dynamical systems by submitting many interesting papers. The Scientific Committee includes the following members: Igor V. Andrianov – Aachen; Jan Awrejcewicz – Ł´od´z; Jose M. Balthazar – Rio Claro; Denis Blackmore – Newark; Iliya Blekhman – Sankt Petersburg; Roman Bogacz – Warsaw; Tadeusz Burczy´nski – Gliwice; Dick van Campen – Eindhoven; Czesław Cempel – Pozna´n; Lothar Gaul – Stuttgart; J´ozef Giergiel – Cracow; Katica Hedrih – Niˇs; Janusz Kowal – Cracow; Vadim A. Krysko – Saratov; Włodzimierz Kurnik – Warsaw; Claude-Henri Lamarque – Lyon; Nuno M. Maia – Lisbon; Leonid I. Manevitch – Moscow; Krzysztof Marchelek – Szczecin; Yuriy Mikhlin – Kharkov; Jan Osiecki – Warsaw; Wiesław Ostachowicz – Gda´nsk; Victor Ostapenko – Dnepropetrovsk; Zbigniew Peradzy´nski – Warsaw; Ladislav P˚ust – Prague; Giuseppe ´ Rega – Rome; Tsuneo Someya – Tokyo; G´abor St´ep´an – Budapest; Jerzy Swider – Gliwice; Ryszard Tadeusiewicz – Cracow; Hans True – Lyngby; Andrzej Tylikowski

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– Warsaw; Ferdinand Verhulst – Utrecht; J´ozef Wojnarowski – Gliwice; Klaus Zimmermann – Ilmenau, and the invited talks follow: 1. Igor V. Andrianov (Aachen, Germany), “Continuous models for discrete media valid for micro- and nano-scales” 2. Vincenzo P. Castelli (Bologna, Italy), “Kinematic and kinetostatic modelling of human articulations: knee and ankle joints” 3. David Y. Gao (Blacksburg, USA), “New way to understand and control chaos: canonical duality approach and triality theory” 4. Alexander P. Seyranian (Moscow, Russia), “Multiparameter stability theory with mechanical applications” The following topics have been covered by the oral presentations: – – – – – – – – –

Bifurcations and chaos in mechanical systems Control in dynamical systems Asymptotic methods in nonlinear dynamics Stability of dynamical systems Lumped and continuous systems vibrations Original numerical methods of vibration analysis Man-machine interactions Dynamics in life sciences, bioengineering, medicine Other problems

It has been observed that an extensive thematic scope comprising dynamical systems stimulates a wide exchange of opinions among researchers dealing with different branches of dynamics, and results in effective solutions of many problems of dynamical systems in mechanics and physics, both in terms of theory and applications. A brief description of the volume content follows. I.V. Andrianov et al. study rod and beam having two different elasticity moduli and having different cross-sections. First governing equations are derived, and then free frequencies of vibrations are estimated using analytical and numerical approaches. In the first case the solution form is formulated, the compatibility conditions are introduced, and then a standard perturbation procedure is applied. The obtained set of recurrent equations yields the being sought free frequencies of the studied rod and beam vibrations. The Pad´e approximations are applied for matching both estimated frequencies of rod and beam vibrations. The obtained frequencies are compared with those yielded via numerical results, and high efficiency of the Pad´e approximation is exhibited. The Lyapunov exponents assess the sensitivity to initial conditions of dynamical systems. This means that it gives the rate of exponential divergence for perturbed initial conditions when time tends to infinity. If all exponents are strictly negative, stability of the system is ensured. On the contrary, if the real part of one exponent is not strictly negative, chaos is possible. Moreover, although obtained through an asymptotic study of the dynamical system, this constant is often used in order to assess finite-time convergence. C.-H. Lamarque and F. Schmidt use a finite-time

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Lyapounov exponent. In the spirit of its definition, they study the maximum value of tangent increment according to normal or non-normal case for the associated linear operator. The definition of a quasi Lyapunov exponent to quantify (estimate) the stability of a dynamical system in the transient regime is proposed. A simple non-linear system is analysed and this definition is applied to it. It is then obvious that, this constant may be much greater to the Lyapunov exponent than it is normally known due to the non-orthogonality of the eigenvectors of the Jacobian matrix. Previous investigations on the nonlinear dynamics of shape memory oscillators, based on a thermomechanically consistent model with four state variables, highlighted the occurrence of chaotic responses in some ranges of the model parameters. The numerical characterization of chaos in smooth dynamical systems is often carried out via the computation of Lyapunov exponents. In the study of D. Bernardini and G. Rega, however, the phase transformations in the shape memory material induce discontinuities in the vector field and are described by an internal variable, the fraction of martensite that belongs to a complex state space. The computation of the classical Lyapunov exponents does not seem to be a convenient strategy. In the present work the attention is focused on the simpler direct numerical tool represented by the method of wandering trajectories. The method is used to distinguish between responses and quantify them in the present thermo-mechanical framework. Numerical simulations in the excitation frequency-amplitude plane, as well as domains of chaotic behavior built in various model parameter planes are presented in order to characterize the richness and robustness of the non-regular dynamical behavior of shape memory oscillators. L.I. Manevitch and V.V. Smirnov present a study of energy exchange in the system of two weak coupled oscillatory chains. The mechanism of energy transfer appears as the interchain hopping of localized nonlinear excitations – breathers. It is supposed that along side with coupling, the amplitudes of vibration are also small enough to restrict ourselves by third and fourth degrees in the power expansion of elastic potential. To reveal spatially localized nonlinear excitations with oscillatory degree of freedom one can derive nonlinear equations with respect to complex combinations of the displacements and velocities. These coupled nonlinear equations were analyzed using multiple scale expansion. As this takes place, three characteristic times may be discriminated. The fastest of them corresponding to higher eigenfrequency of linearized system, intermediate time – to interchain energy transfer, and the slow time – to modulate vibrations. As a result, we have obtained asymptotic solution for the breather transferring energy between the chains. Such a transfer turns out to be impossible when amplitude of vibration grows. In this case localization of the breather on the initially excited chain becomes possible. The condition of the transition from interchain energy exchange to confinement of breather on one chain have been formulated. Analytical results are confirmed by computer simulation data. The model of dynamics of a mechanical system mounted on a moving platform is presented by A. Urba and S. Wojciech. Description of such systems can be used to design control systems which allow us to compensate waving. In the paper dynamic analysis of a gantry crane used to transport BOP (Blow Out Preventer) is presented.

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The crane is placed on a drilling platform. Waves cause motion of the platform and the load. The load is considered as a rigid body with 6-DOF with respect to the platform. The load is connected with the frame by means of two flexible ropes. Homogenous coordinates and transformations are used to describe behavior of the system. Equations of motion are derived using the Lagrange equations of the second order. Numerical calculations present the influence of amplitude, frequency and direction of waves on the system displacements and contact forces between the load and the frame guides. K. Zimmermann et al. present some theoretical investigations of the motion of a straight chain of three (equal) point masses interconnected with kinematical constraints. The ground contact can be described by dry (discontinuous) or viscous (continuous) friction. The controls are assumed in the form of periodic functions with zero average, shifted on a phase concerning each other. Thus, there is a spreading wave along the chain of point masses. In the case of small friction a condition for the locomotion of the center of the mass with the help of an average method is derived. In the case of smooth control and control with impacts explicit expressions for the stationary velocity of the motion of the center of mass are obtained. It is shown that, using specified control motion, this is possible not only in the case of isotropic friction, but also in the direction of the non-isotropic friction. Without shift of the phases in the control law and with a linear friction model the locomotion is impossible. Comparisons of these analytical relations with numerical results are carried out. The received theoretical results can be used for the development of mobile robots applying the principles of the motion outlined above. In machine tools of planar parallel structure with two translatory degrees of freedom, a rotatory degree of freedom is kinematically locked. Yet due to geometric faults, for example, assembly errors or different geometries due to production tolerances, such machine tools exhibit an additional rotational behavior. Stresses within the structure occur leading to deflections of the tool center point, and thus, reducing the quality of the workpiece. For compensating these errors an adaptronic strut, which can be implemented within such a machine tool, has been developed by C. Rudolf and J. Wauer. The strut comprises a piezoceramic sensor-actuator unit for controlled correction of those static and quasi-static deflections. Piezoceramic elements were chosen due to their high positioning accuracy and the small installation space required. In addition, a control design for the compensation is presented. Examining the controlled adaptronic strut initially as stand-alone system under external loads which represent process loads or constraint forces, the strut is eventually implemented within the machine tool. The effect of the controlled strut on the operating behavior of the machine tool is investigated. An approximate method for calculation of rigid rotor motion, supported on two tilting pads aerodynamic bearings oscillating with large amplitudes, limited only by the bearing clearance, is presented by L. P˚ust and J. Kozanek. Dynamic characteristics of such type of bearings are very strongly influenced by inertia properties of tilting pads – the stiffness and dynamic matrices are non-symmetric and their elements are non-monotonous functions of angular frequency. The approximate models of stiffness and damping forces – valid for the entire area of journal motion in the

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aerodynamic bearing’s clearance – are based on the transformation of linear stiffness and damping characteristics into the entire area. The assumption that the rheological properties of three-tilting pads bearing are centrally symmetric is also used. Linear dynamic characteristics of aerodynamic bearings, given for the selected positions of journal in the form of linear stiffness and damping matrices were extended onto strongly nonlinear characteristics by using special correction function and by the plane rotation matrix on the entire area of bearing clearance, and for the common angular positions. Used correction function consists of two parts, the first one proportional to the properties of unloaded bearing, the second one to the properties of loaded bearing. Rigid rotor with generally distributed mass along the axis of rotation produces coupling between motions of both bearings. Differential equations of motion of the rotor are numerically solved and resulting time histories of journal motion and their plane trajectories are presented and analysed. Strongly nonlinear stiffness and damping characteristics of aerodynamic bearings cause various kinds of rotor oscillations. Application of this approximate mathematical model of rotor supported on aerodynamic bearings is presented on examples, where the influence of rotor eccentricity and its shift along the axis of rotation is shown. T. Burczy´nski et al. study selected identification problems of dynamical mechanical structures that some parameters are uncertain and modelled in the framework of fuzzy sets. The identification problem is formulated as the minimization of some objective functionals (fitness functions) which depend on measured and computed dynamical fields such as displacements, strains or natural frequencies. In order to obtain the unique solution of the identification problem one should find the global minimum of the objective functional. In the majority engineering dynamical cases it is not possible to determine exactly all parameters of the system. It is necessary to introduce some uncertain parameters which describe granular character of data. The paper deals with identification of the fuzzy parameters of material and shape of the structure. In order to solve the identification problem the fuzzy evolutionary algorithm is applied. Evaluation of the fitness function is performed by means of the fuzzy finite element method. Several numerical examples of identification of elastic and composite structures are presented. Shape memory alloys (SMA) are materials that even when they are submitted to the some type of deformation, possess the ability to recuperate their original form through adequate thermal procedures. The main phenomena associated with these alloys are the effect of pseudoelastic and shape memory. Such phenomena happen due to martensite phase transformation that occur in these alloys. The dynamical response of systems composed by shape memory alloy presents nonlinear characteristics and a very rich nonlinear dynamic behavior. V. Piccirillo et al. present analytical and numerical investigations of a shape memory oscillator, where the restitution force is described by a polynomial constitutive model. The governing equations of the motion through a perturbation multiple scale method, in the case of the primary resonance are derived and analysed. They are studied numerically by means of phase portrait, Lyapunov exponents, frequency power and Poincar´e maps. Frequency–response curves are constructed for shape memory oscillators by various excitation levels and detuning parameter. A rich class of solutions and bifurcations,

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including jump phenomena, saddle-node bifurcations, is found. Good agreement between numerical and analytic solutions is obtained in the considered resonance region. A. Okni´nski and B. Radziszewski study dynamic properties of a material point moving in a gravitational field and colliding with a moving motion-limiting stop (representing unilateral constraints). The motion of the limiter is assumed as periodic with piecewise constant velocity. The Poincar´e map, describing evolution from an impact to the next impact, is derived. Grazing motion as well as periodic solutions are computed analytically and their stability is determined in analytic form. An investigation of transient is important in engineering, in particular, in the problem of absorption. Over the past years different new devices have been used for the vibration absorption and for the reduction of the transient response of structures. It seems interesting to study nonlinear passive absorbers for this reduction. Yu.V. Mikhlin et al. consider the transient in a system containing a linear oscillator, linearly coupled to an essentially nonlinear attachment with a comparatively small mass. A damping is taken into account. It is assumed that some initial excitation implies vibrations of the linear oscillator. The multiple scales method is used to construct a process of transient in the system under consideration. Numerical simulation confirms an efficiency of the analytical construction. A transfer of energy from initially perturbed linear subsystem to the nonlinear absorber can be observed. A similar construction is made to describe the transient in a system which contains a linear oscillator and a vibro-impact absorber with a comparatively small mass. Both an exact integration with regards to impact conditions, and the multiple scales method are used for this construction. The transient in such system under the external periodical excitation was considered too. Numerical simulation confirms an efficiency of the analytical construction. Most formulations on normal impacts of multibody systems use the momentum balance equations. Newton’s hypothesis is usually employed, where the restitution coefficient is defined by the relative normal velocities of the impacting bodies before and after the collision. When friction is present in the impact, different impact modes are possible: sliding, sticking or reverse sliding. The variables in the momentum balance equations are the changes in the velocity and the two components of the impulse, one in the normal direction to the common tangent of the contact surfaces and other in the tangent direction. To solve the equations two additional conditions are needed, one comes from the Coulomb law, and the other from the definition of the restitution coefficient. The use of the Newton hypothesis leads to wrong results in the simulation of impacts with friction. The use of the Poisson hypothesis can be found in the literature. There are situations, where the direction of slip varies during collision, the only energetically consistent definition is the so-called energetic coefficient of restitution. J.M. Mayo addresses problems related to impacts with friction of planar flexible multibody systems. The floating frame of reference formulation is used to model the flexible bodies. The normal and tangential impulses in the contact point are calculated by a computational algorithm based on the graphics techniques developed by Routh. Numerical results using both Poisson and Stronge definitions of the restitution coefficient are shown.

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Inverse simulation techniques are computational methods, in which control inputs to a dynamic system that produces desired system outputs are determined. Such techniques can be powerful tools for the analysis of problems associated with manoeuvring flight. The problem with the inverse simulation problem at hand is, however, that we deal with an underactuated mechanical system – the six-degree-offreedom aircraft is traditionally controlled by at most four control inputs: the aileron, elevator and rudder deflections and, optionally, by thrust changes. The same number of motion restriction can then be imposed on the aircraft motion. In the study of W. Blajer et al. the following complex problem is studied: a specified trajectory in space (two constraints on aircraft position), a demand on fuselage attitude with respect to the trajectory, and optionally a specification on the flight velocity (motion on the trajectory). A tangent realization of trajectory constraints is observed, which yields two additional constraints on fuselage attitude with respect to the desired trajectory. The consequent governing equations of the prescribed trajectory flight arise then as a set of differential-algebraic equations, and an effective method for solving the equations is developed. The solution consists of time-variations of aircraft state variables in the prescribed motion and the demanded control that ensures the realization of the motion. This gives a unique opportunity to study the simulated control strategies and evaluate feasibility of the modeled aircraft maneuvers. Some results of numerical simulations are reported. C. Behn contributes to the adaptive control of nonlinearly perturbed multi-input u, multi-output y, minimum phase systems with strict relative degree two. The author deals with systems, which are not known exactly, only structural information about the system (like relative degree or the minimum phase condition) are available. The consideration of these uncertain systems leads to the use of adaptive control. The aim is to design universal adaptive controllers, which learn from the behaviour of the system, so automatically adjust their parameter and achieve a pre-specified control objective: stabilization of the system or tracking of a given reference signal with any pre-specified, feasible accuracy lambda >0 (so-called lambda-tracking). Almost all already existing controllers in the literature offer the same drawback: though the controllers consist of a feedback strategy and a simple parameter adaptation law, this adaptation law is of the type, that the gain parameter can only monotonically increase. With respect to limited resources in applications, it is necessary to design adaptation laws, which let the gain parameter increase while the control objective is not achieved, and let him decrease, while it is achieved. But, from the analytical point of view and with respect to the mathematical background (theorems and their proofs) the arbitrary adaptation laws (gain parameter models) cannot be chosen. Some simulations of gain parameter models, which exhibit the properties mentioned above, in application to bio-inspired sensors with unknown system parameters to adaptively compensate unknown ground excitements are presented and the results are discussed. In the paper by J. Awrejcewicz et al. the dynamic instability and nonlinear vibrations of isotropic plates with complex form subjected to in-plane periodic compressive load are analyzed. The proposed approach is based on application of R-function method with variational one. The von K´arm´an governing equations

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are reduced to an ordinary nonlinear differential equation regarding time by the Bubnov-Galerkin method. To apply the Bubnov-Galerkin method for plates of an arbitrary shape and with different boundary conditions, a complete system of basis functions is constructed using R-functions theory. To find instability regions linear equations with periodic coefficients, the so-called Mathieu-Hill equation, are used. The amplitude – frequency characteristics and instability regions for plates with cutouts are obtained. The effect of static and dynamic factors of load action, cutouts parameters and different boundary conditions are studied. L.A. Kov´acs et al. present an analytical investigation on the unconditional stability of robots subjected to digital force control. As a benchmark example, a one degree-of-freedom robot model is considered with proportional plus differential force feedback. The results of the passivity based analysis are compared to the exact/analytical stability limits of the investigated controller in the form of stability charts. In addition, the optimization aspects of slightly damped mechanical structures with digital force control are discussed. Simple closed form results include the largest stable proportional gain and the least steady state force error as well as the optimal control parameters that provide the shortest transients for the controlled force signal. The stability charts based on the analytical results are measured and verified by a series of experiments. G.V. Kostin and V.V. Saurin are aimed at modeling and optimization of controlled dynamical systems with distributed elastic and inertial parameters. The general integro-differential method for solving wide class of boundary value problems is developed and criteria of solution quality are proposed. The numerical algorithm for discrete approximation of controlled motions is carried out and applied to design the optimal control law steering an elastic system to the terminal position and minimizing the given objective function. The polynomial control of plane motions of a homogeneous cantilever beam is investigated. Such type of system disturbances can induce essential elastic deflections and lead to sufficient computational difficulties when the conventional approaches are used. The optimal control problem of beam transportation from the initial rest position to given terminal state, in which the full mechanical energy of the system reaches its minimal value, is considered. The obtained numerical results are analyzed and compared with the conventional Fourier’s solution. L.A. Klimina et al. analyse the problem of behavior of aerodynamic pendulum with vertical axis of rotation in connection with the study of dynamics of small-scale vertical axis wind power generator. It is assumed that the aerodynamic load is comprised of quasi-steady part (determined on the basis of wind tunnel experiments), and unsteady part described with the help of the so-called added masses. Mathematical model of free rotation of such pendulum is constructed and includes nonlinear ODEs and transcendental algebraic equations. Qualitative analysis of the phase portrait is performed: all equilibrium positions are found, their stability is studied, characteristics of stable rotational regime are determined (basing on this regime, the working regime of the wind power generator is formed), attraction domains of equilibrium positions and of this rotational regime are found. Numerical investigation confirms the obtained results. The developed mathematical model is used for

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investigation of working regimes of the system “wind turbine + generator”. Estimations of input power depending on external load in the circuit are obtained, optimal power and load are found. A pitch angle control is proposed in order to increase the output power. E.M. Jarze˛ bowska and P.C. Szklarz present a new strategy to design a kinematic feedback controller, which is based on the error function. The error function is predefined by a designer and the resulting feedback controller ensures the convergence of the error to zero or to some specified bound. Usually in a control problem either at a kinematic or dynamic level, a tracking or following error is defined as a difference between desired and actual values of coordinates, or as a distance form a curve, respectively. It concerns both state and output control. A specific controller has to be subsequently designed in such a way that it has to ensure some kind of convergence or boundedness of the error. A few works present tracking control strategy designs, for which the tracking error is defined in a different way, e.g. due to uncertainties in a system dynamics. However, these approaches do not change the controller design process. In authors’ approach this is the error function dynamics, which ensures the convergence to the predefined system motion and the controller is designed based on this function. The feedback control problem is as follows for a given system kinematics find control inputs that ensure a desired ordinary differential equation on the error function value to be satisfied. The error function value is the value on the actual location of the system. The ordinary differential equation ensures the convergence of the error value to the specified bound. The problem is then precisely defined in a set of ordinary differential equations. According to the formulation of the feedback control problem, a definition of the error function is essential. It may reflect a way in which a system converges to a desired curve in order to achieve a desired error function value behavior. Comparing the authors’ design with typical tracking control designs, the differences start from defining the error function for the desired motion and designing of the controller based on this function. This implies that the convergence of the controlled motion to the desired one is guaranteed in the controller design process. The proposed control strategy offers several advantages significant from the practical point of view. The theoretical development is illustrated with an example of a desired motion control of a two-wheeled mobile platform. In the approach this motion control may be either tracking or following, or both. During the last two decades or so there has been a gradually increasing interest in the detection of damage based upon techniques that take into consideration changes in the dynamic properties of the structures, leading to thousands of articles and entire conferences specifically dedicated to the subject. Those techniques occupy quite a wide spectrum in terms of diversity as recent surveys reveal. They do not only address the problem of detection, but often also include the localization and the quantification. In many cases the main difficulty relies on the fact that the damage indicators that are proposed are not sensitive enough to detect damage in a sufficiently early stage. N.M. Maia and R.P. Sampaio have pursued such a sensitive indicator and found out that a variation of a known measure, known as the Frequency Domain Assurance

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Criterion, could give a good contribution to the objective of detecting damage. The simple technique that the authors have been developing recently and that is based on a correlation factor between dynamic responses taken along a period of time and the main objective of which is to serve as an indicator for the existence of damage, although – as it is discussed – it can also provide information about the relative damage quantification. That indicator has been named as The Detection and Relative damage Quantification indicator (DRQ) and some variations of it have been developed to improve its performance. Some numerical simulations are presented as examples and experimental tests in the laboratory as well as in a real working structure are shown to illustrate the method. V.-F. Duma studies the complete angular and linear scanning functions of a 2-D scanner comprising two individually driven galvoscanners. The characteristic functions of a 1-D scanner were developed and the command function was obtained. It has been shown that with a proper programming of the laser utilized in the system, there exist a wide spectrum of industrial, art and/or scientific applications. A. Krysko et al. are aimed on comparing numerical results of two different and widely used in engineering approaches, namely FDM (Finite Difference Method) and FEM (Finite Element Method) on the example of regular and chaotic dynamics of the Euler-Bernoulli beams. Transition scenarios are illustrated and studied, among other. Systems with a pendulum are often applied for reduction of vibrations in many mechanical and civil structures. Special pendulums mounted in buildings and working against earthquake as dynamical dampers or mounted on bridges tower working against river vortexes are classical examples of such systems. Motions of the system can be both regular or, for some parameters, chaotic. Then, the pendulum may worsen the system response, and it can be required to include a control, to fulfill the protection condition. The paper by K. Ke˛ cik and J. Warmi´nski proposes to use a magneto-rheological damper (MR), installed between oscillator and the ground, to provide controllable damping of the system. The effectiveness of the proposed smart base isolation system is studied numerically and experimentally. The results show that MR fluids can be designed for performing a controllable damping force and can be a very effective tool for vibration control. Common mechanical systems in rotor dynamics are rotating shafts of different shapes joined with special, mostly axi-symmetric bodies, which can be bladed disks, geared wheels, fans, etc. Designed rotating systems and operating conditions are still becoming more and more complex and therefore it is necessary to create advanced mathematical and computer models of the studied dynamical systems. This problem is addressed by M. Hajˇzman et al. The contribution is intended to the modeling of the flexible rotors that can be decomposed into shaft and disk subsystems mutually joined together. Rotating shaft subsystems are considered as one dimensional continuum. The original shaft finite element in the rotating coordinate system is presented. Because disks can be of a complex shape they are modeled as three dimensional continuum also in the rotating coordinate system. The coupling matrix is used for the connection of the disk and shaft subsystems. The presented methodology is advantageous mainly due to the possibility of considering

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various effects of rotation that cannot be introduced in the commercial FEM codes. The described model of the flexible rotor is usable especially for the analysis of high-frequency vibrations, where the common assumption of rigid disks in rotor dynamics is not correct. The possibility of a model reduction by the modal synthesis method is a big benefit too. Stochastic dynamics of the systems composed of hereditary elements is analyzed by K.R. Hedrih. First, the definition of a hereditary system is introduced, and then examples of light standard thermo-rheological elements are provided. The stochastic dynamics of the thermo-rheological double pendulum system putting emphasis on its parametric resonance is studied. S.-M. Cretu focused on a study of achievements of the tensegrity theory regarding living forms and some tensegrities obtained from platonic polyhedra. She presents the calculus for the form-finding of a tensegrity system that approximates the human forearm. Problems related to stability, bifurcation and chaotic behavior of a buck converter controlled by lateral pulse-width modulation and zero average dynamics are studied by F. Angulo et al. The authors are mainly focused on the duty cycle computation using a linear approximation of the sliding surface. M. Popescu and A. Dumitrache study stability problems regarding periodic solutions of the quasi-linear control systems in a critical case. The problem is transformed to that of an orbital stability, and then the theorem associated with stability estimation is formulated and proved. Finally, it should be emphasized that the selected invited papers are mainly oriented toward dynamics and control of engineering systems. They have been also reviewed by two independent referees to satisfy the Springer publisher standards. My sincere gratitude is expressed to the numerous referees, the conference participants and the Department of Automatics and Biomechanics staff. Special thanks go to Ms. Nathalie Jacobs for her help and encouragement to the publication of this volume. Poland November 2008

Jan Awrejcewicz

Contents

Load-Transfer from an Elastic Fibre to Isotropic Half-Space with Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Igor V. Andrianov, Jan Awrejcewicz, and Dieter Weichert 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Asymptotic Simplification of Boundary Conditions . . . . . . . . . . . . . 4 4 A Single Fibre Embedded in the Half-Space . . . . . . . . . . . . . . . . . . . 5 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents . . . . Franziska Schmidt and Claude-Henri Lamarque 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 A General Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Duffing-Like Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Lorenz Attractor and Other Dynamical Systems . . . . . . . . . . . . 3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Characterization of the Chaotic Nonregular Dynamics of Pseudoelastic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Davide Bernardini and Giuseppe Rega 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Description of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Method of the Wandering Trajectories . . . . . . . . . . . . . . . . . . . . 4 Characterization of Typical Trajectories . . . . . . . . . . . . . . . . . . . . . . . 4.1 Symmetric Period 1 Solution . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Chaotic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Unsymmetric Period 1 Solution . . . . . . . . . . . . . . . . . . . . . . 5 Comparison with Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 15 17 22 23 25 25 26 27 28 28 29 29 31

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Overall Characterization of the Non-regular Solutions and Effect of the Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Localized Nonlinear Excitations and Interchain Energy Exchange in the Case of Weak Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leonid I. Manevich and Valeri V. Smirnov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linear Chains with Weak Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Nonlinear Chains with Weak Nonlinearity . . . . . . . . . . . . . . . . . . . . . 4 Chains with Nonlinearity, Compatible with Coupling . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Analysis of the Gantry Crane Used for Transporting BOP . . . . . Andrzej Urba´s and Stanisław Wojciech 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical Model of the System . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Kinetic Energy of the Systems . . . . . . . . . . . . . . . . . . . . . . . 2.2 Potential Energy of Gravity Forces . . . . . . . . . . . . . . . . . . . 2.3 Energy of Deformation and Dissipation Energy of Sde . . 2.4 Energy of Deformation and Dissipation Energy of the Ropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Reaction Forces of the Support . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Motion of a Chain of Three Point Masses on a Rough Plane Under Kinematical Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Klaus Zimmermann, Igor Zeidis and Mikhail Pivovarov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Smooth Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Dry (Discontinuous) Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Viscosity (Continuous) Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compensation of Geometric Errors in a PKM Machine Tool . . . . . . . . . . . Christian Rudolf and J¨org Wauer 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Control Concept for Adaptronic Strut . . . . . . . . . . . . . . . . . . . . . . . . 3 Implementation into PKM Machine Tool . . . . . . . . . . . . . . . . . . . . . . 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 38 41 43 46 47 49 49 49 52 52 52 54 54 56 59 59 61 61 62 64 65 68 69 70 71 71 73 74 77

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5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ladislav P˚ust and Jan Koz´anek 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Motivation of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mathematical Model of the Rotor Motion at Small Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Rotor Motion at Large Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 82 83 87 89 93 94

Identification of Dynamical Systems in the Fuzzy Conditions . . . . . . . . . . . 95 Tadeusz S. Burczy´nski, Witold Beluch and Piotr Orantek 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2 Formulation of the Identification Problem of the Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3 The Two-Stage Fuzzy Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.1 The First Stage – Global Optimization . . . . . . . . . . . . . . . . 99 3.2 The Second Stage – Local Optimization . . . . . . . . . . . . . . . 100 4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.1 The Identification of Geometrical Parameters of a Void . . 102 4.2 The Identification of Laminate’s Elastic Constants . . . . . . 102 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 On Nonlinear Response of a Non-ideal System with Shape Memory Alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 V. Piccirillo, J. M. Balthazar, B. R. Pontes Jr. and J. L. P. Felix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2 (SMA) Constitutive Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3 Mathematical Model of the Non-ideal System . . . . . . . . . . . . . . . . . 109 4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Dynamics of a Material Point Colliding with a Limiter Moving with Piecewise Constant Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Andrzej Okni´nski and Bogusław Radziszewski 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2 Motion with Impacts: A Simple Motion of the Limiter . . . . . . . . . . 118 3 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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Periodic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Chattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Multiple Solutions and Discontinuous Dependence on Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Transient in 2-DOF Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Yuri Mikhlin, Gayane Rudnyeva, Tatiana Bunakova and Nikolai Perepelkin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2 Transient in a System Containing an Essentially Nonlinear Oscillator as Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3 Transient in the Vibro-Impact System . . . . . . . . . . . . . . . . . . . . . . . . 133 3.1 Free Oscillations in the Vibro-Impact System . . . . . . . . . . 134 3.2 Transient in a Case of Forced Oscillations . . . . . . . . . . . . . 136 4 Transient in 2-DOF Nonlinear System with Limited Power Supply 137 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 On the Use of the Energetic Coefficient of Restitution in Flexible Multibody Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Juana M. Mayo 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 2 Floating Frame of Reference Formulation . . . . . . . . . . . . . . . . . . . . . 143 3 Generalized Impulse-Momentum Balance Equations . . . . . . . . . . . . 144 4 Coefficients of Restitution and Routh’s Diagrams . . . . . . . . . . . . . . 145 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Modeling of Aircraft Prescribed Trajectory Flight as an Inverse Simulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Wojciech Blajer, Jerzy Graffstein and Mariusz Krawczyk 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 2 Modeling Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3 Prescribed Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4 Governing Equations and the Solution Code . . . . . . . . . . . . . . . . . . . 158 5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Improved Adaptive Controllers for Sensory Systems – First Attempts . . . 163 Carsten Behn and Joachim Steigenberger 1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 2 General System Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 3 Control Objective & Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

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3.1 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 3.2 Current Control Strategy & Theorem . . . . . . . . . . . . . . . . . 167 4 New Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5 Improved Gain Adaptation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.1 Comparative Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.2 Further Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Research of Stability and Nonlinear Vibrations by R-Functions Method . . 179 Jan Awrejcewicz, Lidiya Kurpa and Olga Mazur 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 3 Method of Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5 R-Functions Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Experiments on the Stability of Digital Force Control of Robots . . . . . . . . 191 L´aszl´o L. Kov´acs, P´eter Galambos, Andr´as Juh´asz, and G´abor St´ep´an 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2 Model of Digital Force Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5 Theoretical vs. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 197 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Motion Analysis and Optimization for Beam Structures . . . . . . . . . . . . . . . 201 Georgy Kostin and Vasily Saurin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 3 An Approximation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Parametrical Analysis of the Behavior of an Aerodynamic Pendulum with Vertical Axis of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Liubov Klimina, Boris Lokshin and Vitaly Samsonov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 2 The Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 3 Existence of Auto-Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

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Numerical Analysis of Rotational Modes and Domains of Attraction for β = 0 and Various Values of c . . . . . . . . . . . . . . . . 214 5 Average Trapped Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6 Pitch Angle Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Error Function Based Kinematic Control Design for Nonholonomic Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 El˙zbieta Jarze˛ bowska and Paweł Cesar Sanjuan Szklarz 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 2 A Control Theoretic System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 223 3 An Abstract Feedback Control Strategy Architecture . . . . . . . . . . . . 223 4 Design of a Kinematic Control Strategy Based on the Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 5 Control of a Two-Wheeled Mobile Robot . . . . . . . . . . . . . . . . . . . . . 226 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 A Simple Correlation Factor as an Effective Tool for Detecting Damage . . 233 Rui Sampaio and Nuno Maia 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 2 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 3.1 Sensitivity to Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 3.2 Quantification of Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 4 Experimental Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Mathematical Functions of a 2-D Scanner with Oscillating Elements . . . . 243 Virgil-Florin Duma 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 2 Equations of the 2-D Scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 3 Angular and Linear Scanning Functions . . . . . . . . . . . . . . . . . . . . . . 245 3.1 Scanning Functions of the 2-D System . . . . . . . . . . . . . . . . 245 3.2 θ and x Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 3.3 ϕ and y Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Analysis of Regular and Chaotic Dynamics of the Euler-Bernoulli Beams Using Finite-Difference and Finite-Element Methods . . . . . . . . . . . 255 Anton Krysko, Jan Awrejcewicz, Maxim Zhigalov, and Olga Saltykowa 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Contents

xxiii

3

On the Numerical Solution to Vibration and Stability Beam Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 3.1 FDM with Approximation O(c2 ) . . . . . . . . . . . . . . . . . . . . . 257 3.2 FEM with the Bubnov-Galerkin Approximation . . . . . . . . 258 4 Numerical Results Obtained via FDM and FEM . . . . . . . . . . . . . . . . 259 5 Transition Scenarios into Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Regular and Chaotic Motions of an Autoparametric Real Pendulum System with the Use of a MR Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Jerzy Warmi´nski and Krzysztof Ke˛ cik 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 2 Model of the Vibrating System and Equations of Motions . . . . . . . . 267 3 Experimental System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 4 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 270 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Modeling of Flexible Rotor Vibrations in the Rotating Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 ˇ sek and Vladim´ır Zeman Michal Hajˇzman, Jakub Saˇ 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 2 FEM Model of a Rotating Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 3 FEM Model of a Rotating Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 4 Mathematical Model of the Whole Disk-Shaft System . . . . . . . . . . 282 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Stochastic Dynamics of Hybrid Systems with Thermorheological Hereditary Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Katica R. (Stevanovi´c) Hedrih 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 2 Light Standard Thermo-Rheological Hereditary Element . . . . . . . . 290 3 Thermo-Rheological Double Pendulum System – System of the Averaged Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 4 Stochastic Dynamics of the Thermo-Rheological Double Pendulum system – Parametric Resonance . . . . . . . . . . . . . . . . . . . . 296 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Tensegrity as a Structural Framework in Life Sciences and Bioengineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Simona-Mariana Cretu 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

xxiv

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2

Regular Polyhedra and Their Applications to Life Sciences and Bioengineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 3 Tensegrity and Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 4 Applications of Tensegrity Concepts to Living Forms . . . . . . . . . . . 304 5 Form-Finding Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 6 Transformations of the Tensegrity Systems . . . . . . . . . . . . . . . . . . . . 308 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Study of Nonlinear Dynamics in a Buck Converter Controlled by Lateral PWM and ZAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Fabiola Angulo, Jorge E. Burgos, and Gerard Olivar 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 2 Modeling the DC-DC Buck Converter with LPWM and ZAD . . . . 314 3 Computation of the Duty Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 4 Bifurcational Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 4.1 Period-Doubling Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 318 4.2 Border Collision Bifurcation and Chaos . . . . . . . . . . . . . . . 323 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Stability of the Periodic Solutions for Quasi-linear Controlled Systems in Critical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Mihai Popescu and Alexandru Dumitrache 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 2 Quasi-linear Controlled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 3 Orbital Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

Load-Transfer from an Elastic Fibre to Isotropic Half-Space with Coating Igor V. Andrianov, Jan Awrejcewicz, and Dieter Weichert

1 Introduction We study the problem of load-transfer from fiber inclusion to matrix. Many papers are devoted to the infinite fibre in an elastic space. 3D analog of Melan problem is analysed by Muki and Sternberg [1]. They regard the original fibre as made of two superimposed elastic fibres, the first with the same characteristics as the matrix and treated in the framework of 3D elasticity, the latter with the elastic coefficient equal to difference between those of the actual fibre and of the matrix considered as a 1D continuum. The governing integral equation is obtained by imposing the same average axial strain in the two fictitious bars. Many researches used as asymptotic parameters ratios λ1 = R/L, λ2 = E/E1 or λ3 = EE1 ( RL )2 ln( 2L R ), where E, E1 are the Young modulus of matrix and fibre, respectively; and R, L are the radius and length of the circular fibre, respectively. Freund [2] studied a model describing sliding of circular cylindrical fibre along a hole in an elastic solid, and obtained asymptotic solutions for the cases when the fibre is very stiff or very weak in comparison with the matrix material (λ2 << 1 and λ2 >> 1, respectively). Eshelby [3] and Argatov and Nazarov [4] used parameters λ1 << 1 and λ2 << 1 and matched asymptotics procedure. Phan-Thien and Kim [6] used parameter λ3 . If λ3 >> 1, then the interfacial shear stress remains almost constant, for λ3 << 1 the load transfer occurs over a finite neighbourhood of the fibre end which is near to the free surface and the interfacial shear stress varies as 1/z3 , where z is the distance from the free surface.

I.V. Andrianov and D. Weichert Department of General Mechanics, TU University of Aachen, Germany, e-mail: igor andrianov@ hotmail.com, [email protected] J. Awrejcewicz Department of Automatics and Biomechanics, Technical University of Lodz, Poland, e-mail: [email protected]

J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

1

2

I.V. Andrianov et al.

Many papers are also devoted to the problem of load-transfer from a single fibre to the half-space. Keer and Luk [7] formulated a problem of load-transfer by means of Hankel transforms and reduced it to a system of coupled singular integral equations, where the unknown quantities are the normal and the shear stresses acting on the entire surface of the fibre. In paper by McCartney [9] the equilibrium equations, the interface conditions and other boundary conditions involving stresses are exactly satisfied. Furthermore, two of the four stress-strain relations are satisfied exactly, whereas the remaining two are satisfied in an average sense. Displacement boundary conditions are also satisfied in an average sense. The approach proposed by Rajapakse and Wang [10] is based on the study of interaction between the 1D elastic fibre and the 3D elastic halfspace with a cylindrical cavity. The displacement compatibility is achieved along the contact surface between the fibre and the half-space. A variational technique coupled with a boundary integral equation scheme base on a set of exact Green’s functions is used in the analysis. The boundary conditions on the top end of the fibre are incorporated into the variational formulation through a set of Lagrange multipliers. Lee and Mura [11] obtained the numerical solution in the case of finite length fibre embedded in elastic space and in elastic half-space. Movchan and Willis [12] analyzed case when the fibres are held in place by Coulomb friction. The stress and displacement field in the composite and the length of the slipping region are obtained by solving a model problem for a fibre in an elastic half-space in an ambient stress field generated by all other fibres and the applied loading. Antipov et al. [13] consider a boundary layer problem for an elastic space containing an infinite cylindrical fibre with a frictional interface. In the region where frictional sliding occurs, the transfer of load across the interface is governed by a Coulomb friction law. Outside the slipping region the fibre and the matrix are perfectly bonded. The problem is reduced to a singular integral equation. Lenci and Menditto [14] obtained solution for dilute and highly concentrated fibre composite with a weak interface in the form of improper integrals. In this paper we analyse load-transfer from single fibre to half-space through interface, when boundary of half-space is rigidly joined with thin elastic coating.

2 Governing Equations In this section, we will consider the case of a single fibre weakly bonded to a surrounding half-space (Fig. 1). Fibre is loaded by uniformly distributed across its cross-section load P. We do not take into account body forces; due to the linearity of the problem it can be done using described approach. We will consider the fibre as 1D continuum without transversal deformation and we will suppose perfect adherence in the direction orthogonal to the fibre-matrix interface. First approximation is based on the inequality λ2 << 1. The matrix material is assumed to be isotropic and linear elastic, with elastic constants E and v. The

Load-Transfer from an Elastic Fibre to Isotropic Half-Space with Coating

3

fibre interface matrix

q r

z

Fig. 1 The dilute concentration problem: a single fibre embedded in an elastic half-space

axial Young modulus of the circular fibre with radius R and the interface stiffness are denoted by E1 and k, respectively. We will use circular cylindrical coordinate system (r, θ , z); axis of the fibre coincides with the z-axis. The problem is axially symmetric; the axial displacement of the fibre is denoted by U f (z) and the radial and longitudinal displacement of the matrix by Ur (r, z) and Uz (r, z), respectively. We also denote the interfacial stress by τ (z) and stresses in the matrix by σr (r, z), σz (r, z), σθ (r, z), τrz (r, z); in our case τ (z) = τrz (R, z). The interface between fibre and matrix can play an important role in determining the properties of the composite material. Usually, stresses are continuous across the interface, while the displacements may be continuous or discontinuous. In the former case, the interface is called “strong”, whereas in the latter case, it is called “weak”. We deal with a weak interface described by the spring-layer model which assumes that the interfacial stress is a function of the gap in the displacements. Asymptotic justifications of spring-layer model were obtained by many authors; for example, see [15] and references cited therein. We suppose the material of the interface to be incompressible so Poisson’s coefficient of interface is equal to 1/2. In this case the interface guarantees perfect bonding in normal direction and only tangential sliding is possible [14, 15]:

τ (z) = k(U f (z) − Uz (R, z)).

(1)

The parameter k summarizes the mechanical characteristics of the interface and can be computed from the elastic moduli of the interface [15]. For the case of an incompressible interface one has k = Ei /(3d), where Ei is the Yung modus of the interface and d is the thickness of the interface. We also suppose that fibre is absolutely rigid in radial direction [14]: Ur (r, z) = 0.

(2)

4

I.V. Andrianov et al.

Displacements and stresses in the matrix can be expressed in terms of the Love potential Φ(r, z) as follows:

∂ 2Φ ∂ 2Φ ,Uz (r, z) = 2(1 − ν )∇2Φ − 2 , ∂ r∂ z ∂z ∂ ∂ 2Φ σr (r, z) = 2G ν ∇2 Φ − 2 , ∂z ∂r ∂ ∂ 2Φ (2 − ν )∇2 Φ − 2 , σz (r, z) = 2G ∂z ∂z ∂ ∂ 2Φ (1 − ν )∇2 Φ − 2 , τrz (r, z) = 2G ∂r ∂z

Ur (r, z) = −

(3) (4)

(5)

E where: ∇2 = ∂∂r2 + 1r ∂∂r + ∂∂z2 , G = 2(1+ ν) . In the absence of body force, the function Φ(r, z) is biharmonic: 2

2

∇2 ∇2 Φ(r, z) = 0.

(6)

Now let us suppose that matrix is coated by thin elastic layer with the small thickness H, rigidly bonded to the elastic half-space. This model is valid for polymer material with a metal coating [16]. The coating material is assumed to be isotropic and linear elastic, with elastic constants E2 and v1 . Due to the small thickness of coating we can treat it as a plate. Then boundary conditions for z = 0 can be written as follows: 2 E2 H ∂ ∂ ∂ 2Φ 1 ∂ 2Φ 1 ∂ 2 − = −2G , (7) + ν ∇ Φ − ∂z ∂ r2 1 − ν12 ∂ r2 r ∂ r r2 ∂ r∂ z E H3 ∂ ∂ 1 ∂ ∂ ∂ 2Φ 2 2 r r 2(1 − ν )∇2Φ − 2 = ∂r r ∂r ∂r ∂z 12 1 − ν1 r ∂ r ∂ ∂ 2Φ 2 = 2G (2 − ν )∇ Φ − 2 . ∂z ∂z

(8)

and Ur ,Uz , σr , σz , σθ θ , τrz → 0 for z → ∞.

(9)

3 Asymptotic Simplification of Boundary Conditions Let us introduce nondimensional variables r1 = r/R, ξ = z/R. Then boundary conditions (7), (8) for ξ = 0 can be rewritten as follows: 2 2 ∂ ∂ Φ ∂ ∂ 2Φ 1 ∂ 1 2 (10) + − = − ν ∇ Φ − d1 ∂ξ ∂ r12 r1 ∂ r1 r12 ∂ r1 ∂ ξ ∂ r12

Load-Transfer from an Elastic Fibre to Isotropic Half-Space with Coating

d2 ∂ ∂ ∂ ∂ 2Φ 1 ∂ 2 r1 r1 2(1 − ν )∇1Φ − r1 ∂ r1 ∂ r1 r1 ∂ r1 ∂ r1 ∂ξ2 ∂ ∂ 2Φ (2 − ν )∇21 Φ − , = ∂ξ ∂ξ2

5

(11) (12)

where: ∇21 =

∂2 ∂2 1 ∂ E2 H E2 H 3 , d2 = + + 2 d1 = . 2 2 ∂ r1 r1 ∂ r1 ∂ ξ 2G(1 − ν1 )R 24G(1 − ν12)R3

Thin coating cannot influence sufficiently the normal stresses. On the other hand coating layer has a large rigidity in the tangential direction that is why one can suppose radial displacements on the boundaries equal zero [5]. Taking into account these assumptions, one can simplify boundary conditions (10), (12). In the first approximation one has (for ξ = 0): 2 2 ∂ ∂ Φ 1 ∂ 1 + − = 0, (13) ∂ r12 r1 ∂ r1 r12 ∂ r1 ∂ ξ ∂ ∂ 2Φ 2 (2 − ν )∇1Φ − = 0. ∂ξ ∂ξ2

(14)

From the physical standpoint one has in this case an inextensible membrane ideally bonded to the matrix at the half-space boundary. In the original variables boundary conditions (13), (14) can be written as follows (for z = 0): ∂Φ = 0, (15) ∂z

∂ 3Φ = 0. ∂ z3

(16)

4 A Single Fibre Embedded in the Half-Space Let us use for solving boundary value problem (6), (15), (16) the cosine Fourier transform Φ(r, s) =

∞

Φ(r, z) cos(sz)dz.

(17)

0

Partial differential equation (6) is transformed to the ordinary differential equation ∇22 ∇22 Φ(r, s) = 0, where: ∇22 =

d2 dr2

d + 1r dr − s2 .

(18)

6

I.V. Andrianov et al.

A general solution to the ordinary differential equation (2) is as follows Φ(r, s) = AK0 (sr) + BsrK1 (sr) + CI0 (sr) + DsrI1 (sr),

(19)

where K0 , K1 , I0 and I1 are the modified Bessel functions [8]. From conditions (9) one obtains C = D = 0, while condition (2) yields A=−

sRK0 (sR) B. K1 (sR)

(20)

Then from (1), (5), (19) and (20) one obtains

where:

τ¯ (s) = τ¯rz (R, s) = 4BG(1 − ν )s3K1 (sR),

(21)

R U¯ z (R, s) = τ¯ (s)g(sR), G

(22)

1 1 K0 (sR) − g(sR) = − sR K1 (sR) 4(1 − ν )

K0 (sR) K1 (sR)

2 +

1 . 4(1 − ν )

(23)

In what follows it will be useful to obtain asymptotics of function g(sR) for s → 0 and s → ∞. Using formulas (9.6.8), (9.6.9), and (9.7.2) from [8] one gets g(sR) ∼ ln(sR) + a, for s → 0, g(sR) ∼ −

3 − 4ν , for s → ∞, 4(1 − ν )sR

(24) (25)

1 where: a = γ − ln 2 + 4(1− ν ) > 0, γ = 0.577215649. . . (the Euler constant). Fibre equilibrium condition can be written as follows [14]:

E1

d 2U f 2 + τ (z) = 0. 2 dz R

(26)

Boundary conditions associated with Eq. (26) have the following form E1 =

dU f = P, for z = 0, dz

U f → 0, for z → ∞.

(27) (28)

Application of cosine Fourier transform (17) to the boundary value problem (26)– (28) yields 2 P −s2U¯ f (s) + τ¯ (s) + = 0. (29) E1 R E1 From condition (1) and relations (22), (23) and (29) we find τ¯ (s), U¯ z (R, s) and U¯ f (s). Using then inverse cosine Fourier transform (26) of the form

Load-Transfer from an Elastic Fibre to Isotropic Half-Space with Coating

2 Uz (R, z) = π one obtains P τ (ξ ) = − R

∞

7

U¯ z (R, s) cos(sz)ds,

(30)

M(ϕ ) cos(ϕξ )d ϕ ,

(31)

0

∞ 0

2(1 + ν )PR Uz (1, ξ ) = − π E1 PR U f (ξ ) = − π E1

σr (1, ξ ) = −

σz (1, ξ ) =

P π

P π

∞

∞

∞

M(ϕ )g(ϕ ) cos(ϕξ )d ϕ ,

(32)

0

E1 E1 − g(ϕ ) M(ϕ ) cos(ϕξ )d ϕ , kR G

(33)

0

∞

2ϕ g(ϕ ) +

0

2ϕ g(ϕ ) −

0

ϕ = sR, k12 =

K0 (ϕ ) M(ϕ ) sin(ϕξ )d ϕ , K1 (ϕ )

ν K0 (ϕ ) M(ϕ ) sin(ϕξ )d ϕ , 1 − ν K1 (ϕ )

kR , M( f ) = E1

2 f2 k12

−

E1 f 2 g( f ) G

+2

.

(34)

(35)

(36)

Formulae (31)–(35) differ from formulae for the problem of single fibre embedded in the space obtained in [14] only by factor 2. Now we will estimate integrals (31)–(35). First of all, we rewrite them in the following form: P τ (ξ ) = I1 , (37) R 2(1 + ν )PR Uz (1, ξ ) = I2 , (38) π E1 P PR I2 , (39) U f (ξ ) = − I1 + πk πG P σr (1, ξ ) = − (2I3 + I4 ), (40) π ν P 2I3 − (41) σz (1, ξ ) = − I4 . π 1−ν Asymptotics of function M(ϕ ) are of the following form M(ϕ ) → 1, for ϕ → 0,

(42)

8

I.V. Andrianov et al.

M(ϕ ) →

2k12 , for ϕ → ∞. ϕ2

(43)

Asymptotic expressions (42), (43) give a possibility to obtain the following interpolation function (valid for all values of ϕ ): M(ϕ ) ≈

2k12 , ϕ 2 + 2k12

(44)

√ π Pk1 I1 ≈ − √ exp − 2k1 ξ , 2R

(45)

√ π Pk1 τ (ξ ) ≈ − √ exp − 2k1 ξ , 2R

(46)

and, respectively, one has

π Pk1 τmax (k) ≈ √ . (47) 2R We compared this value of τmax with numerical data (see [6]). Discrepancy between approximate analytical and numerical results is not sufficient. Now we will analyse integral I2 . Asymptotic expressions for function g(ϕ )M(ϕ ) are as follows g(ϕ )M(ϕ ) → ln ϕ + a, for ϕ → 0, (48) g(ϕ )M(ϕ ) → −

2a1k12 , for ϕ → ∞, ϕ3

(49)

where: a1 = (3−4Rν )a . Let us suppose integral I2 as follows: (1)

(2)

I2 = I2 + I2 , (1)

(2)

where: I2 = I2 − I2 , (2)

(2)

I2 =

∞

(50)

f (ϕ ) cos(ϕξ )d ϕ ,

f (ϕ ) =

0

ln ϕ , 0 < ϕ < 1, 0, 1 ≤ ϕ.

Calculate integral I2 , one obtains: (2)

I2 (ξ ) = −

Si(ξ ) , ξ

(51)

where: Si(ξ ) is familiar sine integral [8]. (1) Expression under the integral sign M2 (ϕ ) in the integral I2 has the following asymptotics: M2 (ϕ ) → a, for ϕ → 0, (52) M2 (ϕ ) → −

2a1k12 , for ϕ → ∞. ϕ3

(53)

Load-Transfer from an Elastic Fibre to Isotropic Half-Space with Coating

9

It means that for all values ϕ one can use the following interpolation function for M2 (ϕ ): 2k12 a (1 − a2ϕ ) , (54) M2 (ϕ ) ≈ (1 + ϕ 2)(2k12 + ϕ 2 ) where: a2 = a1 /a. Using residual theorem, one obtains: √ √ ξ 2ak1 √ (1) I2 = 2 [ 2k1 exp(−ξ ) − exp(− 2k1 )]. 2k1 − 1

(55)

Now we will analyse integral I3 . Let us obtain asymptotic expressions for function ϕ g(ϕ )M(ϕ ): ϕ g(ϕ )M(ϕ ) → ϕ (ln ϕ + a), for ϕ → 0, (56)

ϕ g(ϕ )M(ϕ ) → −

2a1k12 , for ϕ → ∞. ϕ2

(57)

Let us divide integral I3 into two following parts (1)

(2)

I3 = I3 + I3 , (1)

(2)

(2)

(58)

∞

where: I3 = I3 − I3 , I3 = e−ϕ ϕ ln ϕ sin(ϕξ )d ϕ . 0 (2)

Computation of integral I3 yields (2)

I3 =

2 2(1 − γ )ξ − ξ ln(1 + ξ 2) + arctan ξ (1 − ξ 2) . 2 2 (1 + ξ )

(59)

(1)

Expression under the integral I3 has the following asymptotics M3 (ϕ ) → aϕ , for ϕ → 0, M3 (ϕ ) → −

2a1k12 , for ϕ → ∞. ϕ2

(60) (61)

Using asymptotics (60), (61) one can construct interpolation function for M3 (ϕ ), valid for all values of ϕ of the form M3 (ϕ ) ≈ 2k12 a

ϕ (1 − a2ϕ ) . (1 + ϕ 2)(2k12 + ϕ 2 )

On the other hand the residual theorem, yields: (1)

I3 =

√ 2ak12 [exp(−ξ ) − exp(− 2k1 ξ )]. 2 2k1 − 1

(62)

10

I.V. Andrianov et al.

Finally let us analyse integral I4 . Function

k0 (ϕ ) k1 (ϕ ) M(ϕ ) has the following asymptotics:

k0 (ϕ ) M(ϕ ) → −ϕ (ln ϕ + a), for ϕ → 0, k1 (ϕ )

(63)

k0 (ϕ ) 2k2 M(ϕ ) → 21 , for ϕ → ∞. k1 (ϕ ) ϕ

(64)

In what follows we assume the following form of integral I4 (1)

(2)

I4 = I4 + I4 ,

(1)

(2)

I4 = I4 − I4 .

(65)

(1)

Expression under the integral sign M4 (ϕ ) in the integral I4 exhibits the following asymptotics: M4 (ϕ ) → −aϕ for ϕ → 0, (66) M4 (ϕ ) →

2k12 for ϕ → ∞. ϕ2

(67)

Interpolation functions valid for all values of ϕ can be written as follows M4 (ϕ ) ≈ −2k12 a

ϕ (1 − ϕ /a) . (1 + ϕ 2)(2k12 + ϕ 2 )

(68)

Therefore, the residual theorem yields the following result (1)

I4 = −

√ ξ 2ak12 [exp(− ξ ) − exp(− 2k1 )]. 2k12 − 1

(69)

5 Conclusions The obtained results can be used for investigation of a composite fracture. Solved problems in the field of Civil Engineering model, the behaviour of piles or piers embedded in soil media, which exhibit a linear elastic response in the working-load range. Analytic solutions presented in this paper will be useful in evaluating test results calculated by boundary elements and finite element methods. Acknowledgements This work was supported by the German Research Foundation (Deutsche Forschungs-gemeinschaft), grant No. WE 736/25-1 (for I.V. Andrianov and D. Weichert).

References 1. Muki R, Sternberg E (1969) On the diffusion of an axial load from an infinite cylindrical bar embedded in an elastic medium, International Journal of Solids and Structures 5(6), 587–605. 2. Freund LB (1992) The axial force needed to slide a circular fibre along a hole in an elastic material and implications for fibre pull-out, European Journal of Mechanics A-Solids 11(1), 1–19.

Load-Transfer from an Elastic Fibre to Isotropic Half-Space with Coating

11

3. Eshelby JD (1982) The stresses on and in a thin inextensible fibre in a stretched elastic medium, Engineering Fracture Mechanics 16(3), 453–455. 4. Argatov II, Nazarov SA (1996) Asymptotic analysis of problems on junctions of domains of different limit dimensions. A body pierced by a thin rod, Izvestiya Mathematics 60(1), 1–37. 5. Phan-Thien N, Kim S (1994) Microstructures in Elastic Media: Principles and Computational Methods, Oxford University Press, New York. 6. Keer LM, Luk VK (1979) Stress analysis for an elastic half space containing an axially-loaded rigid cylindrical rod, International Journal of Solids and Structures 15, 605–627. 7. McCartney LN (1989) New theoretical model of stress transfer between fibre and matrix in a unaxialy fibre-reinforced composite, Proceedings of the Royal Society of London A 425, 215–244. 8. Rajapakse RKND, Wang Y (1990) Load-transfer problem for transversely isotropic elastic media, Journal of Engineering Mechanics 116(12), 2643–2662. 9. Ven-Gen Lee, Mura T (1994) Load transfer from a finite cylindrical fiber into an elastic halfspace, Journal of Applied Mechanics 61, 971–975. 10. Movchan AB, Willis JR (1997) Asymptotic analysis of reinforcement by frictional fibres, Proceedings of the Royal Society of London A 453, 757–784. 11. Antipov YA, Movchan AB, Movchan NV (2000) Frictional contact of fibre and an elastic solid, Journal of Mechanics and Physics of Solids 48, 1413–1439. 12. Lenci S, Menditto G (2000) Weak interface in long fiber composites, International Journal of Solids and Structures 37, 4239–4260. 13. Geymonat G, Krasucki F, Lenci S (1999) Mathematical analysis of a bonded joint with a soft thin adhesive, Mathematics and Mechanics of Solids 4, 201–225. 14. Alexandrov VM, Mkhitaryan SM (1983) Contact Problems for Bodies with Thin Coatings and Inclusions, Nauka, Moscow. 15. Andrianov IV, Danishevs’kyy VV (2007) Load-transfer to an orthotropic fibre-reinforced composite strip via an elastic element, Technische Mechanik 27(1), 28–36. 16. Abramowitz M, Stegun IA (eds.) (1965) Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York.

On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents Franziska Schmidt and Claude-Henri Lamarque

1 Introduction Lyapunov exponents measure the sensitivity of a dynamical system to initial conditions [1,2]. In fact an infinitesimal difference of initial conditions may lead to totally different paths. This is the case if the computed Lyapunov exponent is strictly positive. The system is then theoretically chaotic in infinite time, but practically this may occur at finite time considered as “asymptotic” for applications. Mathematically, the numerical value of the Lyapunov exponents is given by the formula: 1 ||δ x(t, x0 )|| ln , (1) λ = lim lim t→+∞ ||δ x0 ||→0 t ||δ x0 || where x0 is the initial condition (it may be a vector of initial conditions in Rn ), t is the time, .. is a norm for Rn , δ x0 is an infinitesimal divergence in initial conditions (in Rn ) and x (∈ Rn ) is the path followed by the system starting at x = x0 at time t = 0. The maximum of this spectrum is often the only one that is computed to detect chaos. Practically, these numerical values are assessed through the eigenvalues of the Jacobian matrix of the system. Numerically, it is computed using Gram-Schmidt’s reorthonormalization [3], QR-factorization [4] e.g. . . .. Lyapunov spectra can be computed from the study of δ x(t, x0 ) : a survey can be found in [5, 6]. But for applications and investigation of stability or robustness of dynamical processes, it is necessary to predict the finite-time evolution of a deterministic dynamical system from an initial state known with finite accuracy. That’s why we are here interested in the maximum reached by finite-time pseudo-Lyapunov exponents F. Schmidt and C.-H. Lamarque Universit´e de Lyon, Lyon, F-69003, France, Ecole Nationale des Travaux Publics de l’Etat, CNRS, URA 1652, D´epartement G´enie Civil et Bˆatiment. 3, rue Maurice Audin, Vaulx-en-Velin, F-69120, France, http://www.entpe.fr, e-mail: [email protected], e-mail: [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

13

14

F. Schmidt, C.-H. Lamarque

(see for e.g. [7]), which we call here finite-time pseudo-Lyapunov exponents which are functions of time and space:

0 )|| * λ˜ t = max 1t ln ||δ||x(t,x in the continuous case, δ x0 || t,x0

* λ˜ N = max N1 ln ||δ||xδNx(x||0 )|| in the discrete one. t,x0

0

We hightlight the fact that these last ones may be superior to the classical, infinitetime one. This can be due to the initial divergence that is not infinitesimal [8], to the initial condition or the time [9], or even the eigenvectors of the Jacobian matrix [10]. In a first paragraph, we explicit the analytical background in a more general, continuous case. Then we apply this directly to a Duffing-like oscillator. Finally, we make the same study for other continuous systems, among them being the Lorenz system. In this work calculation of Lyapunov exponents for smooth or non-smooth systems is generally done for approximating asymptotic or long (enough) time behavior, especially to point out chaos. On the contrary, when interested in transient tangent behavior, we show that these constants may not be suitable in order to assess the behavior of this system for finite time. Indeed finite-time pseudo-Lyapunov exponents depend on time and initial conditions, and hence may be superior to the “classical” Lyapunov exponent. First we explicit in a general way what our purpose is. Then to apply this we choose a one-ddl, second-order system (a Duffing-like oscillator) and a three-ddl, first-order system (Lorenz system).

1.1 A General Study In this section we assume for simplicity that the considered system is a continuous dynamical system, so that one can write: dX (t) = F(X ,t), dt

(2)

where t is the time, X is the variable of space (it is often a vector), F is a C1 -function (in the general case) in X. Further, in this study, the system is assumed to be autonomous. Therefore the evolution in time of an infinitesimal difference in initial conditions δ X0 can be evaluated through the formula: dδ X (t) = J(X0 )δ X (t), (3) dt where δ X (t) is the divergence at time t, J(X0 ) is the Jacobian matrix of the transformation F evaluated at initial condition X0 : [J(X0 )]i j =

∂ Fi (X = X0 ). ∂ Xj

(4)

On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents

15

So the further path of this quantity is estimated through:

δ X (t) = etJ(X0 ) δ X0 .

(5)

Now we also assume that J(X0 ) can be written in the form: J(X0 ) = PAP−1, where A is a diagonal matrix and P is invertible. So, with the series of the exponential J(X0 )2 n n −1 0) function eJ (X0 ) = I + J(X 1! + 2! + . . . and as J(X0 ) = PA P , Eq. (5) can be written: δ X (t) = PetA P−1 δ X0 . (6) where the matrix etA can be easily found (A is diagonal). So with the definition of finite-time pseudo-Lyapunov exponents, we want to assess: tA −1 ˆλt = max 1 ln ||P.e P δ X0 || = max g(t, X0 , δ X0 ), t,X0 t t,X0 ||δ X0 ||

(7)

where t must be positive. Therefore if we want to search local maxima of this pseudo-Lyapunov exponent with t ≥ 0 and using Lagrange multipliers, the problem of finding the maximum of (7) can be summarized through: ⎛ ⎞ ⎧ α ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎜0⎟ ⎪ ⎨∇g(t, X0 , δ X0 ) − ⎝ ⎠ = 0, .. (8) . ⎪ ⎪ ⎪ ⎪ α ≥ 0, ⎪ ⎪ ⎩ α t = 0.

1.2 Duffing-Like Oscillator We study the following dynamical system: (9) x¨ + ax˙ + bx + cx3 = 0. 0 1 , where x0 is the initial condition in x. In this case : J(X0 ) = −b − 3cx20 −a Diagonalising the matrix J(X0 ) and writing it in the shape : J(X0 ) = PAP−1, the evolution with time of the initial divergence can be evaluated. For example, if a2 > 4(b + 3cx20) then: γ −β 0 γ + β −1 1 γ −β 1 −1 , , P= ,P = A= 0 γ +β 1 γ +β 2β −(γ − β ) 1

16

F. Schmidt, C.-H. Lamarque

and divergence at time t can be assessed through (δ X (t) = PetA P−1 ): e−γ t −γ sinh(β t) + β cosh(β t) sinh(β t) δ X (t) = δ X0 , (β 2 − γ 2)sinh(β t) γ sinh(β t) + β cosh(β t) β where: γ = a2 , β =

1 2

(10)

a2 − 4(b + 3cx20).

A similar formula for δ X (t) can be obtained in the other cases: a2 < 4(b + 3cx20) and a2 = 4(b + 3cx20). We decide to search the maximum of this divergence (10), according to X0 , δ X0 δ x0 . In order to and t. For this, we solve the system of equations (8), with δ X0 = δ x˙0 ρ cos(θ ) , which does not reduce the facilitate our calculations, we take: δ X0 = ρ sin( θ) f1 (x0 ,t, θ ) generality of our study. So if we write δ X (t) = , then the finite-time f2 (x0 ,t, θ ) pseudo-Lyapunov exponents can be assessed by searching the maximum of:

λt = g(x0 ,t, δ X0 ) =

1 ln[ f12 (x0 ,t, θ ) cos(θ )2 + f22 (x0 ,t, θ ) sin(θ )2 ]. 2t

(11)

This finite-time pseudo-Lyapunov exponent (11) may be superior to the Lyapunov exponent. Indeed, for example with a = 0.15, b = −1, c = 1, this last one is negative λ = −0.12 < 0 (see Fig. 1), whereas λt = 0.252718 for x0 = 2.189, t = 6.44065s, θ = −9.4247 (by solving Eq. (7)). 1

0.5

0

−0.5

−1

−1.5

0

20

40

60 t

80

100

120

Fig. 1 Lyapunov exponent time convergence of the dynamical system (9) calculated with Wolf’s algorithm [3] a = 0.15, b = −1, c = 1

On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents

17

3 2 1 1.2

1.4

1.6

x0

1.8

2

2.2

2.4

0 -1 -2 -3

Fig. 2 Maximum of the finite-time pseudo-Lyapunov exponents (Eq. (11)) of the dynamical system (9) with a = 0.15, b = −1, c = 1 along the initial condition x0 . Thus the finite-time pseudoLyapunov exponent may take negative or positive values (decreasing or growing of the divergence), whereas the Lyapunov exponent is negative

This can be generalized by varying the initial condition x0 . To do this, we solve once again Eq. (7) with fixed initial condition X0 . The results are plotted in Figs. 1–3. Thus, we see that the numerical value of the finite-time pseudo-Lyapunov exponents may be superior to the “classical”, infinite-time Lyapunov exponent. Thus, using this last exponent for assessing the evolution of the divergence at finite time is not adapted for quantifying worst divergence. In this case, the calculations are rather simple and it is possible to solve them analytically. This is generally not the case. That’s why we now study some more challenging dynamical systems.

2 The Lorenz Attractor and Other Dynamical Systems The behavior of the Lorenz attractor is given by the formula: ⎧ ⎪ ⎨x˙ = σ (y − x), y˙ = ρ x − y − xz, ⎪ ⎩ z˙ = xy = bz.

(12)

We apply the same procedure to this dynamical system: here the Jacobian matrix is of size (3 × 3). So the potential eigenvalues are solutions of a three-degrees polynomial function, whose coefficients are real functions of the parameters (σ , ρ , b) and the initial conditions (x0 , y0 , z0 ).

18

F. Schmidt, C.-H. Lamarque

80 60 40 20 0 -20

1.2

1.4

1.6

1.8

2

2.2

2.4

x0

-40

Fig. 3 Angle θ of the initial infinitesimal divergences δ X0 = ρ cos(θ ), δ Y0 = ρ sin(θ ) giving the maximum of λt along the initial condition x0 , obtained by solving system of equations (8).

Nevertheless the conclusions are the same as for the case of the Duffing-like oscillator: the finite-time pseudo-Lyapunov exponents may be superior to the classical one, see Figs. 4–6. Untill now we do not clearly analyze a underlying structure. There are several reasons for this behavior: first, this pseudo-Lyapunov exponent does depend on time and initial conditions whereas the classical Lyapunov exponent does not (ergodicity). Moreover, the Jacobian matrix is just an approximation at first order of the phenomenon. Then, the initial divergence taken for calculations is not infinitesimal. Finally, the eigenvectors of the Jacobian matrix may not be orthogonal, what is assumed computing the Lyapunov exponent via Wolf’s algorithm. It is impossible to distinguish the importance of each of these reasons in the final result. But one fact is clear: each of these causes acts. Indeed, if we take a linear oscillator, the finite-time pseudo-Lyapunov exponent appears to be the superior to the infinite-time one even if the initial condition does not intervene in the calculations (Jacobian matrix depends only on the parameters of the system). For example, for a linear oscillator x¨ + ax˙ + bx = 0 with a = 0.25 and b = 0.01, the Lyapunov exponent is equal to −0.05, whereas the finite-time pseudoLyapunov exponent varies and reaches a maximum λt = 0.1309 of t = 8.9971s and (x0 , x˙0 ) = (ρ cos(θ ), ρ sin(θ )) with θ = −4.9597◦. Moreover, this conclusion is also correct if we take a system of which the nonorthogonality of the system eigenvectors 3 not satisfied. For example, we may propose a logistic map whose Jacobian matrix is (1 × 1): xn+1 = kxn (1 − xn).

(13)

On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents

19

5

Lyapunov exponents

0

-5

-10

-15

-20 0

200

400

600

800

1000

Time t

Fig. 4 Spectrum of the Lyapunov exponent along time of the dynamical system (12) with σ = 10.0, ρ = 28.0, b = 83 , calculated with Wolf’s algorithm [3]. The behavior is thus chaotic, with λ ≈ 2.66 > 0

6.2 6 5.8 5.6 5.4 5.2 5 4.8 -12

-11

-10

-9

-8

-7

-6

z0

Fig. 5 Maximum of the finite-time pseudo-Lyapunov exponents of the dynamical system (12) with σ = 10.0, ρ = 28.0, b = 83 along the initial condition z0 = z(0)

20

F. Schmidt, C.-H. Lamarque

8

6

4

2

-12

-11

-10

-9

-8

-7

-6

z0

Fig. 6 Time t when the maximum of λt (see Fig. 5) is reached along the initial condition z0 = z(0)

1 0.8 0.6 0.4 0.2 0 20

40

60

80

100

-0.2 -0.4 -0.6

Fig. 7 Finite-time pseudo-Lyapunov exponent of system (13) with k = 3.56

For k = 3.56, this system is stable (λ = −0.0653929), but the finite-time pseudoLyapunov exponent shows a somewhat different behavior, see Fig. 7. Finally, we want to highlight the fact that these results can be generalized to non-smooth systems. Indeed, a generalized Lyapunov exponent can be defined

On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents

21

x k m

g cos (wt)

c

x xmax

Fig. 8 The linear impact oscillator

(see [7, 11–15]) and thus also a generalized finite-time pseudo-Lyapunov exponent ( [16, 17]). This Lyapunov exponent can be defined using matrices of passage and matrices of jump, see [13, 16]. This makes it possible to define and calculate the spectrum of finite-time pseudo-Lyapunov exponents. For example, considering an impact linear oscillator (see Fig. 8), the equations of motion can be written: x¨ + 2ε w1 x˙ + w21 x = f cos(wt), (14) x = xmax : x+ = x− , x˙+ = −ex˙− . The matrices of passage and jump can thus be written, see [16]. eα t −α sin(bt) + b cos(bt) sin(bt) J= , −(α 2 + b2) sin(bt) α sin(bt) + b cos(bt) b −e 0 δ x− δ t1 = − −1 , S = − e+1 (w2 x − f cos(wt )) −e , max 1 − 1 x2 x2 where α = −ε w1 , b = w1 1 − ε 2.

(15)

In these equations. J is the matrix of passage, S is the matrix of jump and δ t1 is the difference of time between the two impacts: that of the original path and that of the perturbated one. The divergence between the two paths and after n impacts is as follows:

δ X (t) = Jn Sn Jn−1 Sn−1 . . . J0 δ X0 .

(16)

Here some work has to be done in order to define how to calculate the finitetime pseudo-Lyapunov exponent, that is to say the maximum of the spectrum on the whole path. It is possible to bound this divergence on every part of the path, that is to say every matrix Ji or Si . But in this case, the bound obtained may be too coarse (see [17]).

22

F. Schmidt, C.-H. Lamarque 15

1000

10

500

5

0

1

0

2

3

4

5

-500 1

-5

2

3

4

5

-1000 -1500

-10 -15

-2000 -2500

Fig. 9 Motion and finite-time pseudo-Lyapunov exponent along time for the system of Eq. (15), with ε = 0.01, a = 2ε w1 , w1 = 0.05, w = 2.6, f = 20, xmax = 16, x(0) = x(0) ˙ =0

But it is possible for a given initial condition to calculate its

path and behavior of δ x(t,x0 ) divergence along time, according to the formula 1t ln , see Fig. 9: δ x0 The spectrum of the finite-time pseudo-Lypunov exponent can thus be estimated and the conclusion is the same as in the smooth cases. Here, the consequences of this phenomenon are even worse than in the smooth case, because of the grazing bifurcation. Indeed, this case where the speed of impact on the wall is equal to zero, is challenging: the matrix of jump has got a term that tends to infinity and the behavior of the system is complicated (see [18, 19]).

3 Conclusion Lyapunov exponents are a convenient way to define the behavior of a dynamical system if time tends to infinity. The method is also often used to quantify the finitetime response of a dynamical system if uncertainties exist. For a transient behavior and the practical study of divergence or stability, we look for another estimation that could lead to bounds for tangent behavior. That is why finite-time pseudo-Lyapunov exponents have been computed in continuous cases and discrete ones. In all these studies, the conclusions have always been that they may be superior to the Lyapunov exponent. Particularly, they may be positive whereas the Lyapunov exponent is negative. This does not mean that the domain of parameters concerned by chaos is extended. Indeed, chaos is predicted in an infinite time. Here, we just deal with finite-time and we highlight the fact that this indicator called a finite-time pseudo-Lyapunov exponent makes it possible to bound the tangent finite-time divergence of a dynamical system with initial conditions known with finite accuracy, and numerical calculations made with strictly positive small step of time. This indicator is mainly a bound to quantify finite-time maximum of instability. A way to go further in this study would be to examine the influence of time and initial conditions on the Lyapunov exponent of higher derivatives [20].

On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents

23

References 1. A. M. Lyapunov. The general problem of the stability of motion. International Journal of Control, 55(3):531–773, 1992. 2. V. I. Oseledec. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Transactions of the Moscow Mathematical Society, 19:197–231, 1968. 3. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano. Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3):285–317, 1985. 4. F. E. Udwadia and H. F. von Bremen. An efficient and stable approach for computation of Lyapunov characteristic exponents of continuous dynamical systems. Applied Mathematics and Computation, 121(2–3):219–259, 2001. 5. K. Ramasubramanian and M. S. Sriram. A comparative study of computation of Lyapunov spectra with different algorithms. Physica D: Nonlinear Phenomena, 139(1–2):72–86, 2000. 6. G. Tancredi, A. S`anchez, and F. Roig. A comparison between methods to compute Lyapunov exponents. Astronomical Journal, 121(2):1171–1179, 2001. 7. R. Ding and J. Li. Nonlinear finite-time Lyapunov exponent and predictability. Physics Letters A, 364(5):396–400, 2007. 8. E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, and A. Vulpiani. Growth of non-infinitesimal perturbations in turbulence. Physical Review Letters, 77(7):1262–1265, 1996. 9. J. M. Nese. Quantifying local predictability in phase space. Physica D: Nonlinear Phenomena, 35(1–2):237–250, 1989. 10. C. Nicolis, S. Vannitsem, and J. -F. Royer. Short-range predictability of the atmosphere: mechanisms for superexponential error growth. Quarterly Journal – Royal Meteorological Society, 121(523):705–722, 1995. 11. N. Hinrichs, M. Oestreich, and K. Popp. Dynamics of oscillators with impact and friction. Chaos, Solitons & Fractals Nonlinearities in Mechanical Engineering, 8(4):535–558, 1997. 12. L. Jin, Q. -S. Lu, and E. H. Twizell. A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impact-vibrating systems. Journal of Sound and Vibration, 298(4–5):1019–1033, 2006. 13. P. C. Muller. Calculation of Lyapunov exponents for dynamic systems with discontinuities. Chaos, Solitons & Fractals Some Nonlinear Oscillations Problems in Engineering Sciences, 5(9):1671–1681, 1995. 14. A. Stefa´nski and T. Kapitaniak. Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization. Chaos, Solitons & Fractals, 15(2):233– 244, 2003. 15. Q. Wu and N. Sepehri. On Lyapunov’s stability analysis of non-smooth systems with applications to control engineering. International Journal of Non-linear Mechanics, 36(7):1153– 1161, 2001. 16. F. Schmidt and C. -H. Lamarque. Computation of the solutions of the Fokker-Planck equation for one and two dof systems. Communications in Nonlinear Science and Numerical Simulation, 74(2008):529–542. 17. F. Schmidt and C. -H. Lamarque. Un indicateur pour optimiser les calculs trajectographiques. Bulletin de Liaison des Ponts et Chauss´ees, BLPC No. 263–264, znillet - Qoˆut - septembre 2006. 18. A. P. Ivanov. The dynamics of systems near to grazing collision. Journal of Applied Mathematics and Mechanics, 58(3):437–444, 1994. 19. O. Janin and C. H. Lamarque. Stability of singular periodic motions in a vibro-impact oscillator. Nonlinear Dynamics, 28(3–4):231–241, 2002. 20. U. Dressler and J. D. Farmer. Generalized Lyapunov exponents corresponding to higher derivatives. Physica D: Nonlinear Phenomena, 59(4):365–377, 1992.

Numerical Characterization of the Chaotic Nonregular Dynamics of Pseudoelastic Oscillators Davide Bernardini and Giuseppe Rega

1 Introduction Previous studies on the nonlinear dynamics of pseudoelastic oscillators showed the occurrence of chaotic responses in some ranges of the system parameters [1,2]. The restoring force was modeled by a thermomechanically consistent model with four state variables [3]. In comparison with the simpler polynomial constitutive laws considered for example in [4], the present model is characterized by more governing parameters and it is therefore interesting to understand whether nonregular responses only occur in isolated zones or are actually robust outcomes. The relevant analyses need to be carried out through some synthetic measure of non-regularity that has to be reliable and computationally simple in order to allow for systematic investigations in meaningful parameter spaces. Whereas the numerical characterization of chaos in smooth dynamical systems is often carried out via the computation of Lyapunov exponents, in the present case the computation of such exponents, following, for example [5], does not seem to be a convenient strategy. The attention has thus been focused on the simpler direct numerical tool represented by the method of wandering trajectories [6]. The method is based on the numerical evaluation of the separation between pairs of neighboring trajectories normalized with respect to a suitable measure of the motion amplitude. Such normalized perturbations provide a tool to detect the occurrence of nonregular and chaotic responses. The method has been successfully applied in the literature to estimate regular and chaotic responses for non-smooth mechanical oscillators with up to two degrees of freedom [7].

D. Bernardini and G. Rega Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza Universit`a di Roma, Via Antonio Gramsci 53, 00197 Roma, Italy, e-mail: [email protected], [email protected]

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The purpose of this paper is to calibrate and validate the method of wandering trajectories within a thermomechanical framework and to present some results on the overall characterization of the chaotic response of pseudoelastic oscillators.

2 Description of the System The system under consideration is a simple oscillator where the restoring force is provided by a device with pseudoelastic behavior. The model used for the restoring force fits into the family of models introduced in [3] that are derived from the assignment of two constitutive functions: the free energy and the dissipation function. The specific version used in this work has been introduced in [1] and is characterized by a new form for the dissipation function with respect to those in [3]. The evolution of the system is described, besides by the displacement, velocity and temperature (x, v, ϑ ), also by the martensitic phase fraction ξ ∈ [0, 1]. The typical pseudoelastic loops occur as a consequence of the Forward (FwT) and Reverse (RvT) Transformation, respectively associated to increasing and decreasing of ξ . The nondimensional dynamics of the oscillator is modeled by the following piecewise smooth system of four ordinary differential equations in the variables x := [x, v, ξ , ϑ ] (for details see [1]) x˙ = v, v˙ = −x + (sλ )ξ − ζ v + γ cos ατ , ξ˙ = H[sv − Jh(ϑe − ϑ )],

(1)

Λ + Jλ ϑ ϑ˙ = L H[sv − Jh(ϑe − ϑ )] + h(ϑe − ϑ ), Jλ where H and Λ are constitutive functions that take different expressions depending on the kind of transformation [2] as well as on the value ξ0 of the martensite fraction at the end of the last transformation process. The vector field (1) can thus take three different forms depending on the expressions of H and Λ. However, once activated, each kind of behavior is smooth. The parameters λ , L, J, h represent respectively: the length of the pseudoelastic plateaus, the latent heat of transformation, the linear temperature dependence of the transformation forces and the coefficient of convective heat exchange with the environment. Moreover s = sgn(x) whereas ζ , γ , α denote respectively the viscous damping and the excitation amplitude and frequency. The initial conditions (i.c.) required to compute the evolution of the system is a 5-ple of numbers Σ := (x, v, ξ , ϑ , ξ0 ). The time evolution of ξ0 is almost everywhere constant as it jumps from a value to another whenever there is a switch between different kinds of behavior. For the following analyses it is important to note that not every 5-ple of numbers Σ represents a physically admissible state of the system. A procedure to determine admissible values of Σ is described in [2].

Numerical Characterization of the Chaotic Nonregular Dynamics

27

3 The Method of the Wandering Trajectories The Method of Wandering Trajectories (MWT) is a tool for the characterization of the asymptotic behavior of dynamical systems under periodic forcing excitation. The basic idea is very simple: a motion is classified as non-regular if the separation with a neighboring trajectory starting from an admissible state overcomes a given threshold. Of course, a key issue is the proper definition of the auxiliary trajectory and of the threshold. Let the fiduciary trajectory to be characterized be denoted as u(t) with initial condition u(0) = u0 . For any other trajectory u(t) ˜ such that u(0) ˜ = u˜ 0 initial and current separations are defined as follows h0 := | u0 − u0 | ,

h(t) := | u(t) − u(t)| .

(2)

The MWT proceeds as follows: • Integrate u(t) over T := [0, T] and compute, over the subinterval T1 := [t1 , T] where transients expired, for each component, the vibration amplitude of the fiduciary trajectory, defined as follows Ai :=

1 |ai − bi | , 2

where ai := max ui (t) and bi := min ui (t); t∈T1

t∈T1

(3)

this yields a vector A measuring the scale of the motion. • Another trajectory u(t) is defined by the initial condition u0 derived from u0 by perturbing each component proportionally to the corresponding vibration amplitude h0 := ε A. • Define the normalized separations relative to the vibration amplitudes

αi :=

|hi (t)| . Ai

(4)

Provided they correspond to an admissible state of the system, the initial values αi (0) are thus equal to the chosen parameter ε . If the motion is regular the normalized separations either take values of the same order of magnitude as ε or decay to zero. By contrast, non-regular motions may lead, after the transients, to normalized separations much higher than ε . More specifically, trajectories initiated from two nearby points on a chaotic attractor separate away from each other until the separation levels off at the size of the attractor. The main assumption of the MWT is that, with some preliminary knowledge of the system dynamics, it is possible to determine a threshold level α¯ for the normalized separation that characterizes the occurrence of non-regular motions. In particular, a trajectory u(t) is characterized as non-regular if, at some t ∈ T1 , the normalized separation with respect to a trajectory with initial separation h0 = ε A, exceeds the chosen threshold αi (t) > α . (5)

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This test certainly detects the sensitivity to initial conditions of the trajectory. However this is only a necessary but not sufficient condition for the motion to be chaotic. In facts, the sensitivity to initial conditions alone only indicates that the perturbation may have taken the trajectory outside the basin of attraction of the attractor. A chaotic motion, besides being sensitive to initial conditions, is also wandering in the sense that it attempts to fill a bounded region K of the phase space (for any q ∈ K there is a time t such that u(t) = q). Therefore the MWT tends to overestimate the number of non-regular trajectories, which turns out to be in favor of safety from an engineering viewpoint.

4 Characterization of Typical Trajectories The constitutive model for the restoring force covers a great variety of situations. The following set of parameters, as in [1], is considered as reference for the validation procedure and in the following is referred to as RMP Λ = 8.125, J = 3.1742, L = 0.124, q1 = 0.98, q2 = 1.2, q3 = 1.0246,

h = 0.08,

together with a = 0.03 and ζ = 0.03. These parameters correspond to a typical pseudoelastic cycle in a mildly convective environment. For the physical meaning of the parameters see [1].

4.1 Symmetric Period 1 Solution This is the most common type of solution. The system exhibits two pseudoelastic loops for positive and negative displacements. To show an example, the system, with RMP, is integrated for 100 periods with excitation amplitude γ = 1 and frequency α = 0.6 starting from the admissible initial condition u0 = [0, 0, 0, 1] and ξ0 = 0 corresponding to the system at rest in austenitic phase at the environment temperature. The time histories show that after about 20 periods transients expire and the system reaches a periodic attractor. Maxima, minima and vibration amplitudes for each component are computed for the fiduciary trajectory by taking the time interval T1 = [125.6, 628] x

v

Max 5.835 3.782 Min −5.835 −3.782 A 5.835 3.782

Ξ

ϑ

0.571 0 0.286

1.054 0.981 0.036

Numerical Characterization of the Chaotic Nonregular Dynamics 0.06

0.06

aX

0.04

0.04

0.02

0.02

0

29

ax

0 0

125.6

time 251.2 376.8

502.4

628

0

125.6

time 251.2 376.8

502.4

628

Fig. 1 Normalized separations in x and ξ or a periodic solution

In order to determine the perturbed trajectory it is important to ensure that the perturbed initial condition is again an admissible state. An admissible perturbed initial condition obtained by considering ε = 0.01 on all variables except for ξ , is (see [2]) u˜ 0

= [0.05835, 0.03782, 0.0, 1.00036]ξ0 = 0.0.

Integrating along the same time interval T it turns out that the perturbed trajectory u(t) ˜ is again a symmetric period 1 solution with features analogous to the fiduciary one. The absolute values of separation of the x and ξ components normalized relative to the respective vibration amplitudes are plotted in Fig. 1. The other variables exhibit analogous behavior. During the transient the normalized separations slightly overcome the assumed initial values, but then they rapidly decay to zero since the periodic attractor is asymptotically stable. The method of wandering trajectories classifies this solution as regular for any reasonable choice of α¯ .

4.2 Chaotic Solution As an example of chaotic trajectory the system is tested, as above, but at a lower frequency α = 0.245. Perturbing it again by 0.01Ai, the force-displacement cycles in the time interval [314, 628] for the fiduciary and the perturbed trajectories are drawn in Fig. 2 . It turns out that the fiduciary is chaotic whereas after the perturbation the response is periodic. The normalized separations are very high (Fig. 3), see in particular the ξ component, and the MWT classifies the solution as non-regular since, immediately after the transients, separations much higher than the initial values occur.

4.3 Unsymmetric Period 1 Solution It is rather frequent to find symmetry breaking bifurcations. In this case asymmetric periodic solutions occur and it is interesting to test the method on such solutions. To show an example, the system is integrated, again in the same conditions as in

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Fig. 2 Pseudoelastic cycles for a chaotic trajectory and its perturbation

Fig. 3 Normalized separations in x and ξ or a chaotic solution

Fig. 4 Pseudoelastic cycles for an asymmetric periodic solution

previous examples but at a frequency α = 0.27. The fiduciary trajectory involves complete but highly non symmetric transformations (Fig. 4). Perturbing again, the ensuing solution turns out to be still periodic and non symmetric but mirrored. The normalized separations are large (Fig. 5) so the MWT classifies the solution as non-regular. However, by plotting two components of the separation against each other, a regular evolution is observed as opposed to what happens in the chaotic solution [2]. This is a situation in which the MWT may classify a periodic solution as nonregular. However, the same conclusion would be reached by looking at the maximum Lyapunov exponent that turns out to be positive, about 0.0046, as shown in [2].

Numerical Characterization of the Chaotic Nonregular Dynamics

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Fig. 5 Normalized separations in the asymmetric periodic solution

Fig. 6 Bifurcation diagram and normalized separations

5 Comparison with Bifurcation Diagrams In order to have an overall picture of the system behavior, a constant excitation amplitude (γ = 1) bifurcation diagram with the frequency α as control parameter has been computed. The diagram is obtained by decreasing frequency with variable initial conditions taken from the adjacent computation point, and the region α ∈ (0.15, 0.3) is reported in Fig. 6. In the same frequency interval a systematic application of the MWT has been done for comparison. In particular, for each frequency, the response has been computed for T = 200 periods while checking the normalized separations with respect to trajectories perturbed by ε = 0.01 on T1 = 100 periods.

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Fig. 7 Details of Fig. 6

On the same figure a curve depicting the results of the MWT is superposed. For each frequency the curve (to be read with respect to the right vertical axis) shows the maximum value over T1 of the normalized separation of the displacement. It turns out that, whenever the trajectories are periodic, the separation practically vanish. On the contrary, when the separation overcomes values of about 0.1, a slightly chaotic behavior is already observed. Values of the separation above 0.3 are definitely associated with the consolidated chaos. In Fig. 7 two more detailed pictures in the zones denoted above by R and L show that the normalized separation closely follows the occurrence of non-regular responses and that 0.3 may be taken as a reasonable estimate of the threshold α¯ to detect some consolidated chaotic responses.

6 Overall Characterization of the Non-regular Solutions and Effect of the Hysteresis The robustness of the chaotic response within the overall behavior of the system can now be investigated by computing behavior charts in which some control parameters are varied and the MWT is systematically applied to distinguish between regular and nonregular responses. A natural choice for the control parameters is the pair excitation frequency-amplitude at fixed initial conditions and material parameters. In particular, the analysis has been carried out for the above mentioned set of material parameters RMP as well for another set, called MP1, obtained from RMP by decreasing q2 from 1.2 to 1.02. The parameters MP1 correspond to a pseudoelastic loop with lower hysteresis with respect to RMP. The comparison between the two provide information about the effect of the hysteresis on the chaotic response. According to the previous analyses the threshold level for the normalized separations has been chosen as α¯ = 0.3. Integration of the trajectories has been carried out for 200 excitation periods, while restricting the interval T1 to the last 100 periods. Due to the complexity of the trajectories occurring in some parameter regions,

1

1

0.5

0.5 v_0

v_0

Numerical Characterization of the Chaotic Nonregular Dynamics

0

33

0

a = 0.21

a = 0.245

-0.5

-0.5

-1

-1 -1

-0.5

0 x_0

0.5

1

-1

-0.5

0 x_0

0.5

1

Fig. 8 Regions of nonregular response in initial conditions plane (white: regular, black dot: non-regular) 2

1.8

amplitude

1.6

1.4

1.2

1

0.8 0.1

0.2

0.3

0.4

frequency

Fig. 9 Behavior chart in excitation frequency-amplitude plane for RMP (white: regular, black dot: non-regular)

the application of the method requires a rather fine numerical integration. After calibration of various explicit and implicit integration algorithms, a reasonable compromise between accuracy and computational time has been reached by using a standard fourth-order Runge-Kutta algorithm with 2,000 steps per period. Preliminarily, an investigation has been carried out in the initial conditions domain. More specifically, the MWT has been first applied to build a section of

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D. Bernardini, G. Rega

a kind of basin of attraction of chaotic responses in the plane of initial displacement x0 and velocity v0 . Initial conditions x0 ∈ [−1, 1] and v0 ∈ [−1, 1] have been considered together with ξ = ξ0 = 0 and ϑ = ϑ0 = 1. These values can be shown to be all admissible and correspond to the device in elastic, purely austenitic, phase. Two sample domains corresponding to the excitation amplitude γ = 1 and different frequencies α = 0.245 and α = 0.21 are shown in Fig. 8 (with RMP). At both frequencies, nonregular responses occur for various initial conditions. Analogous responses occur at the other frequencies where chaos is found. From consideration of such analyses, the pair (x0 , v0 ) = (−1.0, −1.0) has been selected as fixed initial condition, together with ξ0 = 0 and ϑ0 = 1, for the subsequent investigations. The frequency-amplitude behavior chart for the basic set of parameters RMP is shown in Fig. 9. For γ = 1, two clearly separated regions of non-regular motion are found, a compact one on the right, a more scattered one on the left. They are likely to correspond with the two kinds of chaotic motions highlighted in [1] by bifurcation diagrams. The presence of scattered points, especially at the higher excitation amplitudes, can be eliminated by a finer numerical integration. The same chart has then been computed with MP1 material parameters (Fig. 10). It turns out, as expected, that decreasing hysteresis leads to a significant increase of the size of the regions of irregular motion, with the intermediate region tending to cluster in nearly vertical stripes at lower frequencies. In-depth understanding of the kind of non-regular motion with respect to the neighbouring regular one would require complementing the chart with a number

2

1.8

amplitude

1.6

1.4

1.2

1

0.8 0.1

0.2

0.3

0.4

frequency

Fig. 10 Behavior chart in excitation frequency-amplitude plane for MP1 (white: regular, black dot: non-regular)

Numerical Characterization of the Chaotic Nonregular Dynamics

35

of bifurcation diagrams with frequency as control parameter (this is left for future investigations). Overall, the charts show that the chaotic motions are robust and persist in significant regions of the excitation frequency-amplitude plane.

7 Conclusions The Method of Wandering Trajectories has been shown to be effective in detecting the sensitivity to initial conditions of the orbits of a thermomechanically based pseudoelastic oscillator. The occurrence of chaotic responses has been characterized via excitation frequency-amplitude charts for two sets of material parameters. The results confirm that, although an increase of the hysteresis in the system tends to reduce chaotic motions, even in the reference case the occurrence of chaos is a robust outcome taking place in large regions of the frequency-amplitude plane.

References 1. Bernardini D, Rega G (2005) Thermomechanical modeling, nonlinear dynamics and chaos in shape memory oscillators, Mathematical and Computer Modelling of Dynamical Systems 11, 291–314. 2. Bernardini D, Rega G (2007) On the characterization of the chaotic response in the nonlinear dynamics of pseudoelastic oscillators, Proceedings of the 18th AIMETA Conference, Brescia (Italy), September 11–14. 3. Bernardini D, Pence TJ (2002) Models for one-variant shape memory materials based on dissipation functions, International Journal of Non-linear Mechanics 37, 1299–1317. 4. Savi MA, Pacheco PMCL (2002) Chaos and hyperchaos in shape memory systems, International Journal of Bifurcation and Chaos 12, 645–667. 5. M¨uller PC (1995) Calculation of Lyapunov exponents for dynamic systems with discontinuities, Chaos Solitons and Fractals 5, 1671–1681. 6. Awrejcewicz J, Dzyubak L, Grebogi C (2004) A direct numerical method for quantifying regular and chaotic orbits, Chaos Solitons Fractals 19, 503–507. 7. Awrejcewicz J, Dzyubak L, Grebogi C (2005) Estimation of chaotic and regular (stick-slip and slip-slip) oscillations exhibited by coupled oscillators with dry friction, Nonlinear Dynamics 42, 383–394.

Localized Nonlinear Excitations and Interchain Energy Exchange in the Case of Weak Coupling Leonid I. Manevich and Valeri V. Smirnov

1 Introduction The problem of energy exchange between weakly coupled nonlinear oscillators is actually far-reaching extension of classical beating problem in linear vibrations theory. Its modern stage goes back to the paper [1] in the field of nonlinear optics (the problem of nonlinear couplers). Then this problem was considered in [2] and [3]. Extension on the case of quasi-harmonic waves in two coupled sin-Gordon chain has been performed in series of papers [4, 5]. A new approach to the problem of non-linear energy exchange was proposed in [6]. In this paper the conception of limiting phase trajectory (LPT), corresponding to complete energy transfer between nonlinear oscillators has been introduced. It was shown that in the framework of this conception an adequate understanding and description of the problem can be obtained with the use of a pair of non-smooth basic functions of time. The proposed approach was also extended on the case of interchain energy exchange by quasiharmonic waves in weakly coupled oscillatory chains [6]. Recently we presented an efficient use of LPT conception in the case of “small” periodic Fermi-Pasta-Ulam (FPU) chains [7]. Contrary to previous papers, both symmetric and asymmetric interparticle potentials of interaction were considered. The interchain energy exchange by breathers was first studied both analytically and numerically in [8, 9]. A real possibility of this phenomenon has been shown. However, assumptions were made in that paper which require a justification. In the present paper we reconsidered the problem of interchain energy exchange using the LPT conception and considering subsequently the cases of weakly coupled chains with various degrees of nonlinearity (linear chains, weakly nonlinear chains, chains with nonlinearity compared with coupling). L.I. Manevich and V.V. Smirnov N.N. Semenov Institute of Chemical Physics, RAS 4 Kosygin street, 119991, Moscow, Russia, e-mail: [email protected], [email protected]

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L.I. Manevich, V.V. Smirnov

2 Linear Chains with Weak Coupling Let us consider a system of weakly coupled (FPU) chains with potential energy, which contains the terms of fourth order alongside with parabolic ones. The respective Hamilton function is: dqn, j 2 c2 β H = ∑ ∑ [( ) + (qn+1, j − qn, j )2 + (qn+1, j − qn, j )4 ] dt 2 4 n j=1, 2 γ (1) + ε (qn, 1 − qn, 2)2 , 2 where qn, j is the dimensionless displacement of particle “n”-th in “j”-th chain, c, β , and γ are dimensionless parameters of interaction and ε is a small coefficient of interchain coupling. It is easy to show that a modulation of particle displacements at the right edge of spectrum of the linearized system (un, j = (−1)n qn, j ) leads to the following continuum equations for envelope functions uj :

∂ 2u j ∂ 2u j + + u j + 16β u3j − εγ u3− j = 0, ∂ τ2 ∂ x2 τ = ω t, ω 2 = 4 + εγ . It is convenient to use the complex variables: 1 ∂uj 1 ∂uj √ √ Ψj = + iu j , Ψ j = − iu j . 2 ∂τ 2 ∂τ

(2)

(3)

The equations of motion (2) are converted to the form: i

1 ∂2 ∂ ¯ j ) − 4β (Ψ j − Ψ ¯ j )3 − ε γ (Ψ3− j − Ψ ¯ 3− j ) = 0. Ψj + Ψj + (Ψ j − Ψ ∂τ 2 ∂ x2 2

(4)

Let us assume the parameter of nonlinearity β is equal to zero. Now we obtain two linear chains with linear coupling. Using the multiscale expansion in ε: Ψ j = ε (ψ j + εψ j, 1 + ε 2 ψ j, 2 + . . .),

τ0 = τ , ξ = εx

τ1 = ετ ,

τ2 = ε 2 τ,

(5)

we obtain the following equations of different orders by small parameter ε :

ε 1: i∂τ0 ψ j + ψ j = 0, ψ j = χ j eiτ0 .

(6)

Localized Nonlinear Excitations and Interchain Energy Exchange

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ε2:

γ i∂τ0 ψ j,1 + i∂τ1 ψ j + ψ j − (ψ3− j − ψ¯ 3− j ) = 0, 2 ψ j,1 = χ j,1 eiτ0 , γ i∂τ0 χ j,1 + i∂τ1 χ j − (χ3− j − χ¯ 3− j e−2iτ0 ) = 0. 2 The last equations lead to the following important relationships: γ i∂τ1 χ j − χ3− j = 0, 2 γ χ j,1 = χ¯ 3− j e−2iτ0 . 4

(7)

(8)

Now we can get the solution of Eqs. (8) in the form: 1 γ γ χ1 = √ [X1 cos( τ1 ) − iX2 sin( τ1 )] 2 2 2 1 γ γ χ2 = √ [X2 cos( τ1 ) − iX1 sin( τ1 )], 2 2 2

(9)

ε3: γ 1 i∂τ0 ψ j, 2 + i∂τ1 ψ j, 1 + i∂τ2 ψ j + ∂ξ2 (ψ j − ψ¯ j ) − (ψ3− j, 1 − ψ¯ 3− j, 1) = 0, 2 2 (10) 1 2 −2i ωτ 0) i∂τ0 χ j, 2 + i∂τ1 χ j, 1 + i∂τ2 χ j + ∂ξ (χ j − χ¯ j e 2 γ −2iωτ0 ¯ − (χ3− j, 1 − χ3− j, 1e ) = 0. 2 It is easy to see that after integrating with respect to fast time τ0 using of second relation (8), equations for different chains turn out to be uncoupled. The main point in the analysis of Eqs. (10) is that the unknown functions depend on “intermediate” time τ1 . The adequate procedure to remove this dependence is the averaging over time τ1 . After simple algebraic manipulations we obtain: γ2 1 i∂τ2 X1 + ∂ξ2 X1 − X1 = 0, 2 8 γ2 1 i∂τ2 X2 + ∂ξ2 X2 − X2 = 0, 2 8

(11)

where X1 and X2 are the functions of slow time τ2 . Eqs. (11) have solution in the form of plane wave X j = A j exp(i(kξ − ωτ2 )) (12) with dispersion relation

γ 1 ω = (k2 + ( )2 ). 2 2

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L.I. Manevich, V.V. Smirnov

a

Energy exchange 1st chain

2nd chain

100

100

80

80

60

60

40

40

20

20

t

0

0 50

100 150 200 250 300 350 400

n

50

100 150 200 250 300 350 400

n

b

Fig. 1 (a) “Map” of total energy of linear chains – bright bands correspond to high energy value, dark bands – correspond to low energy, t – time, n – number of particle in the chain; (b) energy of 200-th particles in the different chains versus time. The plane wave was initiated in left chain at the time t = 0 only

It is very important that the structure of Eqs. (11) allows the wave localization on one chain only. This case corresponds to full energy exchange between the chains, if the solution (12) is considered as initial conditions for the first of Eqs. (8) in the “intermediate” time τ1 . It is obvious that the plane waves (12) migrate from one chain to other in accordance with Eqs. (9). Figure 1 shows an example of full exchange between the chains for the initial conditions A1 = 0.10, A2 = 0.0.

Localized Nonlinear Excitations and Interchain Energy Exchange

41

3 Nonlinear Chains with Weak Nonlinearity For parameter of nonlinearity β = 0 in Eqs. (4), we can study the influence of nonlinearity on the process of energy exchange. Using the series like (5) for a small parameter ε, we obtain the weak nonlinearity asymptotics, because the order of coupling terms is equal to ε2 , while the nonlinear terms give a contribution ∼ε3 . Thus, Eqs. (6–9) are the same as for linear and nonlinear systems, but Eqs. (11) resulting from averaging are changed:

γ2 1 3β i∂τ2 X1 + ∂ξ2 X1 − X1 + (3|X1 |2 X1 + 2|X2|2 X1 − X22X¯1 ) = 0, 2 8 8 (13) γ2 1 2 3β 2 2 2¯ (3|X2 | X2 + 2|X1| X2 − X1 X2 ) = 0. i∂τ2 X2 + ∂ξ X2 − X2 + 2 8 8 Equations (13) describe the pair of nonlinear oscillatory chains with nonlinear coupling contrary to the initial system with the linear coupling. It is very interesting that the structure of nonlinear terms is similar to the case of small FPU-system [7, 8]. These equations allow both anharmonic plane wave solution and a solution in the form of localized vibrations (breathers). The plane wave solution has the form: X j (ξ , τ2 ) = A j exp(−i(ωτ2 − kξ ))

(14)

with dispersion relation

γ 1 ω = (k2 + ( )2 ) − 6β (3A2j + A23− j ). (15) 2 2 Like the case of linear chains Eqs. (13) allow for a wave solution, localized on one chain only. This solution leads to the full energy exchange between chains. Figure 2 shows an example of small amplitude anharmonic plane wave in the weakly nonlinear system. Analysis of “phase plane” in the terms of “angle variables” [2, 5] does not show bifurcation both in-phase and anti-phase stationary point. The solution in the form of plane wave has the phase shift which is equal to π /2. Thus, this trajectory corresponds to the limiting phase trajectory (LPT) in the case of two nonlinear oscillators. Let us consider a localized solution of Eqs. (13): X j (ξ , τ2 ) = A j (ξ − vτ2 ) exp[−i(ωτ2 − qξ )], j = 1, 2.

(16)

Here, Aj are real functions. Substitution of this form into the equations of motion give the relation between wave number q and velocity of the wave v: v = −q. The equations for amplitudes Aj can be written as follows: A j + (ω −

q2 γ 2 − )A j + 6β (3A j 3 + A3− j 2 A j ) = 0, 2 8

(17)

42

L.I. Manevich, V.V. Smirnov

a

Energy exchange 1st chain

2nd chain

100

100

80

80

60

60

40

40

20

20

t

0

0 50

100 150 200 250 300 350 400 n

50

100 150 200 250 300 350 400 n

b

Fig. 2 Plane wave with full energy exchange in the weakly nonlinear system. (a) Energy “map”– bright and dark bands correspond to high and low energy values, respectively. The plane was initiated in right chain at the time t = 0 only. (b) The energy profiles of 200-th particles in both chains versus time

where primes denote differentiation with respect to argument. Let us suppose A1 = kA2 . The conditions of compatibility of Eqs. (17) lead to following values of k: k = +1; k = −1; k = 0. The last value has a principal importance, because of the existence of full energy exchange. Thus, the solution of Eqs. (13) describing the localized oscillations takes the following form:

Localized Nonlinear Excitations and Interchain Energy Exchange 500

400

400

300

300

200

200

100

100

t

500

43

0

0 50

100

150

200 n

250

300

350

400

50

100

150

200 n

250

300

350

400

Fig. 3 The energy “map” for moving small amplitude breather in the case of weak nonlinearity

(q2 + (γ /2)2 ) − 2ω exp[−i(ωτ2 − qξ )] 6(3 + κ 2)β ! (q2 + (γ /2)2) − 2ω (ξ − qτ2 − ξ0 )]X1 , × sch[ 2 X2 (ξ , τ2 ) = κ X1 (ξ , τ2 ), κ = 0, ±1. X1 (ξ , τ2 ) =

(18)

It is worth mentioning that the shape of small-amplitude solution is formed in the time scale that is slower, than the characteristic time of energy transfer between different chains. This statement is valid for both linear and nonlinear systems. An example of energy transfer in the case of moving breather is shown in Fig. 3.

4 Chains with Nonlinearity, Compatible with Coupling Let us return to Eqs. (4) for the case of amplitudes providing compatibility √ of nonlinear and coupling terms by parameter ε. If their values reach a magnitude ε , the expansion (5) turns out to be invalid. Therefore we will use the following multi-scale expansion: √ Ψ j = ε (ψ j + εψ j, 1 + ε 2 ψ j, 2 + . . .),

τ0 = τ , τ1 = ετ , √ ξ = ε x,

τ2 = ε 2 τ ,

(19)

that leads to following equations for different orders of small parameter ε:

ε 1/2 : i∂τ0 ψ j + ψ j = 0, ψ j = χ j eiτ0 ,

(20)

44

L.I. Manevich, V.V. Smirnov

ε 3/2 : 1 γ i∂τ0 ψ j,1 + i∂τ1 ψ j + ψ j + ∂ξ2 (ψ j − ψ¯ j ) − (ψ3− j − ψ¯ 3− j ) 2 2 3 − 4β (ψ j − ψ¯ j ) = 0,

ψ j,1 = χ j,1 eiτ0 ,

(21)

γ 1 i∂τ0 χ j,1 + i∂τ1 χ j + ∂ξ2 (χ j − χ¯ j e−2iτ0 ) − (χ3− j − χ¯ 3− j e−2iτ0 ) 2 2 − 4β (χ j eiτ0 − χ¯ j eiτ0 )3 e−iτ0 = 0. Integrating last equations (21) with respect to “fast” time τ0 , we get two coupled equations: γ 1 i∂τ1 χ j + ∂ξ 2 χ j − χ3− j + 12β |χ j |2 χ j = 0. (22) 2 2 First of all, we can see, that there are two symmetric solutions of Eqs. (22). In the class of localized soliton-like solutions they have the form: in-phase solution

χ1 (ξ , τ1 ) =

1 4

2ω + q2 + γ 1 sch( 3β 4

2ω + q2 + γ (ξ + qτ1 ) 6β

× exp(i(ωτ1 − qξ ),

(23)

χ1 (ξ , τ1 ) = χ2 (ξ , τ1 ) and the anti-phase solution

χ1 (ξ , τ1 ) =

1 4

2ω + q2 − γ 1 sch( 3β 4

2ω + q2 − γ (ξ + qτ1 ) 6β

× exp(i(ωτ1 − qξ ), χ1 (ξ , τ1 ) = −χ2 (ξ , τ1 ).

(24)

The Hamiltonian corresponding to Eqs. (22) is

γ 1 h = − (χ1 χ¯ 2 + χ¯ 1 χ2 ) + (|∂ξ χ1 |2 + |∂ξ χ2 |2 ) 2 2 + 6β (|χ1|4 + |χ1|4 ).

(25)

Similarly to symmetric solutions (23–24 ) we can suppose that localized soliton-like solutions of Eqs. (22) can be represented in the form:

χ j = A(ξ )X j (τ1 ),

(26)

Localized Nonlinear Excitations and Interchain Energy Exchange

45

where a space-dependent amplitude A has the same profile for both chains. Thus, integrating Eq. (25) with respect to space variables ξ, we get the “energy” of the system as a function of time variable:

γ 1 H = − N(X1 X¯2 + X¯1 X2 ) + μ N(|X1 |2 + |X2|2 ) 2 2 + 6β ν N 2 (|X1 |4 + |X2|4 ),

(27)

where new parameters are defined by a soliton profile: N=

ν=

A2 d ξ ,

μ=

(∂ξ A)2 d ξ /

A2 d ξ , (28)

A4 d ξ /( A2 d ξ )2 .

In such a case we get an analog of two nonlinear oscillators, described by functions Xj , which were studied recently in detail [6] by one of authors. Both beating with full energy exchange and confinement of initial excitation in the one of chain can be observed when the value of “occupation number” N grows. It was shown, that the process of energy exchange is defined by trajectories, closed to LPT [6] and pertinent to attractive area of one of two stationary points of the system. There are two stationary points of Eqs. (24) at a small “occupation number” N and four ones exist if N is large enough. It is easy to see from analysis of “phase plane” in the terms of “angle variables”, that the following can be introduced [2]: X1 = cos θ eiδ1 , X2 = sin θ eiδ2 . The parameter controlling a structure of phase plane, is κ = 6βν N/γ . Four typical cases are shown in Fig. 4. The structure of phase trajectories and the conditions of LPT existence were described in detail in [6]. If the parameter κ is smaller than 0.5, only two stationary points exist: in-phase (δ = 0, θ = π /4) and anti-phase (δ = π , θ = π /4) ones (see Fig. 4). Closed trajectories near the LPT describe full energy exchange. At κ = 0.5 anti-phase mode becomes unstable one, that leads to separatrix creation (Fig. 4b). So, if we start from the state near the new asymmetric modes, we can not transfer energy effectively from one chain to another one. But a possibility of full exchange near the LPT is well preserved. The total prohibition of energy exchange appears when κ reaches 1. Then the separatrix coincides with LPT and trajectories closed around anti-phase mode are broken (Fig. 4c). So, full confinement of excitation occurs in the one of chain. The main conclusion is that the full energy exchange is possible up to values of parameter κ does not exceeding unity (Fig. 4d). After that only partial exchange can occur near the asymmetric modes. The computer simulation data, an example of which is shown in Fig. 5, demonstrate a confinement of initial excitations in one the chains at κ ∼1.2.

46

L.I. Manevich, V.V. Smirnov 2p

a

b

c

d

q p

0 2p q p

0

p D

0

p D

2p 0

2p

Fig. 4 Transformation of “phase plane” of Eqs. (24) in the terms of angle variables. θ characterizes the amplitude ratio and Δ = δ1 − δ2 – the phase shift. The occupation number N increases from (a) to (d) fragments: (a) κ < 0.5, (b) 0.5 < κ < 1, (c) κ = 1.0, (d) κ > 1.0 (see text) 1st chain

2nd chain 500

400

400

300

300

200

200

100

100

t

500

0

50 100 150 200 250 300 350 400

n

0

50 100 150 200 250 300 350 400

n

Fig. 5 Confinement of breather in first chain. Breather was initiated in the first chain at t = 0. After exchange with the second chain, breather returns to “parent” chain

5 Conclusions Analytical and numerical studies of wandering excitation both in linear and nonlinear chains coupled by weak linear interaction show an existence of two asymptotic limits of energy transfer between different chains. The first of them is characterized

Localized Nonlinear Excitations and Interchain Energy Exchange

47

by quick energy transfer in comparison to processes of excitation formation. In such a case the waves in the different chains exhibit the phase shift which is equal π /2. It means that the respective trajectory is closed to the LPT. In contrast, excitations with large amplitudes can show both full energy exchange near LPT and partial one near stationary points up to full confinement of excitation in one of the chains. Acknowledgement The work was supported by Program of OXN, Russian Academy of Sciences.

References 1. Jensen SM (1982) The nonlinear coherent coupler, IEEE J Quantum Elect 18, 1580–1583. 2. Kosevich AM, Kovalyov AS (1989) Introduction to Nonlinear Physical Mechanics, Naukova Dumka, Kiev (in Russian). 3. Uzunov IM, Muschall R, G¨olles M, Kivshar YS, Malomed BA, Lederer F (1995) Pulse switching in nonlinear fiber directional couplers, Phys Rev E 51, 2527–2537. 4. Khusnutdinova KR (1992) Non-linear waves in a double row particle system, Vestn MGU Math Mech 2, 71–76. 5. Khusnutdinova KR, Pelinovsky DE (2003) On the exchange of energy in coupled Klein-Gordon equations, Wave Motion 38, 1–10. 6. Manevich LI (2007) New approach to beating phenomenon in coupled nonlinear oscillatory chains, Arch Appl Mech 77, 301–312. 7. Manevich LI, Smirnov VV (2007) Discrete breathers and intrinsic localized modes in small FPU systems, Proc APM 293, St-Petersburg. 8. Manevich LI, Smirnov VV (2007) Intrinsic Localized Modes Mobility in Small Fermi-PastaUlam Systems, this issue. 9. Kosevich YuA, Manevich LI, Savin AV (2007) Wandering breathers and self-trapping in weakly coupled nonlinear chains: classical counterpart of macroscopic tunneling quantum dynamics, e-ArXiv: 0705.1957(1).

Dynamic Analysis of the Gantry Crane Used for Transporting BOP Andrzej Urba´s and Stanisław Wojciech

1 Introduction Dynamic analysis of offshore systems mounted on platforms or vessels is especially difficult since it is necessary to consider phenomena connected with a sea waves [1]. The waving causes additional impulse forces in the system which should be taken into account in design process. The suitable description of dynamic behavior of the system allows its using in design of control systems. The algorithm for generating the equations of motion of the gantry crane using homogenous transformations and joints coordinates is presented in the paper [2, 3]. The Lagrange equations of the second order are used to derive the equations of motion.

2 Mathematical Model of the System The gantry crane used to transport the load called BOP (Blowout Preventer) is analysed. The system is considered as a system of two rigid bodies with 12 DOF (Fig. 1). Frame {F} is treated as a rigid body (6 DOF) connected with platform {D} by means of spring-damping elements (sde). The load is also treated as a rigid body with 6 DOF with respect to platform {D}. The load is connected with the frame by means of two flexible ropes. The motion of the load is limited by guides which are modelled as spring-damping elements with backlash. It is assumed that the motion of the platform is known. Consequently the position of coordinate system {D} with respect to global coordinate system {G} is known.

A. Urba´s and S. Wojciech Department of Mechanics and Computer Science, University of Bielsko-Biała, Poland, e-mail: [email protected], [email protected]

J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

49

50

A. Urba´s, S. Wojciech ˆ (F) {F } Z y (F) A2

frame

A1 B2 C (F)

(

x (F),y (F),z (F) C C C

)

j (F) ˆ (F) X ˆ (L) Y

y (L)

C (L) (x (L),y (L) ,z (L)) C

ˆ (F) Y q (F)

C (F ) ˆ (L) {L} Z

guides C

B1

C(L) {D} ˆ (D) Z

C

load (BOP)

q (L) j (L) ˆ (L) X

y (D)

ˆ (D) Y moving platform

q (D) {G} ˆ Z j (D) ˆ (D) X ˆ Y

ˆ X a heading

Fig. 1 Model of the gantry crane

The position of coordinate system {D} with respect to {G} is defined by: ⎫ x(D) = x(D) (t) ⎬ y(D) = y(D) (t) the coordinates of the origin of coordinate system in, ⎭ z(D) = z(D) (t)

(1)

⎫ ψ (D) = ψ (D) (t) ⎬ θ (D) = θ (D) (t) the Euler angles ZYX [1] which describe ⎭ ϕ (D) = ϕ (D) (t) any possible orientation of frame {D} with respect to {G}. The transformation matrix from coordinate system {D} to {G} has the following form: ⎡

⎤ cψ (D) cθ (D) cψ (D) sθ (D) sϕ (D) − sψ (D) cϕ (D) cψ (D) sθ (D) cϕ (D) + sψ (D) sϕ (D) x(D) ⎢ sψ (D) cθ (D) sψ (D) sθ (D) sϕ (D) + cψ (D) cϕ (D) sψ (D) sθ (D) cϕ (D) − cψ (D) sϕ (D) y(D) ⎥ ⎥, A(D) (t) = ⎢ ⎣ −sθ (D) cθ (D) sϕ (D) cθ (D) cϕ (D) z(D) ⎦ 0 0 0 1

(2)

Dynamic Analysis of the Gantry Crane Used for Transporting BOP

51

where cψ (D) = cos ψ (D) , cθ (D) = cos θ (D) , cϕ (D) = cos ϕ (D) , sψ (D) = sin ψ (D) , sθ (D) = sin θ (D) , sϕ (D) = sin ϕ (D) . In the specific case when there is no motion of platform {D}, transformation matrix A(D) is the matrix with constant coefficients. It is assumed that the motion frame {F} and load {L} with respect to platform {D} is described by independent parameters which are components of the vectors: T

q(F) = x(F) y(F) z(F) ψ (F) θ (F) ϕ (F) , T

q(L) = x(L) y(L) z(L) ψ (L) θ (L) ϕ (L) .

(3)

Frame {F} is connected with platform {D} by means of spring-damping elements. Since angles ϕ (F) , θ (F) , ψ (F) , ϕ (L) , θ (L) , ψ (L) are assumed to be small the transformation matrices from coordinate systems {F} and {L} to {D} can be presented in the form: ⎡ ⎤ 1 −ψ (F) θ (F) x(F) ⎢ (F) 1 −ϕ (F) y(F) ⎥ ⎥, ˜ (F) = ⎢ ψ B (4) ⎣ −θ (F) ϕ (F) 1 z(F) ⎦ 0 0 0 1 ⎡ ⎤ (L) (L) (L) 1 −ψ θ x ⎢ ψ (L) 1 −ϕ (L) y(L) ⎥ ⎥. B˜ (L) = ⎢ ⎣ −θ (L) ϕ (L) 1 z(L) ⎦ 0

0

0

1

The transformation of coordinates from local coordinate systems {F} and {L} to {D} can be written in the form: (D)

(F) = B˜ (F) r˜ P ,

(D) rP

(L) B˜ (L) r˜ P ,

rP

(F)

=

(5)

(L)

where r˜ P , r˜ P are the position vectors of point P in local coordinate systems {F} and {L}. The transformation matrices from local coordinates systems {F} and {L} to global coordinate system {G} are defined as follows: B(F) = A(D) B˜ (F) (q(F) ), B(L) = A(D) B˜ (L) (q(L) ).

(6)

The equations of motion are formulated using the Lagrange equations of the second order: d ∂E ∂E ∂V ∂D − + + = Qk , k = 1, . . . , 12, (7) dt ∂ q˙k ∂ qk ∂ qk ∂ q˙k

52

A. Urba´s, S. Wojciech

where: E is the kinetic energy of the system, V is the potential energy of gravity forces, D is a form describing the dissipation of energy of the system, Qk are non-potential generalised forces, qk , q˙k are generalised coordinates and velocities, respectively.

2.1 Kinetic Energy of the Systems The kinetic energy of the system can be written in the form: E = E (F) + E (L) , (8) , , + + T ˙ (L) H(L) B˙ (L)T are kinetic energy where E (F) = 12 tr B˙ (F) H(F) B˙ (F) , E (L) = 12 tr B of the frame and the load, respectively, H(F) , H(L) are pseudo inertia matrices [3].

2.2 Potential Energy of Gravity Forces The potential energy of gravity forces can be described as follows: (F)

Vg = Vg (F)

(F)

(L)

(L)

+ Vg ,

(9) (L)

where Vg = m(F) g θ3 B(F) rC(F) , Vg = m(L) g θ3 B(L) rC(L) are the potential energies of the frame and the load, respectively: m(F) , m(L) are masses of the frame and

the load, g is the acceleration of gravity, θ3 = 0 0 1 0 , (F)

(L)

r (F ) , r (L) are the vectors of coordinates of the centre of mass of the frame and C C the load in local coordinate systems.

2.3 Energy of Deformation and Dissipation Energy of Sde It is assumed that the load can be in contact with the guides only along lines A(k) , B(k) , where k = 1, 2, 3, 4 (Fig. 2). The guides are modelled as spring-damping ˆ (D) elements (sde) with backlash (E (k, p) ), which limit the motion of the load in X (k) ˆ (D) directions. The number of sdes at each edge is denoted as ne . and Y Sde E (k, p) is described by the following parameters: ˆ and Y ˆ directions, cxE (k, p) , cyE (k, p) – coefficients of stiffness in X

Dynamic Analysis of the Gantry Crane Used for Transporting BOP

53

A (k)

{D} Zˆ (D)

b y(k,p) E

ΔEy (k,p)

cEy (k,p) E (k,p)

ΔEx (k,p)

x c x(k,p) bE (k,p) E

B (k) Yˆ (D) Xˆ (D)

Fig. 2 Spring-damping elements with backlash

ˆ and Y ˆ directions, bxE (k, p) , byE (k, p) – coefficients of damping in X ˆ and Y ˆ directions. Δx (k, p) , Δy (k, p) – backlash in X E

E

Generalised forces arising from sdes can be written as follows: (k)

(F) Qe

=

4 ne

(F)T

(D)

∑ ∑ UE (k, p) FE (k, p) ,

(10)

k=1 p=1

⎡

where

⎢ (F) UE (k, p) = ⎢ ⎣0 1 00 ⎡ 10 ⎢ (L) UE (k, p) = ⎢ 0 ⎣ 1 00 (F)

(F)

(F)

(L)

(F)

(F)

(F) E (k, p)

0

1 0 0 −yE (k, p) zE (k, p)

(L)

0 x 1

0 (L)

(F)

−xE (k, p) (L)

0 −yE (k, p) zE (k, p) (L)

0 xE (k, p) 1

0

0 (L)

⎤ 0 (F) E (k, p) (F) yE (k, p)

−z

⎥ ⎥, ⎦ ⎤

0

(L) ⎥ −zE (k, p) ⎥ ⎦, (L)

−xE (k, p) yE (k, p)

(L)

xE (k, p) , yE (k, p) , zE (k, p) , xE (k, p) , yE (k, p) , zE (k, p) are coordinates of sde E (k, p) in the local coordinate systems {F} and {L}, .T (D) x) (D, y) F (k, p) = F (D, is a force in sde E (k, p) , (k, p) F (k, p) 0 E (D, x)

E (D, x)

E

(D, x)

FE (k, p) = FS,E (k, p) + FT,E (k, p) ,

(D, y)

(D, y)

(D, y)

FE (k, p) = FS,E (k, p) + FT,E (k, p) are stiffness and dampˆ and Y ˆ directions. ing forces seting in each sde E (k, p) in X

54

A. Urba´s, S. Wojciech

2.4 Energy of Deformation and Dissipation Energy of the Ropes The load is connected with the frame by means of two flexible ropes (p = 1, 2). Energy of spring deformation and dissipation of energy of the ropes can be expressed as: (p)

Vs (p)

(p)

where cr , br

1 (p) (p) (p) 2 ΔlA p B p , = c r δr 2

1 (p) (p) ˙ (p) 2 (p) ΔlA p B p , D s = b r δr 2

(11)

are coefficients of stiffness and damping of rope p, (p)

ΔlA p B p is the elongation of rope p, 0 when ΔlA p B p ≤ 0, (p) δr = 1 when ΔlA p B p > 0.

2.5 Reaction Forces of the Support Let P(k) (k = 1, 2, 3, 4) denote the connection points of the frame and the platform. Sdes are placed in these points (Fig. 3). It is assumed that stiffness and damping ˆ (D) , Y ˆ (D) and coefficients have large values and limit the motion of the frame in X (D) ˆ Z directions. Generalised forces arising from those elements are as follows: (F)

Qp =

4

(F)T

(F)

∑ UP(k) FP(k) ,

(12)

k=1

ˆ (D) Z

{D}

y (D) (k)

ˆ (D) Y

P (k) (x (F ) , y (F) , z (F ) )

P

P (k)

P (k)

F (F,(k)y) P

F (F,(k)z) P

x (D) P (k)

ˆ (D) X

Fig. 3 Flexible connecting the frame and the platform

F (F,(k)x) P

P (k)

Dynamic Analysis of the Gantry Crane Used for Transporting BOP SF

y

D

55

[T]

10 t [s] 0 0

5

10

15

20

25

-10 -20 survival operation

-30 -40

a

-50

SF

yD

[T]

60 50 40 30

survival operation

20 10 t [s] 0

b

0

5

10

15

20

25

-10

Fig. 4 The influence of working conditions on sum of forces ∑ FyD : (a) the edge 1 and 2, (b) the edge 3 and 4

(F)

where UP(k) is defined similarly to (10), .T (F) x) (F, y) (F, z) FP(k) = F (F, , F F (k) (k) (k) P P P (F) (F, x) (F, x) = F (k) + F (k) stiffness and P(k) S, P T, P (F, y) (F, z) FP(k) and FP(k) are defined in (10).

F

ˆ direction, damping forces in P(k) , in X

The equations of the system can be written in the following form: ˙ Aq¨ = f(t, q, q),

(13)

where A = A(t, q)is the mass matrix. The model of dynamics of the gantry crane considered is described by 12 ordinary differential equations of the second order. In order to integrate the equations of motion the Runge-Kutta method with constant step-size is used.

56

A. Urba´s, S. Wojciech Fl [T] 280 survival operation

279 278 277 276 275 274 273 272 271

t [s]

a

270

0

5

10

15

20

25

Fl [T] 286 284 282 280 278 276 274 272

survival operation

270 268 266 264

b

t [s] 0

5

10

15

20

25

Fig. 5 The influence of conditions of work on forces in the ropes: (a) the rope 1, (b) the rope 2

3 Numerical Calculations For the mathematical model of the gantry crane the computer program has been developed. The following input data are taken from the technical documentation [4]: the mass of the load and the frame: m(L) = 550 000 kg, m(F) = 110 000 kg, dimensions of the load 4.8 × 5.5 × 20.3 m. Stiffness coefficients of sdes in contact points between the guides and the load were obtained using the simple model of the guides prepared by means of the finite element method package. It has been assumed that damping coefficients are proportional to stiffness coefficients [5]. The large values of the stiffness and damping coefficients of sdes in points P(k) were assumed. These large values caused some numerical problems when integrating the equations of motion of the system. For that reasons we applied the Runge-Kutta method of fourth order with time step equal to 10−4 s. Additionally the damping coefficients were taken of the order which compensates potential problems due to large stiffness.

Dynamic Analysis of the Gantry Crane Used for Transporting BOP SFyD

57

10 0

0

5

25

20

15

10

-10 -20 -30

quasi-statics dynamics

-40 -50 -60 -70

a SFyD

70 60 50 quasi-statics dynamics

40 30 20 10 0

b

0

5

10

15

20

25

-10

Fig. 6 Comparison of results of quasi-static and dynamic analysis - forces ∑ FyD (a) the edge 1 and 2, (b) the edge 3 and 4

Figures 4–7 present calculation results in the case when the motion of the plat(D) form is described in the following way: x(D) = y(D) = ψ (D) = 0, z(D) = z0 + (D)

a3 sin(2π t/T ), θ (D) = a5 sin(2π t/T), ϕ (D) = a6 sin(2π t/T ), where z0 = 36 m, T = 10 s, and a3 , a5 , a6 are amplitudes depending on heading and weather conditions (Table 1). Figure 4 presents a comparison of forces acting between the guides and the load when different conditions of work are considered. The results have been obtained for heading angle equal to 45◦ . The influence of conditions of work on forces in ropes can be observed in Fig. 5. In order to express how dynamic state of the system considered influences the reaction forces, a quasi-static problem has been solved. For the quasi-static analysis it is assumed that in Eq. (13), q¨ = 0 and q˙ = 0. Also the backlash is omitted in that analysis. The resulting set of nonlinear algebraic equations was solved using iterative Newton’s method. Figure 6 presents a comparison of results for quasi-static and dynamic analysis. Forces acting between the guides and the load are presented. It is assumed the heading angle is equal to 90◦ and survival conditions of work.

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A. Urba´s, S. Wojciech SF

yD

[T]

a = 90, a3 = 0, a5 = 0, a6 = 0.0138 (roll)

10 0

t [s] 0

5

10

15

20

25

-10 -20 -30

p) b(k, = by(k, p) = 0 x

-40

p),b(k, p) - 50% b(k, x y p),b(k, p) - 100% (m = 0.1) b(k, x y

-50

p),b(k, p) - 150% b(k, x y

-60

k = 1, 2

p = 1, 2, 3, 4

-70

a

-80

SF

yD

[T]

a = 90, a3 = 0, a5 = 0, a6 = 0.0138 (roll)

80 70 60

p) = b(k, p) = 0 b(k, x y p),b(k, p) - 50% b(k, x y

50

p),b(k, p) - 100% (m = 0.1) b(k, x y

40

p),b(k, p) - 150% b(k, y x

30 k = 3, 4

p = 1, 2, 3, 4

20 10 t [s] 0

b

0

5

10

15

20

25

-10

Fig. 7 The influence coefficients of damping of sdes on sum of forces ∑ FyD : (a) edges 1 and 2, (b) edges 3 and 4 Table 1 Conditions of work for BOP Heading

Operational

Survival

Heave z(D)

Pitch θ (D)

Roll ϕ (D)

Heave z(D)

Pitch θ (D)

Roll ϕ (D)

α

a3

a5

a6

a3

a5

a6

0◦ 45◦ 90◦

0.1343 0.1115 0.1140

0.0023 0.0008 0

0 0.0023 0.0045

0.4458 0.3521 0.3724

0.0061 0.0023 0

0 0.0077 0.0138

The influence of coefficients of damping of sdes on forces acting between the guides and the load is shown in Fig. 7. The motion of the platform is described only by function ϕ (D) in survival conditions of work.

Dynamic Analysis of the Gantry Crane Used for Transporting BOP

59

4 Conclusions The numerical simulations presented prove that dynamic analysis of the BOP is important in the design process. The conditions of work of such an offshore system cause that impulse forces arise which exceed forces obtained in the static analysis. Acknowledgement The investigation presented in the paper has been partially supported by grant 4 T07A049 2B founded by Polish Committee of Science.

References 1. Maczyski A (2005) Positioning and Stabilization of Loads of Luffing Jib Cranes, TechnoHumanistic Academy Press, Theses 14, Bielsko-Biaa (in Polish). 2. Craig JJ (1995) Introduction to Robotics. Mechanics and Control, WNT, Warsaw (in Polish). 3. Wittbrodt E, Adamiec-W´ojcik I, Wojciech S (2006) Dynamics of Flexible Multibody Systems. Rigid Finite Element Method, Springer, Berlin/Heidelberg/New York. 4. Technical documentation for BOP (2007) PROTEA, Gda´nsk-Olesno. 5. Uhl T (1997) Computer-Aided Identification of Models of Mechanical Structures, WNT Warsaw (in Polish).

Motion of a Chain of Three Point Masses on a Rough Plane Under Kinematical Constraints Klaus Zimmermann, Igor Zeidis, and Mikhail Pivovarov

1 Introduction A series of papers have analyzed the rectilinear motion on a rough plane of bodies (mass points) connected by viscoelastic elements (springs and dampers) in the case when the force of normal pressure is not changed. The system is moved by forces that changed harmonically and acting between the bodies. The asymmetry of the friction force, required for a motion in a given direction, is provided by the dependence of the friction coefficient on the sign of the velocity of the bodies which make up the system. This effect can be achieved if the contact surfaces of the bodies are equipped with a special form of scales (needle-shaped plate with a required orientation of scales (needles)). In [1–5], the dynamics of a system of two bodies joined by an elastic element with a linear characteristic were considered. The motion is excited by a harmonic force acting between the bodies. In [3], a magnetizable polymer was employed as an elastic element and the motion was excited by a magnetic field. In the case of small friction, the analytical expression for the average velocity of steady motion of the whole system was found and it is shown, that the motion with this velocity is stable. A similar investigation for a system of two bodies joined by a spring with a nonlinear (cubic) characteristic was shown in [5]. Algebraic equations were obtained for average velocities of the steady motion. It was shown that there exist up to three different motion modes, one or two of them are stable. The limiting case of asymmetric friction is the kinematic condition that admits the motion only in one direction. This condition was considered in [6] in connection with a computer model of an earthworm. In [7], a numerical solution of the motion equations of a chain of bodies joined by viscoelastic elements is presented for the case when each body can move only in one K. Zimmermann, I. Zeidis, and M. Pivovarov Technische Universitaet Ilmenau PF 100565, 98684 Ilmenau, Germany, e-mail: klaus.zimmermann @tu-ilmenau.de, [email protected], [email protected]

J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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direction. In [8] the rectilinear motion of a vibration-driven system on a horizontal rough plane consisting of a carrying body, which interacts with the plane directly, and of internal masses that perform harmonic oscillations relative to the carrying body, is considered. The vertical and horizontal oscillations of the internal masses have the same frequency, but they are shifted in phase. It is shown that by controlling the phase shift of the horizontal and vertical oscillations, it is possible to change the velocity of the steady motion of the carrying body, and it is not necessary to use scales in order to provide friction asymmetry. In [9] the motion of two mass points connected by a linear spring, when the coefficient of friction does not depend on the direction of motion, is discussed. Due to the change of the normal force in dependence on time asymmetry of friction is present. The change of normal force is realized by the rotation of two unbalanced rotors with various angular velocities. In [10], the rectilinear motion of a body with a movable internal mass moving along a straight line parallel to the line of the body motion on a rough plane was investigated. A periodic control mode was constructed for the relative motion of the internal mass for which the main body moves with a periodically changing velocity passing the same distance in a given direction. It is supposed that, at the beginning and the end of each period, the velocity of the main body is zero. The internal mass can move within fixed limits. The control modes relative to the velocity and acceleration of the internal mass were considered. The optimal parameters of both modes which lead to a maximum of the average velocity of motion of the main body for a period were found. In the present paper we consider the motion of a straight chain of three equal mass points interconnected with kinematical constraints. The ground contact can be described by dry (discontinuous) or viscous (continuous) friction. The controls are assumed in the form of periodic functions with zero average, shifted on a phase one concerning each other. Thus, there is a travelling wave along the chain of mass points. It is shown that, using special control algorithms motion is possible by isotropic coefficient of friction and by constant normal force. In the case of non-isotropic friction motion is possible in the direction of the greater friction.

2 Equations of Motion We consider the motion of a system of three mass points with the coordinates xi (i = 1, 2, 3) and with the masses m, connected by kinematical constraints along an axis OX(Fig. 1). The motion of the system is excited by the kinematical constraints setting the distances L1 (t) > 0 and L2 (t) > 0 between mass points L1 (t) = L0 + a1 (t) , where a1 (0) = a2 (0) = 0.

L2 (t) = L0 + a2 (t) ,

(1)

Motion of a Chain of Three Point Masses on a Rough Plane

63

Fig. 1 The schematic view of the system of three mass points

Let us consider L1 (t) and L2 (t) (hence as well a1 (t) and a2 (t)) as periodic functions with period T . The kind of functions a1 (t) and a2 (t) will be considered below. There is the force of friction F (Vi ) acting on each mass point from the surface, directed against motion and depending on the velocity Vi = x˙i (i = 1, 2, 3). The law of friction F (Vi ) will be discussed later. The velocity of the center of masses of system can be represented as V=

1 (x˙1 + x˙2 + x˙3) . 3

The equation of the motion of the center of mass is as follows 3mV˙ = F (x˙1 ) + F (x˙2 ) + F (x˙3 ) ,

(2)

x2 (t) − x1 (t) = L1 (t) , x3 (t) − x2 (t) = L2 (t) .

(3)

where

By substituting the expressions (1) and (3) in (2) the equation of the motion (2) takes the form 1 1 2 1 1 2 3mV˙ = F V − a˙1 − a˙2 + F V + a˙1 − a˙2 + F V + a˙1 + a˙2 . (4) 3 3 3 3 3 3 We assume that in the initial moment t = 0 the velocity of the center of mass V (0) = 0. Let us introduce dimensionless variables in according to the following formulas (the asterisk ∗ is a symbol of dimensional variables): T xi = x∗i / L (i = 1, 2, 3) , V = V ∗ , t = t ∗ / T, L ai = a∗i / L, Li = L∗i / L, (i = 1, 2) , F (V ) =

∗

∗

(5)

∗

F (V ) F (V · L / T ) = . Fs Fs

In the above L is the characteristic linear dimension (for example the greatest value a1 (t) or a2 (t) in period T ), Fs is the characteristic value of the friction force.

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Hereafter we use dimensionless variables. Introducing the dimensionless variables in Eq. (4) and denoting u1 (t) = a˙1 (t), u2 (t) = a˙2 (t), we rewrite Eq. (4) in dimensionless variables (the old symbols are hold) dV ε 2 1 1 1 1 2 = F V − u1 − u2 + F V + u1 − u2 + F V + u1 + u2 , dt 3 3 3 3 3 3 3 (6) 2 where ε = Fms TL . Let us notice, that since a1 (t) and a2 (t) are periodic functions with period T , therefore u1 (t) = a˙1 (t) and u2 (t) = a˙2 (t) are also periodic functions with period T and have zero average value. Further, we assume everywhere that ε 1. The smallness of the parameter ε shows that the value of the friction force Fs is small compared to the amplitude of the “driving” force mL / T 2 . Equation (6) has a so called “standard form” [11]. Averaging the right side of the Eq. (6) relative to the variable t in the period 1 yields

ε dV = G(V ), dt 3

(7)

where G (V ) =

1

2 1 1 1 1 2 F V − u1 − u2 + F V + u1 − u2 + F V + u1 + u2 d t. 3 3 3 3 3 3

0

Now it is necessary to define the functions u1 (t), u2 (t) and the law of friction.

3 Smooth Control Let us consider the functions a1 (t) and a2 (t) composed from the parabolas and shown in Fig. 2, and accordingly marked as a solid and as a dashed curve. These functions have continuous derivatives u1 (t) and u2 (t), and they are shown in Fig. 3. Applied controls have the form ⎧ 0, 0 ≤ t ≤ 1 / 3, ⎪ ⎪ ⎪ ⎨ 2(3t − 1), 1 / 3 < t ≤ 1 / 2, u1 (t) = ⎪ −2(3t − 2), 1 / 2 < t ≤ 5 / 6, ⎪ ⎪ ⎩ 6(t − 1), 5 / 6 < t ≤ 1. (8) ⎧ 6t, 0 ≤ t ≤ 1 6, / ⎪ ⎪ ⎪ ⎨−2(3t − 1), 1 6 < t ≤ 1 2, / / u2 (t) = ⎪ 2(3t − 2), 1 2 < t ≤ 2 / / 3, ⎪ ⎪ ⎩ 0, 2 / 3 < t ≤ 1.

Motion of a Chain of Three Point Masses on a Rough Plane

65

a1, a2 0.2

0.15

0.1

0.05

0

t 0

0.5

1

Fig. 2 The function a1 (t)(solid) and a2 (t)(dashed) u1, u2 1

0.5

0 -0.5 -1

t 0

0.5

1

Fig. 3 The function u1 (t) (solid), u2 (t) (dashed)

They are equal to zero on an interval of length 1 / 3 and are shifted on time for this magnitude one relatively to another.

4 Dry (Discontinuous) Friction We assume that the Coulomb dry friction acts on the mass point i (i = 1, 2, 3). The dimension force of dry friction F ∗ (V ) satisfies the Coulomb law ⎧ ⎪ if V < 0 ⎨F− = k− N, F ∗ (V ) = F0 , if V = 0, ⎪ ⎩ −F+ = −k+ N, if V > 0

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where N is the force of normal pressure (in this case N = m g, where g is the free fall acceleration), k− and k+ are the coefficients of dry friction at the motion in a negative and in a positive direction respectively, F− ≤ F+ (k− ≤ k+ ). Let us denote Fa the resultant of all forces applied to the mass point (except the force of dry friction). Now we have ⎧ ⎪ ⎨

F− , i f V = 0 F0 = −Fa , i f V = 0 ⎪ ⎩ −F+ , i f V = 0

and and and

Fa < −F− − F− ≤ Fa ≤ F+ . Fa > F+

The expression for dimensionless friction force takes the form (i = 1, 2, 3) ⎧ ⎪ ⎨

1, x˙i < 0 F (x˙i ) = μ0 , x˙i = 0. ⎪ ⎩ −μ , x˙i > 0

(9)

In the above the value Fs in formulas (5) is Fs = F− (F− is the magnitude of the friction force at the motion in a negative direction), μ = F+ / F− = k+ / k− ≥ 0, μ0 ∈ [−μ , 1] and the expression for μ0 (x˙i = 0) is ⎧ ⎪ ⎨

1, i f Fa < −1 μ0 = −Fa , i f − 1 ≤ Fa ≤ μ . ⎪ ⎩ −μ , i f Fa > μ

(10)

In what follows we prove the so far obtained result assuming that x˙i (t) is a piecewise continuous function of time. This assumption is quite sufficient for simulating a feasible motion. The first and third conditions (10) are satisfied in case x˙i (t) = 0 and x¨i (t) = 0. This conditions hold simultaneously only at isolated points, which does not effect the system motion. The second condition (10) is connected with sticking (“stick-slip” motions). This effect is characteristic for systems with dry friction. Let us notice that for the given control the velocity of each mass point could not be equal to zero on a finite time interval. Hence, the “stick-slip” effect is absent. After substituting the expression (8) and (9) in the Eq. (7) we obtain ⎧ ⎪ ⎪ ⎪ ⎨

3, V ≤ −2 / 3, dV ε 2 − μ − 3V (1 + μ )/2, −2 / 3 < V ≤ 0, = dt 3⎪ 2 − μ − 6V (1 + μ ), 0 < V ≤ 1 / 3, ⎪ ⎪ ⎩ −3μ , V > 1 / 3. We consider the solution of the Eq. (11) with the initial condition V (0) = 0.

(11)

Motion of a Chain of Three Point Masses on a Rough Plane

67

If μ = 2 (friction in the positive direction is twice more than friction in a negative direction) the system remains in rest. If μ < 2 the chain moves to the right with the velocity V=

. 2−μ · 1 − e−2ε (1+μ )t , 6 (1 + μ )

and tends to a stationary value Vs =

2−μ . 6 (1 + μ )

In case of isotropic friction one gets Vs = 1 / 12. Thus, under this control algorithms, motion is possible in the case of isotropic friction and in the case of non-isotropic friction in the direction of the greater friction. If μ > 2 the chain moves to the left with the velocity V=

. ε 2 (2 − μ ) · 1 − e− 2 (1+μ )t , 3 (1 + μ )

and tends to a stationary value Vs =

2 (2 − μ ) . 3 (1 + μ )

In Fig. 4 the results of the numerical integration of the exact and averaged equations in the case of symmetric friction ( μ = 1) and for parameter ε = 0.3 are shown.

V 0.1 0.08 0.06 0.04 0.02 0

0

5

t 10

Fig. 4 Solutions of the exact and the averaged equations for dry friction

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5 Viscosity (Continuous) Friction In this section we assume that the force of viscous friction is a power function of velocity. For the friction law in dimension we assume F(V ) = ν |V |α sgnV . The expression for dimensionless friction force follows F (V ) = |V |α sgnV,

(12)

where ν is the coefficient of viscous friction, and the value Fs in formulas of α transition to dimensionless variables (5) is Fs = ν VL α , α > 0. After substitution expressions (8) and (12) into Eq. (7), one obtains . dV ε 1+α 1+α 1+α = 4 (1 − 3V ) , − (3V + 2) − 3 |3V | dt 2 (1 + α ) 31+α − 2 / 3 < V < 1 / 3. If V ≤ −2 / 3, then dV / dt > 0 and if V ≥ 1 / 3, then dV / dt < 0. Hence the stationary solution is only possible on an interval −2 / 3 < V < 1 / 3. At linear viscous friction (α = 1) the chain on the average remains in rest. For the other α stationary velocity V = 0 can be found from the equation G1 (V ) = 4 (1 − 3V )1+α − (3V + 2)1+α − 3 |3V |1+α = 0.

(13)

Let us investigate roots of Eq. (13). Observe that G1 (−2 / 3) = 4 · 31+α − 3 · 21+α > 0, G1 (0) = 4 − 21+α , G1 (1 / 3) = −3

1+α

(14) − 3 < 0.

From expression (14) follows that G1 (0) < 0 for α > 1 and G1 (0) > 0 for 0 < α < 1. The expression for the first derivative of the function G1 (V ) for V = 0 has the form

dG1 = − (1 + α ) 4 (1 − 3V )α + (3V + 2)α + 3 |3V |α sgnV . dV On the interval 0 < V < 1 / 3 the derivative of the function G1 (V ) is negative and this means that this function monotonically decreases on this interval. Thus the function G1 (V ) has one positive roots for 0 < α < 1 on the interval 0 < V < 1 / 3. For α > 1 the value G1 (0) is negative and the function G1 (V ) has at the ends of the interval values of identical signs. Thus the function G1 (V ) has no positive roots for α > 1 on the interval 0 < V < 1 / 3. For α > 1 at the ends of the interval −2 / 3 < V < 0 the function G1 (V ) has values of different signs. This implies that the function G1 (V ) has at least one negative root for α > 1 on the interval −2 / 3 < V < 0. In Fig. 5 the results of the numerical integration of the exact and the averaged equations for the case of viscous friction with α = 2 and ε = 0.3 are shown.

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69

V 0

-0.02

-0.04

-0.06

t 0

10

20

30

Fig. 5 Solutions of the exact and the averaged equations for viscous friction

Fig. 6 The prototype of the system of three mass points

6 Conclusions It is shown that using periodical control algorithms, motion is possible in the case of isotropic friction and in the case of non-isotropic friction in the direction of the greater friction. Without a shift of the phases in the control law and with a linear friction model, the locomotion is impossible. In the case of small friction we derived a condition for the locomotion of the center of the mass with the help of an average method. In the case of smooth control we received explicit expressions for the average velocity of the motion of the center of mass. Comparisons of these analytical expressions to numerical results are carried out. A prototype of this system was created (Fig. 6). Acknowledgement This paper is supported by Deutsche Forschungsgemeinschaft (DFG, ZI 540/6-1).

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References 1. Zimmermann K, Zeidis I, Steigenberger J (2002) Mathematical model of worm-like motion systems with finite and infinite degree of freedom, Theory and Practice of Robots and Manipulators, Proceedings of 14-th CISM-IFToMM Symposium. Springer, Wien, NY, 507–516. 2. Zimmermann K, Zeidis I, Steigenberger J, Pivovarov M (2004) An approach to wormlike motion, 21st International Congress of Theoretical and Applied Mechanics, Book of Abstracts. Warsaw, Poland, 371. 3. Zimmermann K, Zeidis I, Naletova VA, Turkov VA (2004) Modelling of worm-like motion systems with magneto-elastic elements, Phys. Status Solidi (c) 1, 3706–3709. 4. Zimmermann K, Zeidis I (2007) Worm-like locomotion as a problem of nonlinear dynamics, Journal of Theoretical and Applied Mechanics, 45, 179–187. 5. Zimmermann K, Zeidis I, Pivovarov M, Abaza K (2007) Forced nonlinear oscillator with nonsymmetric dry friction, Arch. Appl. Mech. 77, 353–362. 6. Miller G (1988) The motion dynamics of snakes and worms, Comput. Graph. 22, 169–173. 7. Steigenberger J (1999) On a Class of Biomorphic Motion Systems. Faculty of Mathematics and Natural Sciences, Technische Universitaet Ilmenau, Germany, preprint 12. 8. Bolotnik NN, Zeidis I, Zimmermann K, Yatsun SF (2006) Dynamics of controlled motion of vibration-driven systems, J. Comput. Sys. Sci. Int. 45, 831–840. 9. Bolotnik NN, Pivovarov M, Zeidis I, Zimmermann K, Yatsun SF (2007) Motion of vibrationdriven mechanical systems along a straight line, 14th International Workshop on Dynamics & Control, Moscow-Zvenigorod, Russia, 19. 10. Chernousko FL (2005) On a motion of a body containing a movable internal mass, Dokl. Akad. Nauk 405, 1–5. 11. Bogolyubov NN, Mitropolskii YuA (1961) Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach, New York.

Compensation of Geometric Errors in a PKM Machine Tool Christian Rudolf and J¨org Wauer

1 Introduction In machine tools of parallel structure with two or three translatory degrees of freedom the rotatory degree of freedom is kinematically locked. Yet due to geometric faults, for example assembly errors or different geometries due to production tolerances, such machine tools exhibit an additional rotational behavior. Stresses within the structure occur leading to deflections of the tool center point, and thus, reducing the quality of the workpiece. For compensating these errors an adaptronic strut which can be implemented within such a machine tool has been developed. The strut comprises a piezoceramic sensor-actuator unit for controlled correction of those static and quasi-static deflections. Piezoceramic elements were chosen due to their high positioning accuracy and the small installation space required. The functional principle of a scale with a vibrating string is used for measuring the external load. A control concept for the adaptronic strut is introduced. Finally, after implementing the strut in a model of an exemplary machine tool the compensation of influences due to specified geometric errors is examined. Figure 1 shows a machine tool with parallel kinematics of three translatory degrees of freedom. Due to geometric errors, such as assembly errors or differing geometries due to production tolerances, stresses within the structure occur resulting in deflections of the tool center point (TCP) of the machine tool. Thus, the quality of the workpiece is reduced. An adaptronic strut as shown in Fig. 2 has been developed for compensating such errors. The strut, similar in shape to conventional struts in machine tools, is cut in two halves and a piezoelectric sensor-actuator unit is implemented in-between, giving the strut an additional degree of freedom. The geometric deflections in focus of this contribution are mostly static or quasistatic, and thus, only inducing static or quasi-static signals on the piezoelectric C. Rudolf and J. Wauer Universit¨at Karlsruhe (TH), Institut f¨ur Technische Mechanik Kaiserstraße 10/Wilhelm-NusseltWeg 4, 76131 Karlsruhe, Germany e-mail: [email protected], [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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struts pillars

headstock

Fig. 1 Parallel kinematics machine tool with three translational degrees of freedom [1]

upper half piezoelectric transducer

lower half

Fig. 2 Adaptronic strut [2]

sensor element. However, due to the internal leakage resistance of piezoceramic materials such signals are not measurable [3, 4]. By adapting the functional principle of a scale with a vibrating string a work-around for this problem was found. A string, which was mounted along the strut, as can be seen in Fig. 2, is excited by a solenoid. A dynamic signal is induced on the strut and onto the piezoelectric sensor. This signal can easily be acquired and, using frequency counters or phase-locked loops, its frequency course can be determined [5]. The equation f0 =

1 2lS

T Aρ

(1)

describes the relation between the eigenfrequency f0 of the string and pre-stress T on the string, with length lS, cross-sectional area A and density ρ of the string. Thus, by measuring the frequency the external load on the strut can be determined. Further information about this functional principle can be found in [2, 5, 6].

Compensation of Geometric Errors in a PKM Machine Tool x1

c2

c1 m1

d2

d1

73

x2

x3 c3

m2 a

m3

F

d3

Fig. 3 Three-body-oscillator as lumped mass approach for modeling the adaptronic strut

2 Control Concept for Adaptronic Strut The simplest mechanical model of the adaptronic shown in Fig. 2 is a three-body oscillator as illustrated in Fig. 3. The upper and lower halves of the strut are approached by lumped masses m1 and m3 , respectively, the piezoelectric element in-between is represented by m2 . The springs ci and dampers di represent corresponding material properties, the force F represents external influences such as constraint forces on the strut. With system matrix A, control vector b and disturbance vector bs , the equations of motion for the three-body oscillator in state space form read d z = Az + bu + bSuS dt

(2)

where state vector z represents positions and velocities of the lumped masses. The controller force of the actuator between bodies 2 and 3 is u = Fr and the external disturbance force is us = F. Using the principle of Least Quadratic Regulator (LQR) the parameters for a state controller for this single variable system can be determined. The controller force Fr becomes Fr = −rT z (3) with state vector z. The control vector r is chosen such that the quadratic cost functional ∞ 1 1 J= zT (t)Qz(t) + Fr2 dt (4) 2 κ 0

is minimized. The scalar κ > 0 is a value for the cost of the controller input whereas Q is a positive, semi-definite matrix weighting the system state. For more information on determining the controller parameters using LQR, see [7–11]. For realizing the state controller described above the system state must be known. However, since not all system states are measured an additional element must be introduced. The so-called Luenberger observer estimates the system states according to ∧ ∧ d ∧ z = A z +bu + l y −y . (5) dt

74

C. Rudolf, J. Wauer F

+ -

C

G H

Fig. 4 Closed-loop system with position feedback with prefilter F = 1, integrator H(s) = 1/s, plant G with state controller and compensation element C [1]

Fig. 5 Simple model of adaptronic strut with flexible bodies [1]

Fig. 6 Model of adaptronic strut by use of CAD data [4]

∧

Here in, y is the measured system state, whereas y is the estimation of this system state by the observer. The observer matrix l is chosen such that the eigenvalues of the observed system are further on the left of the imaginary axis than the eigenvalues of the controlled system. This guarantees the observer to be faster than the controller. Since, according to F¨ollinger [7], the two sets of eigenvalues can be set independently, that can easily be achieved using the pole placement procedure according to Ackermann [8]. However, this control concept proves to be not sufficient. When the statecontrolled system experiences a constant external disturbance force F a steady-state error remains [1]. Therefore, the control concept is extended according to the block diagram depicted in Fig. 4. The controller additionally includes an integrating element H in the feedback path as well as a constant compensator element C. The value for C is determined using the root locus procedure such that the enhanced controlled system is endued with dominating zeroes far away from the imaginary axis and a large stability margin according to the Nyquist criterion [1]. As the simulation results presented in [1, 4] have shown the formerly remaining steady-state error vanishes when the extended control structure is used. This applies for all examined systems, the three-body oscillator shown in Fig. 3 as well as for the flexible multi-body systems shown in Figs. 5 and 6. In the following the influence of this strut when implemented into the machine tool with parallel kinematics shown in Fig. 1 is examined.

3 Implementation into PKM Machine Tool For the examination, the settings shown in Fig. 7 are used. The struts are numbered from 1 to 6 starting from the bottom left when seen from the top, the inertial reference frame is given by axes x and y.

Compensation of Geometric Errors in a PKM Machine Tool

75

y

3

4

2

5 x 1

6

Fig. 7 Inertial reference frame and numbering of struts within machine tool

The strut at position 3 is exchanged for a strut of variable length as described above. Starting from an otherwise ideal system one specified kind of geometric error is introduced into the machine tool at two different positions. The application point of the upper joint of one of the struts is shifted by 15e-6m along the axis of this strut, pointing away from the origin of the introduced inertial reference frame. First, this geometric error is placed on the application point on strut 4, then its influence when on strut 1 is examined. Within the simulations of this contribution, the headstock of the machine tool is at rest and no machining processes are taken into account. The results are shown in Figs. 8–10. The dashed lines represent the static deflections of the tool center point of the machine tool due to the examined error whereas the solid lines show the results when the piezoelectric element in the strut is actuated for compensating the influence of the error. When being on strut 4 the geometric error leads to translational deflections of the TCP of order 1e-8m in x- and of order 1e-5m in y-direction, as depicted in Fig. 8a and b, respectively. The influence of the deflection in x-direction is negligible whereas the deflection in y-direction might result in a reduction of quality of the workpiece during the machining process. Therefore, its compensation is required. For a chosen set value of length of strut 3 for the controller the deflection of the TCP in y-direction can be reduced close to zero, as depicted in Fig. 8b. This actuation has no influence on the x-deflection of the TCP. However, when regarding the rotational deflections of the TCP about these two axes, the influence of the geometrical error even increases when the compensating control unit is used. Looking at the magnitude of deflection, the rotation about the xaxis (1e-7 rad) is negligible. About the y-axis it is in the order of 1e-4 rad. Depending on the machining process this still might be negligible, e.g. for turning of short shafts where there is solely point contact between tool and workpiece. For milling processes, however, this deflection is not acceptable and a different approach should be tried. For completeness, Fig. 9 depicts the translational (a) and angular (b) deflections of the TCP along and about the z-axis. Due to the structure of the examined machine tool these deflections are easily taken into account. By adding an identical offset onto all three driving functions of the guiding skids equal to the occurring deflection

76

C. Rudolf, J. Wauer 10−5

y−deflection TCP [m]

x−deflection TCP [m]

10-8 2

0

−2

0

1 time [s]

0.5 0 −0.5 −1

2

x-deflection of TCP

a

1

b

10−7

0

1 time [s]

2

y-deflection of TCP

−0.5

10−4

0

yy [rad]

yx [rad]

2

−2

−1

−4 0

c

1 time [s]

angular deflection of TCP about global x-axis

−1.5

2

d

0

1 time [s]

2

angular deflection of TCP about global y-axis

Fig. 8 Influence of geometric error on strut 4 on position (a)/(b) and orientation (c)/(d) of the TCP with and without compensation by variable strut on 3

in z-direction, the translational deflection along z can be compensated. The rotational deflection about the z-axis is negligible since is identical to the rotation axis of the spindle. As shown in Fig. 10a–d, similar results occur when the same geometric error occurs in the application point of the upper joint on strut 1 with the machine tool being otherwise ideal. The compensating strut of variable length is again placed on position 3. Now, the geometric error leads to considerable deflections of the TCP in both, xand y-direction. Using the compensation unit of strut 3 the deflection in y-direction can be reduced to zero, shown in Fig. 10b, whereas the deflection in x-direction is mostly unaffected, as illustrated in Fig. 10a. The same applies for the corresponding rotational deflections about x- and y-axis, Fig. 10c and d, respectively. Again, deflections of the TCP along and about the z-axis do not have to be taken into account due to the structure of the machine tool.

Compensation of Geometric Errors in a PKM Machine Tool

2

10-5

8

0

6 4

-2 0

a

10

yz [rad]

z−deflection TCP [m]

10-6

77

1 time [s]

z-deflection of TCP

2

0

b

1 time [s]

2

angular deflection of TCP about global y-axis

Fig. 9 Influence of geometric error on strut 4 on vertical position (a) and orientation (b) of the TCP with and without compensation by variable strut on 3

4 Discussion In both cases examined it was not possible to reduce all deflections of the TCP which were induced by the implemented geometric error. Either one of the deflections was not affected by the actuation of the compensation unit or it even increased when the other one was reduced. Thus, regarding the symmetry of the machine tool under examination, implementing solely one of the presented controlled adaptronic struts into the machine tool is, in general, not sufficient for compensating occurring geometric errors. However, there are exemptions for special cases of geometric errors in combination with position an in dependence of the machining process when one compensation unit is sufficient. In turning processes, for example, there is only point contact between tool and workpiece. Thus, small angular deflections are not significant. Furthermore, assuming one of the struts, e.g. at position 5, to be of different length this can be taken into account by substituting strut 6 by a strut of variable length. Making this pair of struts equal in length and adjusting the driving function of the corresponding guiding skid by an additional offset, the ideal structure of the machine tool can be regained.

5 Conclusions An adaptronic strut for compensating static and quasi-static errors in machine tools with parallel kinematics has been presented. The functional principle of a scale with vibrating string was used for measuring the occurring loads with piezoelectric elements. A control concept for the developed strut was presented. The strut

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C. Rudolf, J. Wauer 10−6

10

y−deflection TCP [m]

x−deflection TCP [m]

-8.18

-8.19

-8.2

-8.21

a

0

0.5 time [s]

-4.96

5

0

-5

1

b

x-deflection of TCP

10-6

0

0.5 time [s]

1

y-deflection of TCP

10-5

10-5 2

yy [rad]

yx [rad]

-4.98 -5 -5.02 -5.04

c

0

-2

0

0.5 time [s]

1

angular deflection of TCP about global x-axis

0

0.5 time [s]

1

angular deflection of TCP about global y-axis

Fig. 10 Influence of deflection of application point of upper joint on strut 1 on the position and orientation of TCP with and without compensation

was implemented into a machine tool with parallel kinematics. Starting from an initially ideal machine tool structure, one of the original struts was substituted by the controlled adaptronic strut. An offset of an application point of a joint connecting the strut with the corresponding guiding skid was implemented into the model of the machine tool, representing a geometric error and resulting in a translational and a rotational deflection of the tool center point. Using the controlled strut, a reduction of this erroneous structure was aspired. However, while compensating deflections along or about a specified axis, deflections along and about perpendicular axes either increased or were not affected by the compensation. Thus, further studies have to be conducted regarding the substitution of one or two more controlled struts of variable length. Additionally, motions of the headstock and, thus, changing positions of the TCP in workspace as well as typical machining processes like turning or milling shall be included in future examinations.

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Acknowledgement Financial support of this research in the frame of the Priority Program No. 1156 “Adaptronik in Werkzeugmaschinen” by the German Research Foundation (DFG) is gratefully acknowledged.

References 1. Rudolf C, Wauer J, Munzinger C, Weis M (2007) Piezoelectric control of a machine tool with parallel kinematics. In: Proceedings of SPIE – Industrial and Commercial Applications of Smart Structures Technologies, 65270G. 2. Rudolf C, Wauer J, Fleischer J, Munzinger C (2005) An approach for compensation of geometric faults in machine tools. In: Proceedings of IDETC/CIE Conference, ASME (DETC 2005–84241). 3. Bill B (2002) Messen mit Kristallen: Grundlagen und Anwendungen der piezoelektrischen Messtechnik, Verlag Moderne Industrie, Landsberg/Lech. 4. Rudolf C, Wauer J, Martin T (2007) Piezoelectric control of a machine tool with parallel kinematics. In: Proceedings of III Eccomas Thematic Conference on Smart Structures and Materials. 5. Rudolf C, Wauer J, Fleischer J, Munzinger C (2006) Measuring static and slowly changing loads using piezoelectric sensors. In: Borgmann H (ed) Actuator 2006 Conference Proceedings, 540–543, HVG, Bremen. 6. Fleischer J, Kn¨odel A, Munzinger C, Weis M (2006) Designing adaptronical components for compensation of static and quasi-static loads. In: Proceedings of IDETC/CIE Conference, ASME (DETC 2006–99461). 7. F¨ollinger O (1994) Regelungstechnik, 8th ed., H¨uthig, Heidelberg. 8. Lunze J (2004) Regelungstechnik 2, 3rd ed., Springer, Heidelberg. 9. Preumont A (2002) Vibration Control of Active Structures – An Introduction, 2nd ed., Kluwer, Dordrecht. 10. D’Azzo JJ, Houpis CH, Sheldon SN (2003) Linear Control System Analysis and Design with Matlab, 5th ed., Taylor & Francis, New York. 11. Lunze J (2003) Regelungstechnik 1, 4th ed., Springer, Heidelberg.

Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics Ladislav Pust ˚ and Jan Koz´anek

1 Introduction Rotor motion in rotating machines supported on oil or gas bearings has been studied in many papers and many theoretical solutions have been published in a number of books, proceedings and journal articles (e.g. [1–5]). During the last decades, great attention has been given to nonlinearities in the entire rotor systems, particularly to the most common sources of nonlinearities in oil-, and also in gas-lubricated bearings. The problem of tilting pad bearings was also solved, but mainly in orientation to oil bearings. One of the first attempts of analytical approximations of stiffness and damping nonlinear characteristics was done in [4], examples of other methods for non-linear dynamic analysis are presented in [5–10]. New trends in developing of fluid dynamics bearings are in the last years focused on elaboration of more exact investigation of dynamic properties of gas and tilting pad bearings, both pressurized and self-acting (aerodynamic), e.g. [6, 11]. In spite of this, there are still many unsolved problems particularly in the field of new bearings, e.g. tilting pad aerodynamic bearings, for which new methods of solution must be elaborated. These bearings work on the aerodynamic principle, where the air at high revolutions is automatically sucked into the bearing clearance and there it forms the supporting layer. They are simpler than classical oil- or airbearings, which need additional equipment of oil or pressed air supply. The properties of aerodynamic bearings with three tilting pads were numerically solved and described by linear matrix expressions in Techlab Ltd., Prague, but only for several loads and at discrete revolutions. Due to the linear form these characteristic matrices can be used for solution of motion only in a small surrounding of stationary equilibrium state. L. P˚ust and J. Koz´anek Institute of Thermomechanics, Academy of Sciences of Czech Republic, Prague, 18200 Dolejskova 5, e-mail: [email protected], [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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In presented contribution, an approximate method for calculation of rigid rotor motion supported on two tilting pad aerodynamic bearings oscillating either with small or with large amplitudes limited only by bearing clearance rh is presented. Dynamic characteristics of such type of bearings are very strongly influenced by inertia properties of tilting pads – the stiffness and dynamic matrices are nonsymmetric and their elements are non-monotonous functions of angular frequency ω . The approximate models of stiffness and damping forces – valid for the entire area of journal motion in the aerodynamic bearing’s clearance – are based on the transformation of linear stiffness and damping characteristics given in two positions of rotor journal into the entire area. The assumption that rheological properties of three-tilting pads bearing are centrally symmetric is also used. Space motion of unbalanced rotor and mutual interaction of two identical aerodynamic bearings is then derived for small oscillations near the equilibrium position. The second part of the article is given to the elaboration of this linear dynamic model to the system modeling the nonlinearity of aerodynamic tilting pad bearings. Several examples show the existence of different types of vibrations forced by rotating unbalance and loaded also by various vertical forces.

2 Motivation of Research The application of air as lubricants in aerodynamic bearings is a modern trend in rotating machinery particularly in chemical and food industry, as it excludes the undesirable damage of products by oil. The main advantage of these bearings is very simple aerodynamic principle, which does not need any auxiliary apparatus (pressed air, oil pumps, etc.) and which is suitable for very high revolutions e.g. for high-speed compressors, working for long time periods with constant revolutions. Therefore several types of aerodynamic bearings were designed and realized in the Institute of Thermomechanics of AS CR in cooperation with CKD – New Energo Ltd. and applied in new rotor structures. An example of this application is the laboratory prototype of rotor supported on two aerodynamic tilting pad bearings shown in Fig. 1.

Fig. 1 Laboratory prototype

Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics

83

This experimental set is based on high-frequency synchronous motor running up to 50,000 rpm. Its rigid rotor is supported in two identical radial aerodynamic bearings (diameter 50 mm, clearance rh = 0.05 mm) with three tilting pads in each bearing. The tilting pads have antifriction layers for safety running up and down. The greater machinery – turbo compressor equipped with aerodynamic bearings (120 mm, 100 kW, 18,000 rpm) is developed in CKD – New Energo Ltd. Theoretical background of this experimental research is provided in the project GACR No.101/06/1787 where the analytical and numerical investigation of simplified mathematical models of rotor excited by centrifugal force from the unbalanced rotor is solved.

3 Mathematical Model of the Rotor Motion at Small Oscillations Schema of experimental rotor IT ASCR is shown in Fig. 2. The rigid rotor is symmetric to the axis of revolution, but slightly asymmetric to the mid-span plane. Its mass is m = 7.6 kg and central inertia moment to the axes x and y is I = 0.10024 kgm2 . Distance between the centres of bearings is l = 0.32 m. Inertia properties defined by mass m and moment of inertia I can be replaced by effects of three masses [12], two of them m1 , m2 in the centres of bearings, the third mass m3 in the mid-span of length l. The centrifugal force meω 2 and the centre of gravity mg are shifted in distances a, b to the right from this mid-span point. The properties of aerodynamics tilting pad bearings (diameter 50 mm) were ascertained by numerical solution in the form of evolutive stiffness K(ω ) and damping B(ω ) matrices calculated in TECHLAB Ltd. For discrete values of revolutions in steps of 2,500 rpm (approx. 250 1/s) up to 50,000 rpm and for different values of loading from F = 0 up to F = 70 N. These values vary very strongly at different revolutions. There are no monotone functions, because the inertia of tilting pads causes resonance phenomena, superimposed on monotone increase or decrease at variation of revolutions. In Fig. 3, there are examples of stiffness and damping properties for unloaded journal (F = 0), when its position is in the centre of bearing. The main stiffness kxx is 1 a kyy m1

m3

m,I

y

m2 z

b x bxx

kxx

Fig. 2 Scheme of rotor IT AS CR

mg mew2 bxx

bxx kxx

x kxx

byy

STIFFNESS K [N/mm]

84

L. P˚ust, J. Koz´anek STIFFNESS, LOAD=0

15000 10000 kxx=kyy

5000

kxy

0 −5000

kyx 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

4000

4500

5000

DAMPING B [Ns/m]

ANGULAR VELOCITY ω [1/s] DAMPING, LOAD=0

6000 bxx=byy

4000

bxy 2000 byx

0 −2000

0

500

1000

1500

2000

2500

3000

3500

ANGULAR VELOCITY ω [1/s]

Fig. 3 Stiffness and damping coefficients of unloaded journal (F = 0)

the same as kyy , the cross stiffness elements are of opposite sign: kyx = −kxy ; matrix K(ω ) is anti-symmetric. The same anti-symmetric properties has also damping matrix B(ω ), which elements are plotted in the bottom part of Fig. 3. The anti-symmetric properties are lost, when the position of the journal is shifted into another equilibrium position due to vertical load, e.g. F = 40 N. The properties of both stiffness and damping matrices change. It is seen from Fig. 4, where the curves kxx , kyy differ each from the other as well as damping curves bxx (ω ) and byy (ω ). The cross stiffness kxy (ω ), kyx (ω ) and cross damping bxy (ω ), byx (ω ) are not yet of opposite sign but they are of general form. Due to the increase of loading from 0 to 40 N, the peaks of curves caused by tilting pads inertia and aerodynamic forces shift from ω ∼ = 1,700 1/s to higher velocities ω ∼ = 2,600 1/s. Knowledge of stiffness matrices K(ω ) and B(ω ) suffices for ascertaining of rotor motion at small oscillations around the equilibrium position. This motion is described by linear differential equation in the form of two matrix equations [7, 8]: ¨ 1 + M3 X ¨ 2 + B1 X ˙ 1 + K1 X1 = meω 2 (1/2 − a/l) O + F, M1 X ¨ 2 + M3 X ¨ 1 + B2 X ˙ 2 + K2 X2 = meω 2 (1/2 + a/l) O + F, M2 X where:

(1)

DAMPING B [Ns/m]

STIFFNESS K [N/mm]

Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics 10−4

85

STIFFNESS, LOAD=40N

2 kyy

1 0

kyx

−1

8000 6000 4000 2000 0 −2000 −4000

kxx

kxy

0

500

1000

1500 2000 2500 3000 3500 ANGULAR VELOCITY ω [1/s]

4000

4500

5000

4000

4500

5000

DAMPING, LOAD=40N bxx byy byx 0

500

1000

bxy

1500 2000 2500 3000 3500 ANGULAR VELOCITY ω [1/s]

Fig. 4 Stiffness and damping coefficients at vertical load (F = 40 N)

mi + m3 /4 m3 /4 , i = 1, 2, m3 /4 mi + m3 /4 m3 /4 0 M3 = 0 m3 /4

Mi =

(2)

Xi = [xi , yi ]T i = 1, 2 O = [cos ω t, sin ω t]T , F = [40, 0]T , kxxi (ω ) kxyi (ω ) bxxi (ω ) bxyi (ω ) , Bi = . Ki = kyxi (ω ) kyyi (ω ) byxi (ω ) byyi (ω )

(3) (4)

Indexes 1, 2 express the values belonging to the different journal equilibrium positions in bearings 1, 2. Multiplying Eqs. (1) by suitable combinations of mass ¨ 1 in the first matrices (2) we get the equations with separate second derivatives X ¨ 2 . SimEq. (1). By means of similar way we get the separate second derivatives X pler expressions are then reached by left multiplying these equations with inverse matrix (M1 M2 − M3 2 )−1 . Derived equations can be used for calculation of small oscillations near equilibrium positions, i.e. for small unbalance e, where the linear stiffness and damping matrices are acceptable. Examples for b = 0 and a = 0; 0.2, are shown in Figs. 5 and 6. Example of oscillations of symmetrical rotor (b = 0, m1 = m2 ) at excitation caused by the unbalance e = 10μ m in the mid-span (a = 0) is shown in Figs. 5a and b. The exact symmetry of rotor and of excitation produces the same motion trajectories in both bearings; as it is seen from time history of motion shown in Fig. 5a. The polar trajectories x1 , y1 and x2 , y2 overlap each other as seen in Fig. 5a. The trajectories in state planes are ellipses, which are influenced by linear properties of bearing characteristics.

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L. P˚ust, J. Koz´anek 1.5

TIME HISTORY OF MOTION x, y, v, vy, F

10-4

MOTION x, y, v, vy, F

1 v1==dx1/dt=v2=dv2/dt,

vy1=dy1/dt=vy2=dy2/dt

0.5

0 x1=x2,

y1=y2

-0.5 -1 F(t) -1.5

0

0.002

0.004

0.006

0.008

0.01 0.012 TIME t [s]

0.014

0.016

0.018

0.02

Fig. 5a Time history of motion

5

10 - 5

JOURNAL MOTION x - y e=35e-6 [m]

4

DISPLACEMENT x [mm]

3 rh

2 1

y

0 -1 x1,y1=x2,y2

-2 -3 -4 -5 -6

x

-4

-2 0 2 DISPLACEMENT y [mm]

4

6 10 -5

Fig. 5b Motion trajectory

Excitation in another point different from the geometric centre changes the form of oscillations. Influence of small shift of unbalance position from the centre (a = 0.04) is shown in Fig. 6a where the time history of stationary oscillations in both bearings x1 (t), y1 (t) (second record) and x2 (t), y2 (t) (forth record) are plotted. Differences are of course also in velocity records first record for bearing 1, third one for bearing 2. Asymmetry of excitation causes different amplitudes and different types of oscillations in both bearings. These properties are markedly seen in x − y plane trajectories (Fig. 6b), where x1 − y1 trajectory is approximately straight line, x2 − y2 trajectory is ellipse.

Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics 1.5

TIME HISTORY OF MOTION x, y, v, vy, F

10-4

1 MOTION x, y, v, vy, F

87

v1=dx1/dt, vy1=dy1/dt)

0.5

x1y1

0 v2=dx2/dt, vy2=dy2/dt) -0.5 x2y2

-1 -1.5

F(t) 0

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 TIME t [s]

0.02

Fig. 6a Time histories of x, y, v, vy and F 10 -5 JOURNAL MOTION x1,y1 and x2,y2, e=40e-6 [m] 5 4

DISPLACEMENT [mm]

3 2 x2,y2

1

y

0 x1,y1

-1 -2 -3 -4 -5

x

-6

-4

-2 0 2 DISPLACEMENT y [mm]

4

6 10 -5

Fig. 6b Planes: (x1 , y1 ), (x2 , y2 ) and (x, y)

4 Rotor Motion at Large Oscillations The method used in previous chapter can be applied due to the linearity of differential equations (1) only for small displacements from the equilibrium position [8,10]. For larger excitations and oscillations, the linear stiffness and damping matrices are not sufficient and therefore a nonlinear form of motion equations has to be derived. No suitable exact mathematical model of tilting pad aerodynamic bearing properties exists as well as appropriate algorithms of calculation. Therefore only approximate methods can be applied for ascertaining large oscillations. The first attempt of extension of force and damping properties on the entire bearing field was done in [7, 8], where the hyperbolic correction function was proposed and applied for solution. This correction function gives zero value in the centre of bearing in order to ensure continuity in this point. The more detailed calculation of stiffness and damping at various loading and various equilibrium positions shows that in the central bearing position at zero loading the stiffness matrix has

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nonzero elements, but it is anti-symmetric (see Fig. 3) and that the application of only one hyperbolic correction function is insufficient. Therefore the more exact correction has to be realized by expression consisting of two parts [9, 10], one of them fcor1 (r)K(ω )o describing properties in the centre of bearing (r = 0) i.e. at zero loading (F = 0), the second part fcor2 (r)K(ω )ro depends on stiffness matrix K(ω ) in the equilibrium point r0 at given loading (e.g. F = 40 N). The first correction function is monotone decreasing from value 1 in the bearing centre fcor1 (r) = (1 − r/rh)2 , (5a) where rh is bearing clearance. The second correction function is selected in the form of hyperbolic function fcor2 = s/(p + rh − r), (5b) where parameters s, p, h are suitably ascertained. This second correction function enables to express well the increase of restoring force at large displacements from the centre of bearing near the contact of journal with bush. More detailed derivation of these correction functions is in [9], where a model of rubbing is also mentioned. The resulting stiffness K(ri , ω ) and damping B(ri , ω ) matrices are calculated either according to the formulae K(ri , ω ) = fcor1 (ri )K(ω )o + fcor2 (ri )K(ω )ro , i = 1, 2, B(ri , ω ) = fcor1 (ri )B(ω )o + fcor2 (ri )B(ω )ro ,

(5c) (5d)

or separately with different correction functions for single elements of stiffness and damping matrices. This latter individual method enables to fit mathematical model better to the experimentally gained data. In this contribution, the transformations (5c) and (5d) of the whole matrices K(ri , ω ), B(ri , ω ) are used. But this r-transformation is not enough. To describe bearing’s properties in the entire plane area of clearance, it is needed to extend the stiffness and damping description from the clearance vertical radius 0 ≤ r < rh onto general position given by the angle ϕ ∈ (0, 2π). This can be realized multiplying stiffness and damping matrices by plane rotation matrix cos(ϕ ) sin(ϕ ) x y /r, r = x2 + y2 . C= , or C = (6) −y x − sin(ϕ ) cos(ϕ ) The motion of rotor journal in the aerodynamic bearing is given in a concise form by ¨ + CT (x, y)K (ri , ω ) C(x, y)X + MX (7) ˙ = O (ω t) meω 2 + F(t), + CT (x, y)B (ri , ω ) C(x, y)X where the matrices and vectors M,

X,

C(x, y),

fcor (r),

K (ri , ω ) ,

B (ri , ω ) ,

must be defined for each bearing (i = 1, 2) individually.

O (ω t) ,

F(t)

Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics

89

5 Examples The Eq. (7) enables to investigate the properties of mathematical model of aerodynamic bearing at various conditions. The next all records are for eccentricity e = 80e-6 m and for the mass distribution (in kg) given by the matrix M = [2.875 0.925; 0.925 2.875] corresponding to the experimental rotor IT AS CR (Figs. 1, 2). If the excitation of symmetrical rotor acts in its mid-span (a/l = 0), both journals vibrate in the same way. Plane trajectories x1 , y1 and x2 , y2 are identical and draw the same curve, see Fig. 7, again for ω = 3, 000 1/s. The vibration is identical in both bearings and periodic with nearly harmonic components x1 (t) = x2 (t) and y1 (t) = y2 (t). Central nonlinearity causes distortions from elliptic trajectory. Area limited by a circle with radius rh depicts the possible position of journal centre. Small shift of exciting unbalance from the mid-span of rotor to the right bearing (a/l = 0.1) causes different vibration of journal 1 and 2. Corresponding plane trajectories x1 , y1 and x2 , y2 differ each from other (see Fig. 8). Increase of eccentricity shift to a/l = 0.2 results in greater differences between trajectories of both journals. Figure 9 shows the transition of x1 , y1 trajectory into slender ellipse; trajectory x2 , y2 of right journal changes its form only in a low degree. The same feature has x2 , y2 trajectory (Fig. 10) also at rising shift of eccentricity position to a/l = 0.3. However the journal trajectory x1 , y1 in the left bearing changes its form markedly into a nearly straight line with opposite inclination in comparison with case a/l = 0.2 (Fig. 9). Small decrease of both trajectories against the centre of bearing (x = 0, y. = 0) is caused by a small static force Fstat = 40 N.

5

JOURNAL MOTION x - y

10 -5

4 rh

DISPLACEMENT x1,x2 [mm]

3 2 1

y

0 -1 x1,y1 = x2,y2

-2 -3 -4 -5

x

-6

-4

Fig. 7 Plane trajectories at a/l = 0

-2 1 2 DISPLACEMENT y1, y2 [mm]

4

6 10 -5

90

L. P˚ust, J. Koz´anek 5

JOURNAL MOTION x - y

10-5

4

a/l = 0.1

rh

DISPLACEMENT x1,x2 [mm]

3 2 1 y

0 x1,y1

-1

x2,y2

-2 -3 -4 -5

-6

-4

x -2 0 2 DISPLACEMENT y1,y2 [mm]

4

6 10-5

Fig. 8 Plane trajectories at a/l = 0.1 5

JOURNAL MOTION x - y

10-5

4

a/l = 0.2

rh

DISPLACEMENT x1,x2 [mm]

3 2

x2,y2

1 0

y

-1

x1,y1

-2 -3 -4 -5

x -6

-4

-2 0 2 DISPLACEMENT y1, y2 [mm]

4

6 10

-5

Fig. 9 Plane trajectories at a/l = 0.2

Let us now see the influence of static load of the individual journals. This load is the same on both journals. The five-times higher static force Fstat = 200 N in comparison with the previous case, results in a great lowering of journal trajectory x1 , y1 and in very imperceptible increasing of its width (Fig. 11a). The second trajectory x2 , y2 comes down only very little and also its deformation does not change. All following records are done with shift of eccentricity position to a/l = 0.3.

Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics 5

91

JOURNAL MOTION x - y

10-5

4

a/l = 0.3

3

Fstat = 38 N

DISPLACEMENT x1,x2 [mm]

rh 2 x2,y2

1

y

0 x1,y1

-1 -2 -3 -4 -5

x -6

-4

-2 0 2 DISPLACEMENT y1, y2 [mm]

4

6 -5 10

Fig. 10 Plane trajectories at a/l = 0.3

DISPLACEMENT x1,x2 [mm]

5

JOURNAL MOTION x - y

10-5

4

Fstat = 200 N

3

a/l = 0.3

rh

2

x2,y2

1 y

0 -1

x1,y1

-2 -3 -4 -5

x -6

-4

-2

0

2

DISPLACEMENT y1, y2 [mm]

4

6 -5 10

Fig. 11a Plane trajectories at Fstat = 200 N

The further increase of static load to Fstat = 400 N changes the form of both trajectories much more, as it is shown in Fig. 11b. The width and also the shift down of trajectory x1 , y1 increases, its length decreases. Trajectory x2 , y2 of the right journal changes its form in a small, but recognisable measure. If the vertical loads on journals increase to Fstat = 700 N, the both plane trajectories of motion change their forms, as shown in Fig. 11c. The upper part of trajectory x2 , y2 shifts down, journal trajectory x1 , y1 shifts only a little, but it changes its form into the drop-shaped one.

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DISPLACEMENT x1,x2 [mm]

5

JOURNAL MOTION x - y

10-5

4

Fstat = 400 N

3

a/l = 0.3

rh

2

x2,y2

1 y

0 -1 -2

x1,y1

-3 -4 -5

x -6

-4

-2

0

2

4

6

DISPLACEMENT y1, y2 [mm]

10

-5

Fig. 11b Plane trajectories, Fstat = 400 N

DISPLACEMENT x1,x2 [mm]

5

JOURNAL MOTION x - y

10-5

4

Fstat = 700 N

3

a/l = 0.3

rh

2 x2,y2

1

y

0 -1 -2

x1,y1

-3 -4 -5

x -6

-4

-2

0

2

DISPLACEMENT y1, y2 [mm]

4

6 -5

10

Fig. 11c Plane trajectories, Fstat = 700 N

The further increase of static load to Fstat = 1,000 N changes the shape of both trajectories fundamentally as it is shown in Fig. 11d. This can be explained by the strongly nonlinear stiffness characteristics near the bearing surface.

Interaction of Two Aerodynamic Bearings with Nonlinear Characteristics 5

93

JOURNAL MOTION x - y

10-5

4

Fstat = 1000 N

3

a/l = 0.3

DISPLACEMENT x1,x2 [mm]

rh

2 1 y

0 -1

x2,y2

-2 x1,y1

-3 -4 -5

x -6

-4

-2

0

2

DISPLACEMENT y1, y2 [mm]

4

6 -5 10

Fig. 11d Plane trajectories, Fstat = 1, 000 N

6 Conclusions The mathematical model of journal motion in aerodynamic bearings including properties of three tilting pads was derived for large amplitudes in the entire domain of clearance of bearing. Dynamic characteristics of such type of bearings are very strongly influenced by inertia properties of tilting pads – stiffness and dynamic matrices are non-symmetric and their elements are non-monotonous functions of angular frequency ω . Dynamic characteristics, given for selected positions of journal in the form of linear stiffness and damping matrices were extended by using special correction function fcor (r) on the entire area of bearing clearance 0 < r < rh , and by plane rotation matrix C for all angular positions −π < ϕ < π. The correction function consists of a part proportional to the stiffness and damping of unloaded bearing and of the second part proportional to the properties of loaded bearing. The approximate mathematical model of stiffness and damping properties of aerodynamic bearing is applied on examples, where influence of various positions of exciting eccentricity and of static loading is shown. Time history of journal motion and its x, y trajectories show the great influence of nonlinearity at large displacements. Due to the sufficient aerodynamic damping, the system was stable in all studied cases. Acknowledgement This work was supported by the Grant Agency of CR No.101/06/1787 “Dynamic properties of gas bearings and their interaction with rotor”.

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References 1. Czolczynski K (1994) Stability of high stiffness journal bearings, Wear 172, 175–183. 2. Lalanne M, Ferraris G (1990) Rotordynamics Prediction in Engineering, Wiley, New York. 3. Strzelecki S (2005) Dynamic characteristics of tilting 5-pad journal bearing. In: Sawicki JT, Muszynska A (eds.) Proceedings of the 3rd ISCORMA 2005, Cleveland, OH, September, 242–249. 4. Tondl A (1965) Some Problems of Rotordynamics, Chapman & Hall, London. 5. Vania A, Tanzi E (2004) Analysis of non-linear effects in oil-film journal bearings, Eighth International Conference on Vibrations in Rotating Machinery, I Mech E, C6231041/2004, 101–110. 6. Martyna M, Kozanecki Z (2007) Non-linear model of a tilting pad gas journal bearing for the power MEMS microturbine. In: Bently DE, Sawicki JT (eds.) Proceedings of the 4th ISCORMA 2007, Calgary, Canada, CD-ROM 309, 1–8. 7. P˚ust L, Koz´anek J (2005) Vibrations of a rigid rotor supported on aerodynamic bearings as an evolutive system. In: Sawicki JT, Muszynska A (eds.) Proceedings of the 3rd ISCORMA 2005, Cleveland, OH, September, 299–308. 8. P˚ust L, Koz´anek J (2005) Nonlinear and evolutive vibrations of rotor supported on aerodynamic bearings. In: Awrejczewicz J, Sendkowski D, Mrozowski J (eds.) Proceedings of 8th Conference on Dynamical Systems Theory and Applications, Lodz, Poland, December, 785–792. ˇ 9. P˚ust L, Simek J, Koz´anek J (2007) Motion of rotor supported on aerodynamic bearings. In: Zolotarev I (ed.) Proceedings of Conference Engineering Mechanics, IT ASCR, Svratka, May, CD-ROM, 116–112. 10. P˚ust L, Koz´anek J (2007) Evolutive and nonlinear vibrations of rotor on aerodynamic bearings, Nonlinear Dynamics 50, 829–840. 11. Hatch ChT et al. (2007) Stiffness and damping results from perturbation testing of externally pressurized radial gas bearings. In: Bently DE, Sawicki JT (eds.) Proceedings of the 4th ISCORMA 2007, Calgary, Canada, CD-ROM 301, 1–11. 12. Brepta R, P˚ust L, Turek F (1994) Mechanical Vibrations, Sobot´ales, Praha.

Identification of Dynamical Systems in the Fuzzy Conditions ´ Tadeusz S. Burczynski, Witold Beluch, and Piotr Orantek

1 Introduction The paper deals with the identification of the fuzzy parameters of material and shape of structures. In many identification and optimization problems for the structures being under dynamical loads one should find some unknown parameters, e.g. materials properties, boundary conditions or geometrical parameters. An identification problem can be formulated as the minimization of some objective functions depending on measured and computed state fields, as displacements, strains, eigenfrequencies or temperature. In order to obtain the unique solution of the identification problem the global minimum of the objective function should be found. In many engineering dynamical cases it is not possible to determine the parameters of the system precisely, so it is necessary to introduce some uncertain parameters which describe the granular character of data. There exist different models of information granularity: interval numbers, fuzzy numbers, rough sets, random variables, etc. In the present paper the granularity of information is represented in the form of the fuzzy numbers. In order to solve an identification problem, some optimization methods have to be used. In the proposed approach the fuzzy version of the evolutionary algorithm (FEA) is used as the first step of the identification procedure. The fuzzy steepest descent method with multilevel artificial neural network (ANN) is used in the second step. The special type of fuzzy ANN for the approximation of the fuzzy fitness function value and the special type of fuzzy fitness function gradient are used. The usage of the ANN enables the reduction of the computation time. The fuzzy finite element method (FFEM) is employed to solve the boundary-value problem.

T.S. Burczy´nski, W. Beluch, and P. Orantek Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, Konarskiego 18a, 44–100 Gliwice, Poland, e-mail: [email protected], [email protected], [email protected]

J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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2 Formulation of the Identification Problem of the Dynamical Systems Consider the first case of an elastic isotropic structure containing a void which size and position are unknown and are described by a vector x. The vector of displacements u(z, t) is described by equation:

μ ∇2 u + (λ + μ )grad div u + Z = ρ u¨ (z,t),

z ∈ Ω,t ∈ T ∈ [0,t f ],

(1)

where: μ , λ – Lame constants, Z – body forces. Equation (1) is supplemented by boundary conditions: u(z,t) = u(z,t), ¯ ¯ p(z,t) = p(z,t),

z ∈ Γ ≡ ∂ Ω, z ∈ Γ ≡ ∂ Ω,

(2)

and initial conditions: u(z,t)|t=0 = uo (z),

o u(z,t)| ˙ t=0 = v (z),

z ∈ Ω.

(3)

It is assumed that boundary conditions and material parameters have the fuzzy character. The identification problem is treated as the minimization of an objective functional (fitness function) J(x), depending on measured uˆ and computed displacements u at sensor points zi , with respect to x: n

min J(x), x

where J(x) = ∑

i=1 T

(u(z,t) − uˆ (z,t))2 δ z − zi dΓdt.

(4)

Γ

Consider the second case of a multi-layered laminate. Multi-layered laminates are the fibre-reinforced composites built of a definite number of the stacked, permanently joined plies [1]. Plies in laminates are usually built of the same material having different fibers’ angles. Laminates have high strength/weight ratio and it is easy to tailor the material properties by manipulating such parameters as: components material, stacking sequence, fibres orientation or layer thicknesses. They can be typically treated as two-dimensional structures with four independent elastic constant: two Young moduli, one shear modulus and one Poisson ratio. The constitutive equation for a laminate’s single layer can be written in the following form [2]: ⎤ ⎡ E1 ν 21 E1 ⎧ ⎫ ⎫ 0 ⎧ 1− ν ν 1− ν ν 12 21 12 21 ⎨ σ 11 ⎬ ⎢ ⎨ ε 11 ⎬ ⎥ ν 12 E 2 E2 σ 22 = ⎢ (5) 0 ⎥ ⎦ ⎩ ε 22 ⎭ , ⎩ ⎭ ⎣ 1−ν 12 ν 21 1−ν 12 ν 21 σ 12 ε 12 0 0 G 12

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where: σij – stress vector, εij – strain vector, E1 , E2 – axial and transverse Young’s moduli, G12 – axial-transverse shear modulus, ν12 , ν21 – axial-transverse and transverse-axial Poisson ratios, respectively (ν21 = ν12 E2 /E1 ). The laminates are often manufactured individually, so the nondestructive methods of determination of the elastic constants must be employed. Indirect identification methods, based on numerical and mixed numerical-experimental techniques, have been developed intensely [3]. The eigenvalue problem for a laminate plate of length a, width b and thickness h in directions x, y and z, respectively, can be presented as [4]:

ρ hω 2 w = D11 w , xxxx + 4D16w , xxxy + 2(D12 + 2D66)w , xxyy + + 4D26w , xyyy + D22 w , yyyy

(6)

where: w – deflection in the z direction, ω – eigenvalue vector, ρ – mass density. The bending stiffness Di j can be obtained from the formula: Di j =

h 2

− h2

(k) (z(k) )2 Q¯ i j dz,

(7)

where: z(k) – the distance from the middle plane to the top of layer k, (k) Q¯ i j – plane stress reduced stiffness component of the layer k. One assumes that material constants in laminates x = (E1 , E2 , G12 , ν) are unknown and have the fuzzy character. In order to find them the functional J(x) which depends on N measured ωˆ j and N calculated ω j natural frequencies has to be minimized [5]: J(x) =

min J(x), x

N

∑ [ω j − ωˆ j ]2 .

(8)

j=1

In the fuzzy representation the objective function is modified to the fuzzy form and the edges of the intervals

J(x) ∈ J(x), J(x) (9) are calculated as follows:

/ / / / J(x j ) = ∑ min /qi − qˆi / , /q¯i − q¯ˆi/ , i

/ / / / ¯ j ) = ∑ max /qi − qˆi/ , /q¯i − q¯ˆi / . J(x

(10)

i

To solve both identification problems, the optimization methods have to be used. In present paper the so called two-stage fuzzy strategy, which merges the evolutionary

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algorithm, gradient methods and artificial neural networks is used as the optimization tool.

3 The Two-Stage Fuzzy Strategy The two-stage fuzzy strategy (TSFS) couples the advantages of gradient optimization methods, evolutionary algorithms (EAs) and artificial neural networks (ANNs) [6]. Gradient methods are fast and give precise results but their application is significantly limited. They can be used for continuous problems and they usually lead to the local optima if some multimodal problems are considered. In such cases the evolutionary algorithms, which are the global optimization methods can be used. As the evolutionary algorithms are time-consuming, it is convenient to couple that both methods (if possible). The block diagram of the TSFS is presented in Fig. 1. In the proposed attitude the fuzzy EA works in the first stage. As a result a set (cloud) of fuzzy points, being the starting point for the next stage, is generated. It is assumed that the considered set of points is located close to the global optimum. The moment of the transition between stages is crucial and depends on the following parameters of the first stage [7]: – the size of the chromosomes’ population – the variability of the best chromosome’s fitness function value – the diversity of the population In second stage the gradient method is used. Sensitivity of the objective function is approximated by means of the ANN.

Fig. 1 The two-stage fuzzy strategy

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3.1 The First Stage – Global Optimization The evolutionary algorithms are global optimization methods which origin from the observation of the natural evolution process [8]. A set (population) of possible solutions (individuals or chromosomes) of the optimization process is processed. Each chromosome consists of genes. Each gene represents one design variable. In order to simulate the evolution process chromosomes are modified. Firstly they are subjected to the selection procedure. Chromosomes’ continuity to exist and reproduction probability depend on their fitness value. Secondly, the evolutionary operators, typically crossover and mutation, are applied to generate new chromosomes which are new possible solutions of the optimization task. The procedure is repeated until the termination condition is fulfilled. The fuzzy evolutionary algorithm (FEA) works on chromosomes consisting of fuzzy genes [9]. Each gene represents one fuzzy number and each chromosome is a potential fuzzy solution of the problem. The j-th fuzzy chromosome ch j in the population consists of N genes and has the following form: ch j (x) = [x1j , x2j , . . . , xij , . . . , xNj ].

(11)

The standard representation of the fuzzy number can be inconvenient from the fuzzy number arithmetic point of view. It is suitable to represent the fuzzy number x as a set of the interval values [x, x] ¯ lying on the adequate levels called α -cuts, as presented in Fig. 2 [10]. In the present paper each gene xij in the chromosome ch j is a vector of five real values representing trapezoidal fuzzy number (Fig. 3): j

j

j

j

j

j

xi = [aL (xi ), aU (xi ), cv(xi ), bL (xi ), bU (xi )],

(12)

where: cv(xij ) – the central value of a fuzzy number;

m

m [0, 1]

1

0.75 0.5 0.25 x

0 x1 x0.75 x0.5 x0.25 x0

Fig. 2 The fuzzy number and corresponding α -cuts

x1 x0.75 x0.5 x0.25 x0

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Fig. 3 The fuzzy gene

ak (xij ), bk (xij ) – distances between the central value and the left and right boundaries of the interval on lower (L) and upper (U) α-cuts, respectively. New fuzzy mutation and crossover operators are introduced to operate on the fuzzy chromosomes [11]. The first type of the fuzzy mutation (FM1) modifies only the central value cv(xij ) of the randomly chosen gene xij , while the second fuzzy mutation operator (FM2) concentrates/deconcentrates the fuzzy gene xij by changing the distances ak (xij ), bk (xij ) for a chosen α -cut. The fuzzy arithmetic crossover (FC) produces two descendants ch1 (x)∗ and ch2 (x)∗ from two parent chromosomes ch1 (x) and ch2 (x) changing selected α -cuts. The fuzzy selection is based on the tournament selection method [12]. The fuzzy fitness function values must be compared in order to select the best individual in the tournament. The better chromosome wins with a probability depending on the introduced parameter β . The fuzzy finite element method (FFEM) is employed in order to calculate fuzzy fitness function values.

3.2 The Second Stage – Local Optimization The block diagram of the second stage of TSFS in presented in Fig. 4. The local optimization method combines the steepest descent method and the artificial neural networks [13]. A set of training vectors is created on the basis of the cloud of points generated in the previous stage. A special multilevel ANN is used as the approximation tool of the fuzzy fitness function. Each level of the ANN multilevel corresponds with a selected parameter of the fuzzy number (Fig. 5). The central level of the multilevel ANN corresponds with j the central value cv(xi ) of the fuzzy number (black colour), while the other levels correspond with other parameters of the fuzzy number: the grey levels correspond with the parameters ai , the white levels correspond with the parameters bi . The coordinates of the points (central value and fuzzy parameters) play the role of the input values while the fuzzy fitness values of these points play a role of the network output.

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Fig. 4 The scheme of the local optimization method

Fig. 5 The scheme of the multilevel ANN

After training the multilevel ANN the optimization process is performed by means of the steepest descent method. As the last step the termination condition is checked. If it is not satisfied, the considered point is added to the training vector set and the next iteration is carried out, otherwise, the point is treated as a result of the optimization process.

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Fig. 6 The structure with identified circular defect

4 Numerical Examples 4.1 The Identification of Geometrical Parameters of a Void The aim of the identification problem is to find the parameters defining the circular defect: x, y and r in a square plate (0.2 × 0.2 m) made of an isotropic material (Fig. 6). The plate is loaded by the fuzzy continuous dynamical loading ¯ p(z,t) = po H(t), where po = 10 kN and H(t) is the Heaviside’s function. 200 time-steps with Δ = 1μs are considered. It is assumed that the Young’s modulus of the plate material E and loading po are also the fuzzy values. The displacements are measured in 21 sensor points on the boundary. The loading po and Young modulus E are described by two α –cuts with the same intervals: po = [99.8; 100.2] kN, E = [2e11 − 2%; 2e11 + 2%] MPa. – – – –

The parameters of the FEA (first stage) are as follows: the population size pop size= 20 the number of generations gen num= 40 arithmetic crossover probability pc = 0.2 Gaussian mutation probability pm = 0.4

In the second stage the local optimization method found the optimum after 156 iterations. The actual and found values of the identified parameters are collected in Tab. 1.

4.2 The Identification of Laminate’s Elastic Constants The aim is to find the elastic constants in a laminate plate consisting of 16 plies of the same thickness h = 0.002 m (Fig. 7).

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Table 1 The identification results for the circular defect Parameter (m)

a1

a2

cv

b2

b1

x y r x y r

0.001 0.001 0.001 0.001 0.001 0.001

0.0005 0.0005 0.0005 0.0005 0.0005 0.0005

0.03 0.03 0.02 0.03 0.03 0.02

0.0005 0.0005 0.0005 0.0005 0.0005 0.0005

0.001 0.001 0.001 0.001 0.001 0.001

Actual

Found

Fig. 7 The laminate plate – dimensions and bearing

The laminate plate with stacking sequence: (0/45/90/-45/0/90/0/90)s is made of the epoxy-glass material. The first N = 10 eigenfrequencies of the plate are taken as the measurement data. Each of identified parameters is described by two α -cuts of different widths. – – – –

The parameters of the FEA are: The population size pop size= 100 The number of generations gen num= 400 Arithmetic crossover probability pc = 0.2 Gaussian mutation probability pm = 0.4

The variations of the identified parameters (for both edges of each interval) in the function of the generation number during the first stage of the strategy are presented in Fig. 8. The number of the local optimization method’s iterations was equal to 1000. The actual and found values of the identified parameters are collected in Tab. 2.

5 Conclusions An intelligent strategy coupling the fuzzy evolutionary algorithm, the multilevel artificial neural networks, and local optimization methods has been presented. This approach can be applied in the optimization and identification of mechanical

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Fig. 8 The identification results during the first stage: (a) E1 , (b)E2 , (c)ν12 , (d)G12 Table 2 The identification results for the laminate plate Parameter (Pa) Actual

Found

E1 E2 G12 ν E1 E2 G12 ν

a1

a2

cv

b2

b1

0.04e10 0.04e9 0.02e9 0.003 0.04e10 0.04e9 0.02e9 0.003

0.02e10 0.02e9 0.04e9 0.001 0.02e10 0.02e9 0.04e9 0.001

3.86e10 8.27e9 7.17e9 0.26 3.86e10 8.27e9 7.17e9 0.26

0.02e10 0.02e9 0.02e9 0.001 0.02e10 0.02e9 0.02e9 0.001

0.04e10 0.04e9 0.04e9 0.003 0.04e10 0.04e9 0.04e9 0.003

structures under dynamic loads. The application of evolutionary algorithms in the first stage of the strategy reduces the possibility of finding the local minimum of the objective function. The gradient method applied in the second stage uses information about the objective functional sensitivity obtained by means of neuro-computing. The identified values and the fitness functions are in the form of fuzzy numbers represented by α -cuts. The fuzzy finite element method has been employed to calculate the fuzzy fitness function values in the first stage of the strategy. The application of the artificial neural networks for approximation of the boundary-value problem in the second stage enables the reduction of the computational time.

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The strategy gives positive results for the identification of various parameters in isotropic as well as in orthotropic (laminates) dynamical structures. As the uncertain conditions have the granular form [14], the models based on the rough sets and random variables can be used instead of the fuzzy and interval approach presented in this paper. Acknowledgement The research is partially financed from the Polish science budget resources as the research project and the Foundation for Polish Science (2005–2008).

References 1. Peters ST (1998) Handbook of Composites. 2nd Edition, Chapman & Hall, London/ Weinheim/New York/Tokyo/Melbourne/Madras. 2. German J (2001) The Basics of the Fibre-Reinforced Composites’ Mechanics, Publications of the Cracow University of Technology, Cracow (in Polish). 3. Bledzki AK, Kessler A, Rickards R, Chate A (1999) Determination of elastic constants of glass/epoxy unidirectional laminates by the vibration testing of plates, Composite Science and Technology 59, 2015–2024. 4. Adali S, Verijenko VE (2001) Optimum stacking sequence design of symmetric hybrid laminates undergoing free vibrations, Composite Structures 54, 131–138. 5. Beluch W, Burczy´nski T (2004) Distributed evolutionary algorithms in identification of material constants in composites. In: 7th National Conference on Evolutionary Computation and Global Optimization, Kazimierz Dolny, 1–8. 6. Burczy´nski T, Orantek P (2005) The fuzzy evolutionary algorithms in optimization problems. In: 8th National Conference on Evolutionary Computation and Global Optimization, Korbiel´ow, 23–30. 7. Orantek P (2005) An intelligent computing technique in identification problems, Computer Assisted Mechanics and Engineering Sciences 13, 351–364. 8. Arabas J (2001) Lectures on Evolutionary Algorithms, WNT, Warsaw (in Polish). 9. Pedrycz W (1998) Fuzzy evolutionary computing, Soft Computing 2, 61–72, Springer, Berlin/Heidelberg. 10. Kacprzyk J (1986) Fuzzy Sets in System Analysis, PWN, Warsaw. 11. Burczy´nski T, Orantek P (2005) The two-stage fuzzy strategy in identification of the uncertain boundary conditions. In: T. Burczyski, W. Cholewa, W. Moczulski (eds.) Methods of Artificial Intelligence, AI-METH 2005, full papers on CD-ROM. 12. Michalewicz Z (1992) Genetic Algorithms + Data Structures = Evolutionary Programs, Springer, New York. 13. Duch W, Korbicz J, Rutkowski L, Tadeusiewicz R (2000) Neural Networks 6, Exit, Warsaw (in Polish). 14. Bargiela A, Pedrycz W (2002) Granular Computing: An Introduction, Kluwer, Boston, MA/Dordrecht/London.

On Nonlinear Response of a Non-ideal System with Shape Memory Alloy Vinicius Piccirillo, Jos´e Manoel Balthazar, Bento R. Pontes Jr., and Jorge L.P. Felix

1 Introduction The study of problems that involve the coupling of several systems, was explored widely in the last year, in function of the change of constructive characteristics of the machines and structures. In this way, some phenomena are observed in composed dynamic systems supporting structure and rotating machines, where are verified that the unbalancing of the rotating parts is the great causer of vibrations. In the study of these systems, for a more realistic formulation is to consider an energy source with limited power supply (non-ideal), that is, to consider the influence of the oscillatory system on the driving force and vice versa. Recently a number of works have been done, in order to investigate the resonant conditions of non-ideal vibrating systems [1], and a number of several of non-ideal systems has been studied, for some examples see [2–5]. On the other hand, the phenomena related to the shape memory alloys (SMA) are associate to the transformations of the phase, which can be caused by the variation of the temperature, as well as, for the variation in the tension level. Basically, the (SMA) presents two stable phases: austenite and martensite. According to the mechanical behavior, the shape memory alloys, can be divided in two categories: shape memory (SME) and pseudoelastic effect. Shape memory effect is related to the ability of the material to recover a great quantity of the residual strain, caused for the action of a loading and unloading, through of the increase of the temperature of the material, in this situation the martensitic phase is stable. Already the pseudoelasticity behavior refers to the ability of the material to obtain V. Piccirillo and B.R. Pontes UNESP – S˜ao Paulo State University, Department of Engineering Mechanics, CP 473, 17033-360, Bauru, SP. e-mail: [email protected], [email protected] J.M. Balthazar and J.L.P. Felix UNESP – S˜ao Paulo State University, Department of Statistics, Applied Mathematical and Computation, CP 178, 13500-230, Rio Claro, SP. e-mail: [email protected], jorgelpfelix@ yahoo.com.br J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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Electric Motor

y Rotating Imbalance

f

m r

SMA Beam

k

x

L

M,J e

a

b

Fig. 1 (a) Diagram of physical model and (b) mathematical model with limited power supply

a very large strain upon loading and fully recover through a hysteresis loop upon loading and it always occurs in high temperatures, in this situation the austenitic phase is stable. Shape memory and pseudoelasticity effects making the (SMA) research very attractive, as material with great potential for diverse applications in engineering science, such as, vibration control, active buckling control, or more generally active structural modifications schemes can be used in these mentioned applications [6, 7]. Diverse approaches and techniques have been developed to describe the thermodynamic behavior of alloys [8]. Here we will analyze a non-ideal (SMA) problem by using a numerical simulations. The main purpose of this paper is to study the possibilities of the existence of regular and irregular motions in a (SMA) non-ideal vibrating problem schematically shown in Fig. 1.

2 (SMA) Constitutive Modeling To describe behavior of the oscillator with a shape memory, the constitutive model proposed by Falk [9] has been adopted in the mathematical modeling of the problem. This model is based on Devonshire theory and defines a free energy of Helmholtz (ψ), in a polynomial form and it is capable to describe the shape memory and pseudoelasticity effects. The polynomial model, as more it is known to deals with one – dimensional cases and it does not consider an explicit potential of dissipation, and no internal variable is considered. On this form, the free energy depends only on the state variable observations (temperature and strain), that is, ψ = ψ (ε, T). The free energy is defined in such a way that, for high temperatures (T > TA ), the energy possesses only one point of minimum corresponding to the null strain representing the stability of the austenite phase (A); for intermediate temperatures

On Nonlinear Response of a Non-ideal System with Shape Memory Alloy

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(TM < T < TA ) it presents three points of a minimum corresponding to the austenitic (A) and two other martensitic phases (M+ and M− ), which are induced by some positive and negative stress fields, respectively upto the low temperature (T < TM ) there are two points of a minimum representation of the two variants of martensite (M+ e M− ), corresponding to the zero value strain. Therefore, the restrictions below are given by the following polynomial equation 1 1 1 ρψ(ε, T) = a(T − TM)ε2 − bε4 + eε6 , (1) 2 4 6 where: a and b are constants of the material, TM corresponds to the temperature at which the martensitic phase is stable and ρ is the mass density. If TA is defined as the temperature above which the austenite is stable and the free energy has only one minimum at zero strain, it is possible to write the following condition TA = TM +

b2 , 4ae

(2)

where constant e may be expressed in terms of other constants of the material. By definition [10], the stress – strains relation is given by

σ = a(T − TM)ε − bε3 + eε5 .

(3)

3 Mathematical Model of the Non-ideal System The adopted model of the vibrating system and the source of disturbance with a limited power supply are illustrated in Fig. 1. The vibrating system, consists of a mass M, a (SMA) element and a linear damping with viscous damping coefficient c. On the object with mass M, a non-ideal DC motor is placed, with a driving rotor of a moment of inertia J and r is the eccentrically of the unbalanced mass. The governing equations of motion of the system have the form ¯ 3 + e¯ x5 = 0, (M + m) x + cx − mr ϕ cos ϕ − ϕ 2 sin ϕ + a¯ (T − TM)x − bx (4) J + mr2 ϕ − mrx cos ϕ = S ϕ − H ϕ , where:

aAr bAr eAr , b¯ = 3 , e¯ = 5 . (5) L L L Let us assume that φ is the angular displacement of the rotor, S (φ ) is the controlled torque of the unbalanced rotor, H (φ ) is the resistant torque of the unbalanced rotor and Ar is the area of the element with a shape memory. Equation (4) may be simplified when the model of the DC motor is taken by removing the effect of the inductance S φ − H φ = g − hφ , (6) a¯ =

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where g represents the control parameter (voltage or constant torque) and h is a constant motor type [11]. It is convenient to work with dimensionless position and time according to u=

x and τ = ω0 t L

in such a way, that Eq. (4) is rewritten in the following form ¨ cos φ − φ ˙ 2 sin φ = 0, u¨ + 2μ u˙ + (θ − 1) u − α u3 + γ u5 − λ φ ¨ − η u¨ cos φ = ξ1 − ξ2 φ, ˙ φ

(7)

(8)

where dots represent the succesive time differentiation. Our dimensionless variables are given below bAr aAr TM T c , α= , θ= , μ= , (M + m)L TM 2(M + m)ω0 (M + m)Lω20 eAr mrL mr γ= , η= , , λ= (M + m)L (J + mr2 ) (M + m)Lω20 g h ξ1 = , ξ2 = . (J + mr2 ) ω0 (J + mr2 ) ω20 ω20 =

(9)

We have assumed by the above that the characteristic curve of the DC motor (energy source) is a straight line. As the shape memory alloys (SMA) present some different temperature dependent properties, in this article we analyse the pseudoelastic behavior considering a higher temperature, where the austenitic phase is stable in the alloy (θ = 2). Therefore the Eq. (7) is cast into the following form ¨ cos φ − φ ˙ 2 sin φ = 0, u¨ + 2μ u˙ + u − α u3 + γ u5 − λ φ (10) ˙ ¨ − η u¨ cos φ = ξ1 − ξ2 φ, φ and finally u˙ 1 = u2 , u˙ 2 =

1 (λ((ξ1 − ξ2 u4 ) cos u3 − u24 sin u3 ) 1 − λη cos2 u3 + αu31 − γu51 − (θ − 1)u1 − 2μu2 ),

u˙ 3 = u4 , u˙ 4 =

ξ1 η cos u3 + 1 − λη cos2 u3 1 − λη cos2 u3 ξ2 u 4 αu31 − γu51 − (θ − 1) u1 − 2μu2 − − λu24 sin u3 . η cosu3

(11)

On Nonlinear Response of a Non-ideal System with Shape Memory Alloy

111

Table 1 Material constants for a Cu-Zn-Al-Ni alloy [12] a(Mpa/K)

b(Mpa)

e(Mpa)

TM (K)

TA (K)

523.29

1.868 × 107

2.186 × 109

288

364.3

4 Numerical Results The objective of this section is to analyze the vibrating problem shown in Fig. 1 taking into account the linear torque defined by the Eq. (9). The numerical simulations were carried out by using the Matlab-Simulink. For numerical simulations we use the variable-step Adams-Bashforth-Moulton method. In all simulations, to analyze the behavior of the non-ideal dynamical system, the spring is assumed to be made of a Cu-Zn-Al-Ni alloy and the properties presents in Table 1. Furthermore, we consider μ = 0.01, η = 0.6, λ = 0.4 and ξ2 = 1.5. Note that the ˙ of the passage through the resonance is obtained by changes in angular velocity φ DC motor. Results presented below were obtained on the dependence of the non-ideal system after taking into account a suitable values of the parameter ξ1 . This fact is related to the greater or lower interaction depending of the value of voltage or constant torque, that is, the dimensionless values ξ1 = 0.3, 1, 1.5; or 2 that we set to investigate the passage through a resonance. Let ξ2 = 1.5 be fixed. The dynamic behavior becomes much richer because some periodic, quasi-periodic or chaotic motions are possible to observe in this non-ideal system. In order to illustrate the response of the non-ideal system, we analyzed the response at higher temperatures (θ = 2), when the alloy is fully austenitic. We also plotted the Poincar´e section which represents the surface of section (u1 (τn ) , u2 (τn )). The points (u1 (τn ) , u2 (τn )) are captured for τn = nT, where n = 1, 2, 3, ..., with period T = Ω2πM [13]. The average angular velocity ΩM is obtained numerically ΩM =

φ (τ1 ) − φ (0) u3 (τ1 ) − u3 (0) = , τ1 τ1

(12)

where τ1 is a long time period for numerical calculation. The highest interaction between the vibrating system and the energy source will occur at a resonance. We define the resonance region as follows dϕ − ω = O (ε) , dt

(13)

where dϕ dt is the angular velocity, ε is a small parameter of the problem of order −3 10 , and ω is the natural frequency of the system. Generally, for a wide range of physical parameters when the system was started from rest, the angular

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4

a

0.23

b

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0.225

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0.22 0.215

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0.21

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10-3

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0 -1

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50 100 150 200 250 300 350 400 450 500

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-0.01

-0.015 -0.01 -0.005

0

0.005 0.01 0.015

Displacement

Fig. 2 (a) Angular velocity time response; (b) phase portrait; (c) power spectrum; and (d) Poincar´e section for ξ1 = 0.3

velocity of the rotor would increase until it reached the neighborhood of the natural frequency ω. Then depending upon physical parameters values dϕ dt would increase beyond the resonance region (pass through) or it remains close to ω (capture). A periodic solution in the case of the angular velocity taken below the resonance at ξ1 = 0.3 is illustrated in Fig 2. Figure 3a illustrates a case when angular velocity is below resonance (ξ1 = 1). The quasi-periodic motion is have reported. The nature of this motion is confirmed in Fig. 3c by the power spectrum.

On Nonlinear Response of a Non-ideal System with Shape Memory Alloy 0.9

0.2

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0.15

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Frequency (rad/s)

4

4.5

5

0.2 0.15

0.1

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0

−0.05 −0.1 −0.15 −0.2

Displacement

Fig. 3 (a) Angular velocity time response; (b) phase portrait; (c) power spectrum; and (d) Poincar´e section for ξ1 = 1

Figure 4 shows an interesting dynamical behavior for ξ1 = 1.5. Figure 4a illustrates a case when angular velocity is captured into the resonance region, causing a chaotic motion (see Fig. 4b). Chaotic dynamics in these kinds of systems can be quantitatively characterized by means of the power spectra (Fig. 4c). A strange attractor on the Poincar´e section (see Fig. 4d) obtained for non-ideal system has the complicated fractal structure with features of chaotic motion. Figure 5a illustrates a case of passage through resonance region for ξ1 = 2. In this case we obtained chaotic motion. The broadband character observed in the power spectrum is a characteristic feature of a chaotic solution. We have used the Poincar´e section to characterize the dynamic of the system. In Fig. 5d a strange attractor behavior of the system for ξ1 = 2 is reported.

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0.4

a

0.2 0.1

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1

0.6

0

−0.1

0.4

−0.2

0.2

−0.3 −0.4 −0.25−0.2−0.15−0.1−0.05 0 0.05 0.1 0.15 0.2 0.25 Displacement

0 0

50 100 150 200 250 300 350 400 450 500 Time

0.3

C

7

d

0.2

6

0.1 Velocity

5 Magnitude

b

0.3

1.2

4 3

0

−0.1

2

−0.2

1 −0.3 0 0

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2

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3

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Frequency (rad/s)

4

4.5

5

−0.15 −0.1 −0.05

0

0.05

0.1

0.15

Displacement

Fig. 4 (a) Angular velocity time response; (b) phase portrait; (c) power spectrum; and (d) Poincar´e section for ξ1 = 1.5

5 Conclusions In this paper, we analyzed the influence of the (SMA) spring on the non-ideal system in the passage through resonance. We present the study on the pseudoelastic behavior, considering a higher temperature, where austenitic phase is stable in the alloy. The torque generated by DC motor is of limited power supply and, according to classical Kononenko theory, is assumed as a straight line. The numerical simulations were performed by using the linear torque as well as by consideration of the primary resonance. During passage through the resonance of the DC

On Nonlinear Response of a Non-ideal System with Shape Memory Alloy 2

0.8

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115

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Frequency (rad/s)

4

4.5

5

−0.2 −0.15 −0.1 −0.05

0

0.05 0.1 0.15 0.2

Displacement

Fig. 5 (a) Angular velocity time response; (b) phase portrait; (c) power spectrum; and (d) Poincar´e section for ξ1 = 2

motor-structure system what is modeled as a (SMA) oscillator, with a non-ideal excitation some “severe” vibrations appear. The interaction between the motor and the oscillating system is confirmed by the different phase portraits. The numerical results presented in this paper show that it is possible to get any regular and chaotic motions for the variation of the control parameter. Rising ξ1 some chaotic motions appear as well.

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References 1. Balthazar JM, Mook DT, Weber HI, Brasil RMFLRF, Fenili A, Belato D, Felix JLP (2003) An overview on non-ideal vibrations. Meccanica 38, 613–621. 2. Belato D, Weber HI, Balthazar JM, Mook DT (2001) Chaotic vibrations of a non-ideal electromechanical system. International Journal of Solids and Structures 38, 1699–1706. 3. Dantas MJH, Balthazar JM (2007) On the existence and stability of periodic orbits in nonideal problems: general results. ZAMM Journal of Applied Mathematics and Mechanics 58, 940–956. 4. Felix JLP, Balthazar JM, Brasil RMFLRF (2005) On tuned liquid column dampers mounted on a structural frame under a non-ideal excitation. Journal of Sound and Vibration 282, 1285– 1292. 5. Pontes BR, Oliveira VA, Balthazar JM (2000) On friction-driven vibrations in a mass blockbelt-motor system with a limited power supply. Journal of Sound and Vibration 234, 713–723. 6. Pietrzakowski M (2000) Natural frequency modification of thermally activated composite plates. Mecanica 1, 313–320. 7. Rogers CA, Liang C, Fuller CR (1991) Modeling of shape memory alloy hybrid composites for structural acoustic control. Journal of Acoustic Society of America 1, 210–220. 8. Paiva A, Savi MA (2006) An overview of constitutive models for shape memory alloys. Mathematical Problems in Engineering 2006, 1–30. 9. Falk F (1980) Model free–energy, mechanical and thermodynamics of shape memory alloys. ACTA Mettalurgica 28, 1773–1780. 10. Savi MA, Braga AMB (1993) Chaotic vibration of an oscillator with shape memory. Journal of the Brazilian Society of Mechanical Science 15, 1–20. 11. Warminski J, Balthazar JM, Brasil RMFLRF (2001) Vibrations of a non-ideal parametrically and self-excited model. Journal Sound and Vibration 245, 363–374. 12. Savi MA, Pacheco PMCL, Braga AMB (2002) Chaos in a shape memory two-bars truss. International Journal of Non-Linear Mechanics 37, 1387–1395. 13. Zucovic M, Cveticanin L (2007) Chaotic response in a stable Duffing system of non-ideal type. Journal of Vibration and Control 13, 751–767.

Dynamics of a Material Point Colliding with a Limiter Moving with Piecewise Constant Velocity ´ Andrzej Okninski and Bogusław Radziszewski

1 Introduction Vibro-impacting systems are very interesting examples of non-linear dynamical systems with important technological applications [1]. Dynamics of such systems can be extremely complicated due to velocity discontinuity arising upon impacts. A very characteristic feature of impacting systems is presence of non-standard bifurcations such as border-collisions and grazing impacts appearing in the case of motion with low velocity after impact, which often leads to complex chaotic motion [1]. The main difficulty with investigating impacting systems is in gluing pre-impact and post-impact solutions. The Poincar´e map, describing evolution from an impact to the next impact, is thus a natural tool to study such systems. In the present paper we investigate motion of a material point in a gravitational field colliding with a moving motion-limiting stop. Typical example of such dynamical system, related to the Fermi model, is a small ball bouncing vertically on a vibrating table. Since evolution between impacts is expressed by a very simple formula the motion in this system is easier to analyze than dynamics of impact oscillators. It is possible to simplify the problem further assuming a special motion of the limiter. The paper is organized as follows. In the second section of this article a onedimensional motion of a material point in a gravitational field, colliding with a limiter, representing unilateral constraints, is considered. In Section 3 results of numerical simulations are described while in Section 4 analytical results are presented. We discuss our results in the last section.

A. Okni´nski and B. Radziszewski ´ ˛ tokrzyska, Wydział Zarza˛ dzania i Modelowania Komputerowego, 25-314 Politechnika Swie Kielce, Al. Tysia˛ clecia P.P 7, Poland, e-mail: [email protected], [email protected]

J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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2 Motion with Impacts: A Simple Motion of the Limiter We consider motion of a material point moving vertically in a gravitational field and colliding with a moving motion-limiting stop, representing unilateral constraints. We shall assume that limiter’s mass is so large that its motion is not affected at impacts. Motion of the material point between impacts is described by equation: mx¨ = −mg,

(1)

where x˙ ≡ dx/dt and motion of the limiter is adopted as: y = y (t),

(2)

with a known function y. We shall also assume that y (y) ˙ is continuous (piecewise continuous) function of time. Impacts are modeled as follows:

+

x˙ τi

x (τi ) = y (τi ), − y( ˙ τi ) = −R[x˙ τi− − y( ˙ τi )],

(3a) (3b)

where impact times are neglected with respect to time intervals of motion between impacts and impacting bodies are considered perfectly rigid. In Eqs. (3) τi stands + for time of i-th impact while x˙− i , x˙i are left-sided and right-sided limits of x˙i (t) for t → τi , respectively, and R is the coefficient of restitution, 0 ≤ R < 1 [2]. − Let us consider Eq. (1) for t ∈ τi+ , τi+1 . General solution of this equation reads: 1 (1) (2) x(t) = − gt 2 + ci t + ci . 2

(4)

Applying to Eq. (4) impact conditions (3) the Poincar´e map is obtained [3]:

γ Y (Ti+1 ) = γ Y (Ti ) − (Ti+1 − Ti )2 + (Ti+1 − Ti )Vi ,

(5a)

Vi+1 = −RVi + 2R (Ti+1 − Ti ) + γ (1 + R) Y˙ (Ti+1 ) ,

(5b)

where Ti , Y (Ti ) , Vi , and γ are nondimensional time, position, velocity and acceleration, respectively: Ti = ωτi , Y (Ti ) = y (τi ) /a, Vi = (2ω / g) x˙ τi+ , γ = 2ω 2 a / g (6) and ω and a determine time and length scales. In our previous paper we have assumed limiter’s motion in form y (t) = a sin (ω t) [3]. This choice leads to serious difficulties in solving Eq. (5a) thus making analytical investigations of dynamics hardly possible. Accordingly, we have decided to choose the limiter’s motion as simple as possible. Let us thus assume that the limiter moves up periodically in time intervals [kT, (k + 1) T), k = 0, 1, . . . , where T = ω −1 , with a constant velocity v from level y (0) = 0 to the level y (T− ) = a and

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119

Y(T) 1.0 0.8 0.6 0.4 0.2

0

1

2

3

4 T

Fig. 1 Motion of the limiter

then immediately falls back, y (T+ ) = 0. In units (6) motion of the limiter is given by: Y (Ti ) = Ti ( mod 1) (7) (see Fig. 1), and Eqs. (5) become:

γ (Ti+1 ( mod 1)) = γ (Ti ( mod 1)) − (Ti+1 − Ti )2 + (Ti+1 − Ti )Vi ,

(8a)

Vi+1 = −RVi + 2R (Ti+1 − Ti ) + γ (1 + R) .

(8b)

Since period of motion of the limiter is equal one the map (8) is invariant under the translation Ti → Ti + 1 and thus the phase space (T, V ) is topologically equivalent to the cylinder. The map (8) must meet several physical conditions to correspond exactly to the original physical problem. First of all, since Eq. (8a) may have multiple solutions, we have to choose the solution Ti+1 with the smallest nonnegative difference Ti+1 − Ti . Moreover, since Vi is the velocity of the material point just after the impact, it must not be smaller than the velocity of the limiter, i.e. the condition Vi ≥ γ must be fulfilled.

3 Computational Results We have computed bifurcation diagrams and basins of attraction for the dynamical system (8). It follows from numerical simulations that, after a stage of transient dynamics, only two modes of dynamics can establish for a given values of control parameters γ , R. Firstly, a periodic motion with m impacts per k periods, referred to as (k/m) attractor, can settle.

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Fig. 2 Bifurcation diagrams in (V, γ )

Fig. 3 Upper boundary for time of the first impact

Secondly, infinite number of impacts in one period of the limiter’s motion ending with state of zero relative velocity is also possible. Dynamics of this kind is referred to as chattering while the state of zero relative velocity is called grazing or sticking. In Figs. 2, 3 bifurcation diagrams in (V, γ ) and (T, γ ) planes are shown, respectively for R = 0.85. Vertical lines in both figures correspond to critical values of γ for 1, 2, 3, 4, ∞ numbers of impacts (from right to the left), horizontal lines in Fig. 2 show the corresponding velocities of the first impact while horizontal lines in Fig. 3 show upper boundary for time of the first impact. 3 Basins of attraction are shown in Fig. 4 for R = 0.85, γ = 37 . The whole rectangular region shown in Fig. 4, V ≥ γ , T ∈ [0, 1], is the basin of periodic attractor (1/1). Wedge like part of the phase space (darker shaded region) depicts initial conditions for which chattering is present in the first period of limiter’s motion.

Dynamics of a Material Point Colliding with a Limiter Moving

121

Fig. 4 Basins of attraction

4 Analytical Results Computational results described in the last section can be fully explained. In what follows we shall at first present analytical computations for periodic motion with m impacts per k periods, denoted as (k/m), then for chattering and, finally, we shall address the problem of finding the solution Ti+1 of Eq. (8a) with the smallest nonnegative difference Ti+1 − Ti in case when multiple solutions are possible.

4.1 Periodic Motion We shall first study periodic solutions of Eqs. (8), for which the following conditions (k/1) (k/1) must be fulfilled: Vn+1 = Vn = V∗ , Tn+1 = Tn + k = T∗ + k, k = 1, 2, . . . , i.e. one impact per k periods. Substituting these conditions into Eqs. (8) we obtain the manifold of fixed points in the (T,V ) space: (k/1)

V∗

(k/1)

= k, γcr

=k

1−R , k = 1, 2, . . . , 1+R

(k/1)

T∗

∈ (0, 1) and arbitrary. (k/1)

(9)

It is important that the solutions appear exactly for γ = γcr (analogous solutions are known for limiter’s motion given by y (t) = a sin (ω t) but are stable in some invariant of parameter γ [3]). (k/1) Stability of solutions (9) can be studied in the standard way. Let Vn = V∗ + (k/1) un , Tn = T∗ + dn where un and dn are small perturbations. Then the linearized equations are of form: 1 (1 + R)/2 dn+1 dn = . (10) 0 R2 un+1 un

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Let us note that one of eigenvalues of the stability matrix in (10) is one (in T eigendirection) while another is R2 < 1. It follows that fixed points (9) are stable for R < 1 and form a manifold (the whole unit interval) on the T axis (in the y (t) = a sin (ω t) case fixed points have sharp values [3]). The results obtained above stress differences between the limiter’s motion (7) and y (t) = a sin (ω t) [3]. However, in the case of the simpler motion of the limiter (7) it is possible to carry out analytical computations much further. Indeed, let us now consider the case of m impacts per one period. This means that the following conditions must be fulfilled: V∗m+1 = V∗1 , T∗m+1 = T∗1 + 1. Substituting these conditions into Eqs. (8) we arrive at the following set of equations: ⎧ ⎪ γ = − (T∗2 − T∗1 ) + V∗1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪γ = − T∗3 − T∗2 + V∗2 ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎨γ (T − T ) = − (T + 1 − T )2 + (T + 1 − T )V ∗1 ∗m ∗1 ∗m ∗1 ∗m ∗m (11) ⎪ V∗2 = −RV∗1 + 2R (T∗2 − T∗1 ) + γ (1 + R) ⎪ ⎪ ⎪ ⎪ ⎪ V∗3 = −RV∗2 + 2R T∗3 − T∗2 + γ (1 + R) ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎩ V∗1 = −RV∗m + 2R (T∗1 + 1 − T∗m ) + γ (1 + R) which can be solved to yield manifold of fixed points:

(1 − R) 1 + R + 2R2 + (1 − R) Rm (1 − R)2 (1 + Rm) (1/m) (1/m) γcr = = , V∗1 , (1 + R)2 (1 − Rm) (1 + R)2 (1 − Rm ) (1/m)

T∗1

∈ (0, τm ) .

(12a)

We still have to demand that all impacts take place in the same period of the limiter’s motion. This condition, T∗m < 1, leads to inequality T∗1 < τm where

τm =

(1 − R) (1 + Rm) . (1 + R) (1 − Rm)

(12b)

We check that for k = m = 1 Eqs. (9), (12) agree. There are also other possibilities of periodic motions, i.e. cycles with m impacts per k periods. For example, to obtain the (3/2) case periodicity conditions V∗i+2 = V∗i , T∗i+2 = T∗i + 3 are imposed in Eqs. (8) to get the following set of equations: ⎧ 2 γ T∗i+1 ( mod 1) = γ (T∗i ( mod 1)) − T∗i+1 − T∗i + T∗i+1 − T∗i V∗i ⎪ ⎪ ⎪ ⎨V ∗i+1 = −RV∗i + 2R T∗i+1 − T∗i + γ (1 + R) 2 , ⎪ T T V γ ( mod 1) = γ ( mod 1) − T − T + T − T ⎪ ∗ ∗ ∗ ∗ ∗ ∗ ∗ i+2 i+1 i+2 i+1 i+2 i+1 i+1 ⎪ ⎩ V∗i+2 = −RV∗i+1 + 2R T∗i+2 − T∗i+1 + γ (1 + R) (13a) (13b) V∗i+2 = V∗i , T∗i+2 = T∗i + 3, T∗i+1 ( mod 1) = T∗i+1 − 1,

Dynamics of a Material Point Colliding with a Limiter Moving

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where we assumed that first impact occurred in the first period, the second impact occurred in the second period and there were no impacts in the third period. After substituting Eqs. (13b) into (13a) we obtain: 9 1 + R2 (3/2) 2 2 2 3 1 + R − 2 (2 + R ) (1 + 2R ) . γcr = (13c) (1 + R)2 (1 − R2) Analysis of the derived formulae and numerical experiments suggest that the following simple expression (k/m) ∼ k 1 − R (14) γcr = m 1+R yields for R ≈ 1 very good approximation for critical value of γ at which the attractor (k/m) arises.

4.2 Chattering Let us consider the case of periodic motion with m impacts per period, c.f. Eqs. (11), (12). In the m → ∞ limit we get: (1 − R) 1 + R + 2R2 (1 − R)2 1−R (1/∞) (1/∞) (1/∞) . γcr = , V = , T ∈ 0, ∗ ∗ 1 1 1+R (1 + R)2 (1 + R)2 (15) (1/∞) (1/∞) (1/∞) For γ = γcr ,V = Vcr and T1 = T∗1 the material point impacts infinite number of times with the limiter in finite time during one period of motion of the limiter (1/∞) and grazes at T∞ = 1 with zero relative velocity, V∞ = γcr , see Fig. 5. Then the limiter moves down (infinitely fast) and the process repeats periodically in time because the conditions T∞+1 = T1 ( mod 1) , V∞+1 = V1 are fulfilled. (1/∞) (1/∞) (1/∞) For γ = γcr ,V = Vcr and T1 < T∗1 the material point impacts infinite number of times with the limiter in finite time during one period of motion of the limiter, grazes and sticks. Then at time T = 1 the limiter escapes infinitely fast

Fig. 5 Material point impacting infinite number of times with the limiter

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and the material point has right initial conditions, T∞ = 1, V∞ = γcr , to perform periodic grazing shown in Fig. 5. Let us now consider chattering without more restrictive assumption that the process be periodic. Assumption that the material point impacts infinite number of times in a period makes possible to write Eqs. (8) in form:

γ = − (Ti+1 − Ti ) + Vi,

(16a)

Vi+1 = −RVi + 2R (Ti+1 − Ti ) + γ (1 + R) .

(16b)

It follows that: wi+1 = Rwi ,

wi = Vi − γ ,

(17)

where wi is the velocity of the material point relative to the limiter. Now for R < 1 we have wi → 0. However, we have to demand additionally that all impacts, which started at time instant T1 , took place within the same period. It follows from Eq. (16a) that: ∞ V1 − γ T∞ − T1 = (V1 − γ ) ∑ Rk = , (18) 1−R k=0 and since the time of grazing must be smaller than length of period of limiter’s motion, T∞ < 1, the condition of chattering ending with graze is obtained: V1 − γ < 1 − T1. 1−R

(19)

Since we have not demanded periodicity the fate of the material point in the next period of the limiter’s motion requires further analysis.

4.3 Multiple Solutions and Discontinuous Dependence on Velocity Equation (8a) may have multiple solutions. More exactly, these equations have one or two solutions for Ti+1 depending on the values of the control parameter γ > 0 and initial conditions Ti , Vi . Let Ti , Ti+1 ∈ [0, 1]. In this case Eq. (8a) takes the form:

γ (Ti + δi ) = γ Ti − δi2 + δi Vi ,

(20)

where δi = Ti+1 − Ti , and has solutions: (1)

δi

= 0,

(21a)

(2)

= Vi − γ .

(21b)

δi

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125

(1)

We note that δi is the trivial solution corresponding to the impact at T = Ti . We (2) thus accept δi as the correct solution provided that the consistency conditions are fulfilled: (2) δi = Vi − γ ≥ 0, Ti+1 = Ti + Vi − γ ≤ 1. (22) When Vi grows the critical condition is reached: Ti + Vi − γ = 1,

(23)

and then, at Vi still growing, the solution (21b) does not fulfill the second of the conditions (22). In this case we assume Ti ∈ [0, 1], Ti+1 ∈ [1, 2] and Eq. (8a) is written as: γ (Ti + δi − 1) = γ Ti − δ 2 + δi Vi . (24) Solutions of Eq. (24) read (3, 4)

δi (4)

where only δi

=

1 Vi − γ ± (Vi − γ )2 + 4γ , 2

(25)

(3)

is meaningful since δi < 0, provided that consistency conditions: 1 Vi − γ ± (Vi − γ )2 + 4γ ∈ [1, 2] (26) Ti+1 = Ti + 2 (2)

(4)

. out that the solutions δi , δi -hold. It turns (1) (2) Vcr ,Vcr where: (1)

Vcr = 1 − Ti + γ −

γ , 1 − Ti

coexist in the velocity interval: Vi ∈

(2)

Vcr = 1 − Ti + γ ,

(27)

(2)

i.e. Vcr fulfills (23). It follows that when the velocity grows from Vi < V (2) to (2) (4) Vi > V (2) the solution for δi changes discontinuously from δi to δi . In general, multiple solutions can exist for Ti+1 ∈ [k, k + 1] , k = 1, 2, . . .. In Fig. 6 the function F(δ , T,V ) was plotted: F(δ , T,V ) = γ [(T + δ ( mod 1)) − (T ( mod 1))]/δ + δ − V,

(28)

where T = 0.63 and V = 0.4 (thin solid line), V = 0.49 (thick solid line, critical value of velocity), V = 0.6 (dashed line). Discontinuity in the figure is at δ = 1 − T (vertical dotted line). Indeed, for increasing value of the velocity number of solutions of Eq. (28) changes from two to one and the smallest solution of this equation changes discontinuously.

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Fig. 6 The shape of function F (δ , T,V )

5 Conclusions In the present paper we have investigated dynamics of a material point moving in gravitational field and colliding with a moving limiter. The motion of the limiter has been assumed as simple as possible, i.e. periodic with piecewise constant velocity. On the basis of analytical as well as computational results described above we shall sketch the overall picture of dynamical behaviour of the system (8). For (1/∞) γ < γcr there is only one asymptotically stable attractor, the grazing manifold Y (T ) = T ( mod 1). For initial conditions T1 , V1 obeying inequality (19) the material point performs infinite number of impacts and sticks after time T∞ and this process repeats in subsequent periods. Otherwise, the material point will approach the attractor asymptotically. (1/∞) In the case when γ = γcr there is only one stable attractor – periodic chattering without sticking, see Fig. 5. (1/∞) When γ > γcr there are two possibilities. Firstly, the parameter γ is exactly equal critical value for which a periodic attractor (k/m) with m impacts per k peri(k/m) ods (with specified sequence of the impacts) is asymptotically stable, i.e. γ = γcr . Then the material point approaches the attractor asymptotically. For initial conditions fulfilling inequality (19) chattering ending with sticking occurs in the first period as a transient process. This has been shown in Fig. 4. Let us stress that also in this case there is only one asymptotically stable attractor (k/m). All these (k/m) (k/m) attractors have sharply defined velocities V∗i while impact times T∗i belong to some manifolds (i = 1, 2, . . . , m), see for example Eqs. (12). We have compared in Section 1 properties of the (1/1) attractor from this work and [3].

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On the other hand, number of periodic attractors (k/m) is infinite but countable. (k/m) Therefore, there is another possibility that γ = γcr for any integer m and k (for a specified sequence of impacts). In this case motion of the material point should be quasi-periodic.

References 1. di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P (2008) Piecewise-Smooth Dynamical Systems. Theory and Applications. Springer, London. 2. Stronge WJ (2000) Impact Mechanics. Cambridge University Press, Cambridge. 3. Okniski A, Radziszewski B (2007) Grazing dynamics and dependence on initial conditions in certain systems with impacts, arXiv:0706.0257.

Transient in 2-DOF Nonlinear Systems Yuri Mikhlin, Gayane Rudnyeva, Tatiana Bunakova, and Nikolai Perepelkin

1 Introduction An investigation of transient is important in engineering, in particular, in problem of absorption. Over the past years different new devices have been used for the vibration absorption and for the reduction of the transient response of structures [1–4]. It seems interesting to study nonlinear passive absorbers for this reduction. In presented paper the transient in a system containing a linear oscillator, linearly coupled to an essentially nonlinear attachment with a comparatively small mass, is considered. A damping is taken into account. It is assumed that some initial excitation implies vibrations of the linear oscillator. The multiple scales method [5] is used to construct a process of transient in some nonlinear systems. A transfer of energy from the initially perturbed linear subsystem to the nonlinear absorber can be observed. A similar construction is made to describe the transient in a system which contains a linear oscillator and a vibro-impact absorber with a comparatively small mass. Both an exact integration with regards to impact conditions, and the multiple scales method are used for this construction. The transient in such system under the external periodical excitation was considered too. Besides, a transient in a system which describes an interaction of some rotating subsystem and the elastic one is constructed by using the multiple scales method too. Such system is known as nonideal system. One has a transient to one of the stationary regimes. Numerical simulation confirms an efficiency of the analytical construction in all considered systems.

Yu. Mikhlin, G. Rudnyeva, T. Bunakova, and N. Perepelkin National Technical University, Kharkov Polytechnic Institute, Kharkov, Ukraine, e-mail: [email protected], [email protected]

J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

129

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x

m

M w2

g

c

Fig. 1 The system with two connected oscillators

2 Transient in a System Containing an Essentially Nonlinear Oscillator as Absorber Let us consider a system with two connected oscillators, namely one linear and one nonlinear with a comparatively small mass, which is considered as absorber of the linear oscillator vibrations (Fig. 1). Here M is a mass of the main linear subsystem, m is a mass of the nonlinear absorber, ω2 ,γ and c characterize elastic springs, δ characterizes a linear dissipation force. To emphasis a smallness of some inertial and elastic characteristics of the absorber, as well a smallness of the dissipation force, the next transformations will be used, namely: m → εm, c → εc, γ → εγ, δ → ε2 δ, where ε is the small parameter (ε << 1). So, the system under consideration is described by the following differential equations εmx¨ + εcx3 + ε2 δx˙ + εγ(x − y) = 0, (1) M y¨ + ω2 y + ε2δy˙ + εγ(y − x) = 0. The solution of the system (1) will be found by the multiple-scale method. One has x = x0 (t0 ,t1 ,t2 , . . .) + εx1 (t0 ,t1 ,t2 , . . .) + ε2 x2 (t0 ,t1 ,t2 , . . .) + . . . , y = y0 (t0 ,t1 ,t2 , . . .) + εy1 (t0 ,t1 ,t2 , . . .) + ε2 y2 (t0 ,t1 ,t2 , . . .) + . . . , Where t0 = t, t1 = εt, t2 = ε2t, . . . , tn = εnt, . . . , d ∂ dt0 ∂ dt1 ∂ dt2 = + + + ... dt ∂ t0 dt ∂ t1 dt ∂ t2 dt ∂ ∂ ∂ ∂ = +ε + ε2 + ε3 + ... ∂ t0 ∂ t1 ∂ t2 ∂ t3 = D0 + εD1 + ε2 D2 + ε3 D3 + . . . etc. One obtains in zero approximation by the small parameter the next equation: ε0 : MD20 y0 + ω2 y0 = 0.

(2)

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131

The solution of this equation is y0 = A1 (t1 ,t2 , . . .) cos ψ0 , y0 = A1 (t1 ,t2 , . . .) cos ψ0 , where

ω2 . M One has in the next approximation by the small parameter the following ODE system: mD20 x0 + cx30 + γ(x0 − y0) = 0, 1 ε : MD20 y1 + 2MD0 D1 y0 + ω2y1 + γ(y0 − x0 ) = 0. ψ0 = Ωt0 + ϕ0 (t1 ,t2 , . . .),

Ω2 =

Let us find the presentation of the x0 in the form x0 = B1 (t1 ,t2 , . . .) cos ψ0 + B2(t1 ,t2 , . . .) cos ψ1 , where ¯ 1 ,t2 , . . .)t0 + ϕ1 (t1 ,t2 , . . .). ψ1 = Ω(t Equating cosine coefficients in the first equation and eliminating secular terms in the second one we get nonlinear algebraic equations: −mB1 Ω2 + c 34 B31 + 32 B1 B22 + γB1 = γA1 , ¯ 2 + 3 cB2 + 3 cB2 = 0, γ − mΩ 2 1 4 2 2MA1 Ω ∂∂ϕt 0 + γB1 − γA1 = 0, ∂ A1 ∂ t1

= 0.

Thus

1

.

A1 = A1 (t2 ,t3 , . . .),

∂ ϕ0 γ (A1 − B1) . = ∂ t1 2MA1 Ω Escaping calculations of the next approximations in the multiple scale method we give expressions for the amplitudes, frequencies and phases of zero-approximation x0 , y0 of (2): δ

B2 = c(t ¯ 2 ,t3 , . . .)e− 2m t1 , A1 =

δ

B1 = c0 (t2 ,t3 , . . .) + c2 (t2 ,t3 , . . .)e− m t1 ,

γ − mΩ2 3 c0 + cc30 , γ 4γ

2

Ω = (γ + (3/4)cB22 + (3/2)cB21)/m. After time-averaging one has the following:

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Fig. 2 Comparison of results of analytical approximation (solid line) and ones obtained by RungeKutta procedure (dash line)

2 Ω ∼ = (γ + (3/2)cc20)/m, γ γ m δ

t1 − ϕ0 = c0t1 − c2 e− m t1 + c∗2 , 2MΩ 2MΩA1 δ

where c2 =

3 2 2 cc¯ c0

mΩ2 − γ − 94 cc20

.

In such a way we have got the zero-approximation of sought solution containing four constants with respect to time t0 , namely c∗1 = c∗1 (t3 ,t4 , . . .), c∗2 = c∗2 (t2 ,t3 , . . .), c∗3 = c∗3 (t2 ,t3 , . . .), c¯ = c(t ¯ 2 ,t3 , . . .). They were found numerically by Newton method from initial conditions: x(0) = x(0) ˙ = 0, y(0) = 0, y(0) ˙ = V. Figures 2 and 3 presents results of comparing the analytical solution (zeroapproximation) with the numerical simulation obtained by using the Runge-Kutta procedure for different initial values.

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Fig. 3 Comparison of results of analytical approximation (solid line) and ones obtained by RungeKutta procedure (dash line)

F cos jt d d c2

g m

M

2

1

x y

Fig. 4 The vibro-impact system under consideration

3 Transient in the Vibro-Impact System One considers the 2-DOF vibro-impact system with the one-sided catch (Fig. 4). This system contains the linear oscillator and the absorber with a comparatively small mass. It is supposed to obtain analytical description of transient, both for free and forced oscillations by using the multiple-scale method. Equations of motion for the system under consideration in a case of the free vibrations are the following: εmx¨ + εγ(x − y) + ε2δx˙ = 0, (3) M y¨ + c2 y + εγ(y − x) + ε2δy˙ = 0,

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where all notations and transformations of parameters are the same as in the Section 2. The small parameter (ε << 1) is introduced to select a smallness of the absorber mass, the connection between oscillators and the dissipation force. It is supposed that an impact here is instantaneous. The restoration coefficient (0 ≤ e ≤ 1) characterizes a lost of velocity in the instant of impact. One has the following conditions of the impact: x(tk+ ) = x(tk− ) = xmax , ˙ k− ), x(t ˙ k+ ) = −ex(t y(tk+ ) = y(tk− ), ˙ k− ). y(t ˙ k+ ) = y(t

(4)

Here: tk is the impact instant, where k is a number of the impact, tk− is an instant before impact, tk+ is one after impact, xmax is a distance between the equilibrium state and the catch.

3.1 Free Oscillations in the Vibro-Impact System To construct an analytical solution by using the multiple scale method, the expansions given in (2) are used. In zero approximation by a small parameter the next solution can be obtained: y0 = A0 (t1 ,t2 ,t3 , ...) cos Ωt0 + B0 (t1 ,t2 ,t3 , ...) sin Ωt0 , where Ω20 = c2 /M,

(5)

x0 = β(A0 (t1 , ...) cos Ωt0 + B0 (t1 , ...) sin Ωt0 ) + √ + A1 (t1 , ...) cos γ/mt0 + B1 (t1 , ...) sin γ/mt0 , where β=

γ . m(γ /m − Ω20)

Conditions of secular terms elimination in the next approximation by the small parameter give us the following expressions for amplitudes of the zero approximation: A0 = −C1 sin Ω1t1 + C2 cos Ω1t1 , B0 = C1 cos Ω1t1 + C2 sin Ω1t1 , where

Transient in 2-DOF Nonlinear Systems

135

Ω1 =

γ(β − 1) . 2MΩ0

Taking onto account the next approximation, one has the approximate solution of the form: x = β(cos Ω2t · (−R1C1 + R2C2 ) + sin Ω2t · (R2C1 + R1C2 )) + eαε t {C3 sin β3t + C4 cos β3t},

y = C1 sin Ω2t + C2 cosΩ2t + εβ1 eαε t {C3 sin β3t + C4 cosβ3t}, εδΩ 2εΩΩ1 , R2 = 1 − , R1 = m (γ / m − Ω2) γ / m − Ω2 β3 = γ / m − β2 ε, Ω2 = Ω − εΩ1 . Impact conditions (4) give the next relations connecting coefficients Ci before (Cik ) and after impact (Cik+1 ): β(cos Ω2t · (−R1C1k+1 + R2C2k+1 ) + sin Ω2t · (R2C1k+1 + R1C2k+1 )) + + , + eαε t C3k+1 sin β3t+ C4k+1 cos β3t = β(cos Ω2t · (−R1C1k + R2C2k ) + + , + sin Ω2t · (R2C1k + R1C2k )) + eαε t C3k sin β3t+ C4k cos β3t .

Ω2 β − sin Ω2t · (−R1C1k+1 + R2C2k+1 ) + cosΩ2t · (R2C1k+1 + R1C2k+1 ) + + + , ,

+ eαε t αε C3k+1 sin β3t+ C4k+1 cos β3t + β3 C3k+1 cos β3t− C4k+1 sin β3t =

= −eΩ2 β − sin Ω2t · (−R1C1k + R2C2k ) + cosΩ2t · (R2C1k + R1C2k ) + + + , ,

+ eαε t αε C3k sin β3t+ C4k cos β3t + β3 C3k cos β3t− C4k sin β3t 0 1 C1k+1 sin Ω2t + C2k+1 cos Ω2t + εβ1 eαε t C3k+1 sin β3t+ C4k+1 cosβ3t = 0 1 = C1k sin Ω2t + C2k cos Ω2t + εβ1 eαε t C3k sin β3t+ C4k cos β3t +

Ω2 C1k+1 cos Ω2t − C2k+1 sin Ω2t + εβ1 eαεt αε C3k+1 sin β3t + C4k+1 cos β3t +

, + β3 C3k+1 cos β3t − C4k+1 sin β3t = +

= Ω2 C1k cos Ω2t − C2k sin Ω2t + εβ1 eαεt αε C3k sin β3t + C4k cosβ3t +

, + β3 C3k cos β3t − C4k sin β3t . The numeric simulation was realized for the next values of parameters: M = 1, m = 1, ε = 0.01, δ = 10, e = 0.9, xmax = 1.4, γ = 1.5, c = 1. Initial values model the instant impact to the linear subsystem: x(0) = 0, x(0) ˙ = 0, y(0) = 0, y(0) ˙ = V˙0 = 1.

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t 1

765 1529 2293 3057 3821 4585 5349 6113 6877 7641 8405 9169 9933

-1 -1.5 -2

Fig. 5 Transient in a case of free oscillations in the vibro-impact system

Comparison of the analytical solution and numerical simulation shows a good exactness of the analytical approximation (Fig. 5).

3.2 Transient in a Case of Forced Oscillations One considers the same 2-DOF vibro-impact system in a case when an external periodic force acts to linear subsystem. The multiple scales method can be successfully used in this case too. In contrast with the solution, obtained in the Sub-section 3.1, the part corresponding to the external excitation, has to be added. One writes the solution of the form: x = β(cos Ω2t · (−R1C1 + R2C2 ) + sin Ω2t · (R2C1 + R1C2 )) + +eαε t {C3 sin β3t+ C4 cos β3t} + (F2 + εF5 ) cos ϕt + εF6 sin ϕt, y = C1 sin Ω2t + C2 cos Ω2t + εβ1 eαε t {C3 sin β3t+ C4 cos β3t} + + (F1 + εF3 ) cos ϕt + εF4 sin ϕt, where F1 = F5 =

F

(Ω2 −ϕ2 )

γF3 , m(γ/ m−ϕ2 )

,

F6 =

F2 = γ m F4 +

γF1 , m(γ/ m−ϕ2 )

(2+ mδ )F2 ϕ

γ/ m−ϕ2

(6) F3 =

−γ (F1 −F2 ) , M(Ω2 −ϕ2 )

F4 =

2ϕF1 , Ω2 −ϕ2

.

Impact conditions (4) give some relations connecting coefficients Ci before (Cik ) and after impact (Cik+1 ). These relations are not presented here. Numerical simulation was made for the same parameters and initial values, as in the preceding Sub-section. Comparison of the analytical solution and numerical simulation (Fig. 6) shows a good exactness of obtained analytical approximation. Vibrations of the linear subsystem with big mass are presented in the Fig. 6.

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y(t) 0,6 0,4 0,2 0 1

t 821 1641 2461 3281 4101 4921 5741 6561 7381 8201 9021 9841

−0,2 −0,4 −0,6

Fig. 6 Transient in a case of forced vibrations in the vibro-impact system

L(j)

D

I

j

r

c1

r

t sinj

j

a m

c0

c1

Fig. 7 The nonlinear system with limited power supply

4 Transient in 2-DOF Nonlinear System with Limited Power Supply One considers a transient in a system which describes an interaction of some rotating subsystem and the linear elastic one. A model of this system is presented in Fig. 7. Equations of motion of this system are the following: mx¨ + βx˙ + cx = c1 r sin(ϕ), (7) ¨ = L(ϕ) ˙ − H(ϕ) ˙ + c1r(x − r sin(ϕ)) cos(ϕ). Iϕ In the above m is a mass of the linear elastic subsystem, I is the inertia moment of the rotating subsystem, the coefficient β characterizes a linear dissipation force, r

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˙ is a controlled torque of the is a radius of the transmission shaft, the function L(ϕ) ˙ is a resistance torque of the rotor, the coefunbalanced rotor of DC motor; H(ϕ) ficient c1 characterizes the rotating and elastic subsystems interaction. The system under consideration is known as non-ideal system [4–6]. It means that the excitation is influenced by the response of the supporting elastic structure and that the energy source has a limited power supply (non-ideal excitation). The Eqs. (7) may be simplified when we accept the torques as linear. One has in this case: ˙ − H(ϕ) ˙ = A − Bϕ. ˙ L(ϕ) We introduce the new dimensionless variables, y = x/r, τ = ωt, and the next parameters M = A/(Iω2 ), N = B/(Iω2 ), εq = c1 r2 /(Iω2 ), εk = c1 /(mω2 ), εh = β/(mω). where ε is a formal small parameter. One can rewrite the Eqs. (7) as follows y + εhy + ω2y = εK sin ϕ, Iϕ = M − Nω + εq(y cosϕ − 0.5 sin 2ϕ).

(8)

A procedure of multiple-scale method which is similar to (2) can be used here. One has the next equations of the zero approximation by the small parameter, and the corresponding solution: D20 y0 + y0 = 0, 0 ε : D20 ϕ0 = M − NωD0ϕ0 , (9) y = A (T , T ) sin(T + Ψ (T , T )), 0

0

1

2

0

0

1

2

M 1 −NωT0 e T0 + F0 (T1 , T2 ) . ϕ0 = Φ0 (T1 , T2 ) + Nω Nω To simplify a construction of solution in the next approximations we will consider the transient after some instant when the exponent in (9) can be negligible. Note that in concrete systems this interval of transient is very small. In this case to eliminate secular terms in the next approximation by the small parameter ε , it must satisfy the following equations:

∂ A0 − A0 = 0 → A0 = A1 (T2 )e−hT1 /2 , ∂ T1 ∂ Ψ0 = 0 → Ψ0 = Ψ1 (T2 ), − A0 ∂ T1 ∂ Φ0 = 0 → Φ0 = Φ1 (T2 ). − Nω ∂ T1 −2

(10)

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139

Then we can write equations of the first approximation by the small parameter, namely: D20 y1 + y1 = K sin(Φ0 + MT0 )/(N ω ), D20 ϕ1 + NωD0 ϕ1 = qA0 sin(T0 + Ψ0 ), − 0.5q sin2(Φ0 + MT0 )/(Nω).

(11)

After some transformation the transient can be presented as −1 y(τ) = A1 (T2 )e−hT1 /2 sin(T0 + Ψ1 (T2 )) + εK 1 − Θ2 sin(Φ1 (T2 ) + ΘT0 ) 1 −NωT0 e + ϕ(τ) = Φ1 (T2 ) + ΘT0 + F1 (T2 ) Nω 2 3 q 1 1 Nω sin((1 − Θ)T0 + Ψ1 − Φ1 + arctg ) + + ε − A1 (T2 )e−hT1 /2 2 1−Θ (N ω )2 + (1 − Θ)2 |1 − Θ| 2 3 1 1 q Nω + ε − A1 (T2 )e−hT1 /2 ) + sin((1 + Θ)T0 + Ψ1 + Φ1 + arctg 2 1+Θ (Nω)2 + (1 + Θ)2 |1 + Θ| 2 3 1 q 1 Nω +ε sin(2ΘT0 + 2Φ1 + arctg ) , (12) 2 (Nω)2 + (2Θ)2 |2Θ| 2Θ

where Θ = M/(N ω ). The solution (12) describes a transfer to a non-resonance stationary solution having frequencies Θ and 2Θ. Vibration amplitudes of this non-resonance regime are not large. The numerical checking calculation shows a very good exactness of the transfer analytical presentation in a region of the non-resonance stationary regime stability. But if this stationary regime is unstable, one has a transfer to the 1:1 resonance stationary regime with large amplitudes.

5 Conclusions The obtained results show an efficiency of the multiple-scale method to describe a transient in different kinds of nonlinear systems, including essentially nonlinear systems. It is important that an exactness of the analytical presentation is good for a sufficiently large time interval. Acknowledgement This work was partly supported by the Foundation of fundamental researches of Ukraine under the project No. Φ25.1/042.

References 1. Shaw J, Shaw S, Haddow AJ (1989) On the response of the non-linear vibration absorber, International Journal of Nonlinear Mechanics 24, 281–293. 2. Frolov KV (ed.) (1995) Vibrations in Engineering, Mashinostroenie, Moscow (in Russian).

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3. Cuvalci O, Ertas A (1996) Pendulum as vibration absorber for flexible structures: experiments and theory, Transactions of the ASME, Journal of Vibration and Acoustics 118, 558–566. 4. Manevitch L, Gendelman O, Musienko AI, Vakakis AF, Bergman LA (2003) Dynamic interaction of a semi-infinite linear chain of coupled oscillators with a strongly nonlinear end attachment, Physica D 178, 1–18. 5. Nayfeh AH, Mook D (1969) Nonlinear Oscillations, Wiley, New York. 6. Kononenko V (1969) Vibrating Systems with Limited Power Supply, Illife, London.

On the Use of the Energetic Coefficient of Restitution in Flexible Multibody Dynamics Juana M. Mayo

1 Introduction The generalized impulse-momentum balance equations are the result of applying impulsive dynamics to the collision of bodies in multibody systems. The impulsemomentum balance equations have been applied to rigid body multibody systems [1], and to multibody systems including rigid and flexible bodies [2]. Flexibility was considered via the approach of floating frame of reference, the component mode synthesis technique was used to reduce the number of flexible co-ordinates and Newton’s rule was used to define the coefficient of restitution. It was shown that when an infinite set of mode shapes is used, the generalized impulse-momentum balance equations result in a velocity jump only at the contact area between the colliding bodies, and the other parts of the system remaining unaltered [3]. The basic assumption in impulsive dynamics is that the period of contact between the colliding bodies is so short that the coordinates remain unaltered during the period. In the approach of floating frame of reference, the coordinates are divided into the reference and flexible coordinates. The reference coordinates describe the rigid body motion in the frame of reference attached to the bodies, while the flexible coordinates account for the flexible degrees of freedom used to describe elastic deformation in the bodies. The assumption of constant coordinates contrasts with the analysis of impact-induced vibrations, which arise from the elastic motion of the flexible bodies during the contact period. It has been shown [4] that the equations are still valid because the collision process is solved a step approach (i.e., simulating the impact process requires the use of several balances). The system’s configuration changes among balances, thus allowing the flexible-coordinates to change as well. The number of times that the generalized impulse-momentum balance equations need to be solved to finish the impact process depends on the number of coordinates J.M. Mayo Department of Mechanical and Materials Engineering University of Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain, e-mail: [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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used to describe flexibility, as well as on the time step employed. The coefficient of restitution, included in the formulation to account for energy losses in the vicinity of the contact area is difficult to interpret under these conditions. This is easily understood because continuous contact is simulated as a virtual succession of instantaneous impacts. At any such impact part of the energy is assumed to be locally lost. But the number and severity of the fictitious instantaneous impacts cannot be controlled in advance and, therefore, different losses of energy can be got for the same coefficient of restitution depending on the number of coordinates or on the time step employed. All the previously mentioned works do not include friction. The presence of friction at the contact points or surfaces makes the problem more complicated as friction may lead to different modes of impact such as sticking, sliding, or reverse sliding. The corresponding generalized impulse-momentum balance equations contain the velocity changes and two impulse components, one in the normal and the other in the tangential directions to the impacting surfaces. Therefore, in order to solve the impulse-momentum equations, two additional conditions are needed. One condition is taken from Coulomb’s law of friction, the other from the definition of the coefficient of restitution. Several authors [5] have shown that the use of Newton’s hypothesis results in the violation of energy conservation principles in some cases. Poisson’s rule has been used by several authors in frictional impacts [6, 7]. Lankarani and Pereira [3] studied the frictional impact of planar rigid multibody systems. They developed an algorithm for numerical generation of Routh’s diagrams [8] in order to calculate the impulse components. Pfeiffer and Glocker [7] use similar equations but differ in calculating the impulse components. They use the linear complementary problem technique, which is computationally very efficient but differs slightly from Coulomb’s law. This technique yields different results when reverse sliding follows sliding or when there is sticking followed by sliding. Stronge [9] states that for situations where the direction of slip varies during collision, the only energetically consistent definition is the energetic coefficient of restitution. This paper proposes a formulation for the frictional impact of planar flexible multibody systems. The floating frame of reference is used to describe flexibility. Stronge’s hypothesis is employed for the definition of the coefficient of restitution. Rouths’s diagrams are used for the calculation of the impulse components, and the continuous impact of finite duration is simulated by successive virtual infinitesimal impacts. The paper is organized as follows. Section 2 explains briefly the floating frame of reference formulation. Section 3 deals with the generation of the generalized impulse-momentum balance equations. Section 4 discusses the definitions of the coefficient of restitution and their representation in the Routh’s diagrams. Section 5 presents a numerical example. Finally some conclusions are presented.

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2 Floating Frame of Reference Formulation The floating reference methods are probably the most widely found in the related literature, numerous authors have contributed to their development. To describe the motion of a flexible multibody system, a coordinate system is associated to each body. This coordinate system describes the rigid body motion. Elastic displacements, calculated using small deformation theory, are superimposed to the rigid motion. Thus the global position of an arbitrary point on body j can be written as:

r j = R j + A j u¯ oj + S j q jf , (1) where R j is a set of Cartesian coordinates that define the location of the origin of the body reference, A j is the transformation matrix between the local and the global j reference systems, u¯ o is the position of the point in the undeformed state expressed in the local system, S j is a space-dependent shape matrix and q jf is the vector of time-dependent elastic generalized coordinates of the deformable body. The separation between the rigid body motion and elastic displacements is not unique. That is, it is possible to select different reference conditions for the same problem, which lead to different floating reference systems. The main advantage of this type of representation is the applicability of the component synthesis method. The application of this method makes it possible to reduce drastically the number of elastic coordinates making it very efficient computationally. Generally, the deformed shape of the body is represented by the superposition of the normal vibration modes of the body constrained by the reference conditions. In most cases, it is not necessary to introduce static deformation modes, although occasionally they can improve convergence or make the formulation easier, as in the case of natural coordinates with fixed frontiers, in which they make it possible to impose some kinematical constraints by simply sharing coordinates. The Lagrange multipliers technique is used to account for constraints on the coordinates. The equations of motion are Mq¨ + Kq + CTq λ = Q, C(q,t) = 0,

(2)

where M and K are the mass and stiffness matrices of the system, respectively; q is the vector of coordinates of the system which contains the reference and the elastic coordinates of all the bodies belonging to the system; C(q, t) are the constraints and Cq is the constraint Jacobian matrix; λ is the vector of Lagrange multipliers; and the generalized forces Q include the external forces, and the quadratic velocity terms associated with the centrifugal and Coriolis accelerations. Further description of the method can be found in the related literature [10].

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3 Generalized Impulse-Momentum Balance Equations The time interval over which impact occurs is assumed to be very short. The reference and flexible coordinates are supposed to be constant over it. Integrating the equations of motion during this time interval yields MΔq˙ + CTq Hλ = Pg ,

(3)

Cq Δq˙ = 0,

where CTq Hλ are the impulses associated to the constraints and Pg are the impulses due to the impulsive forces at the contact points. Hλ can be eliminated from the two previous equations −1 Hλ = Cq M−1 CTq Cq M−1 Pg = HPg .

(4)

Substituting Eq. (4) into Eq. (3) results in Δq˙ = M−1 I − CTq H Pg .

(5)

As the velocity problem in a multibody system is linear, the relative velocity of the contact points can be written as a function of the derivative of the coordinates vector ˙ vr = Dq,

(6)

where matrix D depends on the position at every time step. To get the tangential and normal components of the relative velocity, two unitary vectors are defined. Unit vector n is defined in the normal direction to the contacting surfaces, while unit vector t is along the tangential direction of the contacting surfaces and perpendicular to n. The magnitudes of the component of the relative velocity can be written as ˙ vn = nT vr = nT Dq˙ = cTn q,

(7)

˙ vt = tT vr = tT Dq˙ = ctT q. Using Eqs. (5) and (7), it can be written − − T T −1 ˙ = v− v+ I − CTq H Pg , n = vn + Δvn = vn + cn Δq n + cn M vt+ = vt− + Δvt = vt− + ctT Δq˙ = vt− + ctT M−1 I − CTq H Pg ,

(8)

− where v+ n and vn are the relative normal velocities of the colliding bodies after and before impact, respectively. The same applies to the relative tangential velocities. The generalized impulse can be separated into its normal and tangential components as

Pg =

+

−

DT ( fn n + ft t)dt = cn

Pg = cn Pn + ct Pt ,

+

−

+

fn dt+ct

−

ft dt,

(9)

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where fn is the normal contact force and ft is the frictional force; and Pn and Pt are the normal and tangential impulses due to impulsive forces fn and ft . Substituting Eq. (9) into Eq. (8) yields − v+ n = vn + mnn Pn + mnt Pt ,

vt+ = vt− + mnt Pn + mtt Pt , mnn = cTn M−1 I − CTq H cn , mnt = cTn M−1 I − CTq H ct , mtt = ctT M−1 I − CTq H ct .

where

(10)

(11)

4 Coefficients of Restitution and Routh’s Diagrams In Poisson’s hypothesis, the coefficient of restitution is defined by the ratio between the accumulated normal impulses PnR and PnC corresponding, respectively, to the periods of restitution and compression phases of impact given as: e=

PnR . PnC

(12)

Lankarani and Pereira [6] developed a formulation for planar rigid multibody systems using Poisson’s hypothesis. Seven different cases of impact are identified, shown in Table 1. Stronge [9] states that for situations where the direction of slip varies during collision, the only energetically consistent definition of this coefficient is the energetic coefficient of restitution. The square of the coefficient of restitution, e2∗ , is the negative of the ratio of the elastic strain energy released during restitution to the internal energy of deformation absorbed during compression.

Table 1 Description of the seven cases of impact Cases

Description

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

Sliding and sticking in the compression phase Sliding and sticking in the restitution phase Sliding and reverse sliding in the compression phase Sliding and reverse sliding in the restitution phase Forward sliding Sticking with no relative approach tangential velocity Sliding with no relative approach tangential velocity

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P∗

e2∗

n W (Pn+ ) − W (PnC ) P vn (Pn ) dPn = − PnC =− . nC W (PnC ) 0 vn (Pn ) dPn

(13)

This definition of the coefficient of restitution explicitly separates the dissipation due to hysteresis of contact forces from that due to the friction between the colliding bodies. Routh [8] developed a graphical technique to acquire the impulses. In the diagrams normal impulse is represented versus tangential impulse. Figures 1 and 2 represent Routh’s diagrams for cases 3 and 5, respectively. The horizontal line represent the condition for ending the impact given by the Poisson’s hypothesis, i.e., Pn+ = (1 + e) PnC . The condition given by the energetic definition of the coefficient

Fig. 1 Case 3

Fig. 2 Case 5

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of restitution is not lineal as it depends both on impulses and velocities. Figure 1 shows that each definition yields a different point at the end of the impact for case 3. However, if reverse sliding doesn’t exist, as occurs in case 5, the kinetic and energetic definitions yield the same results, as shown in Fig. 2.

5 Numerical Example The system analyzed, shown in Fig. 3, is a pendulum falling due to gravity which impact against a moving surface. The pendulum is composed of a uniform rod with mass m = 1 kg and length l = 1 m. Gravity is considered g = 10m/s2 . The pendulum is initially at rest in horizontal position (ϕ = 90◦ ). When ϕ = 80◦ , it impacts with the surface. Impact parameters are a friction coefficient μ = 1 and a coefficient of restitution e = 1 (i.e. it is assumed that there are not energy losses due to hysteresis of contact forces). This system, assuming flexibility negligible, has been analyzed [7]. They showed that when the surface was at rest the behaviour of the system was dissipative, the dissipation being introduced by reversed sliding, which means that the frictional impulse acts in two different directions during compression and expansion. Figure 4 shows the results using the kinetic and the energetic definitions of the coefficient of restitution, corresponding to a case 4 (see Table 1). The dissipative behaviour is introduced by reversed sliding, which means the frictional impulse acts in two different directions during compression and expansion. Results using Poisson and Stronge’s definitions are quite different. However, when the surface is moving at 1 m/s, the situation is different. The impact proceeds without transitions to sticking or reverse sliding, so the system is energy preserving. The impact case corresponds to case 5, and both definitions of the coefficient of restitution yield exactly the same results, as shown in Fig. 5. This work includes flexibility. The value EI = 2.1 × 106 Nm2 has been assumed. To simulate correctly the finite duration of a single impact of the flexible pendulum

j j0

V0

Fig. 3 Pendulum impacting a moving surface

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Fig. 4 Rigid pendulum and fixed surface

Fig. 5 Rigid pendulum and v0 = 1m/s

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with the surface moving at 1 m/s many fictitious impacts followed by integration of the motion equations should be performed. Normal modes for the simply-supported beam will be used. Figure 6 shows the time histories for the pendulum. Flexibility has been included through 5, 8, 10 and 11 modes. Solutions for 10 and 11 modes are almost identical, achieving convergence. Under the rigid body assumption there is just one jump for each impact with the fixed surface. However, for the flexible case, each finite duration impact is represented by several fictitious impacts, each of which has a different Routh diagram. Figure 7 magnifies the first continuous impact for the case of 10 modes. Twenty five fictitious impacts have been necessary to simulate the first continuous contact. As the number of modes increases, the absolute value of the impulses in each single fictitious impact becomes smaller, as expected, and therefore, the number of impacts needed to simulate a continuous finite duration impact consequently increases. Figure 8 is analogous to Fig. 7 but using 11 instead of 10 modes. In this case the number of fictitious impacts has increased to 34.

Fig. 6 Flexible pendulum

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Fig. 7 Normal impulses for the first impact with ten modes

Fig. 8 Normal impulses for the first impact with 11 modes

6 Conclusions This works simulates impacts with friction in planar multibody systems. The proposed approach uses the momentum balance equations together with Routh’s diagrams.

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This approach has been previously used for rigid multibody systems including Poisson’s coefficient of restitution. As Poisson’s hypothesis is energetically inconsistent when the direction of slip varies during collision, in this work a formulation including the energetic coefficient of restitution has been developed. Numerical simulations show that both definitions of the coefficients of restitution yield different results if reverse sliding is present during the contact period. Furthermore, the formulation has been extended to flexible multibody systems. The use of impulsive dynamics in flexible systems means that continuous contact is simulated as a virtual succession of instantaneous impacts. Numerical results show how the number of fictitious impacts increases with the number of assumed modes. Acknowledgement The author thanks the CICYT for the support of project DPI2006-15613-C0303, under which this work has been done.

References 1. Wehage RH, Haug EJ (1982) Dynamic analysis of mechanical systems with intermittent motion, Journal of Mechanical Design 104, 778–784. 2. Khulief YA, Shabana AA (1985) Dynamic analysis of constrained systems of rigid and flexible bodies with intermittent motion, ASME Paper 84-DET, 116. 3. Palas H, Hsu WC, Shabana AA (1992) On the use on momentum balance and the assumed modes method in transverse impact problems, Journal of Acoustics and Vibration 114, 364– 373. 4. Escalona JL, Mayo JM, Dom´ınguez J (1998) A critical study of the use of the generalized impulse-momentum balance equations in flexible multibody systems, Journal of Sound and Vibration 217, 523–545. 5. Kane TR (1984) A dynamics puzzle, Stanford Mechanics Alumni Club Newsletter 6. 6. Lankarani HM, Pereira MS (2001) Treatment of impact with friction in planar multibody mechanical systems, Multibody System Dynamics 6, 203–227. 7. Pfeiffer F, Glocker C (1996) Multibody Dynamics with Unilateral Contacts, Wiley series in Nonlinear Science, New York. 8. Routh EJ (1891) Dynamics of a System of Rigid Bodies, 5th ed., Macmillan, London. 9. Stronge WJ (2000) Impact Mechanics, Cambridge University Press, Cambridge. 10. Shabana AA (1998) Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, Cambridge.

Modeling of Aircraft Prescribed Trajectory Flight as an Inverse Simulation Problem Wojciech Blajer, Jerzy Graffstein, and Mariusz Krawczyk

1 Introduction Inverse simulation techniques are computational methods in which control inputs to a dynamic system that produce desired system outputs are determined. Such techniques are finding applications in varied fields like process control, robotics and aerospace engineering. Problems associated with maneuvering flight is another field of application [1–5]. The question of what control commands must the pilot (or automatic control system) take to fly an aircraft along a desired trajectory or perform a specified maneuver can be answered, combined with the dynamic analysis of aircraft in the prescribed motion. This paper is another contribution to this field. In the inverse simulation problem at hand we deal with an underactuated mechanical system in a partly specified motion. The aircraft of n = 6 degrees of freedom is actuated by m = 4 control inputs: the aileron, elevator and rudder deflections and the jet thrust variation, and the same number of m = 4 motion restrictions are imposed on the aircraft motion, m < n. In the present study the latter are: a specified trajectory, which imposes two constraints on aircraft position in space, a predetermined flight velocity, and a demand on fuselage attitude with respect to the trajectory. A tangent realization of the trajectory constraints is observed [6, 7], which yields two additional constraints on fuselage attitude with respect to the desired trajectory. The consequent governing equations of the prescribed trajectory flight arise as a set of index-three differential-algebraic equations (DAEs), and an effective method for solving the DAEs is applied. The solution consists of time-variations of aircraft state variables in the prescribed motion and of the demanded control that W. Blajer Technical University of Radom, Institute of Applied Mechanics, ul. Krasickiego 54, 26-600 Radom, Poland, e-mail: [email protected] J. Graffstein and M. Krawczyk Institute of Aviation, Al. Krakowska 110/114, 02-256 Warsaw, Poland, e-mail: [email protected], [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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ensures the realization of the motion. This gives a unique opportunity to study the simulated control strategies and evaluate feasibility of the modeled aircraft maneuvers. The inverse simulation of landing maneuver of a light training aircraft serves as illustration of the proposed formulation.

2 Modeling Preliminaries Four reference coordinate systems are used in the mathematical description that follows: the inertial (I) frame OxI yI zI , with its origin O at an arbitrary point fixed with respect to the earth, and OzI axis pointed vertical and downward (according to the gravity acceleration vector g); the gravitational (G) frame CxG yG zG with its origin attached at the aircraft’s mass centre C, constantly parallel to (I); the aerodynamic (A) frame CxA yA zA , with CxA axis directed along the velocity vectorv; and the bodyfixed (B) frame CxB yB zB , with CxB zB being the airframe symmetry plane. The three angles that orientate (B) and (A) with respect to (G) are traditionally Bryant angles, respectively, ϕ , θ and ψ (roll, pitch and yaw angles), and φ , γ and χ [8, 9], seen in Fig. 1a. The angular orientation of (A) and (B) is then defined with α (angle of attack) and β (sideslip angle), seen in Fig. 1b. The transformation matrices from (G) to (B), and from (A) to (B) reference systems are: ⎡ ⎤ cθ cψ cθ sψ −sθ ABG (ϕ , θ , ψ ) = ⎣sϕ sθ cψ − cϕ sψ sϕ sθ sψ + cϕ cψ sϕ cθ ⎦ , (1) cϕ sθ cψ + sϕ sψ cϕ sθ sψ − sϕ cψ cϕ cθ ⎡ ⎤ cα cβ −cα sβ −sα cβ 0 ⎦, ABA (α , β ) = ⎣ sβ (2) sα cβ −sα sβ cα where cθ = cos θ , sψ = sin ψ , . . . are used for compactness, and AAG can be obtained from AAG after replacing ϕ , θ , ψ with φ , γ , χ . One can then write ABG (ϕ , θ , ψ ) = ABA (α , β )AAG (φ , γ , χ ), which allows one for determination of ϕ , θ and ψ in terms

a

yB / yA y/c

b

yG xB / xA

yB

q/g j/f

b b trajectory

C v

q/g y/c

y/c xG

yA

j/f

q/g

zB /zA

zG

C

xA a b xB

j/f

a

zB

a zA

Fig. 1 Reference frames (B)/(A) with respect to (G), and frames (B) and (A)

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of φ , γ , χ , α and β . The kinematical relationship that will be used in the sequel is also ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ sβ 0 p 1 0 −sγ φ˙ α˙ (B) (3) ωB ≡ ⎣ q ⎦ = ABA (α , β )(⎣0 cφ sφ cγ ⎦ ⎣ γ˙ ⎦ + ⎣cβ 0 ⎦ ˙ ) β r 0 −sφ cφ cγ χ˙ 0 −1 B = ω A +ω B/A expressed in which is the matrix representation of vector formula ω (B) frame, and where p, q and r are the roll, pitch and yaw rates, respectively. The aircraft is modeled as a rigid body of n = 6 degrees of freedom, symmetrical with respect to CxB zB plane. The m = 4 control inputs are the deflections of T

aileron, elevator and rudder, δ = δa δe δr , and the jet thrust T . It is assumed that δ deflections affect only the aerodynamic moments, and their direct influence on the aerodynamic forces is neglected. The coefficients of aerodynamic forces and

(A) (A) moments, respectively, cF = [cD cS cL ]T and cN = cl cm cTn , are then: cD = cD (α , β , q),

cl = cl (α , β , p, r, δa , δr ),

cS = cS (α , β , p, r), cL = cL (α , β , q),

cm = cm (α , q, δe ), cn = cn (α , β , p, r, δa , δr ).

(4)

It is assumed that the jet thrust vector T is contained in CxB zB plane, it is inclined to CxB axis with angle αT , and d is the distance between C point and the line of T force. For the purpose of the subsequent formulation, it is convenient to use the dynamic equations of translatory motions expressed in (A), and the dynamic equations of rotational motions in (B). In matrix notation these equations are [8, 9]: (A)

m v˙ (A) + m ω˜ A v(A) = F(A) , (B)

(B)

(B)

J ω˙ B + ω˜ B J ωB = N(B) ,

(5) (6)

where m is the aircraft mass, J is the matrix of aircraft moments of inertial in (B),

T v(A) = 1 0 0 v is the representation in (A) of aircraft velocity v with respect to air (see Fig. 1b), equal to the aircraft absolute velocity (with respect to OxI yI zI ) for (K) windless conditions, ωK (K = A, B) is the representation in (K) axes of absolute K of (K) frame, and the superscript ∼ denotes a skew-symmetric angular velocity ω matrix (a cross product operator) so that to representa ×b with a˜ b in matrix notation. The applied forces F(A) and torques N(B) , respectively, in (A) and (B) axes, are: ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ − sin γ cD cos(α + αT ) cos β 1 F(A) = − ρ Sv2 ⎣ cS ⎦ + T ⎣− cos(α + αT ) sin β ⎦ + mg ⎣ sin φ cos γ ⎦ , (7) 2 cL − sin(α + αT ) cos φ cos γ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ b cl 0 −q sin αT 1 N(B) = ρ Sv2 ABA (α , β ) ⎣ca cm ⎦ + T ⎣d ⎦ + JT ωT ⎣ p sin αT − r cos αT ⎦ , (8) 2 0 b cn q cos αT

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where ρ is the air density, S is the lifting surface area, b and ca are the wing span and mean aerodynamic chord, JT and ωT are the moment of inertia and angular (B) velocity of rotating jet parts. The third component of N(B) in Eq. (8), Ngir , denotes the gyroscopic terms due to the rotating parts of the jet mounted on the airframe that

T (B) rotates with angular velocity ωB = p q r , where p, q and r are, respectively, the roll, pitch and yaw rates. Assumed ωT and T are related to each other, i.e. a (B) relationship T = T (ωT ) is known, we can write Ngir (p, q, r, T ).

3 Prescribed Flight The underactuated system (n = 6 degrees of freedom and m = 4 control inputs, m < k) can be constrained to at most m = 4 desired outputs, referenced to as servoconstraints or control constraints [6, 7] on a controlled system, as distinct from passive or contact constraints in the classical sense. For the case at hand, the motion specifications are a desired trajectory in space (two constraints on center of mass position), a specified time history of flight speed (constraint on motion along the trajectory), and a condition on airframe attitude with respect to the trajectory. The desired trajectory is defined using the position vector of center of mass of aircraft in (I) axes (Fig. 2), and is represented in the following parametric form with the arc length s as the parameter

T r(I) = r(s) = x(s) y(s) z(s) ,

(9)

where superscripting with means ‘specified’. The trajectory specification is equivalent to two constraints on aircraft position in space. For the purpose of this formulation, r(s) must be at least twice differentiable function. To this aim, it is first sketched by a set of successive points in space, and then approximated by cubic spline functions, followed by a procedure described in [10], which is not reported here for shortness.

C

s

v r yI prescribed trajectory

Fig. 2 Aircraft in prescribed trajectory flight

O xI

zI

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y" (x)

8 4

4

0 2

-4

first fit second fit

0 0

2

x 4

6

8

first fit second fit

-8 10

0

2

4

6

x 8

10

Fig. 3 The first-fit and second-fit trajectories

The use of cubic splines to approximate the flight path results in piecewise linear second derivatives of the spline functions, r (s), which are then used in the governing equations (reported in the sequel) and spoil smoothness of the obtained required control activity when passing the trajectory sketching points. The effect can be solved by applying higher order splines, but this may be at the expense of large oscillations in the resultant spline. The other possibility, used in the present formulation, is then to use the cubic spline fitting procedure twice, applying a greater number of sketching points from the first fit trajectory as input data for the second approximation. Then, while both the first-fit and the second-fit trajectories r(s) are usually almost the same looking, the second-fit trajectory curvature, r (s), is considerably smoothed, which results in much more gentle maneuvers that are simulated. The problem is illustrated in Fig. 3 by using the simple example of smoothing the function y(x), which was initially sketched by 11 points denoted by black squares. In the second fit the number of sketching points taken from the first-fit trajectory was increased ten times. As seen, while both the first-fit and second-fit trajectories are similar, the smoothness of y (x) for the second-fit trajectory is (numerically) much more improved (it is still piecewise linear, in fact, but due to the much more sketching points the effect is concealed). The aircraft motion on the trajectory r(s) is specified either by v = v(s)

or s = s(t),

(10)

˙ = s. ¨ In simple cases, i.e. v = const, the formulation s = s(t) where v = s˙ and v based on the velocity specification v = v(s) is evident. In a more general case, s = s(t) can be obtained based on v = v(s) as a (numerical) solution to the following integral equation t(t) ds = t. (11) v(s) 0

The specification on fuselage attitude with respect to the trajectory is

β = β (s).

(12)

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Most often β = 0 is used to model a wide range of maneuvers that assume coordinated turns.

4 Governing Equations and the Solution Code With the available control, the requirement (10) is directly regulated by changes in T value, and (11) is actuated by coordinated δ deflections (one condition involving three actuators). The realization of these two servo-constraint is orthogonal, i.e. it is directly influenced by the available control [6, 7]. The realization of the trajectory constraint (9) is different, referred to as tangent realization as the available control reactions have no direct effect on changing the trajectory curvature, and note that T , whose projection into the plane orthogonal to the trajectory is small, is already “used” to actuate the condition (10). The coordinated δ deflections must thus be used to accommodate the airframe attitude with respect to the trajectory so that to produce aerodynamic forces needed to regulate the required trajectory curvature changes. This means two additional requirements on the aircraft angular position in space, subsequent to the tangent realization of trajectory constraints. The condition of tangency ofv to the trajectory r(s) involves ⎡ ⎤ ⎡ ⎤ cos γ cos χ x (s) ⎣ cos γ sin χ ⎦ = ⎣y (s)⎦ , (13) z (s) − sin γ which constitute two independent demands on γ and χ that orientatev with respect (G), and ‘prime’ denotes differentiation with respect to s. The conditions for balance of the applied and centrifugal inertial forces in the plane orthogonal to the trajectory and in the tangential direction, respectively, along CyA and CzA axes and along CxA axis, are then: 1 ρ Sv2 cS + T cos(α + αT ) sin β − mg sin φ cos γ 2

+ m v2 x (sin φ sin γ cos χ − cos φ sin χ )

(14)

+y (sin φ sin γ sin χ + cos φ cos χ ) + z sin φ cos γ = 0, 1 ρ Sv2 cL + T sin(α + αT ) − mg cos φ cos γ 2

+ mv2 x (cos φ sin γ cos χ + sin φ sin χ )

(15)

+y (cos φ sin γ sin χ − sin φ cos χ ) + z cos φ cos γ = 0, 1 ρ S v2cD − T cos(α + αT ) cos β + mg sin γ + mv˙ = 0. 2

(16)

By gathering the angular velocities in z = [p q r]T , the attitude coordinates in y = [α β φ γ χ ]T , and the control variables in u = [T δ T ]T (u = [T δa δe δr ]T ), the

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governing equations for the aircraft dynamic analysis and synthesis of control in the prescribed trajectory flight are: z˙ = E(y, z, δ , T ), z = F(y, y˙ ), 0 = G(y,t), 0 = H(y, z, T,t).

(17)

The three differential equations z˙ = E(y, z, δ , T ) are the resolved form of the dynamic equation (6) of rotational motion, z = F(y, y˙ ) represents the three kinematical relationships of Eq. (3), 0 = G(y,t) represent the attitude constraint (12) and the condition (13) of tangency of aircraft velocity to the trajectory, counted as two independent conditions on γ and χ , and 0 = H(y, z, T,t) are three algebraic ˙ and v = s(t), ¨ equations composed of (14), (15) and (14) after setting v = s(t) which express the balance of applied and inertial forces along CxA , CyA and CzA directions. Equations (17) form 12 index-three DAEs [7] in the same number of variables y, z and u. The solution to DAEs (17) are time histories of the aircraft state variables y(t) and z(t) in the prescribed trajectory flight, and of control u(t) required for realization of the motion. A range of DAE solvers can be used to solve the DAEs. Most of the methods originate from Gear’s approach [11, 12]. The simplest algorithm of this type is based on Euler backward differentiation approximation method in which the time derivatives z˙ and y˙ at time tn+1 = tn + Δt are approximated by their backward differences (for example, z˙ is approximated by (zn+1 − zn )/Δt). Using this approach to solve DAEs (17), given yn and zn at time tn (note that un are not involved), the values yn+1 , zn+1 and un+1 at time tn+1 = tn + Δt can be found as a solution to the following nonlinear algebraic equations: zn+1 − zn − E(yn+1 , zn+1 , δn+1 , Tn+1 ) = 0, Δt yn+1 − yn+1 ) = 0, zn+1 − F (yn+1, Δt G(yn+1 ,tn+1 ) = 0,

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H(yn+1 , zn+1 , Tn+1 ,tn+1 ) = 0, and we used the Newton-Raphson method for this solution. In this way the solution is advanced from time tn to tn+1 = tn + Δt. In order to improve numerical accuracy of solution to DAEs (18), the Euler method as above can possibly be replaced by a higher-order backward difference approximation method [11, 12]. It may be worth noting, however, that the scheme (18) leads to stable solution. More strictly, y(t) and T(t) are explicitly determined from the algebraic equations 0 = G(y,t) and 0 = H(y, z, T,t) with a numerical accuracy of the Newton-Raphson method. Then, only the solutions z(t) and δ (t) are burdened with the truncation error of the rough backward difference scheme used. The errors do not accumulate in time, however, as the solution is based on

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the numerically exact solutions y(t) and T(t). The simple code (18) leads thus to stable and, for small Δt values, reasonably accurate results. In particular, the obtained inverse simulation control commands, used as input data in direct simulation, enforce the modeled aircraft to replicate the specified maneuvers with some numerical accuracy.

5 Case Study The described technique was applied to the inverse simulation of a space maneuver of the light aircraft PZL I-23 Manager. The desired trajectory was initially sketched by 30 points, and the cubic spline approximation routine described above was used to define the trajectory as in Eq. (9). As motivated before, we used the spline fitting procedure twice, applying 500 points from the first-fit trajectory as the input data for the second approximation fit. The resulted trajectory is illustrated in Fig. 4. Constant aircraft velocity v = 60 m/s during the whole maneuver was assumed, and β = 0 was applied. The integration time step was Δt = 0.25 s. x [m] 7000

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Selected results of numerical simulations are reported in Fig. 5. As seen, the roll angle ϕ increases up to 60◦ and −35◦ during the two modeled turns on the trajectory, followed by the required increase in the angle of attack α and the required control surface commands δ . Appropriate changes in the thrust value T are also required to keep the assumed velocity constant, and in particular negative T is required when the trajectory slopes down.

6 Conclusions A mathematical model for aircraft prescribed trajectory flight has been presented. The motion specifications are a specified trajectory, a specified velocity/motion on the trajectory, and a specified airframe attitude with respect to the trajectory. The governing equations were formulated as index-three DAEs, and an effective numerical code for solving the DAEs was described. The solution consists of timevariations of state variables and the required control in the specified motion. Due to the differential flatness of the problem [10], the solution is unique. A wide range of aircraft maneuvers, including extreme flight conditions and aerobatic maneuvers, can be simulated using the formulation reported above. Simulation

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of unmanned aerial vehicle missions is also possible. This gives a unique opportunity to study simulated control strategies for different constrained flight scenarios. The obtained inverse control u(t) can then be used as an open-loop tracking control in aircraft predetermined trajectory flight. It should then be enhanced by a feedback control to provide stable tracking of the reference trajectory in the presence of external perturbations and modeling uncertainties. The influence of wind drift should also be included into the mathematical model. Acknowledgement The work was financed in part from the government support of scientific research for years 2006–2008, under grant No. 4 T12C 062 30.

References 1. Avancini G, de Matteis G (1999) Two-timescale-integration method for inverse simulation. Journal of Guidance, Control, and Dynamics 22, 395–401. 2. Bottasso CL, Ragazzi A (2001) Deferred-correction optimal control with applications to inverse problems in flight mechanics. Journal of Guidance, Control, and Dynamics 24, 101–108. 3. Hess RA, Gao C, Wang SH (1991) Generalized technique for inverse simulation applied to aircraft maneuvers. Journal of Guidance, Control, and Dynamics 14, 192–200. 4. Kato O, Sugiura I (1986) An interpretation of airplane general motion and control as inverse problem. Journal of Guidance, Control, and Dynamics 9, 198–204. 5. Snell SA, Stout PW (1998) Flight control law using nonlinear dynamic inversion combined with quantitative feedback theory. Journal of Dynamic Systems, Measurement, and Control 120, 208–215. 6. Blajer W (1997) Dynamics and control of mechanical systems in partly specified motion. Journal of the Franklin Institute 334B, 407–426. 7. Blajer W, Kołodziejczyk K (2007b) Control of underactuated mechanical systems with servoconstraints. Nonlinear Dynamics 50, 781–791. 8. Abzug MJ (1998) Computational flight dynamics. American Institute of Aeronautics and Astronautics, Reston, VA. 9. Etkin B (2005) Dynamics of atmospheric flight. Dover Publications, Mineola, NY. 10. Blajer W, Kołodziejczyk K (2007a) Motion planning and control of gantry cranes in cluttered work environment. IET Control Theory & Applications 1, 1370–1379. 11. Ascher UM, Petzold LR (1998) Computer methods for ordinary differential equations and differential-algebraic equations. Society for Industrial and Applied Mathematics, Philadelphia, PA. 12. Gear CW, Petzold LR (1984) ODE methods for the solution of differential/algebraic equations. SIAM Journal on Numerical Analysis 21, 716–728.

Improved Adaptive Controllers for Sensory Systems – First Attempts Carsten Behn and Joachim Steigenberger

C+ , C− is the open Notations: Let R≥0 , R>0 := [0, ∞), (0, ∞), respectively; √ right-, left-half complex plane, respectively; x := xT x, x ∈ Rn ; C (I; Rn ) is the set of continuous functions x : I → Rn , I ⊂ R; L ∞ (I; Rn ) is the space of measurable essentially bounded functions x : I → Rn , I ⊂ R with norm x∞ := ess sup x(t); R is the set of differentiable functions yref : R≥0 → Rm with y˙ref t∈I

absolutely continuous on compact intervals and yref , y˙ref , y¨ref ∈ L ∞ (R≥0 ; Rm ).

1 Motivation The reception of vibrations is a special sense of touch. It is important for many insects, especially for the group of arachnids. Spiders have various sensilla to notice these vibrations (see Fig. 1). We do not want to distinguish the different forms of sensilla, but we state the following common properties: ‘The stimulus of this sense of touch is mechanical oscillation energy which is transmitted to the receptor cells in the case of direct contact with an oscillating object’ [2]. Moreover, ‘the cells for reception of vibrations adjust their sensibility to a continuing excitation in such a way that despite this permanent excitation the whole system tends to the rest position’ [3]. Hence, this biological paradigm offers a fundamental principle: adaptation. Motivated by these biological observations and the scheme of a spider sensillum (Fig. 2) we consider a simple sensory system in form of a spring-mass-dampersystem within a rigid frame, which is forced by an unknown time-dependent C. Behn Department of Technical Mechanics, Technische Universit¨at Ilmenau, Max-Planck-Ring 12, Building F, 98693 Ilmenau, Germany, e-mail: [email protected] J. Steigenberger Institute of Mathematics, Technische Universit¨at Ilmenau, Weimarer Straße 25, 98693 Ilmenau, Germany, e-mail: [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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Fig. 1 A spider leg with sensilla [1]

Fig. 2 Schema of spider sensillum [1]

displacement a(·). Moreover, the mass is under the action of an internal control force u(·) to compensate the unknown ground excitations, see Fig. 3, where x is the absolute coordinate. The parameters of this sensory system are m (the forced seismic point mass), the damping factor d and the spring stiffness c. If y(t) = x(t)− a(t), for all t ∈ R≥0 , as the relative coordinate of the point mass, is the measured output of the system, we arrive at the following system of differential equations in normalised form: • ⎫ 0 0 1 y(t) y(t) 0 + 1 u(t) + ,⎬ = y(t) ˙ y(t) ˙ −a(t) ¨ − mc − md (1) m ⎭ y(0) = x0 − a(0), y(0) ˙ = x1 − a(0). ˙

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d u(t) m y(t) c x(t)

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In general, one cannot expect to have complete information about a sophisticated mechanical or biological system, but instead only structural properties are known. Hence, for the further analysis of the sensory system, we suppose that both mass, spring stiffness and damping factor are unknown. Further-more, with respect to an uncertain environment, we assume no knowledge of the ground excitations. Summarising, we have to deal with a highly uncertain (control) system of known structure. The consideration of uncertain systems leads us to the usage of adaptive control. By the above mentioned adjustment of the sensilla we are given the task to adaptively compensate the unknown ground excitation: we have to design an adaptive controller, which learns from the behaviour of the system, so automatically adjusts its parameters in such a way that the seismic point mass tends to the rest position in spite of the continuing excitation. Remark 1. Most adaptive control mechanisms first attempt to identify or to estimate certain parameters of the system, and then design a feedback controller on the basis of this information. In this paper, we consider adaptive controllers, which are not based on any parameter identification or estimation algorithms. The objective is not to obtain information about the system, but simply to control the uncertain system utilising the high-gain property of this system [5].

2 General System Class The equations of motion (1) fall into the category of quadratic, finite-dimensional, nonlinearly perturbed, m-input u(·), m-output y(·) systems (MIMO-systems) of relative degree two of the form y(t) ¨ = A2 y(t) ˙ + f1 (s1 (t), y(t), z(t)) + Gy(t), ˙ + f2 (s2 (t), y(t)), x(t) ˙ = A2 z(t) + A0 y(t)

y(t0 ) = y0 , y(t ˙ 0 ) = y1 , z(t0 ) = z0 ,

(2)

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with y(t), y0 , y1 , u(t) ∈ Rm , z(t), z0 ∈ Rn−2m , A2 , G ∈ Rm×m , A5 ∈ R(n−2m)×(n−2m) , A0 ∈ R(n−2m)×m , n ≥ 2m, and, for q1 , q2 ∈ N, and (i) σ (G) ⊂ C+ , i.e., the spectrum of the “high-frequency gain” lies in the open right-half complex plane. (ii) s1 (·) ∈ L ∞ (R≥0 ; Rq1 ), s2 (·) ∈ L ∞ (R≥0 ; Rq2 ) may be thought of as (bounded) disturbance terms, where si (t) = ψi (t, y(t), y(t), ˙ z(t)) is also possible with ψi (·, ·, ·, ·) ∈ L ∞ (R≥0 × Rm × Rm × Rn−2m; Rqi ). (iii) f1 : Rq1 × Rm × Rn−2m → Rm and f2 : Rq2 × Rm → Rn−2m are continuous functions and, for compact sets C1 ⊂ Rq1 and C2 ⊂ Rq2 , there exist c1 , c2 ≥ 0 such that f1 (s, y, z) ≤ c1 [1 + y + z] f2 (s, y) ≤ c2 [1 + y]

for all (s, y, z) ∈ C1 × Rm × Rn−2m , for all (s, y) ∈ C2 × Rm ;

(iv) σ (A5 ) ⊂ C− , i.e., the system is minimum phase, provided f1 = 0, f2 = 0. The matrix A0 is necessary for under-actuated systems, see [6]. Relative degree two means that the control u(·) directly influences the second derivative of each output component.

3 Control Objective & Controllers 3.1 Control Objective Because the sensory system is excited by an unknown ground excitation and the cells for reception adjust their sensibility in such a way that the whole system tends to the rest position, we try to design a universal feedback controller to adaptively compensate this unknown ground excitation. Since we deal with uncertain, nonlinearly perturbed (ground excitation or the continuous functions fi , respectively) MIMO-systems, which are not necessarily autonomous, particular attention is paid to the adaptive λ -tracking control objective. Precisely, given λ > 0, a control strategy y → u is sought which, when applied to any system of the presented system class, achieves Υ-tracking for every reference signal yref (·) of a certain class, i.e., the following: (i) Every solution of the closed-loop system is defined and bounded on R≥0 . (ii) The output y(·) tracks yref (·) with asymptotic accuracy quantified by λ > 0 in the sense that max{0, y(t) − yref (t) − λ } → 0 as t → ∞, see Fig. 4. Choosing yref (·) ≡ 0, λ = 0 we arrive at the so-called adaptive stabilization control objective, see [4] and [5].

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3.2 Current Control Strategy & Theorem A preferred control strategy is the following: e(t) := y(t) − yref (t), u(t) = − k(t)e(t) + dtd (k(t)e(t)) , ˙ = γ (max {0, e(t) − λ })2 . k(t)

⎫ ⎬ k(0) = k0 ∈ R.

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(3)

with λ > 0, yref (·) ∈ R, u(t), e(t) ∈ Rm , k(t) ∈ R and γ > 0. This controller consists of a very simple feedback mechanism and adaptation law, and is only based on the output of the system and its time derivative – no knowledge of the system parameters is required. For brevity the adaptor above, in (3), will be called ‘classical’ adaptor. For the control strategy (3) the governing theorem is the following, see [4]. Theorem 1. Let λ > 0. Then the controller (3) applied to any system of class (2) yields, for any reference signal yref (·) ∈ R and any initial data (y0 , η0 , z0 , k0 ) ∈ Rm × Rm × Rn−2m × R, a nonlinear closed-loop system of first order differential equations (a feedback controlled initial-value problem) ⎫ y˙ = η (t), y(0) = y0 , ⎪ ⎪ ⎪ ⎪ η˙ (t) = A2 η (t) + f1 (s1 (t), y(t), z(t)) ⎪ ⎪ ⎪ ⎬ −G[k(t)(y(t) − yref (t)) + k(t)(η (t) − y˙ref (t)) . (4) 2 + γ (max {0, y(t) − yref (t) − λ }) (y(t) − yref (t)) , η (0) = η0 , ⎪ ⎪ ⎪ z(0) = z0 , ⎪ ⎪ z˙(t) = A5 z(t) + A0 η (t) + f2 (s2 (t), y(t)), ⎪ ⎪ 2 k(0) = k0 , ⎭ ˙k(t) = γ (max {0, y(t) − yref (t) − λ }) , which has a solution (y, η , z, k) : [0,t ) → Rm × Rm × Rn−2m × R with the following properties: (i) t = ∞, i.e. there does not exist a finite escape time. (ii) lim k(t) exists and is finite. t→∞

(iii) The solution, η˙ (·), z˙(·), and u(·) as in (3), are bounded. (iv) lim sup y(t) − yref (t) ≤ λ , i.e., (3) is an adaptive λ -tracker. t→∞

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For the proof the reader’s attention is invited to [4]. Remark 2. The limit k∞ in Assertion (ii) of Theorem 1 is unknown and might be unfeasibly big. The parameter γ strongly determines the growth of the gain parameter k(·). In [4] and [6] the case γ = 1 was dealt with. With small γ (e.g., γ = 1 as formerly) k(·) grows often too slowly as to achieve a good tracking behaviour. Therefore, a sufficiently large γ 1 should be used. But, if we choose γ too large, we arrive at high feedback values and/or the sensor is not really sensitive to extraordinary impulses. Example 1. sensor: m = 1, c = 10, d = 5, (y(0), y(0)) ˙ = (−a(0), −a(0)); ˙ reference signal: yref (·) ≡ 0 (rest position to be tracked); λ -tracker: λ = 0.2 with k0 = 0;

2 −0.5(t−20.5) cos(t) - amplitude peak around ground excitation: t → a(t) = 1 + 3e t = 20.5. We choose γ = 300: The large γ = 300 shrinks the system’s response to the permanent excitation a(t) = cos(t) close to zero – at the price of a permanently extremely big gain parameter k(·) (see Fig. 6). Furthermore, the system is not really sensitive anymore to notice the excitation peak around t = 20.5 (see Fig. 5).

4 New Control Strategies In this section we introduce some modifications of the control strategy (3). These will show up with an altered feedback law, various new adaptors, and allow for input and output noise. The main feature is that all the adaptors to be used in simulations

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appear as specializations of a general adaptor that is a differential equation of 1st order for k(·). Again, let the control system be of class (2). Then we shall consider controllers of the general form ⎫ e(t) := y(t) − yre f (t) + n(t), ⎬ u(t) = −k(t)(e(t) + κ e(t)) ˙ + uin(t), (5) ˙ = (k(t), e(t),t), k(0) = k0 ∈ R, ⎭ k(t) λ with λ > 0, κ > 0. The unknown functions uin (·) ∈ L ∞ (R≥0 : Rm ) and n(·) ∈ R are noise corruptions of input and output, respectively. Sometimes κ = 0 is sufficient for λ -tracking of SISO-systems of class (2), see [4]. Remark 3. We have to claim a bound for the output measurement noise signal n(·). This fact is often neglected in papers about tracking in the presence of noise, see for example [7]. Figure 7 shows a λ - and a n(·)∞ -tube along the reference signal. On the left we have n(·)∞ < λ and the right part shows the constellation n(·)∞ ≥ λ . One can easily conclude that λ -tracking only makes sense if the bound of the output measurement noise signal is smaller than λ : n(·)∞ < λ .

(6)

If (6) is violated then the controller cannot distinguish between reference and noise signal, and so is λ -tracks the wrong signal, yref (·) − n(·). As to the adaptor function fλ we shall assume fλ (·, e,t) ∈ C1 (R; R) and fλ (k, ·, ·) measurable (or piecewise continuous). These properties will guarantee

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(by Carath´eodory’s theory) an absolutely continuous solution (provided e(·) is of ˙ = γ (max {0, e(t) − λ })2 , belongs to this class, too). The ‘classical’ adaptor, k(t) this class. The modifications to be presented below are aimed at avoiding the monotonic increase of k(·). All the respective simulations realized until now exhibited promising results. Some of them are presented in the following. Although there is still a lack of proof for λ -tracking when using the modified adaptors we anticipate that, mutatis mutandis, there hold propositions like Theorem 1. What obviously had to be changed in the conclusions is: (ii) to be replaced by lim sup k(t) < ∞, and in t→∞

(iii) drop boundedness of y(·) and z(·) – they could represent increasing angles of rotating bodies. Remark 4. Still, lim sup k(t) might be unfeasibly big. In practice, indeed, one does t→∞

not steer a system with t → ∞. Therefore, an adaptor has to ensure a feasible bound for k(·) on some sufficiently large compact time interval [0,t ].

5 Improved Gain Adaptation Laws ˙ ≥ 0 for all t Once again we remind of the fact, that the ‘classical’ adaptor yields k(t) thus implies monotonic increase of k(·), cf. any of the preceding simulation results. Now we propose some new adaptation laws, which let k(·) decrease when e is in the tube (and thus a big k(·) is not needed): • A very simple modification of the adaptation law is the so-called σ -modification ˙ = −σ k(t) + γ (max {0, e(t) − λ })2 , σ > 0, γ 1. k(t)

(7)

This adaptor achieves damping and increase of the gain k simultaneously when e is outside the tube. This law often leads to chaotic behaviour of the system [8]. • A similar adaptation law is the following ˙ = −a (1 − b sign (max {0, e(t) − λ })) k(t), k(t)

(8)

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with a > 0 and b > 1, which offers exponential decrease if e is in the λ -tube and exponential increase otherwise. • It could happen that e rapidly traverses the λ -tube. Then it would be inadequate to immediately decrease k(·) after e entered the tube. Rather we should distinguish three cases: 1. increasing k(·) while e is outside the tube, 2. constant k(·) after e entered the tube – no longer than a pre-specified duration td of stay, and 3. decreasing k(·) after this duration has been exceeded. Moreover, in order to make the attraction of the tube stronger, we should use different exponents for large/small distance from the tube. For instance: ⎧ γ (e(t) − λ )2, e(t) ≥ λ + 1, ⎪ ⎪ ⎪ ⎨γ (e(t) − λ )0.5, λ + 1 > e(t) ≥ λ , ˙ = k(t) (9) ⎪ 0, (e(t) < λ ) ∧ (t − tE < td ), ⎪ ⎪ ⎩ −σ k(t), (e(t) < λ ) ∧ (t − tE ≥ td ), with σ , γ ,td ,tE as before. One way to guarantee that e will not leave the λ -tube after entering the tube and k(·) is going to be decreased, is tracking of a smaller value than the desired λ , for example ε := 0.7 λ , what we will call ε -safe λ -tracking. Remark 5. Tuning of the adaptors (i.e., to find favourable values of γ , σ , a, b, td , ε ) is still a matter of simulation experiments. In practice, it might be dictated by the system to be controlled and the tools of implementation.

6 Simulations We point out, that the adaptive nature of the controllers is expressed by the arbitrary choice of the system parameters. Obviously numerical simulation needs fixed (and known) system data, but the controllers adjust their gain parameter to each set of system data. We exemplarily choose the following parameters (as well as for the simulations before): sensor: m = 1, c = 10, d = 5, (y(0), y(0)) ˙ = (−a(0), −a(0)); ˙ reference signal: yref (·) ≡ 0 (rest position to be tracked); λ -tracker: λ = 0.2 with k0 = 0.

6.1 Comparative Simulations Here we present simulations with the above set of system parameters, but with different adaptors: the ‘classical’ one and adaptors (7)–(9). We choose γ = 300, σ = 1, td = 3, ε = 0.7λ , κ = 1, a = 0.2, b = 20 (b > 1 and b a) and t → n(t) = 0.1 sin(2π t), uin (·) ≡ 0 and t → a(t) = sin(2π t). We receive the following figures.

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The output is not captured by the tubes yet (Fig. 8), hence k(·) has to grow further on (Fig. 9). Simulation with adaptor (8): It is obvious that we have to choose an initial value k(0) > 0. Let k(0) = 1. Again, k(·) is immediately decreasing (Fig. 11) when e entered the λ -tube (Fig. 10). Hence, we arrive at a jumpy behaviour, too. But compared to the previous

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simulation we have exponential decrease if e is in the λ -tube and exponential increase otherwise, see Fig. 11. Simulation with adaptor (9): Here, one clearly sees the advantage of two alternating exponents: e is forced very fast back into the λ -tube, see Fig. 12, because the attraction of the tube is

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stronger for small deviations e(t) − λ > 0. The gain parameter k(·) decreases if e stays in the tube for a duration larger than td (Fig. 13). Simulation with adaptor (9), but ε -tracking: In Fig. 12, the output apparently periodically leaves the λ -tube. Then ε -tracking with ε := 0.7λ makes e not to leave the desired λ -tube, see Figs. 14 and 15. The high value of k(·) at the beginning is natural because of the ε -tracking with ε < λ . It could possibly be diminished by choosing κ < 1 and some k0 > 0.

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6.2 Further Simulations ε -tracking with adaptor (9) turns out to be the best. Therefore, we apply this ε adaptor in further simulations of the sensor to show some influences of the parameter of the controller. We use the same parameters introduced above.

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ε-tube

λ-tube

0.2 0 − 0.2 y(t) − 0.4 − 0.6 − 0.8 −1

t 0

5

10

15

20

30

25

Fig. 16 Output and tubes 45 40 35 30 25 k(t)

20 15 10 5 0

t 0

5

10

15

20

25

30

Fig. 17 Gain parameter

Example 2. We choose

γ = 50, σ = 1, td = 2, ε = 0.7 λ , κ = 1 and t → a(t) = 2 −0.5(t−20.5) 1 + 3e cos(t) with no input and output disturbances. The new controller enables one to notice the peak at t = 20.5 by observing y(t) or k(t), see Figs. 16 and 17. If we insist on λ -tracking we could try with a smaller ε or (see Figs. 18 and 19) by using a smaller σ = 0.1, say. Now, the peak could be detectable via k(t). The technological set-up at hand might decide for one of these ways.

Improved Adaptive Controllers for Sensory Systems – First Attempts 0.4

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λ-tube

0.2 0 − 0.2 y(t) − 0.4 − 0.6 − 0.8 −1

t 5

0

15

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30

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Fig. 18 Output and tubes 45 40 35 30 k(t)

25 20 15 10 5 0

t 0

5

10

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25

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Fig. 19 Gain parameter

7 Conclusions Using a simple linear model of a sensory system adaptive controllers have been considered which compensate unknown permanent ground excitations. Classical adaptors suffer from a monotonic increase of the control gain parameter, thereby possibly paralysing the sensor’s capability to detect future extraordinary excitations. The paper proposes some modified adaptors which aim at avoiding this drawback.

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Despite of a present lack of stringent theory all simulation results encourage one to continue these investigations in particular in application to more involved sensory systems.

References 1. 2. 3. 4. 5. 6. 7. 8.

Barth FG (2004) Current Opinion in Neurobiology 14:415–422. Penzlin H (1991) Lehrbuch der Tierphysiologie. 5. Edition, Gustav-Fischer, Jena Dudel J, Menzel R, Schmidt RF (1996) Neurowissenschaft. Springer, Berlin Behn C (2005) Ein Beitrag zur adaptiven Regelung technischer Systeme nach biologischem Vorbild. Cuvillier, G¨ottingen Ilchmann A (1993) Non-identifier-Based High-Gain Adaptive Control. Springer, London Behn C, Zimmermann K (2006) Robotics and Autonomous Systems 54:529–545 Ilchmann A, Ryan EP (1994) Automatica 30(2):337–346 Mareels I, Polderman JW (1996) Adaptive Systems. Birkh¨auser, Boston, MA

Research of Stability and Nonlinear Vibrations by R-Functions Method Jan Awrejcewicz, Lidiya Kurpa, and Olga Mazur

1 Introduction The most modern constructions used in building, aerospace and other fields are modulated by plate and shell structures. Vibration research of plates loaded by compressive pulsating force has received particular interest, since in such system dynamic instability may occur, due to certain combinations of parameters of load and eigenfrequency. Given problem was reviewed by Sahu and Datta [1]. Dynamic instability was investigated by Bolotin [2], Hutt and Salam [3] and many others. Investigation of nonlinear parametric vibrations was carried out in references [2, 4–6], and others. However, in the mentioned works dynamic behavior of rectangular plates with some classical types of boundary conditions has been studied. On the other hand, many components of modern engineering constructions have different shape, and in particular they may be with cutouts. In the present work stability and nonlinear vibrations of plates with central cutout are investigated. R- functions method (RFM) [7] and variational one are applied to get over the mathematical difficulties related to complex form of plates. The main idea of offered approach is focused on reducing the von K´arm´an equations governing dynamics of isotropic plates to an ordinary differential equation regarding time by the Bubnov-Galerkin method. Coefficients of this equation are found by R-functions theory. Instability regions and nonlinear response characteristics are determined. The influence of size of cutout [8, 9], boundary conditions and parameters of load [11] on investigated characteristics is studied. J. Awrejcewicz Department of Automatics and Biomechanics, Technical University of Lodz, Poland, e-mail: [email protected] L. Kurpa and O. Mazur Department of Applied Mathematics, NTU “Kharkov Politechnical Institute”, Ukraine, e-mail: [email protected]., [email protected]

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2 Formulation The isotropic plate with constant thickness h is subjected to an uniformly distributed in-plane load of the form p = p0 + pt cosθt, (1) where p0 is the static component of p(t), pt is the amplitude of the dynamic component of p(t), and θ is the frequency of excitation. The governing equations of nonlinear dynamics of plate have the form [6]

∂ Nx ∂ T + = 0, ∂x ∂y ∂ Ny ∂ T + = 0, ∂y ∂x 2 ∂ w ∂ 2w ∂ 2w ∂ 2w 4 T + − ρh D∇ w = N + 2 N x y ∂ x2 ∂ x∂ y ∂ y2 ∂ t2

(2) (3)

In Eqs. (2), (3) Nx , Ny , T are membrane stress resultants [6], u, v, w are the displacements regarding x, y and z directions, respectively; E is elasticity modulus, μ is Possion’s ratio, ρ is density of plate, D = Eh3 /(12(1 − μ2 )) is flexural rigidity. Different types of the boundary conditions for deflection w are considered: 1. Clamped plate (C)

∂w = 0, (x, y) ∈ ∂ Ω; ∂n

(4)

w = 0, Mn = 0, (x, y) ∈ ∂ Ω;

(5)

w = 0, 2. Simply supported plate (SS)

3. Mixed boundary conditions (type 1 – C-SS)

∂w = 0, (x, y) ∈ ∂ Ω1 , ∂n w = 0, Mn = 0, (x, y) ∈ ∂ Ω2 ; ∂ Ω1 ∪ ∂ Ω2 = ∂ Ω; w = 0,

(6)

4. Mixed boundary conditions (type 2 – SS-F, F – free edge) w = 0, Mn = 0, (x, y) ∈ ∂ Ω1 , Mn = 0, Qn = 0, (x, y) ∈ ∂ Ω2 ,

∂ Ω1 ∪ ∂ Ω2 = ∂ Ω.

(7)

Displacements u and v have to satisfy the following conditions: Namely, and on loaded part of the boundary Nn = −p, Nτ = 0,

(8)

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on unloaded part of the boundary Nn = 0, Nτ = 0.

(9)

In Eqs. (4)–(9) Mn is bending moment, Qn is transverse force, Nn , Nτ are normal and tangent forces [6], and n, τ are normal and tangent to domain boundary ∂ Ω. The initial conditions have the following form w/t=0 = w0 , w /t=0 = 0. Later, we use nondimensional form of equations. Relations between dimensional and nondimensional values are defined as follows x y w ua va , y¯ = , w ¯ = , u = 2, v = 2, a a h h h Nx a 2 Nx a 2 Ta2 Nn a 2 Nτ a 2 Nx = , N = , T = , N = , N = x n τ Eh3 Eh3 Eh3 Eh3 Eh3 ! ! pa2 h E a2 ρ a2 ρ θ, ωL = ωL . p¯ = ,t= 2 t, θ = 3 a ρ h E h E Eh x¯ =

(10)

where ωL denotes linear frequency. In what follows, bars over nondimensional values are omitted. Substituting expressions for Nx , Ny , T into Eqs. (2), (3) and taking into account Eq. (10), system Eqs. (2), (3) in displacements u, v, w may be presented in the following form AU = NL (w), (11)

∂ 2w 1 ∇4 w = Nx (u, v, w) + 2 12(1 − μ ) ∂ x2 +2

∂ 2w ∂ 2w ∂ 2w T(u, v, w) + 2 Ny (u, v, w) − 2 , ∂ x∂ y ∂y ∂t

where the vector U and nonlinear operator Nl(w) have the following form ⎛ 2 ⎞ ∂ 1+μ ∂2 1−μ ∂2 ⎜ ∂ x2 + 2 ∂ y2 2 ∂ x∂ y ⎟ u ⎜ ⎟ ,A = ⎜ U= ⎟, 2 2 v ⎝ 1+μ ∂2 ∂ 1−μ ∂ ⎠ + 2 ∂ x∂ y ∂ y2 2 ∂ x2 ⎛

⎞ ∂ w ∂ 2w 1 + μ ∂ w ∂ 2w 1 − μ ∂ w ∂ 2w ⎜ ∂ x ∂ x2 + 2 ∂ y ∂ x∂ y + 2 ∂ x ∂ y2 ⎟ ⎜ ⎟ Nl(w) = − ⎜ ⎟. 2 ⎝ ∂ w ∂ 2w 1 + μ ∂ w ∂ 2w 1−μ ∂w ∂ w⎠ + + ∂ y ∂ y2 2 ∂ x ∂ x∂ y 2 ∂ y ∂ x2

(12)

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3 Method of Solving Supposing that a plate is in the inhomogeneous subcriticality state, then at the first step of the proposed method the related problem of elasticity theory have to be solved: AU1 = 0 (13) with the associated boundary conditions on loaded part of the boundary NLn (u1 , v1 ) = −1, NLτ (u1 , v1 ) = 0,

(14)

and on unloaded part of boundary NLn (u1 , v1 ) = 0, NLτ (u1 , v1 ) = 0,

(15)

where NLn , NLτ are normal and shear linear forces (linear part of Nn , Nτ ). The problem Eqs. (13)–(15) are solved by variational Ritz’s method. The corresponding functional has the following form ∂ u1 2 ∂ v1 2 1 − μ ∂ u1 ∂ v1 2 ∂ u1 ∂ v1 I1 (U1 ) = + dΩ + + + + 2μ ∂x ∂y 2 ∂y ∂x ∂x ∂y Ω

+ 2(1 − μ2 )

(u1 l + v1m)d∂ Ω,

∂Ω

where l, m are directional cosines of normal n. Construction of system of the basic functions is carried out using the R-functions theory [7]. Our further investigation is reduced to linear vibration problem for the unloaded plate. To find the natural frequencies and related modes, Ritz’s method and Rfunctions theory are used. The functional regarding this problem is constructed in the following form 2 2 2 2 3 ∂ 2w ∂ 2w ∂ w 2 I2 (w) = (Δw) − 2(1 − μ) − ∂ x2 ∂ y2 ∂ x∂ y Ω 2 2 2 −12(1 − μ )ωL w dΩ · A solution to nonlinear system of Eqs. (11)–(12) is being sought for in the following form w(x, y, t) = f(t)w1 (x, y), u(x, y, t) = (p0 + pt · cos θt)u1 (x, y) + f2(t) · u2 (x, y), v(x, y, t) = (p0 + pt · cos θt)v1 (x, y) + f2(t) · v2 (x, y) .

(16)

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In the above functions u1 (x, y), v1 (x, y), w1 (x, y) (eigenfunction, which corresponds to main frequency ) are known functions, whereas functions u2 (x, y), v2 (x, y) are found by solving inhomogeneous linear system of differential equations of the form AU2 = Nl(w1 ),

(17)

which simulates the plane elastic problem. Right hand-side of Eq. (17) can be considered as a fictitious force. Equation (17) is supplemented with the following boundary conditions Nn (u2 , v2 ) = 0, Nτ (u2 , v2 ) = 0. (18) In order to solve the problem governed by Eqs. (17), (18) in the case of plates with complex form, the Ritz’s method in combination with R-functions theory is applied. The functional used to solve the problem has the following form ∂ u2 2 ∂ v2 2 1 − μ ∂ u2 ∂ v2 2 ∂ u2 ∂ v2 dΩ − I3 (U2 ) = + + + 2μ + ∂x ∂y 2 ∂y ∂x ∂x ∂y Ω − 2 (Nl(w1 ), U2 ) dΩ. It is easy to see, the displacements u, v, w represented by Eqs. (16) satisfy the system Eqs. (11), (12) with the corresponding boundary conditions. Substituting Eq. (16) into Eq. (12) and applying the Bubnov-Galerkin method the following nonlinear differential equation is obtained. f (t) + ω2L(1 − αp0 − αpt cos θt)f(t) + βf3 (t) = 0, where the coefficients α, β are defined as follows 2 ∂ w1 L ∂ 2 w1 L 1 T (u1 , v1 ) + Nx (u1 , v1 ) + 2 α= 2 2 2 ∂x ∂ x∂ y ωL w1 Ω ∂ 2 w1 L N (u1 , v1 ) w1 dΩ, + ∂ y2 y 2 ∂ w1 ∂ 2 w1 1 T(u2 , v2 , w1 ) + N (u , v , w ) + 2 β=− x 2 2 1 w1 2 ∂ x2 ∂ x∂ y Ω ∂ 2 w1 Ny (u2 , v2 , w1 ) w1 dΩ. + ∂ y2

(19)

(20)

In the above NLx , NLy , TL are linear membrane stress resultants (linear part of Nx , Ny , T). Equation (19) can be cast into the following form f (t) + Ω2 (1 − 2k · cos(θt))f(t) + βf3 (t) = 0,

(21)

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where Ω = ωL (1 − p0 α)1/2 is vibration frequency of a plate loaded by static component p0 , and k = pt α/(2 − 2p0α) is excitation coefficient.

4 Stability Analysis To determine instability regions, the Bolotin’s method [2] is used. If one takes in (21) the coefficient β = 0, then the Mathieu equation, governing linear plate parametric vibrations plate is obtained of the form f (t) + Ω2(1 − 2k · cos(θt))f(t) = 0.

(22)

It is well known that the solutions of the Eq. (22) can be bounded or unbounded. The boundaries between stable and unstable solutions are formed by periodic solutions with period T and 2T, where T = 2π/θ. Two solutions of the same period confine regions of instability (in the vicinity of θ = 2Ω/r, r = 1, 2, 3, . . .), two solutions with different periods confine the regions of stability. Equations of curves, bounding regions of stability and instability are known. The first region of instability is defines by the curves √ √ θ1 = 2Ω 1 − k, θ2 = 2Ω 1 + k.

5 R-Functions Method R-functions theory is used to construct the system of basic functions for solving problems governed by Eqs. (13)–(15), Eqs. (17), (18), and the problem of linear vibrations of unloaded plate. According to the RFM, to create structures of solutions, it is necessary to construct the equation of the domain boundary. Method of these equations construction is described in [7]. Equation of boundary ω(x, y), constructed in such a way, satisfy the following conditions ω(x, y) = 0, ω(x, y) > 0,

∂ω = −1, (x, y) ∈ ∂ Ω. ∂n

Structures of solutions, which satisfy only principal boundary conditions can be constructed in the following way: 1. For the boundary conditions defined by (4), (8), (9): w1 = ω2 P0 , ui = Pi , vi = Pi+2 , i = 1..2;

(23)

2. For the boundary conditions defined by (5), (8), (9): w1 = ω1 P0 , ui = Pi , vi = Pi+2 , i = 1..2.

(24)

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3. For boundary conditions defined by (6), (8), (9): w1 = ω1 ω P0 , ui = Pi , vi = Pi+2 , i = 1..2;

(25)

4. For boundary conditions defined by (7), (8), (9): w1 = ω1 P0 , ui = Pi , vi = Pi+2 , i = 1..2.

(26)

In Eqs. (23)–(26) ω = 0 defines the equation of domain boundary, in Eq. (25) ω1 = 0 corresponds to equation of clamped part of the domain boundary, and in Eq. (26) ω1 = 0 is the equation of simply supported part of the boundary domain. Uncertain components Pi , i = 0,..,4 are presented as decomposition in the series with a help of some complete system of functions. In this work the system of power polynomials is used.

6 Numerical Results Numerical results presented in the work concern a plate with a central cutout (Fig. 1) subjected to load (23) for different boundary conditions. In this case the equation of domain boundary is constructed in the following way: ω(x, y) = (f1 ∧0 f2 ) ∧0 (f3 ∨0 f4 ).

(27)

The required R-operations, used in Eq. (27), are as follows x ∧0 y = x + y − x2 + y2 , x ∨0 y = x + y + x2 + y2 , x = −x. Functions fi , i = 1,. . . ,4 appeared in Eq. (27) are determined as follows f1 = 1/a((a/2)2 − x2), f2 = 1/b((b/2)2 − y2), f3 = 1/c((c/2)2 − x2), f4 = 1/d((d/2)2 − y2). In order to taste the offered method, calculations of plate eigenfrequencies for different size of cutout are carried out. Boundary conditions SS-F (Fig. 2) are considered. These boundary conditions are associated with the structure of solution to Eq. (26). The obtained results are presented in Table 1. The values of parameters used in our further analysis follow: a/b = 1, c = d, μ = 0.3. In the present work only primary region of instability is considered. Investigation of influence of boundary conditions (Fig. 2) on instability regions (Fig. 3) and fundamental frequency (Table 2) is carried out. Instability regions are presented for p0 = 1, c/a = 0.2. The obtained results exhibit the following effect: for conditions SS-F instability domain is located lower then other ones and has a smaller area. Instability domains of plates with boundary conditions SS-SS, C-SS

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Fig. 1 The analyzed plate

Fig. 2 Boundary conditions Table 1 Comparison of non-dimensional fundamental frequencies λ = ωL a2 (ρ h/D)1/2 of a simply supported square plate with a free cutout c/a

Present

[8]

[9]

[10]

0 0.2 0.4 0.5 0.6 0.8

19,742 19,708 21,108 23,791 28,719 57,948

19,734 19,134 20,739 23,422 28,307 56,949

19,739 18,901 20,556 23,329 28,491 58,847

19,740 18,762 20,785 23,664 28,844 58,062

and C-C correspond to a larger value of excitation frequency θ and occupy a larger area. The effect of the static load factor on instability region is investigated for a plate satisfying boundary conditions SS-F (Fig. 3). Static load factor takes the values p0 = 0.5, 1, 1.5 (c/a = 0.2). Owing to increase of static component, instability regions remove to lower value of the excitation frequencies. For a given plate, effect of cutout size is studied for different values of ratio c/a(0 ≤ c/a ≤ 0.4, SS-F, p0 = 1). Instability region (see Fig. 4) is shifted to the lower value of excitation frequencies up to c/a = 0.28. Instability regions tend to a larger value of excitation frequencies from c/a = 0.28 to c/a = 0.4 (for p0 = 2 behavior of the system is similar).

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Fig. 3 Effect of boundary conditions and static component on instability regions Table 2 Non-dimensional fundamental frequencies λ = ωL a2 (ρ h/D)1/2 of a square plate with a cutout for different boundary conditions

11,5 11,0 10,5

θ

p0=1

c/a

C-C

SS-SS

C-SS

0 0.2 0.4 0.5 0.6

123.649 230.334 315.753 511.139 1833.052

73.249 125.148 167.648 260.824 994.522

109.307 188.225 255.669 397.456 153.941

p0=2

boundary conditions - ss-f

boundary conditions - ss-f 9,0

θ

8,5 8,0 7,5

10,0 7,0 9,5 9,0

6,5

c/a = 0 c/a = 0.2 c/a = 0.24 c/a = 0.28 c/a = 0.32 c/a = 0.36 c/a = 0.4

6,0

k 5,5 kk 8,5 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40

Fig. 4 Effect of size of cutout on instability regions

Next, the dynamic nonlinear response for chosen values of load parameters p0 = pt = 1 with initial amplitudes w0 /h = 0.0001 and t ∈ [0..500] is studied (see Fig. 4). It should be emphasized, that instability region predicated by the linear theory almost coincides with results obtained by nonlinear analysis. It can also be seen that the amplitudes are comparable with initial conditionsoutside of instability region. In a critical zone the amplitude values increase. The influence of the size of cutout on response curve are analyzed for different values of ratio c/a(0 ≤ c/a ≤ 0.5, SS-F, p0 = pt = 1) (Fig. 5). The extension of cutout leads to

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Fig. 5 Non-dimensional amplitudes for different size of cutout

Fig. 6 Non-dimensional amplitudes for different types of boundary conditions

increase of the vibration amplitudes and movement of response curves (zone of resonance are located between critical frequencies θ1 and θ2 ). The values of amplitudes for various boundary conditions (C-C, C-SS, SS-SS, SS-F, c/a = 0.2, p0 = pt = 1) are also studied (Fig. 6). The amplitude values for the last considered boundary conditions are essentially larger.

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7 Conclusions The obtained results can be summarized as follows: instability regions are shifted into lower frequencies with increase of a static load component. Change of the cutout size yields movement of instability regions: first to lower frequencies, and then to higher frequencies. On the other hand, nonlinear vibration analysis of plates allows to note that the extension of cutout causes an increase of vibration amplitudes.

References 1. Sahu SK, Datta PK (2007) Research Advances in the dynamic stability behavior of plates and shells: 1987–2005. Part1: Conservative system. Applied Mechanics Reviews 60, 65–75. 2. Bolotin VV (1956) The Dynamic Stability of Elastic Systems, Gos. Tech. Izdat., Moscow (in Russian). 3. Hutt JM, Salam AE (1971) Dynamic stability of plates by finite element. Journal of the Engineering Mechanics Division 97, 879–899. 4. Awrejcewicz J, Krys’ko AV (2003) Analysis of complex parametric vibrations of plates and shells using Bubnov-Galerkin approach. Applied Mechanics 73, 495–504. 5. Ganapathi M, Patel BP, Boise P, Touratier M (2000) Non-linear dynamic stability characteristics of elastic plates subjected to periodic in-plane load. International Journal of Non-linear Machanics 35, 467–480. 6. Volmir AS (1972) Nonlinear Dynamics of Plates and Shells, Nauka, Moscow (in Russian). 7. Rvachev VL, Kurpa LV (1987) R-functions in Problems of Theory of Plates, Naukova Dumka, Kiev (in Russian). 8. Sahu SK, Datta PK (2002) Dynamic stability of curved panels with cutouts. Journal of Sound and Vibration 251(4), 683–696. 9. Lee HP, Lim SP, Chow ST (1990) Prediction of natural frequencies of rectangular plates with rectangular cutouts. Computers and Structures 36, 861–869. 10. Lam KY, Hung KC (1990) Orthogonal polynomials and subsectioning method for vibration of plates. Computers and Structures 34(6), 827–834. 11. Mundkur G, Bhat RB, Neriya S (1994) Vibration of plates with cutouts using boundary characteristics orthogonal polynomial functions in the Rayleigh-Ritz method. Journal of Sound and Vibration 176, 136–144.

Experiments on the Stability of Digital Force Control of Robots L´aszl´o L. Kov´acs, P´eter Galambos, Andr´as Juh´asz, and G´abor St´ep´an

1 Introduction Digital force control has a great importance in many robotic applications. For example, in automated manufacturing processes the contact force between the robot and the workpiece has to be controlled in order to achieve the desired product quality. Deburring and polishing of castings can be mentioned as typical examples for this situation [1, 2]. In addition, force control has an important role in contour following tasks, where the robot recognizes the contour of a given object by applying a prescribed contact force normal to the object’s surface [3]. Similarly, for on-line trajectory generation of robots (teaching-in) by a human operator, force control is a possible and obvious solution. In this case, the operator grasps the handle of the teaching-in device and leads the robot through the desired trajectory, which requires the compensation of the contact force to be zero between the handle and the robot’s flange. This control technique can successfully be applied for the teaching in of robots used in medical rehabilitation, where the physiotherapist having no robot programming skills and he/she has to teach-in complex motion trajectories to the robot depending on the patient’s actual condition [4]. Nowadays, service robotic applications come to the front of force control related research [5, 6]. These applications are ranging from simple cleaning robots to humanoids, which operate in the vicinity of humans. Hence, they have to be safe and dependable, which requirements may be provided via force feedback based control.

L.L. Kov´acs and G. St´ep´an Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest H-1521, Hungary, e-mail: [email protected], [email protected] P. Galambos and A. Juh´asz Department of Manufacturing Engineering, Budapest University of Technology and Economics, Budapest H-1521, Hungary, e-mail: [email protected], [email protected]

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In the domestic area, service robots have to avoid obstacles and may also manipulate a variety of small objects with very different material characteristics, which is still a difficult task. Even in case of advanced robotic hands (like [7]) the grasping force control may have stability problems due to the destabilizing digital effects, which are often compensated by making the fingertip of the robotic hand softer (e.g., using a silicon coating). In general, the stability of robotic systems subjected to digital force control is seriously affected by the controller’s nonlinearities like sampling, roundoff errors and control signal saturation, but also by the nonlinear effect of Coulomb friction. The digitally implemented control algorithms can drive the force controlled manipulator into a limit cycle, even for very high sampling frequencies that by far satisfies Shannon’s sampling theorem [8]. The goal of this work is to contribute to the understanding of the intricate dynamic behavior of robotic devices with digital force control. The presented results are based on modeling the fundamental characteristics of a 1 degree-of-freedom (DoF), force controlled robotic manipulator. That is, the controller’s sampling time, the mechanical impedance of the robot in contact with the environment, and the friction loss in the driving system. For the case of simple proportional force control the stable domain of mechanical and control parameters are given in the form of stability charts. In order to support the theoretical conclusions, experiments were performed with a HIRATA (MB-H180-500) DC drive robot. The experimental results reveal also the existence of stable and unstable self-excited vibrations due to friction.

2 Model of Digital Force Control Consider the mechanical model shown in Fig. 1. This is a 1 DoF model that can give a good approximation of the behavior of a robotic arm with force control in one direction. The modal mass m and equivalent stiffness k represent the inertia and stiffness of the robot and the environment in the force controlled direction, while b denotes the viscous damping originated either in the contacted environment or the servo motor characteristics. The generalized force Q represents the effects of the joint drives, while C denotes the magnitude of the effective Coulomb friction force.

Fd

Controller (P) & Power supply

Fm k x 0 q

Q(t) C

Fig. 1 Mechanical model

0 m

qd

k b

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Similar models are frequently used in some papers and books to analyze force control [9–12]. This simplified modeling gives a good representation if the system is under uni-directional force control only or in the case of hybrid position-force control where the time scale (fundamental frequency) associated with the position controlled directions are much lower than that of the force controlled direction. Considering a proportional force controller, the equation of motion of the model shown in Fig. 1 has the form mq(t) ¨ + bq(t) ˙ + Csgnq(t) ˙ + kq(t) = Q(t) with Q(t) = Fm (t) − P(Fm(t) − Fd ),

(1)

where P the proportional feedback gain. In addition, Fm (t) = kq(t) denotes the measured force, while Fd = kqd stands for the desired contact force. Variable x denotes a small perturbation around the constant desired position qd = Fd /k is to be used later in Section 3.

3 Stability Analysis For the case of analog force control, it can easily be shown that the model presented in Fig. 1 always results in an asymptotically stable behavior for P > 0. The only limiting factor for the tuning of the control parameters is the power constraint of the driving system. On the other hand, digital force control for the same model can have a very different behavior. According to [10] and [13, 14] the stability analysis of the discrete-time system can be carried out via the construction of a discrete map of the state variables. For this calculation, we neglect the dry friction from the model. Hence, only the stability boundaries of the linearized system are calculated. In addition, a uniform sampling with zero-order-hold (ZOH) is used to model the basic time characteristics of the digital controller. Consequently, the equation of motion for the jth sampling interval can be written as q(t) ¨ = 2ζ ωn q(t) ˙ + ωn2q(t) = ωn2 q j − ωn2 P(q j − qd ), t ∈ [t j ,t j+1 ] (2) where ωn = k/m is the natural angular frequency of the uncontrolled undamped mechanical system and ζ = b/(2mωn) is the damping ratio, while t j = jΔt, j = 0, 1, 2, . . . denotes the jth sampling instant with the sampling time Δt of the closed control loop. Then introducing the dimensionless time as T = ωnt, and the notation T j = jΔT to denote the jth dimensionless sampling instant, Eq. (2) simplifies to x (T ) + 2ζ x (T ) + x(T ) = (1 − P)x j , T ∈ [T j , T j+1 )

(3)

where x = q − qd is a small perturbation around the desired position, ΔT = wn Δt is the dimensionless sampling time, and d(•)/dT = (•) , d(•)/dt = ωn d(•)/dT .

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This equation has a central role in the characterization of the stability of digital force control. To shorten the analytical calculations, Eq. (3) is arranged into the matrix form x = Ax + Bx j , T ∈ [T j , T j+1 ), j = 0, 1, 2 . . . (4) where x = col(x(T ) x (T )) is the vector of state variables, while the corresponding coefficient matrices are 0 0 0 1 . (5) , B= A= 1−P 0 −1 −2ζ This form of (3) gives the state space model of the system between the consecutive sampling instants in the dimensionless time domain. Thus, the stability of the system is determined by the subsequent piecewise solutions of (4). Accordingly, the solution of the non-homogeneous system (4) for T ∈ [T j , T j+1 ) with the initial condition x(T j ) = x j is x(T ) = eA(T −Tj ) c − A−1Bx j , where c = (I + A−1 B)x j .

(6)

Thus, the state variables at the end of the jth sampling interval can be calculated as x j+1 = Hx j , with H = eAΔT + (eAΔT − I)A−1 B,

(7)

which represents a generalized three-dimensional geometric series as a discrete mapping between the consecutive states. The convergence of (7) is equivalent to the asymptotic stability of the force control described by (4) [15]. The stability of the system is determined by the eigenvalues z1,2 of the transition matrix H, i.e., det(zI − H) = 0, |z1,2 | < 1. The corresponding stability charts are calculated numerically and presented in Fig. 2 in the parameter space of the dimensionless time, the proportional gain and the damping ratio. Clearly, this charts show that the effect of sampling significantly limits the selection of the proportional gain even in case of very high sampling frequencies. When the robots comes contact with a stiff environment, the effective damping ratio decreases as well as the maximum stable proportional gain. An important consequence of this that the minimum of the steady state force error of the digitally controlled system will increase, since it inversely proportional to the controller’s gain in the presence of dry friction.

4 Experimental Setup For the experimental validation of the theoretical results a HIRATA (MB-H180500) DC drive robot was used. As shown in Fig. 3, the first axis of this robot was fixed during the experiments, while the second axis was connected to the robots’s

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30

Proportional gain − P

25

z=2

20

z = 1.5

15 z=1

10

z = 0.5

5 z=0 0

0.5

0

2 1.5 1 Dimensionless time − wn D t

2.5

3

Fig. 2 Stability charts for different damping ratios

Moving robot axis

DC motor Fixed robot axis

Load cell HIRATA robot

Spring

Fig. 3 Experimental setup

base (environment) via a helical spring with known stiffness k = 7, 460 N/m. The contact force was measured by a bending beam load cell (Tedea-Huntleigh Model 355) mounted between the spring and the robot’s flange. The driving system of the moving axis consisted of a HIRATA HRM-020-100-A DC servo motor directly connected to a ballscrew with a 20-mm-pitch thread. The moving axis of the robot was controlled by a self developed micro-controller based control unit providing

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Table 1 Electrical and mechanical data of the HRM-020-100-A DC servo Description

Notation

Value

Torque constant Back EMF constant Terminal resistance Torque loss due to friction Inertia of rotor

kM kE R MF Jrot

0.118701 Nm/A 0.11841 Vs 1Ω 0.03924 Nm 1.733 · 10−4 kgm2

the maximum sampling frequency 1 kHz for the overall force control loop. This controller made it possible to vary the sampling time as integer multiples of 1ms and to set the control force by the pulse with modulation (PWM) of supply voltage of the DC motor. Since DC electric motors have typically low torque performance, they require high ratios when used in (force controlled) robotic applications. For example, in the present experimental setup the gear ratio of the ballscrew is i = 100π . Considering ideal DC motor characteristics in stable operation, and using the motor data collected in Table 1, the constant reduced inertia, damping and friction of the motor gives the approximate values of the mechanical parameters in (1) as follows m = Jrot i2 = 17.1 kg, kM k E 2 i = 1387 Ns/m and b= R C = MF i = 12.3 N

(8)

However, the accurate modeling of the dynamics of the driving system requires the experimental identification of the parameters of the whole transmission line including also the clucth, the ballscrew and the robot’s flange with the force sensor. To measure these parameters the following simple experiment was conducted. The spring modeling the environment was dismounted in order to make possible the unconstrained motion of the robot. Then the robot was accelerated by a constant force for a while (along a 135 mm length path) applying PWM = 50% at supply voltage U0 = 30 V. After this, the PWM value was immediately set to be zero (causing short-cut at the motor’s terminals) and the robot was stopped by the friction and damping existing in the driving system. The results of this experiment are presented in Fig. 4, where v∞ denotes the maximum (steady-state) velocity of the robot at the constant force excitation, and tbreak is the breaking time of the free system. By measuring the Coulomb friction force C = 16.5 N using the force sensor, the solution of (1) with k = 0 and constant excitation force gives 1 kM PWM U0 b= i − C = 1447 Ns/m and v∞ R (9) btbreak m=− = 29.57 kg. C ln C+bv ∞

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0.2

Position [m]

0.145

tbreak = 72 ms

0.15 0.14

0.1

fitted data points measured time history

0.05 0.135

0

t = 381ms 0

0.2

0.4

0.4

0.6

0.45

0.5

0.8

1

0.8

1

Time [s]

Velocity [m / s]

0.5 v = 0.375 m / s

0.4 0.3 0.2

tbreak = 72 ms

0.1 0

0

0.2

0.4

0.6 Time [s]

Fig. 4 Measurement data used for system identification

These results corresponds well to those are calculated in (8). In addition, Fig. 4 shows that the measured time history of braking is in good agreement with the fitted curve obtained by using the parameters presented in (9).

5 Theoretical vs. Experimental Results By using the identified model parameters presented in (9), the stability chart of the HIRATA robot with digital force control can be poltted at ζ = 1.54 (and ωn = 15.88 rad/s). This chart together with the experimental results are shown in Fig. 5. In this figure the theoretical limit of linear stability is denoted by the thick solid line, while the black dots denote the corresponding experimental results. During the experiments, the sampling time of the digital force control loop was varied between 10–100 ms. At each selected sampling times, the proportional gain was slowly increased till the system started oscillating due to the effect of small perturbations applied at the set gain values. Subsequently, we decreased the controller’s gain to that value, where these oscillations were stopped and the system could stably provide the desired Fd = 50 [N] contact force. These values of the proportional gain are presented as white circles in the stability chart. The hysteresis between the maximum (black dots) and minimum (white dots) values of the measured proportional

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stable 35

boundary of linear stability at ζ= 1.54

bistable unstable

Proportional gain − P

30 25 measured boundaries of the bistable region

20 15 10 5 0

0

0.5

1 1.5 Dimensionless time − ωnΔ t

2

2.5

Fig. 5 Theoretical vs. experimental stability charts

gain indicates evidently the existence of an unstable limit cycle due to the nonlinear effect of Coulomb friction. Compared to the theoretical results of linear stability, this means, that some unexpected vibrations may occur when the proportional gain is in the bistable region of the stability chart (see Fig. 5). When the proportional gain is increased in order to improve the accuracy of digital force control, even small disturbances may cause large amplitude vibrations.

6 Conclusions The intricate stability properties of digital force control has been studied as the function of the proportional gain, the controller’s sampling time, and the mechanical parameters (natural frequency, damping ratio) characterizing the robot in contact with its environment. The theoretical predictions on linear stability were confirmed by a series of experiments. In addition, the the experimental results revealed the existence of unstable self-excited vibrations as the proportional gain approaches to the boundary of linear stability. Acknowledgement This research was supported by the Hungarian Scientific Research Foundation under grant No. K68910, the European project RESCUER (contract No. 511492), and by the HAS-BME Research Group of Dynamics of Machines and Vehicles.

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References 1. St´ep´an G, Steven A and Maunder L (1990), Design Principles of Digitally Controlled Robots, Mechanism and Machine Theory 25:515–527 2. Pires JN, Ramming J, Rauch S and Arajo R (2002), Force/torque sensing applied to industrial robotic deburring, Sensor Review 22(3):232–241 3. Pires JN, Afonso G and Estrela N (2007), Force control experiments for industrial applications: a test case using an industrial deburring example, Assembly Automation Journal 27(2):148–156 4. Arz G, T´oth A, Fazekas G, Bratanov D and Zlatov N (2003), Three-dimensional Anti-Spastic Physiotherapy with the Industrial Robots of “REHAROB”, in Proceedings of the Eight International Conference on Rehabilitation Robotics (ICORR 2003), April 22–25, Daejeon, Korea, 215–218 5. Marrone F, Raimondi FM and Strobel M (2002), Compliant Interaction of a Domestic Service Robot with a Human and the Environment, in Proceedings of the 33rd International Symposium on Robotics (ISR 2002), October 7–11, Stockholm, Sweden 6. Alami R, Albu-Schffer A, Bicchi A et al. (2006), Safe and Dependable Physical HumanRobot Interaction in Anthropic Domains: State of the Art and Challenges, in Proceedings of the 2006 IEEE/RSJ Conference on Intelligent Robots and Systems (Workshop on Physical HRI), Beijing, PRC 7. Borst Ch, Fischer M, Haidacher S, Liu H and Hirzinger G (2003), DLR Hand II: Experiments and Experiences with an Anthropomorphic Hand, in Proceedings of the 2003 IEEE International Conference on Robotics & Automation, September 14–19, Taipei, Taiwan, 702–707 8. Gonzalez JJ and Widmann GR (1992), Force control of robots with nonlinearities, Journal of Intelligent and Robotic Systems 6(2–3):219–240 9. Gorinevsky DM, Formalsky AM and Schneider AYu (1997), Force Control of Robotics Systems, CRC Press LLC, Boca Raton, FL 10. St´ep´an G (2001), Vibrations of machines subjected to digital force control, International Journal of Solids and Structures 38:2149–2159 11. Quian HP and Schutter J (1992), The Role of Damping and Low Pass Filtering in the Stability of Discrete Time Implemented Robot Force Control, in Proceedings of the 1992 IEEE International Conference on Robotics and Automation, May, Nice, France, 1368–1373 12. Craig JJ (1986), Introduction to Robotics Mechanics and Control, Addison-Wesley, Reading, MA ˚ om KJ and Wittenmark B (1990), Computer-Controlled Systems: Theory and Design, 13. Astr¨ Prentice-Hall, Upper Saddle River, NJ 14. Slotine JJE and Li W (1991), Applied Nonlinear Control, Prentice-Hall, Englewood cliffs, NJ 15. Kuo BC (1977), Digital Control Systems, SRL Publishing Company, Champaign, IL

Motion Analysis and Optimization for Beam Structures Georgy Kostin and Vasily Saurin

1 Introduction The elastic properties of structural elements have significant influence on their dynamical behavior. For a large number of mechanical structures their elements have a special geometrical feature that one of characteristic dimensions is mach larger than the other two. Therefore, the beam theories occupy a special place among approximate approaches in mechanics. The investigation of beam systems leads to a wide class of initial-boundary value problems for which a large number of approaches is developed. The regular perturbation method (the small parameter method) for analysis of nonuniform rod dynamics with arbitrary distributed bending stiffness and linear density and various boundary conditions is proposed by Akulenko and Kostin [1]. Based on the classical Rayleigh–Ritz approach, a numerical-analytic method of fast convergence that allows one to obtain eigenvalues and eigenfunctions of nonuniform beams with given accuracy [2]. In elastic system modeling to reduce an initial-boundary value problem for partial differential equations to a system of ordinary differential equations the methods based on finite-dimensional approximation of unknown functions, for example, the decomposition method and the regularization method, are developed. It is worth to notice the method of separation of variables using to solve beam motion problems [3]. In [4] the comparison analysis was performed for the method of integrodifferential relations and the Fourier approach. The direct discretisation methods in optimal control problems are well known (see, e.g. [5]). In this paper the method of integrodifferential relations (MIDR) proposed by Kostin and Saurin [4,6–9] is applied to finding the optimal control for the movement of elastic beams under linear boundary conditions. In Section 2 a variational principle for these initial-boundary value problems is formulated and grounded. In the G. Kostin and V. Saurin Institute for Problems in Mechanics of the Russian Academy of Sciences, Pr. Vernadskogo 101-1, 119526, Moscow, Russia, e-mail: [email protected], [email protected]

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next section an optimization algorithm for motion of uniform beams is constructed based on the MIDR and spline technique. In Section 4 analysis of the numerical results obtained by using this method for a polynomial control and a quadratic cost functional is performed.

2 Statement of the Problem Consider plane lateral motions of a rectilinear elastic beam under the distributed load u as an unknown control function. Taking into account the assumption about smallness of elastic deformations the controlled motion of the beam can be described by the following system of partial differential equations [4]: p˙ + m = u, ˙ p = ρ w,

y ∈ (0, L),

m = EIw ,

t ∈ (0, T ).

(1) (2)

Here p is the linear momentum density; m is the bending moment in the beam cross section; w are the lateral displacements (deflections); L and ρ are the length and linear density of the beam, respectively; EI is its flexural rigidity; and T is the terminal instant of the control process. The dotted symbols denote the partial derivatives with respect to the time t, and the primed symbols stand for the partial derivatives with respect to the beam coordinate y. We confine ourselves to the case when the homogeneous boundary conditions at the beam ends y = 0, L may be written in the following linear form:

α0, l w + μ0, l m y=0, L = 0, β0, l w + ν0, l m y=0, L = 0, (3) where α0, l , μ0, l , β0, l , and ν0, l are given time dependent functions defining the type of boundary conditions. In particular, if α0, l = 1 and μ0, l = 0 on the corresponding end of the beam then, according to Eq. (3), the displacements w are equal to zero, whereas if α0,l = 0 and μ0,l = 1 then the zero external forces m are defined. Similarly, the pairs β0, l = 1, μ0, l = 0 and β0, l = 0, μ0, l = 1 fix either the angle w = 0 or the moment m = 0, respectively. Various combinations of linear elastic supports are also included in conditions (3) if α0, l , μ0, l or β0, l , μ0, l are simultaneously nonzero (α0 μ0 < 0, αl μl > 0, β0 ν0 > 0, βl μl < 0). The shape of the beam lateral displacements w and its relative linear momentum density p are given at the initial time t = 0 w(0, y) = f (y),

p(0, y) = g(y).

(4)

It is worth noting that the initial conditions (4) and boundary conditions (3) should be compatible. The problem is to find an optimal control u(t, y) that moves the beam from its initial state (4) to a terminal set of states w(T, y) ∈ WT ,

p(T, y) ∈ PT

(5)

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in the given time T and minimizes a objective function J[u] in the class U of admissible controls: J[u] → min . (6) u∈U

To solve the initial-boundary value problem (1)–(4), we apply the MIDR, described by Kostin and Saurin [4, 6–9], in which some strict local equalities are replaced by an integral relation. In the case under consideration it is possible to reduce problem (1)–(4) to a variational problem. If a solution p∗ , m∗ , and w∗ exists then the following functional Φ reaches its absolute minimum on this solution over all admissible functions p, m, and w under local constraints (1), (3), (4) Φ(p∗ , m∗ , w∗ ) = min Φ(p, m, w) = 0, p,m,w

Φ=

T L

ϕ (p, m, w)dydt,

ϕ=

0 0

˙ 2 (m − EIw )2 (p − ρ w) . + 2ρ 2EI

(7)

Note that the integrand ϕ in (7) has the dimension of the energy linear density and is nonnegative. Hence, the corresponding integral is nonnegative for any arbitrary functions p, m, and w (Φ ≥ 0). Denote the actual and arbitrary admissible linear momentum densities, bending moments, and displacements by p∗ , m∗ , w∗ and p, m, w, respectively, and specify that p = p∗ + δ p, m = m∗ + δ m, w = w∗ + δ w. Then Φ(p, m, w) = Φ(p∗ , m∗ , w∗ ) + δ p Φ + δm Φ + δw Φ + δ 2 Φ, where δ p Φ, δm Φ, δw Φ are the first variations of the functional Φ with respect to p, m, w and δ 2 Φ is its second variation. Integrating the first variations by parts and taking into account Eqs. (1), (3), and (4) result in the following relations

δw Φ = ρ

−1

T L

( p˙ − ρ w) ¨ − (m − EIw ) δ wdydt−

0 0

L

(p − ρ w)| ˙ t=T t=0

δ wdy−

0

δ p Φ = ρ −1

T L

T

/y=L /y=L (m − EIw )/y=0 δ w dt + (m − EIw ) /y=0 δ wdt,

0

0

T

(p − ρ w) ˙ δ pdydt,

0 0

δm Φ = (EI)−1

T L

(m − EIw )δ mdy dt.

(8)

0 0

It is seen from Eq. (8) that the fist variation of the functional Φ is equal to zero for any admissible variations δ p, δ m, δ w if the following equations are valid: p − ρ w˙ = 0,

m − EIw = 0.

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Hence, the stationary conditions (8) of the functional Φ are equivalent to relations (2) and together with constraints (1), (3), and (4) constitute the full system of dynamical equations for the beam. The second variation

δ 2Φ =

T L

ϕ (δ p, δ m, δ w)dydt ≥ 0,

0 0

which is quadratic with respect to the variations δ p, δ m, δ w, is nonnegative because the integrand ϕ (δ p, δ m, δ w) ≥ 0.

3 An Approximation Algorithm To find an approximate solution of the optimization problem for cantilever beam motions (a special case of problem Eqs. (1), (3)–(7)) a polynomial representation of the unknown functions have been used in [4, 8, 9]. In this work the functions p, m, and w are approximated by bivariate piece-wise polynomial splines defined on rectangular meshes. Let one divided the time-space domain Ω = (0, T ) × (0, L) on N × M rectangles Ωkl which vertices have coordinates Qk−1, l−1 , Qk−1, l , Qk, l−1 , Qk, l , where Qk, l = (tk , yl ); tk > tk−1 , k = 1, . . . , N; yl > yl−1 , l = 1, . . . , M; t0 = 0, tN = T , y0 = 0, yM = L. Let also the boundary edges of these time-space rectangles be named Lkl = (Qk,l−1 , Qkl ), k = 0, . . . , N, l = 1, . . . , M, and Tkl = (Qk−1,l , Qkl ), k = 1, . . . , N, l = 0, . . . , M. For all rectangles Ωkl the polynomial approximating functions are given p

p˜kl =

Nkl

∑

(i j) pkl t i y j ,

m˜ kl =

i+ j=0

m Nkl

∑

(i j) mkl t i y j ,

w˜ kl =

i+ j=0

The control function u is restricted to a set of splines ⎧ u Nkl ⎨ (i j) u : u = uk t i y j ; ∑ UNM = i=0 ⎩ t ∈ (tk−1 ,tk ), k = 1, . . . , N; y ∈ (yl−1 , yl ), (i j)

(i j)

(i j)

(i j)

w Nkl

∑

(i j)

wkl t i y j .

(9)

i+ j=0

⎫ ⎬ l = 1, . . . , M

⎭

.

(10)

Here pkl , mkl , wkl , and ukl are unknown real coefficients. The basis functions are chosen so that the approximations can exactly satisfy boundary and piece-wise polynomial initial conditions (3), (4), and the equilibrium equation (1) on the rectangles Ωkl by suitably selected integers Nklp , Nklm , Nklw , and Nklu . In addition, to apply the variational formulation given above the following conformed interelement relations must be satisfied

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w˜ kl (tk , y) = w˜ k+1, l (tk , y), p˜kl (tk , y) = p˜k+1, l (tk , y), (tk , y) ∈ Lkl , k = 1, . . . , N − 1, l = 1, . . . , M; w˜ kl (t, yl ) = w˜ k, l+1 (t, yl ),

w˜ kl (t, yl ) = w˜ k, l+1 (t, yl ),

m˜ kl (t, yl ) = m˜ k, l+1 (t, yl ),

m˜ kl (t, yl )

(t, yl ) ∈ Tkl ,

k = 1, . . . , N,

(11)

= m˜ k, l+1 (t, yl ),

l = 1, . . . , M − 1.

These approximations give one the possibility to obtain numerical solutions and analyze its quality and convergence rate for resulting finite dimensional systems by using the distribution of the discretisation error ϕ and the value of the functional Φ. After satisfying constraints (1), (3)–(5) the resulted finite-dimensional unconstrained minimization problem (7) yields an approximate solution p˜ ∗ (t, y, u), m˜ ∗ (t, y, u), w˜ ∗ (t, y, u) for an arbitrary control u ∈ UNM , where UNM is the set of piece-wise polynomial functions with a given degree Nklu on the rectangles Ωkl . The optimal control u∗ (t, y) is found by minimizing the objective function J in Eq. (6). Let us consider a functional J[u] quadratic with respect to the control parameters ui . In this case if U ⊂ UNM in Eq. (6) then the corresponding optimization problem is reduced to a system of linear equations with respect to unknown control parameters ui . The value of the functional ˜ = Φ( p˜∗ (t, y, u∗ (t, y)), m˜ ∗ (t, y, u∗ (t, y)), w˜ ∗ (t, y, u∗ (t, y))) Φ is an absolute integral criterion of the optimal solution quality. To define relative integral errors the following nonnegative functionals with the dimension of action can be used Ψ=

T L

ψ (p, m, w)dydt,

0 0 T L

Ψ pm =

Ψw =

0 0 T L

ψ=

ψ pm (p, m, w)dydt,

ψw (p, m, w)dydt,

pw˙ EImw + ; 2 2EI

ψ pm =

ψw =

p2 m2 ; + 2ρ 2EI

(12)

ρ w˙ 2 EI (w )2 + ; 2 2

0 0

Φ = Ψ pm + Ψw − 2Ψ. After substituting optimal functions p˜ ∗ (t, y, u∗ (t, y)), m˜ ∗ (t, y, u∗ (t, y)), w˜ ∗ (t, y, u∗ (t, y)) in the functionals Ψ, Ψ pm , Ψw various dimensionless values can be generated, for example,

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Ψ pm + Ψw 2Ψ − 1, Δ2 = 1 − , 2Ψ Ψ pm + Ψw Φ Φ Δ3 = , Δ4 = . 2Ψ pm 2Ψw Δ1 =

(13)

and so on. The integrand ϕ in Eq. (7) can serve as a local criterion of solution quality, whereas the functions ψ˜ (t, y) = ψ ( p˜∗ , m˜ ∗ , w˜ ∗ ), ψ˜ pm (t, y) = ψ pm ( p˜∗ , m˜ ∗ , w˜ ∗ ), ψ˜ w (t, y) = ψw ( p˜∗ , m˜ ∗ , w˜ ∗ ) at u = u∗ (t, y) are the estimates of energy linear density.

4 Numerical Example As a numerical example, consider plane controlled motions of a homogeneous rectilinear elastic beam. One end of the beam L is free, and the other is clamped on a truck that can move along a horizontal line with the velocity v (see Fig. 1). In the undeformed state, the beam is fixed in a vertical position. The control action on the beam is the horizontal acceleration −u(t)/ρ of the clamped beam end, where the dimension of control function u is the linear force density. Initially, the shape of the beam lateral deflection (displacement) w and its relative linear momentum density p are given in a coordinate system Oxy moving with the truck. The location of the clamped beam point O is specified by the coordinate x in a stationary coordinate system O XY ; hereinafter, x˙ = v and v˙ = −u/ρ . Without loss of generality, it can be assumed that the coordinate and velocity of the truck are initially equal to zero. Let the zero initial functions f (y) = 0 and g(y) = 0 in Eq. (4) are given. Boundary conditions (3) in the example under consideration take the form w(t, 0) = w (t, 0) = 0,

m(t, l) = m (t, l) = 0.

(14)

For numerical modeling, the following dimensionless parameters are used: L = 1, ρ = 1, EI = 1, and T = 2. The optimal control problem of beam transportation

Fig. 1 Beam clamped on a truck

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from the initial rest position to given terminal state, in which the total mechanical energy of the system reaches its minimal value, is considered. The quadratic objective function J in the proposed variational formulation can be expressed in terms of the energy density function ψw defined in Eq. (12) J=

L

ψw (T, y)dy,

ψw (t, y) =

ρ w˙ 2 EI(w )2 + . 2 2

(15)

0

The control u is chosen as a time polynomial of the fixed degree Nu = 5 Nu

u = ∑ ui t i ,

t ∈ (0, T )

(16)

i=0

The unknown coefficient ui must satisfy terminal conditions v(T ) = 0, x(T ) = 1 at the time T = 2 and minimize the functional J in Eq. (15). In the case when Nu > 1, the control u in (16) contains Nu − 1 unknown optimizing parameters. To demonstrate the effectiveness of the numerical algorithm proposed in Section 3 the following mesh and approximation parameters are assigned: N = 3, M = 1, Nklp = 15, Nklm = 15, Nklw = 15. To improve the integral quality of numerical solution a time nonuniform mesh with the parameters t1 = 0.1875 and t2 = 0.75 was found for this optimal problem. To estimate the polynomial optimization resources for beam dynamic problems an admissible truck acceleration (of the lowest number of degree Nu = 1) u0 = 3 (1 − t) /2 is specified as a sample control (the dash line in Fig. 2). As the number of free parameters of the polynomial control in the optimization problem (1), (3)– (7) increases, the total mechanical energy of the beam at the terminal time reduces considerably. The optimal control obtained by the MIDR for Nu = 5 are shown in

Fig. 2 Optimal control u

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Fig. 3 Distribution of mechanical energy density ψ pm

Fig. 4 Distribution of local solution error ϕ

Fig. 2 by the solid curve. The values of J are approximately equal to 0.306 and 3.02 × 10−4 for the sample and optimal control, respectively. In the case of four free control parameters, the value of J can be reduced by more than 103 times. The distribution of energy density characterizes the dynamic process taking place in the beam during the optimal control. In Fig. 3 the function ψ pm (t, y) defined in Eq. (12) is presented. As it is seen from the picture the mechanical energy is reduced noticeably at the end of the optimal process. The value of the functional Φ can be considered as an integral performance criterion for the optimal solution whereas the integrand ϕ in (7) is a local quality characteristic. Figure 4 shows the distribution of the function ϕ (t, y) for Nu = 5.

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It can be seen that its value is small almost everywhere, except for the vicinity of t = 0 with its maximum at the point y = 0. For the defined parameters the value of the functional is equal to Φ = 3, 22 × 10−5 whereas its relative value, for example, Δ3 = 0.06%.

5 Conclusions The presented method of integro-differential relations to solve mechanical initialboundary value problems can be considered as an alternative to conventional approaches. Based on the MIDR a variational principle which stationary conditions are equivalent to the constitutive beam relations was formulated. For this principle the nonnegative quadratic functional under minimization can serve as integral criteria of the solution quality, whereas its integrand characterizes the local error distribution. The principle proposed is also applicable to the beam problems with the mixed boundary conditions (e.g., elastic support). The finite element algorithm developed enables one to construct effective bilateral estimates for various integral characteristics (elastic energy, displacements, etc.). The polynomial splines technique allows one to take into account nonhomogeneous inertial and elastic properties. This FEM realization gives one the possibility to work out various strategies of p-h adaptive mesh refinement by using the local error estimates. The computational cost of the algorithm proposed is stipulated by the efficiency of particular FEM realization. Acknowledgement This research was supported by the Russian Foundation for Basic Research (grants Nos. 05-01-00563, 05-08-18094, 08-01-00234, 08-01-00362) and the Program “State Support for Leading Scientific Schools” (NSh-1245.2006.1, NSh-9831.2006.1).

References 1. Akulenko LD, Kostin GV (1992) The perturbation method in problems of the dynamics of inhomogeneous elastic rods, J. Appl. Math. Mech. 56, 372–382. 2. Akulenko LD, Kostin GV, Nesterov SV (1995) Numerical-analytic method for investigation of free oscillations on nonhomogeneous beams, Mech. Solids 30, 173–182. 3. Chernousko FL (1996) Control of elastic systems by bounded distributed forces, Appl. Math. Comput. 78, 103–110. 4. Kostin GV, Saurin VV (2006) Modeling of controlled motions of an elastic rod by the method of integro-differential relations, J. Comput. Sys. Sci. Int. 45, 56–63. 5. Leineweber D, Bauer EI, Bock H, Schloeder J (2003) An efficient multiple shooting based reduced SQP strategy for large dynamic process optimization. Part 1: Theoretical aspects, Comput. Chem. Eng. 27, 157–166. 6. Kostin GV, Saurin VV (2005) Itegro-differential approach to solving problems of linear elasticity, Dokl. Phys. 50, 535–538.

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7. Kostin GV, Saurin VV (2006) The method of integrodifferential relations for linear elasticity problems, Arch. Appl. Mech. 76, 391–402. 8. Kostin GV, Saurin VV (2006) The optimization of the motion of an elastic rod by the method of integro-differential relations, J. Comput. Sys. Sci. Int. 45, 217–225. 9. Kostin GV, Saurin VV (2006) Modeling and optimization of elastic system motions by the method of integro-differential relations, Dokl. Math. 73, 469–472.

Parametrical Analysis of the Behavior of an Aerodynamic Pendulum with Vertical Axis of Rotation Liubov Klimina, Boris Lokshin, and Vitaly Samsonov

1 Introduction Parametric analysis of the behavior of an aerodynamic pendulum is carried out, motivated by the study of a small wind power generator with a vertical axis. A mathematical model for the free rotation of this pendulum is constructed, leading to a system of nonlinear ODEs and transcendental algebraic equations. A qualitative analysis of the phase portrait is carried out: all equilibrium solutions are found, their stability is studied, characteristics of a stable rotational regime are determined; and domains of attraction for equilibrium solutions and for the rotational regime are also found. The mathematical model is used to study the operational regimes of the system “wind turbine + generator”. Estimations of the trapped power as a function of the external load in the circuit are obtained; optimal values for the power and the load are found. A pitch angle control mechanism is proposed in order to increase this power. Humanity used wind power for ages (e.g. windmills, pumps, sailboats). 1920s and 1930s saw a sharp increase of interest in wind power stations, both in science and engineering, against the backdrop of rapid development in research of propelled aviation (for both airplanes and helicopters). A recurrence of this interest became a feature of the last decades, as more attention is paid to ecology and sustained development. We can now observe increased numbers of new wind power stations and more use of wind energy, as well as larger amounts and volumes of research publications on the subject. While a comprehensive review of such literature is beyond the scope of the present work, we should mention a few examples [1–5]. While most authors engage in detailed descriptions of the units, we would like to present a sufficiently simple model which can be used not only to adequately describe the observed phenomena and effects, but also to optimize the parameters of the construction. L. Klimina, B. Lokshin and V. Samsonov Institute of Mechanics of Lomonosov Moscow State University, Russian Federation, 119192, Moscow, Michurinsky pr., 1 e-mail: [email protected], [email protected], [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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Fig. 1 The aerodynamic pendulum (top view)

In our approach, an aerodynamic pendulum is considered as a model for the principal element of the wind-capturing unit of a wind generator for a Vertical Axis Wind Turbine (VAWT). A VAWT has a number of advantages over the horizontal axis wind turbine. For a VAWT an orientation device is not required; the generator can be set on the ground; the construction is simpler; and gyroscopic loads are lower. The invention of Darrieus turbine in 1926 in France can be marked as the beginning of the modern age of scientific research on VAWT. Since then the VAWT have been actively studied in many countries (see for example [1, 3, 5–9]). We consider an aerodynamic pendulum with a vertical rotation axis in a steady horizontal wind flow as the model of the wind-receiving element of a straight winged VAWT. Assume that the pendulum consists of a thin rectangular flat plate and a horizontal weightless holder OA, which is much longer than the width of the plate. The plate is attached to the holder at its geometric center A, so that the plate stays vertical and forms the angle β with the vertical plane orthogonal to the holder (Fig. 1). In the first part of this paper we assume that β ≡ const, so that the plate and the holder form a single rigid body that can rotate about the fixed vertical axis O. The center of mass of the plate coincides with its geometric center A. Suppose that the flow around the plate is planar. Assume that the aerodynamic force acting on the plate consists of two components. The quasi-steady component is determined in stationary wind tunnel experiments (e.g. Fig. 2, where e(α ) is given for r = 2l) and the non-steady component is described by the tensor of apparent masses.

2 The Equations of Motion Since the system has just one degree of freedom, we take ϑ as the generalized coordinate. Then the equations of the pendulum motion in steady flow can be represented in the following form according to the method used in [6–8, 11]:

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Fig. 2 Functions Cx (α ), Cy (α ), e(α ) for flat aspect-ratio-8 plate ( [10])

ϑ = ω, Aω = u2 (Cy (α )(sin(α + β ) − e(α ) cos α ) − Cx (α )(cos(α + β ) + + e(α ) sin α )) + bω (cos(ϑ + 2β ) + e(α ) sin(ϑ + β )) + + bω 2 (e(α ) cos β − 0.5 sin(2β )) − cω .

(1)

In the above u and α are determined from the kinematic relations: u cos α = ω cos β + sin(ϑ + β ), u sin α = e(α )ω − ω sin β + cos(ϑ + β ).

(2)

Terms, containing the coefficient b, are due to the effect of apparent masses. The term −cω models the load upon the axis of rotation due to the operation of the generator. Observing the kinematic relations (2) we can note the following. If e(α ) ≡ 0 (no shift of the center of pressure), then u and α are uniquely determined at each point (ϑ , ω ) of the phase plane, except for (−π /2, 1) and (π /2, −1). At those two points we have u = 0, and any α satisfies the Eqs. (2). If we account for the shift of the center of pressure, then there are two small domains of ambiguity in the vicinities of points (−π /2, 1) and (π /2, −1). For each point (ϑ , ω ) in these domains there exist multiple pairs u, α satisfying the relations (2). If any trajectory of the system enters the domain of ambiguity, we choose α in this domain by continuity. Then at the point of exit from the domain of ambiguity α can have discontinuity.

3 Existence of Auto-Rotation Existence of rotational modes for the pendulum motion is of special interest, because a stable rotational mode can serve as an approximation to the operating regime of a wind turbine.

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A rotational mode of motion corresponds to a periodic trajectory of the system (1)–(2) encircling the phase cylinder. For such a trajectory, the following property holds: any value ϑ1 satisfies the equation ϑ1+2π / / ω (ϑ , ω (ϑ ))d ϑ = 0, meaning that ω /ϑ =ϑ = ω / . ϑ =ϑ1 +2π

1

ϑ1

The following claim takes place for the system (1)–(2) (see [12]): Let the following equations hold for some value Ω0 = const = 0: ⎛ ⎞/ ϑ1+2π ϑ / 1 +2π / d ⎝ ⎠ ω (ϑ , Ω0 )d ϑ = 0; ω (ϑ , Ω)d ϑ // = 0 dΩ / ϑ ϑ 1

(3)

Ω=Ω0

1

Then for a sufficiently large value of A the system (1)–(2) has a periodic phase trajectory initiated from the straight line ω = Ω0 . For e(α ) ≡ 0 and certain assumptions concerning Cx (α ), Cy (α ) it can be shown that there are at least two values Ω0 (with different signs) satisfying (3), if quantities β and c satisfy the inequality 3[cos 2β ·

π /2

π /2

0

0

(Cy (ϑ ) sin 2ϑ − Cx (ϑ ) cos 2ϑ )d ϑ −

Cx (ϑ )d ϑ ] − π c > 0.

(4)

For specific functions Cx (α ), Cy (α ) (as in Fig. 2) (4) assumes the following form: 3[1.237 cos2β − 1.196] − π c > 0.

(5)

Thus, (5) is a sufficient and realizable condition for existence of rotational modes in the motion of the pendulum in question, with large moment of inertia (when e(α ) ≡ 0). Moreover, these rotational modes will be attracting. Consider the equality (3) as an equation for Ω0 . If β = 0, an approximate solution can be found for the case Ω0 > 1: Ω20 ≈ 0.5Cy (0)/Cx (0) (see [6]). It can be shown, that Ω0 is an even function of β for small β (for b = 0 and e(α ) ≡ 0).

4 Numerical Analysis of Rotational Modes and Domains of Attraction for β = 0 and Various Values of c For the plate described above (Fig. 2) in the case β = 0, let the system be characterized by the fixed parameters A and b. Here we take the following values A = 8.14, b = 0.1 π /2. For example, these values correspond to the case, when r = 1.2 m, l = 0.06 m, h = 0.96 m, Jo = 1 kg· m2 , i = 0.0125 kg· m2 , m1 = 0.0136 kg, ρ = 1.25 kg/m3 . As estimates for m1 and i we take: m1 = ρπ hl 2, i = 0.0625ρπ h4l. Let V = 10 m/s.

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Consider some features of the phase portrait of the system (1)–(2). Let’s start by looking for periodic trajectories. Denoting the dimensionless aerodynamic torque in the system (1) by q(ϑ , ω ), one can rewrite equations as follows: ϑ = ω ; Aω = q(ϑ , ω ) − cω . Introduce the function f (Ω) = 02π q(ϑ , Ω)d ϑ /2π . Then the Eq. (3) can be rewritten as f (Ω0 ) − cΩ0 = 0 (6) First assume c = 0. It is possible to show numerically, that f (Ω) has a unique nonzero root Ω0 (for sufficiently large A, the lines ω = ±Ω0 give birth to attracting periodic trajectories) and that there are no other periodic trajectories, not just originating from straight lines, but also emerging for other reasons. Besides, for c = 0 there exists an unstable cycle Γ. Inside this cycle Γ there are nine rest points, two unstable cycles and one stable one. Outside Γ there is only one equilibrium point – the saddle point (0, 0). All phase points lying outside of the cycle Γ belong to the domain of attraction of one of the periodic trajectories found. As the coefficient c increases, the unstable cycle Γ expands and bifurcates into two unstable periodic trajectories, and this bifurcation is not described by the origination of periodic trajectories from straight lines. To explain of the further evolution of the phase portrait with the parameter c we give a qualitative picture of the dependence between the values of the coefficient c and the values ω (0) = ω |ϑ =0 at the periodic phase trajectories (Fig. 3). We give a schematic picture of the zones of attraction of steady periodic trajectories. Unstable periodic trajectories (thin lines) limit domains of attraction of the stable ones (bold lines). Note that the similar figure was obtained for aerodynamic pendulum for which the plate is fixed along the holder (see [7]).

Fig. 3 Diagram of values ω |ϑ =0 at periodic trajectories as functions of c

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5 Average Trapped Power Consider the stable periodic trajectory of the system (1)–(2) originating from the straight line ω = Ω0 > 0. Average trapped power in the corresponding regime is P = ρ V 3 lh 02π ω q(ϑ , ω )d ϑ /2π . Its magnitude is determined by the parameters β and c. Let us estimate this power as P˜ = ρ V 3 lhΩ0 f (Ω0 ). The value Ω0 and the behavior of the function f (Ω) are also determined by β and c, so P˜ is a function of β and c. It is interesting to find the maximum of the function p(c, β ) = Ω0 (c, β ) f (Ω0 (c, β )). Using (6), it can be written as follows p(c, β ) = cΩ20 (c, β )

(7)

Fix β and determine the value of c for which the maximum of p(c, β ) = Ω0 f (Ω0 ) is achieved. As Ω0 , for fixed β , is a function of c as given by (6), the search for the optimal c can be replaced by searching for the optimal value Ω∗ of variable Ω0 . This value is determined from the equation / d f // f (Ω∗ ) + Ω∗ =0 (8) dΩ /Ω=Ω∗ Then c∗ , the optimal value of the coefficient c, is given by the expression: c∗ = f (Ω∗ )/Ω∗ . For example, for β = 0 in the system with fixed parameters as outlined above, a numerical solution of (8) gives Ω∗ ≈ 4.18, that is ϑ˙ ∗ = 34.8 1/s. At this regime we have C∗ ≈ 0.056 kg · m2 /s, P ≈ 68W. Alternatively, the Eq. (8) allows for an approximate analytical solution (for Ω∗ > 1) at β = 0 [6]: Cy (0) . (Ω∗ )2 = 6Cx (0) In the considered case we have Cy (0) = 4.1826, Cx (0) = 0.04, and the last formula gives the value Ω∗ = 4.17, corresponding to ϑ˙ ∗ = 34.75 1/s. A direct search for c∗ by determining the values of P for various values of c via numerical integration of the equations of motion gives Ω∗ ≈ 4.14, C∗ ≈ 0.058 kg · m2 /s, P ≈ 70W. Comparison of the results shows that in practical tasks it ˜ is sufficient to maximize the function P. Note: numerical calculations showed that the domain of attraction of the obtained stable periodic mode is limited by an unstable periodic trajectory in the neighborhood of the straight line ω = 3.5. The approximate values of maximum power at the rotational mode, obtained numerically, are given in the Table 1 for several values of pitch angle β and for wind speed V = 10 mps. From the data of the Table 1 we can conclude that the output power depends essentially on the pitch angle. Note that for values of β large enough, the sufficient condition (5) of the existence of a rotational mode becomes unrealizable. Numerical integration of the equations of motion showed that for β = −7◦ and c = 0, an

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Table 1 Estimations of maximum trapped power Angleβ (degree)

Coefficient C∗ (kg · m2 /s)

Average value of ϑ˙ ∗ (1/s)

Average power P∗ (W)

−5 −4 −3 −2 −1 0 1 2 3

0.011 0.022 0.031 0.040 0.054 0.058 0.049 0.038 0.010

23.0 25.1 29.3 34.0 34.4 34.5 34.4 34.3 29.6

6 14 27 46 65 70 59 38 8

attracting periodic trajectory still exists, whereas for β = ±10◦ there no longer are any periodic trajectories.

6 Pitch Angle Control We investigate the possibility of increasing the average trapped power by varying the pitch angle β , taking it as a control function. Rewrite the equations of motion and the kinematic relations for the case of variable β :

ϑ = ω, Aω = u2 (Cy (α )(sin(α + β ) − e(α ) cos α ) − Cx (α )(cos(α + β )+ + e(α ) sin α )) + bω (cos(ϑ + 2β ) + e(α ) sin(ϑ + β ))+ 1 + bω 2(e(α ) cos β − sin(2β )) − cω + b[−ω sin(2β )+ 2 + cos(ϑ + 2β ) + ω e(α ) cos β + e(α ) sin(ϑ + β )]β , u cos α = ω cos β + sin(ϑ + β ), u sin α = e(α )ω − ω sin β + cos(ϑ + β ) + e(α )β .

(9)

(10)

Assume c is fixed. Then due to (7) the task of increasing of the average trapped power is reduced to increasing of the average angular velocity of the rotational mode, which in its turn is reduced to the increase of the aerodynamic torque. To simplify the realization of the control we choose β to be a function of ϑ only. We illustrate this approach to selection of β in a simplified setting, where we assume m1 = 0, e(α ) ≡ 0, ω = Ω, with Ω = const is the average angular velocity of the rotational mode obtained. It can be shown that under these assumptions the dimensionless aerodynamic torque can be written as follows: Ma = (Ω + sin ϑ )2 + cos2 ϑ (Cy (α ) cos ϑ − Cx (α )(Ω + sin ϑ )),

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Fig. 4 Attracting periodic trajectory for variable β

where

cos ϑ . Ω + sin ϑ First we search for the value α = α ∗ where the maximum of Ma is achieved. Setting to zero the derivative of Ma with respect to α , we obtain: Cy (α ) cos ϑ −Cx (α )(Ω + sin ϑ ) = 0. Assume that Cy (α ) ≈ Cy (0); Cx (α ) ≈ Cx (0)α . These simplifications are based on the specific properties of the aerodynamic functions: Cy (α ) is close to a linear function and Cx (α ) is close to a quadratic function when |α | < αm << 1 (for the functions shown in Fig. 2 αm = 0.2). With these assumptions the solution of the last equation is given by the formula:

α = −β + arctg

α∗ =

Cy (0) cos ϑ . Cx (0)(Ω + sin ϑ )

If the condition |α ∗ | > αm holds, we set α ∗ equal to −αm or αm , respectively. Now we can approximately solve for the desired β via the formula:

β = −α ∗ +

cos ϑ . Ω + sin ϑ

After substituting the corresponding values of β and β into the system (9)–(10), one can find a value Ω for which the trajectory with ω |ϑ =0 = Ω is periodic. For the following values of the parameters: C = 0.17 kg · m2 /s, Ω = 3.2 at the obtained rotational mode, the average trapped power is approximately equal to 120W. The corresponding periodic trajectory of the system (9)–(10) is shown in Fig. 4. The pitch angle and the angle of attack as functions of the angle ϑ in this regime are shown in Fig. 5. The domain of attraction of the obtained trajectory is bounded below by an unstable periodic solution with ω ≈ 2.6.

7 Conclusions In this paper, an aerodynamic pendulum is considered as a model for the principal element of the wind-capturing unit of a wind generator for a Vertical Axis Wind Turbine VAWT. Existence of auto-rotational regimes is studied analytically

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Fig. 5 Dependence of angles α and β on the angle ϑ

and numerically. Numerical analysis of the phase portrait is performed. Domains of attraction of stable periodic trajectories are founded. Estimates of the magnitude of the power of a VAWT, neglecting losses in the electric generator, are obtained numerically for several values of the pitch angle. The possibility of using the variable pitch angle as the control function is considered. It is shown that the trapped power can be increased by 70% using pitch control. The work is supported by RFBR (grants NN 06-01-00079, 08-08-00390).

Notation β = pitch angle (the angle between the plate and the plane orthogonal to the holder); r = length of the holder; 2l = width of the plate (chord); h = height of the plate; S = 2lh = surface area of the plate; JO = moment of inertia of the pendulum about the axis of rotation; m1 , m2 = 0, i = diagonal elements of the central tensor of apparent masses of the plate; J = JO + i + m1 r2 sin β = the moment of inertia of the system about the rotation axis; ρ = density of air; V = wind velocity vector; U = VF − V, where VF is the velocity of the center of pressure F; α = the effective angle of attack (this is the angle between the vector U and the plate, it is called “angle of attack” below); e(α ) = AF/r = shift of the center of pressure; Cx (α ),Cy (α ) = coefficients of the drag and the lift forces; X = 0.5Cx (α )ρ SU 2 ,Y = 0.5Cy (α )ρ SU 2 = magnitudes of the drag and the lift; C = coefficient of electro-mechanic coupling; ϑ = angle between the holder and the vector V; ϑ˙ = angular velocity of the rotation of the pendulum (here a dot denotes the derivative with respect to time); τ = Vt/r = dimensionless time; u = U/V = dimensionless velocity of the center of pressure;

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ϑ = ω = rϑ˙ /V (prime denotes the derivative with respect to τ ); m1 J C A = ρ lhr 3 ; b = ρ lhr ; c = V ρ lhr2 = dimensionless parameters.

References 1. Gorelov DN (2003) Problems of Darrieus wind turbine aerodynamic (in Russian). Journal of Termophysics and Aeromechanics 1, 47–51. 2. Kharitonov VP (2006) Autonomous Wind-Electrical Plants. Publishers of the Research Institute of Agricultural Electrification, Moscow (in Russian). 3. Klimas PC, Sheldahl RE (1978) Four Aerodynamic Prediction Scheemes for VerticalAxis Turbines: A Compendium. Issued by Sandia Laboratories, operated for United States Department of Energy by Sandia Corporation. 4. Leishman JG (2002) Challenges in modelling the unsteady aerodynamics of wind turbines. Journal of Wind Energy 5, 85–132. 5. Paraschivoiu I (1983) Predicted and experimental aerodynamic forces on the Darrieus rotor. Journal of Energy 6(7), 610–615. 6. Dosaev MZ, Kobrin AI, Lokshin BY, Samsonov VA, Seliutsky YD (2007) Constructive Theory of Small-Scale Wind Power Generators. Study guide. Parts I–II. Publishers of the Mechanical and Mathematical Faculty of the MSU, Moscow. 7. Lokshin BYa, Okunev YuM, Ryzhova VE, Samsonov VA (1996) Parametric analysis of motion of the aerodynamic pendulum. Proceedings of the 2nd European Nonlinear Oscillations Conference (ENOC) 1, Prague, 261–264. 8. Lokshin BYa, Samsonov VA (1998) On heuristic model of aerodynamic pendulum (in Russian). Journal of Fundamental and Applied Mathematics 3(4), 1047–1063. 9. Parshin DYe, Samsonov VA (1994) Aerodynamical pendulum as a model of the vertical axis wind turbine. Programme and abstracts of the 1st European Nonlinear Oscillations Conference (ENOC), 116. 10. Tabachnikov VG (1974) Stationary characteristics of wings at small speeds for full-range of angles of attack. Transactions of TsAGI 1621, 79–92 (in Russian). 11. Lokshin BYa, Privalov VA, Samsonov VA (1986) Introduction to the Problem of the Motion of a Rigid in Resistant Medium. Study guide. Publishers of the MSU, Moscow (in Russian). 12. Bautin NN, Leontovich EA (1990) Methods and Techniques of the Qualitative Research of the Dynamic Systems in the Plane. Science, Moscow (in Russian).

Error Function Based Kinematic Control Design for Nonholonomic Mechanical Systems El˙zbieta Jarze˛ bowska and Paweł Cesar Sanjuan Szklarz

1 Introduction Tracking or following a trajectory at a kinematic level is the most common approach to control nonholonomic systems. The classical feedback approaches may be either local or global. The local approach to feedback control is achieved through the standard linear control design. Convergence is obtained provided that a system, e.g. a vehicle, starts its motion sufficiently close to a desired trajectory. In the global approach, feedback linearization is pursued and it provides asymptotic stability of tracking errors for arbitrary initial states. Both static and dynamic nonlinear feedback linearization may be used in this approach. In this classical kinematic control design using feedback linearization, some additional properties of kinematic models are used. The first property is the chained-form representation. Transformation of the kinematic control model to the chained form simplifies the control design and provides a framework for the direct extension of the controller to vehicles with more complex kinematics [1, 2]. However, this transformation is not strictly necessary. The second property is that the kinematic control model of a system is Chaplygin. Most theoretic control results at the kinematic level are developed for Chaplygin systems; most of them fail for non-Chaplygin systems, which are recognized as systems hard to control [2]. Feedback linearization enables obtaining full-state or input-output linearization. In the latter method an internal dynamics may be left and stability of the internal dynamics must be analyzed separately. Using either linearization, a controller has to be selected in such a way that it provides some kind of the tracking error convergence. E. Jarze˛ bowska Warsaw University of Technology, Institute of Aeronautics and Applied Mechanics, 00-665 Warsaw, Nowowiejska 24 street, e-mail: [email protected] P.C.S. Szklarz PM–Soft, 01-460 Warsaw, G´orczewska 226/31 street, e-mail: [email protected]

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Feedback linearization may require a redefinition of a system output and a subsequent dynamic extension for a system. The redefined system output may require, in turn, a conversion of a desired motion generated in, e.g. Cartesian coordinates, to a reference motion of the redefined output. Also, the selection of the linearizing outputs is not unique. There are other drawbacks of the dynamic feedback linearization approach. Some of them may be eliminated by an additional transformation of the kinematic model, e.g. tracking trajectories with linear velocities vanishing to zero is not allowed, unless the separation of geometric and time information for the system in the control law is performed [3]. Some limitations are structural for nonholonomic systems, e.g. the decoupling matrix becomes singular in the process of the dynamic extension of a system model [3, 4]. A path following task is obtained as an intermediate step in the point stabilization. Due to the Brockett condition, smooth static state feedback does not apply to stabilization of nonholonomic systems. Instead, time-varying feedback control either smooth or non-smooth may be pursued to achieve asymptotic stability [2, 4]. These are different control strategies than the ones applied to tracking [5]. In the paper we develop a control theoretic framework for a control strategy design. The theoretic framework itself yields an abstract feedback control strategy. This abstract strategy supplemented by a control problem specification and a definition of an error function yields a kinematic feedback controller that can be applied to both tracking and following of a desired motion. The strategy is based on an error function. The error function is predefined by a designer and the resulting feedback controller ensures the convergence of the error to zero or to some specified bound. Usually in a control problem either at a kinematic or dynamic level, a tracking or following error is defined as a difference between desired or actual values of coordinates, or as a distance from a curve, respectively. It concerns both state and output control. A specific controller has to be subsequently designed in such a way that it has to ensure some kind of convergence or boundedness of the error. In our approach this is the error function dynamics, which ensures the convergence to the predefined system motion and the controller is designed based on this function. Main motivations to develop the error function based strategy are an attempt to eliminate some of the drawbacks of the feedback linearization approach and to design a unified control strategy for both tracking and following. Herein, we present preliminary results towards these attempts. The paper contributes to control of nonholonomic systems. Firstly, our approach allows us to solve the tracking or following control problems, or mixed trackingfollowing control problems. Secondly, the convergence to the desired curve in space is predefined. Also, the control law, which is determined based on the error function, does not need any further convergence proof. The strategy does not need any more data and state measurement than other tracking or following strategies do. The strategy is developed at the kinematic level; yet the control theoretic framework it is based on enables its extension to the dynamic level. The paper is organized as follows. In Section 2 an abstract control oriented model of a mechanical system is developed. In Section 3 a control theoretic framework that yields an abstract feedback control strategy is developed. In Section 4 a unified

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control strategy is designed. The theory is illustrated with examples of a desired motion control for a two-wheeled mobile robot. The paper closes with conclusions and the list of references.

2 A Control Theoretic System Model To develop an abstract feedback control strategy, we formulate a control theoretic model of a nonholonomic system. It is an abstract model that can relate motion, i.e. velocity, and control. At the abstract model level, control means any action which is able to regulate the velocity, i.e. to execute the system predefined motion. The control theoretic model consists of the following: 1. Q – differential manifold which is the configuration space of a system with q ∈ Q 2. U – set of controls that consists of u(t) 3. v – function that relates velocity and control We assume that the system velocity at any state is uniquely determined by control. This relation is specified by v : Q × U → T Q, where T Q is a tangent bundle to Q. For a given q ∈ Q, vq = v (q, ·) : U → T Qq . A nonholonomic constraint specification on a system is included into v.

3 An Abstract Feedback Control Strategy Architecture A general structure of a control action can be developed as follows. The system motion is uniquely determined by an initial condition q(0) = q0 ∈ Q and a function C of the form C : Q × T → U, where T = R denotes time. The function C is referred to as a controller. Then, the system velocity is uniquely determined as q˙ = v (q,C(q,t)). For each real model we need to check the assumptions of the standard ordinary differential equation theorems to establish the existence and uniqueness of the solution. Architecture of the abstract feedback control strategy is presented in Fig. 1. t

Controller q

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Fig. 1 Architecture of the abstract feedback control strategy

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The controller requires information about time t and the system position q. It generates the control action u, which is translated into the system velocity. The strategy is abstract since the controller may be designed based on a specific system and a control problem formulation, e.g. on the error function.

4 Design of a Kinematic Control Strategy Based on the Error Function To design a feedback controller, we formulate the control problem as follows: given a system kinematics, find control inputs that ensure a desired ordinary differential equation on the error function value to be satisfied. The error function value is the value on the actual location of the system. The ordinary differential equation ensures the convergence of the error value to the specified bound. The problem is then precisely defined in a set of ordinary differential equations. According to our formulation of the feedback control problem, a definition of the error function is essential. It relates the control objective and the controller structure. It reflects a way in which a system converges to a desired curve. The error function has the form B : Q × T → R+ . For the selected error function we formulate a control strategy: find a control input u to satisfy DB · (v(q, u), dt) = −B(q,t) (1) Equation (1) implicates that the error function value decreases exponentially B (q(t),t) = K0 e−t ,

(2)

where K0 = B (q(0), 0). Equation (1) may not determine control uniquely. Then, we can take some extra control constraints. Architecture of the kinematic control strategy based on the error function is presented in Fig. 2. Comparing our design with typical tracking control designs, the difference consists in that we start from defining the error function for the desired motion. The control goal is to ensure the desired motion tracking, which is equivalent to ensure a predefined behavior of the error function values in time. This implies that the convergence of the controlled motion to the desired one is guaranteed. We will show that the result (2) allows solving classical tracking and following control problems. In tracking, the control problem is to track motion of a system in the configuration space, which is given by a function p such that p: T → Q. The following lemma can be formulated. Lemma 1. If the controller ensures (1) and the error function satisfies the conditions B (p(t),t) = 0,

(3)

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Control Problem

Error dynamics equations

Problem level

Error Function Error function level

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B (q(t),t) > 0 ∀q = p(t),

(4)

and the system initial configuration is q0 = p(0),

(5)

then the system trajectory is q(t) = p(t). Proof. Based on (1) and (2) we obtain that B (q(t),t) = K0 e−t and from (3) and (5) we know that B (q(0), 0) = B (p(0), 0). What follows is that K0 = 0 and B (q(t),t) = 0. The condition (3) implicates that the latter equality is satisfied if and only if q(t) = p(t). In following, the control problem is to follow a specified curve in the configuration space, and the curve is given as k : [0, 2π ] → Q, where k[0] = k[2π ]. The following lemma can be formulated. Lemma 2. If the controller ensures (1) and the error function satisfies the conditions B (q(t),t) = 0

∀q ∈ k (0, 2π ) ,

(6)

B (q(t),t) > 0

∀q ∈ / k (0, 2π ) ,

(7)

and the system initial configuration is q0 = k(0), then the system trajectory belongs to the curve k, i.e. q(t) ∈ k ([0, 2π ]). The proof is similar to the proof of Lemma 1.

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5 Control of a Two-Wheeled Mobile Robot Let us illustrate the application of the error function based kinematic control strategy for a two-wheeled mobile robot model presented in Fig. 3. The configuration space is Q = R5 with q = (x, y, ϕ , ϕl , ϕr ), where (x, y) are coordinates of the mass center C. Angles ϕ , ϕr , ϕl denote the heading angle, wheel angles due to rolling for the right and left wheel, respectively. The distance between the wheels is equal to 2b and the robot mass center C is located at a distance d from the geometric center O. Material constraint equations specify conditions that the robot does not slip sideways and its driving wheels do not slip either, i.e. ycos ˙ ϕ − xsin ˙ ϕ − ϕ˙ d = 0, ˙ ϕ + ϕ˙ b − ϕ˙ r r = 0, x˙ cos ϕ + ysin

(8) x˙ cos ϕ + ysin ˙ ϕ − ϕ˙ b − ϕ˙ l r = 0.

Relations for velocities follow from (8) and (9) r cos ϕ (ϕ˙ r + ϕ˙ l ) dr sin ϕ (ϕ˙ r − ϕ˙ l ) − , x˙ = 2 2b r sin ϕ (ϕ˙ r + ϕ˙ l ) dr cos ϕ (ϕ˙ r − ϕ˙ l ) r (ϕ˙ r − ϕ˙ l ) y˙ = + , ϕ˙ = . 2 2b 2b

(9)

(10)

Due to (10) it is enough to specify ϕ˙ r and ϕ˙ l . Selecting the control inputs to be ϕ˙ l = u1 := ul and ϕ˙ r = u2 := ur we have that U = R2 . The function v: Q ×U → T Q results uniquely from (10). To design a controller let B(q,t) = x2 + y2 − R, where R ≥ 0. Then 1 2xx˙ + 2yy˙ x˙ ˙ . = (x y) B= 2 2 2 2 y˙ 2 x +y x +y

y1

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Fig. 3 Two-wheeled mobile robot model

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Substituting (10) into the relation above we obtain that B˙ = (kr kl ) (ϕ˙ r where (b sin ϕ + d cos ϕ ) ry + (b cos ϕ − d sin ϕ ) rx kr = , 2b x2 + y2 (b sin ϕ − d cos ϕ ) ry + (d sin ϕ + b cos ϕ ) rx . kl = 2b x2 + y2 Equation (1) may now be written in the form

u x2 + y2 − R = −B B˙ = (kr kl ) r = − ul

ϕ˙ l )T ,

(11)

It can be seen that (ul , ur ) cannot be uniquely determined from (11). Control constraints can supplement (11). The control constraints usually arise from design, operation or safety conditions, or from actuator power limitations which are specific for a given mechanical system. Take the simplest case and select the control constraint as ul (q,t) = ur (q,t). Then

− x2 + y2 − R u l = ur = (12a) kr + kl for all points (x, y) for which kr + kl = 0. We may select the second option and set the control constraint as min (ul (q,t), ur (q,t)2 ) (ul , ur ) = −

(k , k ) r l x2 + y2 − R kr , kl 2

(12b)

for all points (x, y) = (0, 0). Selection of either control constraint completes the controller design. Tracking a circular trajectory by the two controllers with the control constraints (12a) and (12b) is illustrated by simulations in Figs. 4–6 run for a series of initial motion conditions. Selected error functions are listed in Table 1. Table 1 Definition of error functions and their constraints of control Error function B1 (q,t) = x2 + y2 − R B1 (q,t) = x2 + y2 − R B2 (q,t) = (x − R sin(ω t))2 + (y − cos(ω t))2

2 + x2 + y2 − R

Control constraints ul (q,t) = ur (q,t) min (ul (q,t), ur (q,t)2 )

min (ul (q,t), ur (q,t)2 )

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6 Conclusions In the paper the kinematic control strategy whose design is based on the error function is presented. The error function is predefined by a designer according to a desired convergence of the controlled motion to the reference one. The resulting feedback controller ensures the desired convergence of the error to zero or to a pre-specified bound. The strategy consists of three levels: the problem level at which

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a kinematic control model and control constraints for a real system are formulated, the error function level, at which a designer specifies a desired tracking or following performance, and the execution level. In our approach motion control may be either tracking or following, or both. The strategy may be extended to model-based control, i.e. a controller design is based on a dynamic model.

References 1. Alexander JC, Maddocks JH (1989) On the kinematics of wheeled mobile robots, Int. J. Robot. Res. 8, 15–27. 2. Bloch AM (2003) Nonholonomic mechanics and control, Springer, New York. 3. De Luca A, Benedetto MD (1993) Control of nonholonomic systems via dynamic compensation, Kybernetica 29, 593–608. 4. De Luca A, Oriolo G, Samson C (1998) Feedback control of a nonholonomic car-like robot. In: Laumond J-P (ed.) Robot motion planning and control, Springer, London. 5. Samson C (1995) Control of chained systems: Application to path following and time-varying point stabilization of mobile robots, IEEE Trans. Automat. Contr. 40, 64–77.

A Simple Correlation Factor as an Effective Tool for Detecting Damage Rui Sampaio and Nuno Maia

1 Introduction In previous works [1–3] the authors proposed the generalization of some wellknown methods from mode shapes to operational deflection shapes, and their respective spatial derivatives. Such a generalization and necessary normalization of the maximum occurrences proved to be effective in the task of locating the damage with the additional advantage of avoiding the modal identification process. In previous papers [4, 5] the authors proposed a new method, the Detection and Relative damage Quantification indicator, DRQ, based upon a simplified form of the Frequency Domain Assurance Criterion [6–8], the Response Vector Assurance Criterion [9]. This method aims at detecting damage, not locating it, simply identifying its existence. The advantage is that is very simple, just using the measured frequency response functions without any need for identification of natural frequencies or mode shapes. In this paper new assessments are made concerning the ability of the indicator to detect and relatively quantify damage. Numerical and experimental work is presented to illustrate the effectiveness of the indicator.

2 Theoretical Description To measure the degree of correlation between the Operational Deflection Shapes (ODS), the Frequency Domain Assurance Criterion (FDAC) seems appropriate [6, 7]: R. Sampaio Av. Engo Bonneville Franco, Pac¸o d’Arcos, 2780-572 Pac¸o D’Arcos, Portugal e-mail: [email protected] N. Maia IDMEC/IST, Technical University of Lisbon, Department of Engineering Mechanics, Av. Rovisco Pais, 1049-001 Lisboa, Portugal e-mail: [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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/2 / / /N d / ∑ αi j (ω2 )αi j (ω1 )/ / /i=1 FDAC jd (ω1 , ω2 ) = N . N . ∑ d αi j (ω2 )d αi j (ω2 ) ∑ αi j (ω1 )αi j (ω1 ) i=1

(1)

i=1

where represents the conjugate operator, αi j (ω ) is the receptance at frequency ω , measured at co-ordinate i for a force applied at co-ordinate j and N is the total number of co-ordinates or measuring points. This is the most general form of FDAC, as it allows for the comparison amongst all the different pairs of ODSs that are possible to consider. In fact, for each damage case, d, and coordinate of force application, j, one obtains an ODS for each frequency, ω2 , which may be compared with the respective cases of undamaged ODS with force application at coordinate j and frequency ω1 . For the purpose of this paper one has considered a simplified form of FDAC, referred to as RVAC or Response Vector Assurance Criterion [9], with only one applied force (so that the receptance matrix turns to be just a vector) and pairs of ODSs at the same frequency, ω : /2 / / /N d / ∑ αi (ω ) αi (ω )/ / /i=1 RVACd (ω ) = N . N .. ∑ d αi (ω ) d αi (ω ) ∑ αi (ω ) αi (ω ) i=1

(2)

i=1

From this definition one can formulate the Detection and Relative damage Quantification indicator, as: ∑ RVACd (ω ) ω DRQd = , (3) Nω where Nω is the number of frequencies and so, DRQ will vary between 0 and 1. A further improvement can be made with a normalization of the maximum occurrences, DRQid . In fact, it is possible that the RVAC at a small number of frequencies mask the DRQ calculated for the entire range of frequencies. The normalization of the maximum occurrences gives the same weight to each RVAC. This indicator is obtained sorting out, in descending order, at each frequency the different RVAC calculated for each damage situation and adding to the respective indicator not the RVAC but its order in the mentioned sorting. At the end the indicator is normalized to the range [0, 1] by dividing each indicator, calculated for each damage case, by the product Number of Frequencies times Number of Damage Cases. Finally and because some of the best known damage localisation methods, like the Damage Index and Mode Shape Curvature [3], use the second spatial derivative of the ODSs to locate the damage, one can also calculate the DRQ based on these derivatives: ∑ RVACd (ω ) DRQd = ω , (4) Nω

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as well as the corresponding DRQ id , where /2 / / N d / / ∑ α (ω ) α (ω )/ i /i=1 i / RVACd (ω ) = N . N .. d d ∑ αi (ω ) αi (ω ) ∑ αi (ω ) αi (ω ) i=1

(5)

i=1

The second derivative of the ODSs can be obtained by a central difference approximation: αi (ω ) = αi−1 (ω ) − 2αi (ω ) + αi+1 (ω ) (8) considering an equal distance between measurement points i. Because of their best performance only the DRQs based on the second derivatives will be addressed in this paper.

3 Numerical Examples For the numerical simulations an FE model of 20 beam elements and 21 nodes of a free-free beam was used. The model has three degrees of freedom at each node (ux , uy , θz ). The beam dimensions are 1000 × 60 × 6mm (L × b × h) (Fig. 1), the Young’s modulus is E = 196GPa, the shear modulus is G = 80GPa, the shear deflection constant is Fs = 6/5, the density is ρ = 7.917 × 103kg/m3 , the second moment of area is I = b · h3 / 12 and the cross-section area is A = b · h. The damage is simulated with a reduction in the second moment of area of the considered element. Modal and harmonic analyses for different frequency ranges were performed in a Mathcad program. A force was applied at each coordinate, one at a time, in the direction of the smallest dimension, h. To simulate the measurement noise, the real and imaginary components of αi j (ω ) were polluted with 0.5%, 1% and 3% of uniform distribution random noise in their magnitudes. Together with the 0% noise case, one has a total of four noise cases. z

Element 12

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L

Fig. 1 Beam in transverse vibration

nodes x

responses y

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3.1 Sensitivity to Noise For this numerical simulation the frequency range was 1–1600 Hz with 1600 frequency lines, the force was applied at location 5 and 12 cases of damage inflicted to element 12 were considered (see Tab. 1). The damage cases 2–4 simulate three measurements of the beam without damage but the damage case 1 is the reference measurement. The results for 0%, 0.5%, 1% and 3% added noise are presented in Fig. 2. From the results presented in Fig. 2 both indicators detect and recognize the different levels of damage. As expected the DRQi is less sensitive to noise than DRQ .

3.2 Quantification of Damage In this numerical simulation the frequency range was 1–1,600 Hz with 1,600 frequency lines, the force was applied at location 5 and 36 cases of damage inflicted to element 12 were considered (Fig. 3).

Table 1 Damage levels at element 12 Damage level

% of the 2nd moment of area of the elem. 12

Damage level

% of the 2nd moment of area of the elem. 12

100 100 100 100 90 85

7 8 9 10 11 12

80 75 70 65 60 55

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0 % noise 0.5 % noise 1 % noise 3 % noise

Fig. 2 DRQ and Drqi for 12 damage cases at element 12 and 4 situations of added noise

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Reduction 2º area moment elem. 12

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RVAC

0.8 0.6 0.4 0.2 0

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600 800 1000 Frequency [Hz]

undamage element damage level 12 at damage level 26 at damage level 36 at

1200

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Fig. 4 RVAC versus frequency from FRFs

The damage cases 2–4 simulate three measurements of the beam without damage and the damage case 1 is the reference measurement. The results, for 3% added noise, are presented in the next figures. In Fig. 4 one can see that the correlation between ODSs is poor below the 500 Hz because one can not distinguish the four different levels of damage. This is a consequence of the added noise. In the rest of the frequency band one can see the damage evolution, as well as the most sensitive regions in terms of frequency range. This data display can be a first damage assessment. From Fig. 5 one can see that the DRQ is specially indicated to quantify relatively the damage because it is possible to reproduce the pattern of damage variation and the DRQi is specially indicated to detect the presence of damage because is more

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Fig. 5 DRQ (left) and Drqi (right) for a frequency band of 1–1,600 Hz 1

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Fig. 6 DRQ (left) and Drqi (right) for a frequency band of 550–650 Hz

sensitive than the DRQ to damage. If one chooses a favorable band, namely where one notices the biggest variation of the RVAC, one obtains better results (Fig. 6):

4 Experimental Examples Beam. This test item is a steel beam, with constant rectangular cross-section, with dimensions 997 × 60 × 6mm (see Fig. 7). The test item is suspended by soft springs to simulate free end conditions. Twenty-one equally spaced points for translation response measurements were considered and the test specimen was excited by a Bruel&Kjaer 4809 shaker, powered by an Bruel&Kjaer 2706 power amplifier, at location 8. The force was transmitted through a stinger and measured by a Bruel&Kjaer 8200 force transducer; the responses were measured by 21 Bruel&Kjaer 4507 CCLD accelerometers. The signals were fed into the Multichannel Data Acquisition Unit Bruel&Kjaer 2816 (PULSE) and analysed directly with the Labshop 6.1 Pulse software from the attached laptop. To simulate the

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z

f(t) x y1(t)

y2(t)

yi(t)

y

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RVAC

Fig. 7 Beam in transversal vibration

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undamage element 14 damage level 3 at element 14 damage level 6 at element 14 damage level 9 at element 14

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undamage element 14 damage level 3 at element 14 damage level 6 at element 14 damage level 9 at element 14

Fig. 8 Beam-RVAC versus frequency (the graph on the right is the zoomed left one)

damage it was decided to cut the beam with various depths in the middle of locations 14 and 15. The frequency range was 0–400 Hz (800 lines) (Fig. 8). For this test the beam was measured in nine conditions – the undamaged or reference, the undamaged but different from the reference and the seven levels of damage at element 14 (Tab. 2). In Figs. 9 and 10 one can distinguish the nine damage level situations. Because we only had one measurement of the undamaged beam it was not possible to differentiate it from a damaged situation. In real monitored structures, one would have, without difficulty, many measurements of the undamaged structure before a damage occurrence. Irvine Column. The data used in this experimental example, the Irvine column tests, was downloaded from the website of Structural Health Monitoring (http://ext.lanl.gov/projects/damage id/data.htm) of Los Alamos National Laboratory, USA. The test structure consisted of a 24-in.-dia (61-cm-dia) concrete bridge column that was subsequently retrofitted to 36-in.-dia (91-cm-dia) column. A hydraulic

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Fig. 9 Beam-DRQ (left) and Drqi (right) with frequency band 0.5–400 Hz 1

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Fig. 10 Beam-DRQ (left) and Drqi (right) with frequency band 325–375 Hz Table 2 Damage levels at element 14 Damage level 3 4 5 6 7 8 9

Δx (mm)

Δy (mm)

0.2 0.8 2.0 2.2 3.0 4.0 5.0

0.20 0.40 0.80 1.00 1.50 1.90 2.00

actuator was used to apply lateral load to the top of the column in a cyclic manner. The loads were first applied in a force-controlled manner to produce lateral deformations at the top of the column corresponding to 0.25ΔyT, 0.5ΔyT , 0.75ΔyT and ΔyT . Here ΔyT is the lateral deformation at the top of the column corresponding to the theoretical first yield of the longitudinal reinforcement. The structure was cycled three times at each of these load levels. Based on the observed response, a lateral deformation corresponding to the actual first yield, Δy, was calculated and the structure was cycled three times in a displacement controlled manner to that deformation level. Next, the loading was applied in a displacement-controlled manner, again in sets of three cycles, at displacements corresponding to 1.5Δy, 2.0Δy, 2.5Δy, etc. until the ultimate capacity of the column was reached. This kind of loading put

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incremental and quantifiable damage into the structure. Five damage levels were considered. The axial load was applied during all static tests. For the experimental modal analyses the excitation was provided by an APS electromagnetic shaker mounted off-axis at the top of the structure. Forty accelerometers were mounted on the structure. Data acquisition parameters were specified such that frequency response functions (FRFs) and coherence functions in the range of 0–400 Hz could be measured (Figs. 11–13). Each spectrum was calculated from

1 0.8 RVAC

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Fig. 11 Irvine column – RVAC versus frequency

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Fig. 12 Irvine column – DRQ (left) and Drqi (right) with frequency band 0.5–400 Hz

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Fig. 13 Irvine column – DRQ (left) and Drqi (right) with frequency band 100–150 Hz

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30 averages of 2-s-long time-histories discredited with 2,048 points. These sampling parameters produced a frequency resolution of 0.5 Hz. Hanning windows were applied to all measured time-histories prior to the calculation of spectral quantities. From the figures above one can distinguish the five damage level situations.

5 Conclusions In this paper new assessments of the DRQ indicator were presented with numerical and experimental data. It was found that the DRQ indicator is relatively adequate to quantify damage and that the DRQi is adequate for an early detection of damage. It was also found that the RVAC characteristics versus frequency can be a helpful tool for damage detection. Acknowledgement The current investigation had the support of FCT, under the project POCI 2010.

References 1. Sampaio RPC, Maia NMM, Silva JMM (1999) Damage detection using the frequencyresponse-function curvature method, Journal of Sound and Vibration 226(4), 1029–1042. 2. Maia NMM, Silva JMM, Sampaio RPC (1997) Localization of damage using curvature of the frequency-response-functions, Proceedings of IMAC XV, Orlando, FL, 942–946. 3. Sampaio RPC, Maia NMM, Silva JMM, Almas EAM (2003) Damage detection in structures: from mode shape to frequency response function methods, Mechanical Systems and Signal Processing 17(3), 489–498. 4. Sampaio RPC, Maia NMM, Silva JMM (2003) The frequency domain assurance criterion as a tool for damage detection, Key Engineering Materials 245(3), 69–76. 5. Sampaio RPC, Maia NMM (2004) On the detection and relative damage quantification, 2nd European Workshop on Structural Health Monitoring, Munich, Germany, 757–766. 6. Pascual RJC, Golinval J-C, Razeto M (1997) A frequency domain correlation technique for model correlation and updating, Proceedings of IMAC XV, Orlando, FL, 587–592. 7. Pascual RJC, Golinval J-C, Razeto M (1999) On-line damage assessment using operating deflection shapes, Proceedings of IMAC XVII, Kissimmee, FL, 238–243. 8. Fotsch D, Ewins DJ (2000) Application of MAC in the frequency domain, Proceedings of IMAC XVIII, San Antonio, TX, 1225–1231. 9. Heylen W, Lammens S, Sas P (1998) Modal Analysis Theory and Testing (Section A.6), K. U. Leuven – PMA, Belgium.

Mathematical Functions of a 2-D Scanner with Oscillating Elements Virgil-Florin Duma

1 Introduction The bi-dimensional (2-D) scanners are used in a variety of applications, e.g. marking, cutting, and welding, in art applications, not to mention security and military applications. The 2-D scanners were developed using different combinations of uni-dimensional (1-D) scanners [1, 2]: polygonal [3, 4], or galvanometric [5]. Our present approach refers to a 2-D scanner consisting of two 1-D galvoscanners, of a type we previously developed [5] in order to obtain an enhanced duty cycle with regard to the state-of-the-art [6,7], which means a higher efficiency in using the available time for the scanning process. The main advantage of this 2-D system is the fast, precise positioning of the laser beam reflected by each of the mirrors of the two individually driven scanners in any point of the two axis of the scanned plane.

2 Equations of the 2-D Scanner The scan of the xOy plane (Fig. 1) is done with two oscillating scanners, the first one (Sc.1) performing the scan on the Oy axis, and the second one (Sc.2), the scan on the Ox axis. Although this particular device, consisting of two scanners of oscillating type, was chosen, a combination of rotating and oscillating devices can be used as well [1, 8]. With the ϕ and θ rotation angles of the two 1-D scanners, the expressions of the scanning functions of the 2-D device, that characterize the position of the spot I(x, y) of the laser beam (Fig. 1), are:

V.-F. Duma Aurel Vlaicu University of Arad, 77 Revolutiei Ave., Arad 310130, Romania, e-mail: [email protected]

J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

243

244

V.-F. Duma y x

0

x

y

L-l

I(x,y) Sc.2

Sc.1

l

ϕ

θ

Laser

Fig. 1 Bi-dimensional (2-D) scanning system with two galvanoscanners M

2q

p/4

q

q

Fig. 2 Reflection of the rays on the oscillating mirror (M) of the x-scan

x = l · tan 2θ , θ ∈ (−θa , θa ) , y = L · tan 2ϕ , ϕ ∈ (−ϕa , ϕa ) ,

(1)

where the θ = 0 position was chosen according to Fig. 2; similar for ϕ = 0. The origin of the scanned plane is I (ϕ = 0, θ = 0) = O (x = 0, y = 0) and the angular amplitudes θa and ϕa are chosen from Eq. (1) in order to have a complete scan of the surface of a rectangle characterized by the dimensions: 2xa = 2l · tan 2θa , (2) 2ya = 2L · tan 2ϕa . The scan can be performed in two different ways: with a continuous scan on the horizontal direction and with a step-by-step procedure on the vertical axis or the other

Mathematical Functions of a 2-D Scanner with Oscillating Elements

245

A Dy

2ya

2xa

B

Fig. 3 Scan pattern in the object plane

way round, in that concerns the functioning laws of the two individual scanners. The first solution will be considered in this study (Fig. 3).

3 Angular and Linear Scanning Functions 3.1 Scanning Functions of the 2-D System The problem that has to be solved is the ascertainment of the angular scanning functions θ (t) and ϕ (t), respectively of the linear ones, x(t) and y(t), that result from the angular ones through Eq. (1), and then to correlate the two aspects, in order to obtain a convenient, efficient scan of the plane. The initial conditions are chosen with regard to the start point of the scan. With the start in the superior point A in the left, and the scan on the Ox axis from left to right, and then the other way round, and on the Oy axis from top to bottom, with a rapid return from the final point B to the initial one A, the initial conditions are: θ0 = −θa x0 = −xa t = t0 : , , (3) y0 = ya ϕ0 = +ϕa and the time histories of functions θ , ϕ , x and y are presented in Fig. 4.

3.2 θ and x Functions A study is required in order to obtain the functions of the two 1-D scanners, as shown through the diagrams in Fig. 4.

246

V.-F. Duma J T

2Ja

j

Dj1

ja=j1

2T

Dj2

Dj3

j2

j3

4T

3T

tn

T

2T

3T

4T

T

2T

3T

4T

3T

4T

t Ty

jn

x

2xa

ta

t

2t T

y T

2ya

2T

t Ty

Fig. 4 The movement laws of the laser spot: the 2-D scanning functions x t 0 2t

a

T x

ta t

0.7 T

ta 0.4 T

2 t

0

b

1 T

0.33 T

Fig. 5 Scanning functions x(t): (a) proposed solution (detail of Fig. 4); (b) state-of-the-art solution

For the Ox scan (Sc.2, in Fig. 1), one may use a commercially available galvoscanner. Yet, as pointed out in Fig. 5, there are still certain drawbacks of the existing solutions. This is why, in the following, we shall theoretically propose an improved galvoscanner, with a linear + polynomial or a linear + sinusoidal function x(t) [5] (see Figs. 4 and 5a and also Tab. 1). The oscillating mirror scanner [6, 7] emerged as one of the best solutions with a better duty cycle η (defined as the amount of time available for the scanning

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Table 1 The scanning function x(t) Time interval ( j ∈ N)

Scanning function x(t) Linear + polynomial Linear + sinusoidala

ν t − xa jT, jT + T2 − 2τ T T νt 2 νT νT 2 − 2 τ , jT + jT + − 2 2 2τ + 2τ t + xa − 8τ

T jT + 2 , ( j + 1) T − 2τ xa (t + τ − T /2)/τ 2 [( j + 1) T − 2τ , ( j + 1) T ] ν2tτ − xa

ν t − xa

x (t) = H − b 1 − cos Ω T2 − τ − t xa (t + τ − T /2)/τ x (t) = −H − b [1 − cos Ω (T − τ − t)]

a b and Ω, respectively b’ and Ω’ are the coefficients of the sinusoidal portions, defined by Eqs. (13) and (15), respectively (20) and (21)

process) than the one of the polygon scanners [1, 2]. High scanning frequencies and velocities are achieved in compact constructive solutions, while positioning speed as fast as 150 μs and positioning resolution of 1μrad are achievable. However, for the devices in the state of the art characterized by the scanning functions presented in Fig. 5b, the duty cycle η reaches 66% for curve 2, and 70% for curve 1 [6,7]. Although these devices may be used for our 2-D application, improvements may still be made to reach a higher η , as the Ox-scan efficiency determines the velocity of the entire process. A 1-D galvoscanner that may achieve a duty cycle closer to 100% (Fig. 5a) it is possible with a command function that results from the scanning function that has to be found. In order to reach this η , we have to choose τ (2τ = returning time interval) much shorter than ta (2ta = active time interval), but also to correlate the characteristic equations of the device as it will be further demonstrated. The Ox scan is desired to be performed linearly, that is, with a constant scanning velocity v of the laser spot on the 2xa portion (Fig. 3). Turning portions are still necessary for both x(t) and θ (t). The curves in Figs. 4 and 5a are of the minimum duration 2τ (Fig. 5a) and the amplitude H (slightly higher than xa ), that would complete the active parts of the horizontal trajectory of the spot being achieved with some alternative solutions of any polynomial or sinusoidal functions.

3.2.1 The Linear + Polynomial x(t) Function The time origin of the process considered in Figs. 4 and 5a will divide the period of both x(t) functions in the following four cases (see Tab. 1). Case 1. For t ∈ [jT, jT + T/2 − 2τ ], j ∈ N the scanning function on the Ox axis is the useful/active part: x(t) = ν t − xa . (4) Case 2. For t ∈ [jT + T/2 − 2τ , jT + T/2], a polynomial portion will be considered: x(t) = at 2 + bt + c;

x˙ = 2at + b,

(5)

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V.-F. Duma

where a, b, c are the constants to be ascertained. From Fig. 5a, the following conditions are necessary: T T T x − 2τ = x = xa , x − τ = H, 2 2 2 T T T x˙ − τ = 0, x˙ − 2τ = −x˙ = ν. 2 2 2

(6)

where without altering the generality of the result, the only first period was considered. We derive from Eq. (6) the coefficients of Eq. (5): vT ν T 2 ν T − a=− , b=v − 1 , c = xa + . (7) 2τ 2τ 2 8τ Case 3. For t ∈ [jT + T/2, (j + 1)T − 2τ ], x(t) is the active portion: x(t) = xa (t + τ − T /2)/τ .

(8)

Case 4. For t ∈ [(j + 1)T − 2τ , (j + 1)T], the function x(t) is provided by Eq. (5) with the following conditions (Fig. 5a): x(T − 2τ ) = x(T ) = −xa ; x(T − τ ) = −H, ˙ ) = v; x(T ˙ − τ ) = 0. − x(T ˙ − 2τ ) = x(T

(9)

From Eq. (5) and using assumptions (9), the coefficients of the scanning function result: v T vT 2 a = , b = v 1− , c = −xa − vT − . (10) 2τ τ 2τ From Eqs. (6)2, (5)1 and (9)2 the designing equation is obtained: vτ = 2(H − xa),

(11)

and from the condition of performing of the linear part of the characteristics, given in Eq. (11), one finds a second designing equation: 4 T 2 − 2τ ν = 2xa => vT = 4(2H − xa ). (12) The application theme imposes the time interval T and the linear domain to be scanned. By choosing an appropriate value for H close to xa , the scanning velocity v and the time interval τ are obtained from Eqs. (11) and (12), respectively. From the theory of mechanisms [9], polygonal movement functions produce high acceleration steps. For the galvoscanner, this produces high inertia torques J θ¨ , where θ (t) is obtained form x(t) – Eq. (1)1 . This means that, despite of the easierto-design polygonal function, a sinusoidal one is more advantageous with regard to the mechanical functioning of the device.

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3.2.2 The Linear + Sinusoidal x(t) Function Case 1. For t ∈ [jT, jT + T/2 − 2τ ], x(t) is also provided by Eq. (4). Case 2. For t ∈ [jT + T/2 − 2τ , jT + T/2] a function is considered: x(t) = a + b sin(Ωt + ϕ ).

(13)

From Eqs. (6) valid for this case as well, the coefficients are:

ϕ = π /2 − Ω (T /2 − τ ) , a = H − b, while from Eqs. (6) and (14) ⎧ ⎨ b sin Ωτ = ν ν2 H − xa Ω , + => b = ⎩ b cosΩτ = H − x − b 2 (H − xa ) Ω2 2 a

(14)

(15)

and respectively tan Ωτ =

2 (H−xa )Ω ν

− (H−xν a )Ω

.

(16)

The Ω-unknown Eq. (16) may be numerically computed with regard to the desired velocity v on the Ox axis. Let for the example 2xa = 30 mm H = 16 mm and v = 150 · 103 mm/s, then for this turning portion one obtains Ω = 2.2756 s−1 with the time interval τ = 2.6 · 10−5 s that was evaluated from Eq. (11). For a double value of the velocity v = 300 · 103 mm/s, Ω = 2.5606 s−1 results, with τ = 1.3 · 10−5 s from Eq. (11). In conclusion, from Eqs. (13) and (14), the scanning function T x (t) = H + b −1 + cosΩ −τ −t (17) 2 results, where the coefficient b is provided by Eq. (15). Case 3. For t ∈ [jT + T/2, (j + 1)T − 2τ ], with Eqs. (9) and (13), the coefficients are: 4 ϕ = π 2 − Ω (T − τ ) , a = −H − b , (18) where in this case b = −

ν2 H + xa , − 2 (H + xa ) Ω 2 2

(19)

while Ω is provided by the equation tan Ω τ =

2 (H+xa )Ω ν

− (H+xνa )Ω

.

Similarly to the previous example, Ω is obtained from Eq. (20).

(20)

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V.-F. Duma

With the new estimated coefficients b and Ω , the scanning function for this portion results:

x (t) = −H − b 1 − cosΩ (T − τ − t) . (21) Case 4. For t ∈ [(j + 1)T − 2τ , (j + 1)T], x(t) has Eq. (8). The synthesis of the results for both type of functions is presented in Table 1. The command function of the 1-D scanner will result from the very equation of the mobile element of the galvanometer: J θ¨ + cθ˙ + kθ = M (t) .

(22)

where, J is the inertia momentum of this mobile element, c is the damping coefficient of the galvanometer, k is the elastic coefficient of the spring system, and M(t), the magneto-electric active torque applied in order to produce the required movement. From Eq. (1)1 , one obtains θ˙ and θ¨ and by replacing them in Eq. (22), M (t) =

cx˙ + J x¨ x xx˙2 1 k −J · 4 2 + arctan . 4 2 .2 + 2L · L3 2 l 1+ x l 1+ x l

(23)

Therefore, with Eqs. (23), the current function i(t), that is, the necessary command in order to obtain the required scanning function x(t), results: i (t) = M(t)/BNS,

(24)

where, B is the magnetic induction, N is the number of spires, and S is the surface of the mobile coil of the galvanometer.

3.3 ϕ and y Functions The main problem is to achieve the correlation between the values of these two functions. Imposing y(t), with its characteristic steps, the corresponding ϕ (t) function has to be ascertained, and this will be approached in the following. The scan resolution the number of lines on Oy is: N = 1/Δy.

(25)

With respect to Δy, there will be “n + 1” lines in order to scan the entire 2ya distance where

4 n = 2ya Δy . (26) One has then to calculate ϕ j and y j (ϕ0 = ϕa ; y0 = ya ). For y j it is obvious that, as Δy = ct., y j = ya − j · Δy, j = 0, n. (27)

Mathematical Functions of a 2-D Scanner with Oscillating Elements

jj

jj

Djj

2t

Djj

jj+1

a

251

jj+1

2t

b

Fig. 6 Joining curve for two consecutive levels of the ϕ (t) function

For ϕ j , one must have in mind that the function y = y(ϕ ) – Eq. (1)2 – is non-linear, therefore, at constant steps on Oy, one will have variable steps on ϕ . From Eq. (1)2 :

ϕ = 0.5 arctan y/L.

(28)

From the “j” step to the “j+1” step, one obtains: y j = L · tan 2ϕ j

(29)

y j+1 = L · tan 2 (ϕ j − Δϕ j ) = y j − Δy from which:

1 Δy . Δϕ j = ϕ j − arctan tan 2ϕ j − 2 L

(30)

Passing from one level of the curve ϕ to another (and this also goes for the function y) cannot be accomplished with steep portions, as in Fig. 6a. Instead, one has to impose a profile of the curve tangent to the two levels, of coordinates ϕj and ϕj+1 , for the time interval 2τ , as in Fig. 6b. The profile of the curve ϕ (t) on the connecting portion is convenient to be:

ϕ (t) =

π

Δϕ j · cos ·t −tj , 2 2τ

t ∈ (t j ,t j + 2τ ) .

(31)

A particular problem is the returning trajectory of the laser spot to the initial point A (see Fig. 7), from the final point B (Fig. 3), after the completion of a scan of

ϕa

ϕn<ϕa Ty
Fig. 7 Returning trajectory of the laser spot to the initial point A at the end of a plane scan

252

V.-F. Duma

the entire plane surface characterized by Eq. (2). One may see from Fig. 4 that the available time interval for this process is: Ty − tn = 2τ + 2ta + 2τ ,

(32)

where the following notations were made: Ty− period of the function y(t), tn− the time of scanning the object space following the Oy axis, ya − ymin = n · Δy the total displacement of the spot in the vertical direction. The movement of the mirror of the Sc.1 (Fig. 1) will therefore have to be performed in this time interval according to the equation ϕa + ϕn π 2π · sin ϕ (t) = · t − − tn , t ∈ (tn , Ty ) , (33) 2 Ty − tn 4 where

/ / / / n / / ϕn = /ϕa − ∑ Δϕ j / / / j=1

(34)

obviously, with ϕn < ϕa .

4 Conclusions The study ascertains the complete angular and linear scanning functions of a 2-D scanner comprising two individually-driven galvoscanners. For the one that provides the linear, continuous scan (Ox-scan, in the study), the solution we propose aims at an enhanced duty cycle. The characteristic functions of this 1-D scanner were developed (linear + polynomial and linear + sinusoidal) and the command function was obtained. For the 1-D scanner that provides the incremental movement of the laser spot (Oy-scan, in the study), the necessary steps were deduced and the angular, variable ones were deduced, in order to obtain linear constant steps for the scan of the proposed object surface. With a proper programming of the laser utilized in the system, various applications, e.g. marking, cutting and welding are thus made possible, for industry, as well as for other fields: art (e.g. architectural restoration) or scientific applications. The prime target of the system we have presented, by example, was a 2-D scanner for a confocal microscope. Future studies are envisaged on the use of the system in rapidly developing fields, i.e. medical imaging [10] and rapid prototyping.

References 1. Bass M (1995) Handbook of Optics I, McGraw-Hill, New York. 2. Duma VF (2004) Scanning, Politehnica, Timisoara.

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3. Duma VF (2005) On-line measurements with optical scanners: metrological aspects. In: Osten W, Gorecki C (eds) Optical Measurement Systems for Industrial Inspection IV. Proceedings of SPIE 5856, Bellingham, 606–617. 4. Duma VF (2006) Novel approaches in the designing of the polygon scanners. In: Vlad VI (ed) 8th Conference on Optics, Proceedings of SPIE 6785, Bellingham, 6785-1Q (invited paper). 5. Duma VF (2007) Theoretical approach on a galvanometric scanner with an enhanced duty cycle with vibration control. In: Awrejcewicz J (ed) 9th Dynamic Systems Theory and Applications, Ł´od´z, Dec 17–20, 711–718 6. Aylward RP (1999) Advances and technologies of galvanometer-based optical scanners. In: Beiser L, Sagan SF (eds) Optical Scanning, Proceedings of SPIE 3787, Bellingham, 158–164. 7. Gadhok JS (1999) Achieving high-duty cycle sawtooth scanning with galvanometric scanners. In: Beiser L, Sagan SF (eds) Optical Scanning, Proceedings of SPIE 3787, Bellingham, 173– 180. 8. Richter B (1992) Laser scan devices for industrial application, WIRE 42(6), Meiscubach GmbH, D8600 Bamberg, Germany. 9. Perju D (1991) Mechanisms for fine mechanics, Politehnica, Timisoara. 10. Podoleanu AGh, Dobre GM, Cucu RG (2004) Sequential optical coherence tomography and confocal imaging, Optics Letters 29(4), 364–366.

Analysis of Regular and Chaotic Dynamics of the Euler-Bernoulli Beams Using Finite-Difference and Finite-Element Methods Anton Krysko, Jan Awrejcewicz, Maxim Zhigalov, and Olga Saltykowa

1 Introduction Owing to remarkable development of aeronautics, astronautics and ship-building industry, the problem of an accurate and engineering-accepted beam dynamics (taking into account various boundary conditions and sign changeable loads) is of high importance. It is well known that the problems yielded by mechanical engineering require construction and analysis of their mathematical models. Modeling of flexible beam vibrations subjected to transversal and longitudinal sign-changeable loads belongs to one of the hottest problems of today’s mechanics. Key targets of modeling and analysis of beams, plates and shells include studies of transition from regular to chaotic dynamics and vice versa, and the methods of dynamics control via external load action (see, for instance, references [1–5]). Our aim in this work was to compare results of two different methods of mathematical modeling, i.e. FDM and FEM, using the example of Euler-Bernoulli type flexible beams.

2 Problem Formulation A mathematical model of transversal Euler-Bernoulli beam vibrations with various boundary conditions is derived in this work. 0 The Cartesian coordinates system 1 XOZ is introduced, and then in the space Ω = x ∈ [0, a] ; −h ≤ z ≤ h; − b2 ≤ y ≤ b2 a thin A. Krysko The Saratov State Technical University, Department of Applied Mathematics, Saratov, Russia, e-mail: [email protected] J. Awrejcewicz Technical University of Lodz, Department of Automatics and Biomechanics, Lodz, Poland, e-mail: [email protected] M. Zhigalov and O. Saltykowa The Saratov State Technical University, Department of Higher Mathematics, Saratov, Russia, e-mail: [email protected], olga [email protected]. J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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2 elastic beam with its middle surface deformation εx = ∂∂ ux + 12 ∂∂wx is studied, where w(x,t) denotes beam deflection, and u(x,t) is the middle surface displacement along the ox axis. It is assumed that owing to the Euler-Bernoulli hypothesis a normal to the beam middle surface is still normal after the beam deformation:

εxx = εx − z ∂∂ xw2 , where εx is the middle surface deformation, Nx = 2

longitudinal force, and Mx =

h −h

3

h

−h

σxx dz is the

∂ w σxx z dz = − (2h) 12 E ∂ x2 denotes the bending moment. 2

Dynamics governing equations have the following form [6]: ⎧ + , ⎨E (2h) ∂ 2 u2 + L3 (w, w) − (2h) γ ∂ 22u − ε2 (2h) γ ∂ u = 0, g ∂t g ∂t , + ∂x 2 ⎩E (2h) L1 (u, w) + L2 (w, w) − (2h) ∂ 4 w4 + q − (2h) γ ∂ 2 w2 − ε1 (2h) γ ∂ w = 0, 12 ∂ x g ∂t g ∂t ∂ 2u ∂ w ∂ x2 ∂ x

∂ u ∂ 2w ∂ x ∂ x2 ,

3 ∂ 2w 2 ∂ x2

∂w ∂x

2

(1) ∂ 2w ∂ w , ∂ x2 ∂ x

where L1 (u, w) = + L2 (w, w) = , L3 (w, w) = ε1 , ε2 – dissipation coefficients; q = q(x,t) – transversal load, E – Young modulus, ρ , γ – density and weight density, respectively, and g – acceleration of gravity. The following non-dimensional variables are introduced w¯ =

ua a4 w x a , u¯ = , q¯ = q , , x¯ = , λ = 2 (2h) (2h) a (2h) (2h)4 E t a t¯ = , τ = , c = τ c

Eg a , ε¯i = εi , i = 1, 2. γ c

(2)

Taking into account (2), system (1) takes the form ∂ 2u 2 + L3 (w, w) − ∂∂ t 2u − ε2 ∂∂ ut = 0,, ∂ x2+ 2 1 1 ∂ 4w L2 (w, w) + L1 (u, w) − 12 − ∂∂ t w2 − ε1 ∂∂wt + q = 0, λ2 ∂ x4

(3)

where in the above bars over non-dimensional quantities are omitted. The following boundary conditions at the beam ends are attached to Eqs. (3): Problem 1. “Clamping – clamping”:

∂ w(0,t) ∂ w(a,t) = = 0. ∂x ∂x

(4)

w(0,t) = w(a,t) = u(0,t) = u(a,t) = 0; Mx (0,t) = Mx (a,t) = 0.

(5)

w(0,t) = w(a,t) = u(0,t) = u(a,t) = Problem 2. “Hinge – hinge”:

Problem 3. “Hinge – clamping”:

Analysis of Regular and Chaotic Dynamics of the Euler-Bernoulli Beams

257

∂ w (a,t) = 0. ∂x

(6)

w(0,t) = Mx (0,t) = u(0,t) = 0; Mx (a,t) = Nx (a,t) = Qx (a,t) = 0.

(7)

w(0,t) = w(a,t) = u(0,t) = u(a,t) = 0; Mx (0,t) = 0; Problem 4. “Hinge – free”:

Additionally, the following initial conditions are attached to Eqs. (3) through (7): / / ∂ w(x,t) // ∂ u(x,t) // w(x,t)|t=0 = = u(x,t)|t=0 = (8) t=0 = 0. ∂ t /t=0 ∂t /

3 On the Numerical Solution to Vibration and Stability Beam Problems Investigation of nonlinear vibrations of constructions with various dynamic states (regular and/or chaotic) requires highly accurate computational algorithms and implementation of numerical methods. Since analytical methods devoted to the analysis of non-linear models can be rarely applied, the only way is to apply various numerical approaches to verification of reliability of the obtained results. In this work, various numerical approaches are applied, namely direct one (FDM) and variational one (FEM) in the Bubnov-Galerkin form. A comparison is made for various boundary conditions and for various dynamic regimes. In all investigated cases the beam geometric and physical parameters are taken as the same.

3.1 FDM with Approximation O(c2 ) The infinite dimensional problem (3)–(8) can be reduced to the finite dimensional one via the finite difference method (FDM) with approximation O(c2 ). Namely, at each mesh node the following system of ordinary differential equations is obtained: L1, c (wi , ui ) = ε1 w˙ i + w¨ i , L2, c (wi , ui ) = ε2 u˙i + u¨i , (9) (i = 0, . . . , n), where n denotes the partition numbers regarding spatial coordinates, and

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1 1 1 (ui+1 − 2ui + ui−1 ) + (wi−1 − wi+1 ) 2 (wi+1 − 2wi + wi−1 ), c2 2c c 1 1 1 (wi−2 − 4wi−1 + 6wi − 4wi+1 + wi+2 ) + L1,c (wi , ui ) = 2 (− λ 12 c4 1 1 1 1 + (wi−1 − wi+1 ) 2 (ui+1 − 2ui + ui−1 ) + (wi−1 − wi+1 ) 2 (ui+1 − 2ui + ui−1 ) + 2c c 2c c 2 1 1 (wi−1 − wi+1 ) (wi+1 − 2wi + wi−1 ) + + 2c c2 1 1 1 + 2 (wi+1 − 2wi + wi−1 ) (ui+1 − ui−1 ) + 2 (wi−1 − wi+1 )(wi−1 − wi+1 ) + q). c 2c 8c

L2,c (wi , ui ) =

For i =1, i = n − 1 one has to take into consideration the so-called out of contour points, which are defined by the following boundary conditions: for problem 1 w−i = wi , whereas for problem 2 w−i = −wi . The following additional equations are supplemented to Eqs. (9) for Problems 1–3: w0 = 0;

wn = 0;

u0 = 0;

un = 0,

(10)

and for Problem 4 w0 = 0;

u0 = 0;

Mx = 0;

Nx = 0;

Qx = 0.

(11)

The initial conditions (8) for the considered cases have the following difference form w(xi )|t=0 = 0; u(xi )|t=0 = 0; w(x ˙ i )|t=0 = 0; u(x ˙ i )|t=0 = 0, i = 0, . . . , n.

(12)

3.2 FEM with the Bubnov-Galerkin Approximation The so far defined problem (3)–(8) is solved now via FEM. Owing to the FEM theory, in order to construct a beam element we need to introduce the testing functions. The following four degrees of freedom (w1 , w2 , θ1 , θ2 ) are associated with the element and the following approximation polynomial is applied: w (x) = a1 + a2 x + a3x2 + a4 x3 , θ (x) = −

dw = − a2 + 2a3x + 3a4x2 . dx

After defining the constant values, an approximation function has the following form: w = Nw {W}

– form where [Nw ] = 1 − 3ξ 2 + 2ς 3 ; −l ξ (ξ − 1)2 ; 3ξ 2 − 2ξ 3; −l ξ ξ 2 − ξ matrix; {W} = (w1 , θ1 , w2 , θ2 )T – node displacement matrix; ξ = x/l – nondimensional quantity (local coordinate).

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Displacement approximation u(x) has the following form: u = Nu {U} , where Nu = (1 − ξ ; ξ ) {U} = (u1 u2 )T . Applying the Bubnov-Galerkin procedure and taking into account the introduced approximations, the following equations of FEM are obtained

¨ + C1 W˙ + K1 [W ] = F1 (q,U), M1 W

(13) M2 U¨ + C2 U˙ + K2 [U] = F2 (p,W ), where Mi , Ci , Ki are the matrices of mass, damping and stiffness, respectively.

4 Numerical Results Obtained via FDM and FEM The considered beam is subjected to the action of the following transversal load q = q0 sin(ω pt),

(14)

where ω p is the excitation frequency, and q0 is its amplitude. The studied system is dissipative, and the damping coefficients denoted by ε1 , ε2 correspond to deflection w and displacement u, respectively. Next, we study numerically the beam dynamics and stability. Any method of beam partition allows us to approximate PDEs by ODEs. Integration of the latter ones can be divided into two groups, i.e. explicit and implicit methods. The explicit methods are mainly realized via the Runge-Kutta schemes, and they are sufficient to solve our beam problem. It is mainly motivated by an observation that the considered Cauchy problem does not belong to stiff one, since in the frequency spectrum of eigen values of the Bernoulli-Euler type equations there are no frequencies differing in the order of magnitude (see, for instance, considerations in reference [7]). In order to verify the validity and accuracy of beam vibration simulations, both mentioned methods (FEM and FDM) are applied in problem 4, and the following fixed damping coefficients ε1 = 1, ε2 = 0, where ω p = 5.1 is the excitation frequency, and λ = a/2h = 50 denotes the relative beam length. The beam is subjected to the harmonic load action with the amplitude q0 . The computation step regarding spatial coordinate equals c and time step is Δt. Both of them are yielded by the Runge principle. The stated problem is solved for beam partitions n = 40, c = 1/40, and with the time step Δt = 0.9052 · 10−3. In order to compare the numerical results, power spectra and time histories (signals) w(t)are reported in Table 1 for q0 = 100 (it corresponds to regular dynamics), and for q0 = 3,200 (it corresponds to chaotic dynamics). From Table 1 one may conclude that signals obtained via FEM and FDM practically coincide for the case of regular dynamics. In the case of chaotic dynamics, a

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Table 1 Power spectra and time histories w(t) for q0 = 100 and for q0 = 3, 200

Signal w(t)

Frequency spectra −5

0.003

ωp

FDM n=40

−10

0.002

q0=100

S,db

w

ω

0.001

−15

Mkp 0 Mk3 −0.001

ωp

FEM n=40

t 10

10.2 10.4 10.6 10.8

11

11.2 11.4 11.6 11.8

12

−15 0

1

5 S,db

−10

−0.002 −0.003

0

−5

ω 0

5 ωp

S,db

w

q0=32200

−10 FDM n=40

0.5

−20 Mkp 0

0

2

ω 4

0 S,db

Mk3 − 0.5

−1

ωp

−10

10

10.2 10.4 10.6 10.8

11

t 11.2 11.4 11.6 11.8 12

−20

FEM n=40 0

2

ω 4

signal produced by FDM is slightly delayed in comparison to that produced by FEM and possesses smaller amplitude. Frequency power spectra of vibrations practically either coincide in the case of regular dynamics or are close to each other in the case of chaotic dynamics. Hence, owing to the results included in Table 1, the results obtained via the FEM and FDM methods are reliable for either regular or chaotic beam dynamics analysis. In order to investigate beams dynamics driven by harmonic loads a special program package has been developed enabling construction of vibration type charts vs. control parameters {q0 , ω p }. For instance, in order to construct a chart with the resolution of 200 × 200 points, one needs to solve a problem of dynamics, to analyze frequency power spectrum and finally to compute the Lyapunov exponents for each choice of the control parameters. The developed algorithm enables also separation of the periodic dynamic zones, the Hopf bifurcation zones, quasi-periodic zones, as well as chaotic zones.

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Table 2 Vibration type charts vs. Control parameters {q0 , ω p } for problem 4 FEM

FDM q0

7500 6750 6000 5250 4500 3750 3000 2250 1500 750 0 2.55

I

3.4

II

4.25

5.1

q0

III

5.95

6.8

7.65

ω

7500 6750 6000 5250 4500 3750 3000 2250 1500 750 0 2.55

I

3.4

II

4.25

5.1

III

5.95

6.8

frequency independent oscillations

ωp / 2 frequency oscillations

damped oscillations

chaos

harmonic oscillations

bifurcations

ω 7.65

In Table 2, the vibration type charts vs. the control parameters {q0 , ω p } for problem 4 are reported. Charts are constructed either with the application of FEM or FDM with the following fixed parameters ε1 = 0.1, ε2 = 0 for the beam length partition n = 40, and for the beam relative length λ = a/2h = 50. The excitation frequency changes from ω0 /2 (chart I) to 3ω0 /2 (chart III), where ω0 (chart II) denotes free frequency of the associated linear system (for problem 4 we have ω0 = 5.1). A maximal excitation amplitude corresponds to the beam deflection of 5(2h), and the charts are built with resolution 300 × 300. Analysis of the obtained vibration type charts also supports reliability of the results obtained for various vibration regimes. Observe that the zones of chaotic vibrations vs. frequency obtained via FEM are wider than those obtained via FDM, whereas they coincide regarding the amplitude of vibrations. In order to get a vibration character chart vs. control parameters with resolution 300 × 300 one has to carry out 9·104 computational variants. In the case of FEM, the computational time increases about 1.5 times comparing to the FDM application (for n = 40). The notation introduced in Table 2, regarding vibration type, is also used further. Computation of such a chart with the use of a Celeron 1700 processor takes 400 days. However, the knowledge of such charts enables a full system control. In order to confirm reliability of the results obtained for other types of boundary conditions, in Table 3 scales of vibration type beam character depending on the excitation amplitude q0 ∈ 0.6 × 104 and for one value of ω p are reported, and also dependences wmax (q0 ) are shown. The problems are solved for the following parameters: ε1 = 1, ε2 = 0, λ = a/2h = 50, ω p = 6.9, and beam partition regarding spatial coordinate n = 40. We show how the boundary conditions essentially influence the system dynamics. For Problem 1, the beam exhibits periodic and bifurcation type dynamics (either for FDM or for FEM). In this case there is no transition to chaotic dynamics. In graph wmax (q0 ) sudden jumps do not occur, and the function is smooth. In

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Table 3 Beam vibrations depending on excitation amplitude q0 ∈ 0.6 ∗ 104 Mkp

Problem 1

Mk3 1

wmax

Mkp 0.5 Mk3 q0

0 0

1.104

2.104

3.104

4.104

5.104

1.104

2.104

3.104

4.104

5.104

1.104

2.104

3.104

4.104

5.104

6.104

Problem 2

Mkp Mk3 1 wmax Mkp 0.5 Mk3 q0

0 0

6.104

Mkp

Problem 3

Mk3 4 w max Mkp 2 Mk3 0

q0 0

6.104

Problem 2, one may observe chaotic zones matched with bifurcation zones, but periodic dynamics is not exhibited. A function presenting maximal deflection vs. excitation amplitude is smooth only at the beginning (for q0 = 0.1·104), where sudden jumps of wmax are not observed. Transition of the system from periodic to chaotic vibrations and vice versa, is characterized by sudden changes of wmax even for a small change of the excitation amplitude, and this is understood as stability loss of the system dynamics. In the case of non-symmetric boundary conditions (Problem 3) one may observe that the system transition into chaotic state occurs for q0 > 2.5·104. For the given boundary conditions periodic dynamics occurs for q0 ∈ (1.1, 2.5)·104 . It is remarkable that within beam chaotic regime in the graph wmax (q0 ) not only sudden jumps appear but also the functions are discontinuous.

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5 Transition Scenarios into Chaos As the earlier results of local chaos investigations show, there are a few typical transition scenarios leading a dynamic system from periodicity into chaos, which sometimes are also combined. On the other hand, as it will be shown further, such transitions, however, understood globally, may differ for the same system (here beam) for various boundary conditions. Mainly four typical transitions are well understood, namely the Landau-Hopf scenario, the Ruelle-Takens-Newhouse scenario, the Feigenbaum scenario and the Pomeau-Manneville scenario. Below, we investigate and define a beam scenario of transition into chaos for Problem 2. The numerical investigation is carried out by two methods: FEM and FDM. Table 4 shows the fundamental steps helping in the scenario detection. Observe that for q0 = 100, in a frequency power spectrum, only frequency of excitation ω p = 6.9 is exhibited. An increasing amplitude of excitation causes the occurrence of two independent frequencies (quasi-periodicity), which are evidenced by FEM and FDM, and their estimated values are the same. A further increase of the excitation amplitude causes the occurrence of linear combinations of the earlier mentioned frequencies ω p , ω1 , ω2 . For example, let us study the system behavior for q0 = 11,000 applying FDM. It is remarkable that the system dynamics is governed by a linear combination of frequencies ω p , ω2 , ω4 . The following three frequency groups are distinguished: ω4 , ω7 , ω9 – the first group, where frequency values differ by the amount of frequency ω4 ; ω1 , ω2 , ω10 , ω11 – the second group, where the frequencies differ from each other either by ω4 , or by ω4 · 2; ω p , ω3 , ω5 , ω8 – the third group, where the linear combination of frequencies is preserved. Observe that an analogous system behavior is also monitored for q0 = 8,700 in the case of FEM application. A further increase of q0 yields more evident changes of the earlier mentioned frequencies, and finally all of the frequencies become linearly dependent. For q0 = 20,000 (FDM) and for q0 = 19,900 (FEM) all frequency distances are almost equal, and the difference between them achieves 1.062.

6 Conclusions An increase of the amplitude of external excitation causes variation of frequencies. The mentioned frequencies again appear and disappear. As a result, in the frequency spectra, either for FDM or for FEM, one may distinguish six linearly independent groups of frequencies, each group containing linearly dependent frequencies which differ by the amount of 0.29. Then, when all of the born frequencies become linearly dependent, the system dynamics is transited into chaotic state, which is clearly manifested by the system frequency spectra for q0 = 4 · 104 (FDM) and q0 = 4.9 · 104 (FEM).

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Table 4 Some fundamental steps helping in the scenario detection Signal

S(wp)

w(05, t)

q0

Power spectrum

100

FDM

-5 -10

100

-5

w 0

5

wp w = 6.9 p

S,db

11000

FDM

S,db w4

w w5 p

w7

w3 w2 w1 w8

w9 w 10

-15

w11 w

0

wp = 6.9 w10 = 3.22 w1 = 2.73 w11 = 4.65 w2 = 3.68 w3 = 6.4 w4 = 0.48 w5 = 5.93 w7 = 1.45 w8 = 4.96 w9 = 0.96

5

wp = 6.9 w10 = 0.96 S,db wp w1 = 6.4 w11 = 4.65 w2 w2 = 0.48 w 6 -5 w3 = 3.68 w4 = 2.73 w10 w6 = 5.93 w w 3 4 -10 w11 w7 = 3.22 w1 w8 w7 w w8 = 1.45 w9 = 4.96 w9 -15 0 1 2 3 4 5 6 7

8700

FEM

0

20000

FDM

0

S,db w4 w7

-10

-20

wp = 6.9

w2

w1

0

w8

wp w5 w1 = 2.65 w2 = 3.69 w4 = 0.53 w5 = 5.83 w w7 = 1.59 w8 = 4.77

5

0

19900

FEM

-5

S,db w2 w8

-10 -15

wp = 6.9

wp w6

w1 = 0.53 w3 = 3.69 w4 = 2.65 w6= 5.83 w8 = 1.59 w9 = 4.77

w9 w4 w 3

w 0 1 2 3 4 5 6 7

40000

FDM

0 S,db

wp

-10

-20 0 0

0.001 w

0.001 w

5.10-4 0 t

2540

2545

0.2 w 0.1

t 2540

2545

FEM

39000

w 5

0.1

0 . w

- 0.2 - 0.4 - 0.2

2545

0

0.2

0.4

0.2 w

0.2 w

0

0 t 2540

2545

- 0.2 - 0.2 - 0.1

0

0.1

. w 0.2

0.2 w

w

0

0 t

- 0.2

2540 0.5 w

2545

. w

- 0.2 -0.5 0.5

0.5

0

w

0 t

- 0.5 2540

2545

. w

-0.5 -0.5

0

0.5

0.5 w

0

0

0

- 0.1 t

wp

20

. w

- 0.2 -0.1 0.2 w 0.1

0.5 w

10

0.002

0

0.2 w 0.1 0 - 0.1 - 0.2 2540

- 0.2 0.2

. w 0

- 0.1

- 0.2

5

S,db

-5.10-4 -0.001 -0.002 0.2 w 0.1

0 - 0.1

0

w

. w - 0.001 -0.001 -5.10-4 0 -5.10-4 0.001

t 2545

5.10-4 0

-5.10-4 -0.001

w -20 0 1 2 3 4 5 6 7

-10

0

-5.10 -0.001 2540

-15

-5

0.001 w

-4

-10

0

0.001 w 5.10-4 0

-15 -20

FEM

wp wp = 6.9

S,db

Phase portrait . w(w&)

0

-0.5

t 2540

2545

-0.5 -2

. w -1

0

1

2

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Finally, taking into account the previous description and comments regarding the scenario of transition of our beam into chaotic dynamics monitored via FEM and FDM, the detected scenario fits to the well-known Ruelle-Takens-Newhouse scenario, where in the latter classical case the transition is realized via two independent frequencies and their linear combinations.

References 1. Wang Du, Guo Z, Hagiwara I (2002) Nonlinear vibration control by semi-active piezoactuator damping. JSME Int. J. C. 45(2), 442–448. 2. Awrejcewicz J, Krysko VA (2001) Feigenbaum scenario exhibited by thin plate dynamics. Nonlinear Dynam. 24, 373–398. 3. Awrejcewicz J, Krysko V, Krysko A (2002) Spatial-temporal chaos and solitons exhibited by Von Karman model. Int. J. Bifurcat. Chaos 12(7), 1465–1513. 4. Krysko VA, Shchekutorova TV (2004) Chaotic vibrations of conical shells. Izvestia RAS MTT 4, 140–150 (in Russian). 5. Krysko VA, Zhigalov MV, Saltykova OA, Desatova AS (to appear) Dissipative dynamics of geometrically non-linear Euler-Bernoulli beams. Izvestia RAS MTT. 6. Volmir AS (1972) Nonlinear Dynamics of Plates and Shells. Nauka, Moscow (in Russian). 7. Krysko VA, Awrejcewicz J, Saltykova OA, Chebotyrevskiy YuV (to appear) Nonlinear vibrations of the Euler-Bernoulli beam subject to transversal load and impact actions. Math. Probl. Eng.

Regular and Chaotic Motions of an Autoparametric Real Pendulum System with the Use of a MR Damper ´ Jerzy Warminski and Krzysztof Ke˛ cik

1 Introduction Autoparametric vibrations, which may occur within mechanical structures and machinery, are studied by several researches [1, 2]. An oscillator with an attached pendulum is an example of the autoparametric system which is used in many mechanical and civil engineering applications. Gantry cranes, lifts or special dynamical absorbers, mounted in buildings and working as dynamical dampers against earthquake, are classical examples where interactions between the support and the pendulum occurs [3]. The phenomenon of vibrations absorption of the mass–spring oscillator can be achieved due to proper pendulum swinging. However, for some parameters the situation may worsen and pendulum vibrations may increase dramatically, and then the protection of the structure (modelled as a mass-spring oscillator) is lost. Response of the system can be regular or, under some circumstances, may become chaotic [4]. To avoid such dangerous situations a special suspension of the system is analysed in this paper.

2 Model of the Vibrating System and Equations of Motions The considered mechanical mass spring model, presented in Fig. 1, is composed of two subsystems: a nonlinear oscillator and a pendulum, made up of two masses m p and m2 , attached in a pivot to the mass m1 . The oscillator, supported by a nonlinear spring, is forced by a harmonic motion of the bottom of the classical linear spring (kinematical excitation). J. Warmi´nski and K. Ke˛ cik Lublin University of Technology, Nadbystrzycka 36, 20-618, Poland, e-mail: [email protected], [email protected]

J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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268

Fig. 1 Scheme of the vibrating model

Damping of the model presented in Fig. 1 is studied in two variants: (a) as linear viscous and (b) nonlinear magnetorheological damping. Magnetorheological fluids are materials with properties which can be substantially and reversibly altered in milliseconds when exposed to a magnetic field. One of the applications of this fluid are MR dampers, which allow for precise and instantaneous control by continuously variable damping. Low voltage (12 V DC) and current (2 A maximum) requirements, industrial durability, real-time control, simple electronics, immediate implementation are additional advantages of MR dampers applications. To describe the nonlinear behaviours of the MR damper the Bingham’s model is adopted. This model is combined by Coulomb’s friction and viscous damping. A magnetic viscous-dry friction damping force can be represented by [5]. x˙ x Fd = d(I) tanh , (1) + vo d o where: d(I) is an experimentally determined dry friction force as a function of the input magnetic field, while parameters vo , do denote, respectively, velocity of a damper’s piston and the piston’s displacement. The motions of the system is described by two generalised coordinates: x – displacement of the oscillator in vertical direction, and ϕ – angle of the pendulum rotation. Differential equations of motion are derived by applying the second kind of Lagrange equations: s

(2)

1 1 (m2 + m p )l 2 ϕ¨ + cϕ ϕ˙ + (m2 + m p )l (x¨ + g) sin ϕ = 0 (3) 3 2 Introducing dimensionless time τ = ω0 t, where ω0 = (k + k2)/(m1 + m2 + m p) is the natural frequency of the oscillator, X = x/xst , ϕ ≡ ϕ are dimensionless

Regular and Chaotic Motions of an Autoparametric Real Pendulum System

269

coordinates and xst = (m1 + m2 + m p)g/(k + k2) is a static displacement of the linear oscillator, the Eqs. (2) and (3), are expressed in dimensionless form: ˙ + X + γ X 3 + μλ (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ ) = q cos ϑ τ X¨ + α1 X˙ + α3 tanh(10X) ϕ¨ + α2 ϕ˙ + λ X¨ + 1 sin ϕ = 0

(4) (5)

where dimensionless parameters α1 , α2 , α3 , μ , γ , λ , q, ϑ take the following definitions: cφ c ω α1 = , α2 = ,ϑ = , 1 2 (m1 + m2 + m p) ω0 ω0 m2 + 3 m p l ω 0 m2 + 12 m p l 2 d α3 = ,μ = , (m1 + m2 + m p) ω02 xst (m1 + m2 + m p ) x2st

λ =

(m2 + 12 m p )xst (m2 + 13 m p )l

,q =

k2 Q k1 2 ,γ = x . (k + k2 )xst k + k2 st

(6)

The dimensionless natural frequency of the linear oscillator is reduced to one. Substituting the α3 = 0 we get the model with the linear viscous damping.

3 Experimental System The experiment has been performed on an especially prepared two degree of freedom system. Figure 2 shows a photo of the real mechanical system with pendulum. Mechanical system consists of two components: (the first) a pendulum (4) which allows for full rotation, (the second) an oscillator (1) together with additions masses (2). Mass moment of inertia of the pendulum can be fitted for required dynamical conditions by changing of the pendulum masses (5) or the length. The pivot of the pendulum is connected with encoder (3) which has an accuracy 2π/2000. Motion of the system is realized by mechanisms: motor (7), inverter (6), a system changing rotation of the DC motor into translational motion (8). The dynamics of the system is investigated for two variants of dampers: a linear viscous damper (10) which is controlled by a hydraulic valve connected to an oil tank (11), or a nonlinear composite damper RD 1097-01 (9) with a suitable control system. The frequency of the vertical oscillations are controlled by an inverter, while an amplitude of the kinematical excitation is set by a change of a pitch of the drive shaft. The spring (12) which connects masses m1 with the foundation is considered also in two variants: linear or nonlinear with different stiffness characteristics. National Instrument card NI- DAQPad 6015 with DasyLab version 9 and Measurement Studio development environments with features such as DAQ Assistant and a single programming interface for all device functions are used for data acquisition and for control of the system.

J. Warmi´nski, K. Ke˛ cik

270

a

b

(3) (4) (11)

(1)

(6) (9) (12) (7) (10) (8) (5) (2)

Fig. 2 Photo of experimental system (a) and a view of damper RD 1,097–01(b)

The magnetorheological fluid damper (9), manufactured by Lord Corporation, may perform maximal damping force Fmax = 100 N and is suitable for low force, light duty suspension and isolation applications. Friction of the damper is controlled by the increase in yield strength of the MR fluid in response to magnetic field strength. More information about technical specification is provided on website www.lord.com. The damper is fixed on special strain gauge KMM30, it permit to measure damping force in damper. An additional strain gauge is mounted under spring which connects mass m1 with ground (Fmax = 200 N). The same damper has been used in [6] for control cables vibrations and by [7] for active suspension of the body. The angle of rotation ϕ of the pendulum and the displacement x of the oscillator are measured in the considered system. Derivatives calculated from received signals give velocity and acceleration of the pendulum and oscillator m1 . Application of additional sensors allows for measuring a damping force (described by Eq. 1) and force transmitted on the foundation. Experimental tests have been performed for both, the turned on and off the magnetorheological damper.

4 Numerical and Experimental Results Analysis of the system is based on numerical here been performed integrations of the set of differential Eqs. (4) and (5). Numerical analyses in Matlab-Simulink and Dynamics environments [8]. Because the damping force generated by a MR damper is approximated by a smooth function (1), the fourth order Runge-Kutta method is used in the numerical integration. All numerical results are obtained for there initial conditions: x = 0, x = 0, ϕ = 0.1 and ϕ = 0. Numerical investigations of the autoparametric model have been carried out on the basis of some data taken from

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Fig. 3 Resonance curve for pendulum (a) and bifurcation diagram (b) for α3 = 0

Fig. 4 Chaotic motions of the pendulum: numerical (a) and experimental (b) time histories for ϑ = 0.7, α3 = 0

the realistic system pictured in Fig. 2. Dimensionless parameters are determined by Eq. (6) and take values: α1 = 0.305392, α2 = 0.1, q = 2.32395, μ = 14.6863, λ = 0.134165, γ = 0. In this type of the autoparametric system some unstable region in the resonance case may be observed [9]. If the system works in this region the motions becomes not an periodic ones but quasi-periodic or chaotic. Figure 3 presents the analytical resonance curve of the pendulum obtained from Eq. (6) in see a description papers [10, 11] for the viscous damper when α3 = 0. For the system presented in this paper this region appears in the main parametric resonance in the frequency interval ϑ ≈ (0.6 − 1.15), (dashed line in Fig. 3a). Exemplary time histories plotted for the linear system (γ = 0) and when MR damper is switched off (α3 = 0), what corresponds to the typical viscous linear damper are presented in Fig. 4. Numerical and experimental time histories exhibit a similar behaviour, both are composed either of the positive or negative rotations with a small oscillations (swinging).

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Fig. 5 Irregular motions of the pendulum and force transmitted on the foundation; composed of rotation and swinging (α3 = 0.105634, ϑ = 0.7, (a), (b)), and composed of swinging ((α3 = 0.316902, ϑ = 0.7 (c), (d))

The region of instability can be reduced or eliminated by introduction of the system larger damping but then the area of effective vibrations absorption caused by the pendulum swinging is reduced as well. An alternative proposal for the instability avoidance can be achieved by application MR damping special control. Figure 5c presents time histories where the introduction of higher MR damping stops rotation of the pendulum. The nonlinear dynamical force transmitted on foundation is described by function: F(τ ) = x(τ )+ γ x3 (τ )+ α1 x( ˙ τ )+ 10α3 x( ˙ τ ). When the pendulum does not execute full rotation, the force transmitted on the base reaches values presented in Fig. 5d. Then the force is smaller comparing with fully rotating pendulum (Fig. 5b). If the system works near the unstable region, switching on the MR damper with proper damping value, reduces the instability effect. Magnetorheological damper can be used for the control of the system behaviour (adaptive system). If amplitude of the pendulum becomes too large, then suitable MR damping would lead to elimination (reduction) of the dangerous situation (Fig. 6b). To check the influence of the MR damping the MR damper has been on and off at different time instances. When we activate the device (parameter α3 is

Regular and Chaotic Motions of an Autoparametric Real Pendulum System

273

Fig. 6 Time histories of the α3 MR damper’s parameter (a) and pendulum (b) for frequency ϑ = 0.7 and α3 = 0.316902, numerical simulations

Fig. 7 Force transmitted on the ground (a) and damping force Fd (b), for frequency ϑ = 0.7 and α3 = 0.316902, numerical simulations

switched on), then the damping force jumps suddenly (point B, Fig. 7b) and the dynamical force transmitted on the ground increases (Fig. 7a). When we turn the MR device off, force in damper decreased gently (point A, Fig. 7b) and the force transmitted on the ground increases. Figure 8a presents experimental time histories of the pendulum while the MR damping is turn on and turn off in similar time intervals lasting about 10 s. In Fig. 8b time histories of the force transmitted on the ground are shown. Comparing this results with numerical simulations (Fig. 6b) we can conclude that in the real experiment the pendulum amplitude has smoother course. It results from time lag of the real apparatus and additional damping (e.g. damping of the oscillator in the sideway). An experimental force in the damper and hysteretic loop of the RD 1,097-01 damper are presented in the Fig. 9. Comparing experimental damping force (Fig. 9a) with numerical results (Fig. 7b) we may notice that a force jump in experimental history is almost invisible, but value of force amplitude is similar.

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Fig. 8 Experimental time histories (a) and force transmitted on the ground (b) during turn on and turn off of MR damper, α3 = 0.316902 and ϑ = 0.7

Fig. 9 Experimental histories of damping force (a) and hysteretic loops (b) for α3 = 0.316902 and ϑ = 0.7

Figure 10a presents a bifurcation diagram, in which the increase of damping coefficient up to α3 ≈ 0.33 eliminates the chaotic dynamics and the system transits to the quasi periodic and then to periodic are. If α3 ranges between [0.26, 0.33] 0.26–0.33, the chaotic motion consists only of the irregular pendulum swinging. Near the unstable region the centre of pendulum vibrations is shifted (see Fig. 11a, α3 ≈ 0.3). This shift may be located symmetrically around the lower static position of the pendulum. Introduction of suitable MR damping reduces the shifting effect. To complete the analysis the phase portraits and hysteretic loops of the damping force Fd of the MR damper (Eq. 1) received for different MR dampers are presented in Fig. 11. The area of the loop lets us determine the energy amount dissipated by the damper during oscillation, for different MR parameters. However, if oscillations are irregular (Fig. 11a, for α3 ≈ 0, and α3 ≈ 0, 105, 634) the determination of the energy dissipation is more complicated (the loop has an irregular complex shape).

Regular and Chaotic Motions of an Autoparametric Real Pendulum System

275

Fig. 10 Bifurcation diagram (a) and Lyapunov exponent (b) for ϑ = 0.7

Fig. 11 Phase portraits (a) of the pendulum and hysteretic loops of the damping force (b) for different MR damper parameters and frequency ϑ = 0.7

5 Conclusions The vibration absorption of the autoparametric system with pendulum employing MR fluid dampers has been investigated in the paper. Obtained results show that the application of the nonlinear magnetorheological damper may be an effective method of elimination of vibration transmitted on the foundation. Motion in unstable region can be eliminated by introduction of MR damping in the system suspension. Increase of MR damping reduces amplitudes in the resonance region but also reduces the area of the effective dynamical vibration absorption generated by pendulum swinging. To avoid such a situation it is planned to apply feedback control to

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the system in future investigations. MR damper can also be used for a reduction of the centre shifting of the pendulum vibrations. Current research is focused on the development of control strategy for the considered system. Numerical and experimental results indicate that MR damper can be successfully applied for an active vibration absorption and for control of the system response. Acknowledgment The work has been supported by grant NR 65/6. PR UE/2,005/7 and by grant NR N502 049 31/1449 from the Ministry of Science and Higher Education.

References 1. Cartmell MP, Lawson J (1994) Performance enhancement of an autoparametric vibration absorber by means of computers control. Journal of Sound and Vibration 2, 173–195. 2. Tondl A, Ruijgrok T, Verhulst F, Nabergoj R (2000) Autoparametric Resonance in Mechanical Systems, Cambridge University Press, Cambridge. 3. Spencer BF, Sain MK (1997) Controlling Buildings: A New frontier in feedback. Special Issue of the IEEE Control Systems Magazine on Emerging Technology 17(5), 19–35. 4. Hatwal H, Mallik AK, Ghosh A (1983) Forced nonlinear oscillations of an autoparametric system. Part 2. Chaotic responses. Journal of Applied Mechanics. Transactions of the ASME, 663–668. 5. Tang D, Gavin H, Dwell E (2004) Study of airfoil gust response alleviation using on electromagnetic dry friction damper. Part1: Theory. Journal of Sound and Vibration 269, 853–874. 6. Sapinski B, Snanima J, Maslanka M, Rosoł M (2006) Facility for testing magnetorheological damping system for cable vibrations. Mechanics 45(2), 136–142. 7. Kromulski J, Kazimierczak J (2006) Damping of vibrations with using magnetorheological fluid devices. Journal of Research and Applications in Agricultural Engineering 51(2), 47–49. 8. Nusse HE, Yorke JA (1994) Dynamics: Numerical Explorations, Springer, New York. 9. Song Y, Sato H, Iwata Y, Komatsuzaki T (2003) The response of a dynamic vibration absorber system with a parametrically excited pendulum. Journal of Sound and Vibration 259(3), 747– 759. 10. Kecik K, Warminski J, Szabelski K (2007) Regular and chaotic vibrations of a mechanical system with pendulum. Ist Congress of Polish Mechanics, 28–31 August. 11. Kecik K, Warminski J, Szabelski K (2007) Instability area in the main parametric resonance of mechanical system with pendulum. Proceedings 114–119. 2nd International Conference of Nonlinear Dynamics at the National University, 25–28 September.

Modeling of Flexible Rotor Vibrations in the Rotating Coordinate System ˇ sek, and Vladim´ır Zeman Michal Hajˇzman, Jakub Saˇ

1 Introduction This contribution is intended to the modelling of the flexible rotors that can be decomposed into a shaft and disk subsystems mutually joined together. Rotating shaft subsystems are considered as 1D continuum. The original shaft finite element in the rotating coordinate system is presented. Because disks can be of a complex shape they are modelled as 3D continuum also in the rotating coordinate system. The coupling matrix is used for the connection of the disk and shaft subsystems. The presented methodology is advantageous mainly due to the possibility of considering various effects of rotation that cannot be easily introduced in the commercial FEM codes. It is usable especially for the analysis of high-frequency vibrations, where the common assumption of rigid disks in rotor dynamics is not correct. Common mechanical systems in rotor dynamics are rotating shafts of different shapes joined with a special, mostly axi-symmetric bodies, which can be bladed disks, geared wheels, fans etc. Designed rotating systems and operating conditions are still becoming more and more complex and therefore it is necessary to create some advanced mathematical and computer models of the studied dynamical systems. The vibration analysis of rotating systems is commonly performed with the assumption of an ideal rigid disks. However, there are cases in which the vibrations of disks become important and the rigid body assumption is too rough for detailed dynamic analysis. Classic monographs, like [1, 2], describe the modelling of rotors considering disks as rigid bodies with their mass and inertia moments. Effects of rotation are presented in the form of the gyroscopic matrix. Rotating shafts are modelled usually on the basis of Bernoulli-Euler or Timoshenko theories. Special elastic shaft finite element based on these theories is introduced for example ˇ sek, and V. Zeman M. Hajˇzman, J. Saˇ Department of Mechanics, Faculty of Applied Sciences, University of West Bohemia in Pilsen, Univerzitn´ı 22, Plzeˇn, 306 14, Czech Republic, e-mail: [email protected], [email protected], [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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in [2], but rigid disks are still supposed. Many publications are dedicated to the dynamic analysis of thin rotating disks. One of the newest and most comprehensive monographs intended to dynamics of rotating systems is the monograph [3], where the issue of the rotating disks is described in more detail. Starting from the classical membrane theory for thin disks there are shown some possible approaches for modelling general three dimensional rotors and disks considering gyroscopic and centrifugal effects. This paper is aimed at the modelling of rotating systems, which can be decomposed into aˇ shaft and disˇc subsystems. The models of the particular rotating subsystems are described in the paper. Further, the methodology for the modelling of coupling between the disk and the shaft is shown. The proposed theory is tested using simple numerical example.

2 FEM Model of a Rotating Shaft It is supposed that the subsystems are rotating with constant angular velocity ω0 around their X -axis. The rotating shaft subsystem can be modelled as a one dimensional continuum on assumption of the undeformable cross-section that is still perpendicular to the shaft centre-line. The shaft is discretized using shaft finite elements (see Fig. 1 ) with two nodes. The deformations of the arbitrary point of the shaft finite element are described by six generalized coordinates in the rotating frame xyz – three displacements u(x), v(x), w(x) and three rotations ϕ (x), ϑ (x), ψ (x). In order to formulate the equations of motion of the rotating shaft vibrations the motion of the infinitesimal element of the length dx (see Fig.1 ) can be decomposed into the sliding motion of the element centre and into the spherical motion of this element. The sliding velocity is

Fig. 1 Shaft finite element

Modeling of Flexible Rotor Vibrations in the Rotating Coordinate System

⎡

279

⎤

u(x) ˙ ˙ − w(x)ω0 ⎦ , νS (x) = ⎣v(x) w(x) ˙ + v(x)ω0

(1)

and the angular velocity of the spherical motion is (parameter x is omitted in the expression) ⎡ ⎤ ⎡ ⎤ ϕ˙ + ω0 ϕ˙ + ϑ˙ sin ψ + ω0 cos ϑ cos ψ ω S (x) = ⎣ ϑ˙ cos ψ − ω0 cos ϑ sin ψ ⎦ = ˙ ⎣ϑ˙ − ω0 ψ ⎦ , (2) ψ˙ + ω0 ϑ ψ˙ + ω0 sin ϑ on assumption of the small angles ϕ (x), ϑ (x), ψ (x) and small appropriate angular velocities. The kinetic energy of the rotating shaft finite element can be written as (e) Ek

(e) (e) Ek sli + Ek sph

=

1 = 2

l

A(x)ν TS ν S ρ dx +

0 (e)

1 2

l

A(x)ω TS Jω S ρ dx,

(3)

0 (e)

where Ek sli is the kinetic energy of the sliding motion and Ek sph is the kinetic energy of the spherical motion. In expression (3) A(x) denotes the element cross-section, ρ is the material density and J = diag(J p , J, J) is the diagonal inertia matrix of the cross-section consisted of polar J p and area J moments of inertia. Substituting Eqs. (1) and (2) into the kinetic energies given in Eq. (3) one has (e) Ek sli

1 = 2

l

A(x) u˙2 + v˙2 + w˙ 2 − 2vw ˙ ω0 + 2wv ˙ ω0 + (w2 + v2 )ω02 ρ dx

(4)

0

for the kinetic energy of sliding motion and (e)

Ek sph =

l

˙ 0 + ω02 ) ρ dx + J p (ϕ˙ 2 + 2ϕω

1 2

0

+

1 2

l

˙ 0 ϑ + ω02 ϑ 2 ) ρ dx (5) J(ϑ˙ 2 − 2ϑ˙ ω0 ψ + ω02ψ 2 ) + J(ψ˙ 2 + 2ψω

0

for the kinetic energy of the spherical motion. The potential (deformation) energy of the shaft finite element can be expressed in the form (e)

Ep =

1 2

l 0 (A(x))

2 2 2 E εx (x) + G γxy (x) + γxz (x) dA(x)dx,

(6)

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where E and F are the Young an shear moduli, respectively. Considering the undeformable cross-section and assuming that ϕ -angle is small the elements of the necessary components of the strain tensor are [4]:

εx = u (x) − yv (x) − zw (x), γxy = −zϕ (x), γxz = yϕ (x),

(7)

while the prime denotes the differentiation with respect to x. After the substitution of Eqs. (7) in the potential energy (6) it yields (e) Ep

1 = 2

l

+ , 2 E u (x) − yv (x) − zw (x) + Gϕ 2 (x)(y2 + z2 ) dAdx.

(8)

0 (A(x))

The deformations of the shaft element are approached by cubic (for bending) and linear (for torsional and axial deformations) polynomials

ν (x) = Φ(x)c1 , w(x) = Φ(x)c2 , u(x) = Ψ(x)c1 , ψ (x) = ν (x) = Φ (x)c1 , ϑ (x) = −w (x) = −Φ (x)c2 , ϕ (x) = Ψ(x)c4 ,

Φ(x) = 1, x, x2 , x3 , Ψ(x) = [1, x] .

(9)

The complete deformation state of the shaft finite element of length l can be described by the nodal deformations ⎡ ⎡ ⎤ ⎤ w(0) ν (0) ⎢ψ (0)⎥ ⎢ ⎥ ⎥ , q2 = ⎢ϑ (0)⎥ , q3 = u(0) , q4 = ϕ (0) , (10) q1 = ⎢ ⎣ ν (l) ⎦ ⎣ w(l) ⎦ u(l) ϕ (l) ψ (l) ϑ (l) composed in the element vector q(e) = [qT1 , qT2 , qT3 , qT3 ]T of generalized coordinates. By using expressions (10) it results in q1 = S1 c1 , q2 = S2 c2 , q3 = S3 c3 , q4 = S3 c4 , ⎡ ⎡ ⎤ ⎤ 10 0 0 1 0 0 0 ⎢0 1 0 0 ⎥ ⎢0 −1 0 0 ⎥ 10 ⎢ ⎥ ⎥ . S1 = ⎢ = = , S , S ⎣1 l l 2 l 3 ⎦ 2 ⎣1 l l 2 l 3 ⎦ 3 1l 0 1 2l 3l 2 0 −1 −2l −3l 2

(11)

(12)

Expressing vectors ci of time dependent coefficients from (12) the relations between deformations of the shaft finite element and nodal generalized coordinates can be calculated

Modeling of Flexible Rotor Vibrations in the Rotating Coordinate System

281

−1 −1 u(x) = Ψ(x)S−1 3 q3 , ν (x) = Φ(x)S1 q1 , w(x) = Φ(x)S2 q2 , −1 ϕ (x) = Ψ(x)S−1 3 q4 , ϑ (x) = −w (x) = −Φ (x)S2 q2 ,

(14)

ψ (x) = v (x) = Φ (x)S−1 1 q1 , Equations (14) can be substituted into the expressions (4), (5) and (6) for the kinetic and potential energies, so the energies can be rewritten in the subsequent matrix forms T 1 (e) T (e) (e) 1 T (e) (e) Ek = q˙ M q˙ + ω0 q˙ (e) C(e) q(e) + ω02 q(e) K d q(e) + 2 2 T 1 2 (e) (e) 2 (e) + ω0 q f 1 + ω0 I , 2

1 (e) T (e) (e) (e) q Ep = Ks q , (15) 2 (e)

where M(e) is the element mass matrix, C(e) is the element Coriolis matrix, K d (e)

is the element dynamic stiffness matrix, f 1 is the vector of the element gyro(e) scopic forces, I (e) is the inertia moment of the element and K s is the element static stiffness matrix. The kinetic and potential energies (15) of all elements can be summarized in the total kinetic and potential energies Ek and E p , which can be used in Lagrange’s equations [4] to formulate the mathematical model of the rotating shaft discretized by the special 1D shaft finite elements. Without consideration of damping and external loading the model of the shaft subsystem S in the rotating coordinate system is of the form MS q¨ S (t) + ω0GS q˙ S (t) + K sS − ω02 K dS qS (t) = 0, (16) where qS (t) is the global vector of shaft subsystem generalized coordinates and ω0 GS = ω0 C − CT is global gyroscopic matrix of the shaft subsystem.

3 FEM Model of a Rotating Disk Similar methodology as for the rotating shaft modelling is used for the formulation of the mathematical model of the rotating disk subsystem described in the rotating frame (see Fig. 2 ). It is based on the assumption of the disk as a three dimensional continuum and on its discretisation by solid finite elements with eight nodes. The deformations of an arbitrary point of the disk are described by three displacements u(t), v(t), w(t) defined in the rotating coordinate system. After the formulation of kinetic and potential energies and after the application of Lagrange’s equations the whole rotating disk model can be written [5] as MD q¨ D (t) + ω0 GD q˙ D (t) + K sD − ω02 K dD qD (t) = 0, (17)

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Fig. 2 Rotating disk considered as the 3D continuum

while the meaning of the arising matrices is identical as of those in Eq. (16). A centrifugal force vector presented in [5] is omitted in Eq. (17) with respect to the further application for the rotor modal analysis.

4 Mathematical Model of the Whole Disk-Shaft System In order to model complex rotor systems composed of various disk and shaft subsystems the methodology of the modelling of subsystems coupling by special stiffness matrix was introduced in [6]. The modelling method is shown for the system composed of two subsystems but the generalization for the more complex systems is straightforward. The models of the coupled subsystems can be written using vector

T T T f C = f CS , f CD of internal coupling forces as M S q¨ S (t) + ω0GS q˙ S (t) + K sS − ω02 K dS + KB qS (t) = f CS , M D q¨ D (t) + ω0 GD q˙ D (t) + K sD − ω02 K dD qD (t) = f CD .

(18)

The stiffness matrix K B , that represents bearings, were introduced. It holds f C = −KC q

(19)

for the coupling force vector, while KC is the global coupling matrix between the

T shaft and disk subsystems and q = qTS , qTD is the global vector of rotor generalized coordinates. The whole model of the rotor intended for a modal analysis is

Modeling of Flexible Rotor Vibrations in the Rotating Coordinate System

MS 0 0 MD

283

q¨ S GS 0 q˙ S + ω0 + 0 GD q˙ D q¨ D 0 qS 0 K S (ω0 ) . + KC = + 0 K D (ω0 ) 0 qD

(20)

The advantageous approach based on the modal synthesis method suitable for the reduction of the degrees of freedom number is also shown in [6].

5 Numerical Example The proposed method can be demonstrated using the test rotor (disk-shaft-bearing system) shown in Fig. 3 . The shaft radius is r = 0.025 m, the disk radius R = 0.08 m, the disk width h = 0.04 m and the shaft lengths a = b = 0.14 m. The shaft was discretized using 16 shaft 1D elements and the disk was discretized using 576 solid elements. The isotropic bearings (stiffness kB = 109 N/m) were considered in the outside nodes of the shaft (left bearing – radial and axial direction, right bearing – radial direction). Standard steel material properties were considered. The original in-house software was created in MATLAB programming environment based on the developed modelling methodology. The modal analyses of the rotor were calculated for a range of angular velocities and some results are illustrated by the chosen parts of the Campbell diagram in Figs. 4 and 5. There are shown chosen eigenfrequencies [Hz] in dependence on the rotor angular velocity [revolutions per minute]. The subscript of eigenfrequencies denotes the number of the mode shape. The character of the rotor mode shape is noted in the brackets. In order to verify the new methodology for the flexible rotor modelling the comparison with the results obtained by the standard methodology using the rigid disk assumption was made. The model based on the representation of the disk as a rigid

Fig. 3 Scheme of the simple test rotor

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Fig. 4 Chosen part of the Campbell diagram for the test rotor (lower eigenfrequencies)

Fig. 5 Chosen part of the Campbell diagram for the test rotor (higher eigenfrequencies)

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285

Table 1 Eigenfrequencies [Hz] of the test rotor obtained using various models Reference model

Flexible model

Note

0 (1) 651 (2, 3) 1410 (4) 2222 (5, 6) 5758 (7) 5912 (8) 6083 (9, 10) 6248 (11, 12) – – 9760 (13) 11997 (14) 12627 (15, 16) 12810 (17, 18) – – 17558 (19) –

0 (1) 708 (2, 3) 1402 (4) 2226 (5, 6) 5878 (8) 4564 (7) 6291 (11, 12) 5939 (9, 10) 7065 (13) 7072 (14) 7432 (15) 10254 (16) 11599 (19, 20) 11443 (17, 18) 12899 (21) 12900 (22) 13919 (23) 14278 (24, 25)

Free rotation Bending Axial Bending Torsional Torsional Bending bending Bending of the disk Bending of the disk Axial Axial Bending Bending Disk radial deformation Disk radial deformation Torsional Bending of the disk

body with calculated inertia properties mounted in chosen shaft node will be called reference model. The comparison of the eigenfrequencies [Hz] obtained by means of two different models (reference and flexible disk model) for non rotating rotor (ω0 = 0) is shown in Tab. 1. The numbers in brackets in Tab. 1 denote the number of the corresponding eigenmode. The character of the eigenmodes of vibration is described in the last column. The first eigenfrequency is zero because the rotor can freely rotate around its axis of rotation. The most important result of this analysis is the presence of the eigenmodes characterized by the pure flexible disk vibration (bending or radial deformation) that cannot be catched by the original reference model. These eigenmodes can be excited by high-frequency excitation and the proposed new modelling methodology can be more accurate than the original one based on the assumption of rigid disks.

6 Conclusions The original methodology of the rotor vibration modelling in the rotating coordinate system is presented in the paper. The special shaft 1D finite elements and the 3D solid finite elements are used. The whole model composed of several shaft and disk subsystems can be assembled by means of the derived coupling stiffness matrix. The presented methodology is advantageous mainly due to the possibility of considering various effects of rotation that cannot be easily introduced in the commercial FEM codes. The described model of the flexible rotors is usable especially for the analysis

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of high-frequency vibrations, where the common assumption of rigid disks in rotor dynamics is not correct. The presented method is verified using the simple test rotor example shown in the last section. The results of the modal analysis of the rotor with flexible disk are compared with the results obtained for the reference model characterized by the assumption of the ideal rigid disk. The character of the chosen eigenmodes of vibration of the whole flexible rotor is shown in Figs. 6–9.

Fig. 6 The second eigenmode of vibration of the flexible test rotor (bending)

Fig. 7 The fourth eigenmode of vibration of the flexible test rotor (axial)

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Fig. 8 The seventh eigenmode of vibration of the flexible test rotor (torsional)

Fig. 9 The 14th eigenmode of vibration of the flexible test rotor (disk bending)

The advantage of the proposed modelling method is also in the possibility of the usage of the modal reduction. Then the models can be easily employ in the complex dynamical analysis of the rotor of rotor system. Acknowledgement This work was supported by the research project 101/07/P231 of the Czech Science Foundation and by the research project MSM4977751303 of the Ministry of Education, Youth and Sports of the Czech Republic.

References 1. Kr¨amer E (1993) Dynamics of Rotors and Foundations. Springer, Berlin. 2. Yamamoto T, Ishida Y (2001) Linear and Nonlinear Rotor Dynamics, A Modern Treatment with Applications. Wiley, New York.

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3. Genta G (2005) Dynamics of Rotating Systems. Springer, New York. 4. Slav´ık J, Stejskal V, Zeman V (1997) Elements of Machine Dynamics, Czech Technical University, Prague (in Czech). ˇ sek J, Zeman V, Hajˇzman M (2006) Modal properties of rotating disks. In: Vimmr J (ed) 5. Saˇ Proceedings of Computational Mechanics 2006. University of West Bohemia in Pilsen, Hrad Neˇctiny, 593–600. ˇ sek J, Zeman V (2007) Vibrations of rotors with flexible disks. In: Zolotarev 6. Hajˇzman M, Saˇ I (ed) Proceedings of Engineering Mechanics 2007. Institute of Thermomechanics AS CR, Svratka, 1–11.

Stochastic Dynamics of Hybrid Systems with Thermorheological Hereditary Elements Katica R. (Stevanovi´c) Hedrih

1 Introduction Coupled subsystems [1–8] switch between many operating modes where each mode is governed by its own characteristic dynamical laws. The scientific literature of coupled systems with coupled disparate nature fields covers almost all branches of science. Research interaction between subsystems in the hybrid system is current topic. The multi-pendulum system is hereditary if material particles are interconnected by one or more standard hereditary elements. In [9–12] dynamics of a thermo-rheological hereditary pendulum is mathematically described. Research results in area of mechanics of hereditary discrete systems obtained by Goroshko and Hedrih are generalized and presented in the monographs [9] and [13], which contain of first presentation of the analytical dynamics of the hereditary discrete systems. We can conclude that these monographs contain complete foundation of the analytical dynamics theory of discrete hereditary systems and by using these results, numerous examples are obtained and solved. In current literature term “hereditary” and “rheological” systems are equivalent. In opinion of Rabotnov Yu.N. the name “hereditary” system or continuum proposed by Voltera B., is more precise as well as suitable. The discrete hereditary system is a system of discrete material particles interconnected by standard constraint light hereditary elements. When we talk about coupled systems we must define different kinds of coupled subsystems [1–4, 6, 8]. Firstly, we can define that we are talking about coupled discrete and continuous system coupled by some elastic, visco elastic and creep property elements [7, 8] and [1–3, 6] or by dynamical constraints. Than we obtain coupled structures of subsystems. These systems can be with linear, rheolinear or nonlinear properties. Also we can define that processes in the subsystem are coupled. Than we have systems with rheolinear, rheological or nonlinear elements, K.R. Hedrih Faculty of Mechanical Engineering University of Niˇs, Mathematical Institute SANU, 18 000-Niˇs, ul. Vojvode Tankosi´ca 3/22, Serbia, e-mail: [email protected], [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

289

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or nonlinear properties of the basic structure materials, or geometry these many interactions and intercoupling appear.

2 Light Standard Thermo-Rheological Hereditary Element For stress strain state of the standard hereditary element, the relation between generalized coordinate of hereditary element deformation ρ − ρ0 and corresponding force of deformation P (t) [13, 14] is nP˙ (t) + P (t) = ncρ˙ (t) + c[ ˜ ρ (t) − ρ0 ] ,

(1)

where n is time of relaxation, and c, c˜ an instantaneous rigidity and a prolonged one of an element. In Fig. 1 series of schematic notations of a thermo-visco-elastic light simple and complex elements are presented: a∗ schematic notations of a thermo-elastic light simple element; b∗ the thermo-modified Maxwell elastic-viscosic light hereditary element; c∗ the thermo-modified Kelvin-Foight visco-elastic light hereditary element; d∗ the thermo-modified Burgers light hereditary element. When standard hereditary element is modified by two temperatures TK (t) and TM (t), which are introduced by thermo-modification of visco-elastic properties by temperature TK (t), and by thermo-modification of elastic-viscosic properties by temperature TM (t), than constitutive relation between stress and strain state of the thermo-rheological hereditary element [9] is governed by the following equation ˙ + P(t) + nF˙M (t) + FK (t) = ncρ˙ (t) + c[ nP(t) ˜ ρ (t) − ρ0 ],

(2)

where (see in [13]) FM (t) = cM αM TM (t) , FK (t) = cK αK TK (t)

(3)

are thermo-elastic forces, ρ (t) is rheological coordinate, cM , cK are coefficients of thermo-elastic rigidity, and αM , αK are coefficients of thermo-elastic dilatations. The constitutive differential Eq. (2) of the thermo-rheological hereditary element can be rewritten in the two integro-differential following forms: a∗ explicit with respect to the force P (t) ⎡ ⎤ P (t) = c ⎣ρ (t) − ρ0 −

t

[ρ (τ ) − ρ0]R (t − τ ) d τ ⎦ − FM (t)

0

+

c c − c˜

t

[FM (τ ) − FK (τ )] R (t − τ ) d τ ,

0

or b∗ explicit with respect to the coordinate ρ (t)

(4)

Stochastic Dynamics of Hybrid Systems with Thermorheological Hereditary Elements a*

d*

b*

aTK

EU

aT

mK

TK

EM

T(t)

mM

s(t),e(t),T(t) c*

aTM

EK

ETK EM

TM(t)

mM

aTM ETM EK

TK ETK

aTK

s M (t),e M (t),T M (t)

mK

291

aTM ETM T M(t)

s K (t),e K (t),T K (t)

s (t),e (t),T K (t),T M (t)

Fig. 1 a∗ Schematic notation of a thermo-elastic light element; b∗ Schematic presentation of the thermo-modified Maxwell elastic-viscosic light hereditary element; c∗ Schematic presentation of the thermo-modified Kelvin-Foight visco-elastic light hereditary element; d∗ Schematic presentation of the thermo-modified Burgers light hereditary element

⎤ ⎡ t 1⎣ ρ (t) − ρ0 = P (t) + P (τ ) K (t − τ ) d τ ⎦ + FM (t) c 0

c˜ − c − c˜ t−τ

t 0

c FM (τ ) K (t − τ ) d τ + c − c˜

t

FK (τ ) K (t − τ ) d τ , (5)

0 c(t− ˜ τ)

c˜ − n c˜ − nc where: R (t − τ ) = c− is a kernel of relaxation and K (t − τ ) = c− is nc e nc e a kernel of rheology. In Figs. 2 and 3 two thermo-rheological systems containing finite numbers of coupled pendulums [9] are presented. We take into consideration finite number of coupled mathematical pendulums presented in Fig. 3, all with material particles of mass m with length and with finite numbers n degrees of freedom defined by generalized coordinates ϕi , i = 1, 2, 3, . . . m, and the standard light thermo-visco-elastic elements thermo-modified by temperature T (t), coupling pendulums at distance and coupled, in parallel, but temperature isolated and with the standard light nonlinear springs with coefficients of the linear and nonlinear rigidity respectively denoted by cT , and cˆ = ε χ c, where ε is a small parameter (see Fig. 4). Now, we take into account that these standard light thermo-visco-elastic elements thermo modified by temperature T (t) are in the dynamic state, and that we didn’t neglect thermo modification of the element strain. Then the constitutive relation of the thermo-visco-elastic stress-strain state has the following form

292

K.R. Hedrih 02

01 c,cˆ = ecc

lT

01

lc

lT

cT,b,aT,T(t)

l1

PT (t)

j1

h1

j2

l2

l2 h1 h2

m1

m2

j1

PT (t)

02

l j2

l2

j1

l2 h1

m 1g

m1

m2g

01

lc

cT,b,aT,T(t)

l1

m1g

02

c,cˆ = ecc

c,c ,a ,T(t.,) T T

h2

m2

l2 m1g

m1

m2g

Fig. 2 System with four modified hereditary element

02

01

l

l j1 h1

cT,b,aT,T(t)

m1

l2

j2

c,cˆ = ecc m2

m2g

h2 m1

m2g

b*

and two

(b∗ )

pendulums inter coupled by standard light thermo-

01

02

j1

j2

01

l

l h1

m1g

h2

m2

a*

(a∗ )

j2

l2

cT,b,aT,T(t)

c,cˆ = ecc m2

j1

m2g

h2 m1

01

l

l h1

m1g

02

cT,b,aT,T(t)

l2

c,cˆ = ecc m2

j1 h1

m2g

l

l

j2

m1g

02

h2 m1

cT,b,aT,T(t)

l2

j2

m1g

ˆ = ecc m2 c,c

m2g

Fig. 3 System with “chain” pendulums interconnected by standard light thermo-modified hereditary elements g 2 w~ 0 = l

0

x1 m

c0 = mg / l

Fig. 4 Thermo-rheological rheolinear system first partial oscillator 1

Δ0 = αT T (t) (0 + x) , Pher(k) (t) = −cT (k) Δ0(k) + T (k) (ϕk+1 − ϕk ),

(6)

Pher(k) (t) = −cT (k) T (k) (ϕk+1 − ϕk ) 1 + αT(k) T(k) (t) − cT (k) αT (k) 0(k) T(k) (t),

(7)

and that forces of nonlinear spring and damping are defined as follows Pnolinelstic(k) (t) = −c(k) c(k) (ϕk+1 − ϕk ) + ε χk 3c(k) (ϕk+1 − ϕk )3 , Pdamp(k) (t) = −bk b(k) (ϕ˙ k+1 − ϕ˙ k ) .

(8)

Differential equations of the thermo-rheological coupled pendulums presented in Fig. 3 are:

Stochastic Dynamics of Hybrid Systems with Thermorheological Hereditary Elements

2 ϕ¨ 1 + ω˜ 02 ϕ1 − ω02 (ϕ2 − ϕ1 ) − ω0T (ϕ2 − ϕ1 ) 1 + γ T˜ (t) − 2δ (ϕ˙ 2 − ϕ˙ 1 ) 3 ϕ1 ϕ15 ϕ17 ϕ19 2 − + − + . . . .. + εω02 χ˜ (ϕ2 − ϕ1 )3 , h0 T˜ (t) + ω˜ 02 = ω0T 3! 5! 7! 9!

293

(9)

2 ϕ¨ k + ω˜ 02 ϕk + ω02 (ϕk − ϕk−1 ) − ω02 (ϕk+1 − ϕk ) + ω0T (ϕk − ϕk−1 )

2 1 + γ T˜ (t) − ω0T (ϕk+1 − ϕk ) 1 + γ T˜ (t) + (10) 2 2 + 2δ (ϕ˙ k − ϕ˙ k−1 ) − 2δ (ϕ˙ k+1 − ϕ˙ k ) = −ω0T h0 T˜ (t) + ω0T h0 T˜ (t) + 3 2 3 5 9 7 ϕ ϕ ϕ ϕ k − k + k − k + . . . .. − εω02 χ˜ (ϕk − ϕk−1 )3 + ω˜ 02 3! 5! 7! 9!

+ εω02 χ˜ (ϕk+1 − ϕk )3 ,

2 ϕ¨ n + ω˜ 02 ϕn + ω02 (ϕn − ϕn−1 ) + ω0T (ϕn − ϕn−1 ) 1 + γ T˜ (t) + 2δ (ϕ˙ n − ϕ˙ n−1 ) = 3 (11) ϕn ϕn5 ϕn7 ϕn9 2 − + − + . . . .. − εω02 χ˜ (ϕn − ϕn−1 )3 , h0 T˜ (t) + ω˜ 02 = −ω0T 3! 5! 7! 9! 2 = cT , ω ˜ 02 = g , γ = αT T0 , 2δ = mb , h0 = αT 0 T0 , where: k = 2, . . . , n − 1, ω02 = mc , ω0T m ↔ 1 T˜ (t) = T0 T (t), χ = χ 2 . Basic linear equations of the previous system for homogeneous case have the form 2 2 ϕ¨ 1 + ω˜ 02 + ω02 + ω0T ϕ1 − ω02 + ω0T ϕ2 = 0, 2 2 2 2 2 2 ϕ¨ k − ω0 + ω0T ϕk−1 + ω˜ 0 + 2ω0 + 2ω0T ϕk − ω02 + ω0T ϕk+1 = 0, 2 2 2 2 2 ϕ¨ n − ω0 + ω0T ϕ + ω˜ 0 + ω0 + ω0T ϕn = 0. (12)

Using the trigonometric method (see Refs. [7, 8, 15]) for the previous system Ak = C sin kϕ we can find eigennumbers and eigenfrequencies in the following forms ! 2 ωs2 − ω˜ 02 2 sπ 2 sin2 sπ and ωs = ω˜ 02 + 4 ω0c = 4 sin + ω0T us = 2 2n 2n ω˜ 0cT s = 0, 1, 2, 3, 4, . . .n − 1,

(13)

and for three pendulum system with solutions: ϕ1 (t) = ξ1 + ξ2 + ξ3 , ϕ2 (t) = ξ1 − 2ξ3 and ϕ3 (t) = ξ1 − ξ2 + ξ3 , where: ξ1 = C1 cos (ω1t + α1 ), ξ2 = C2 cos (ω2t + α2 ), ξ = C3 cos (ω3t + α3 ) are normal coordinates and C1 , C2 , C3 , α1 , α2 and α3 are constant values. For the three mass pendulum system governing nonlinear equations expressed by normal coordinates of the basic corresponding linear system are obtained:

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2

3

ω˜ 2 (ξ1 + ξ2 + ξ3 )3 (ξ1 − 2ξ3 )3 (ξ1 − ξ2 + ξ3 )3 , ξ¨1 + ω˜ 02 ξ1 ≈ 0 + + 3 3! 3! 3!

2 2 ξ¨2 + ω˜ 02 + ω02 ξ2 + ω0T ξ2 1 + γ T˜ (t) + 2δ ξ˙2 = ω0T h0 T˜ (t) + 2 3 . ω˜ 2 (ξ + ξ + ξ )3 (ξ − ξ + ξ )3 εω02 χ˜ 1 2 3 1 2 3 3 3 0 (−3ξ3 − ξ2 ) + (−ξ2 + 3ξ3 ) + − , + 2 2 3! 3!

3 2 ξ 1 + γ T˜ (t) + 6δ ξ˙ = 1 εω 2 χ ξ¨3 + ω02 + 3ω02 ξ3 + 3ω0T 3 3 0 ˜ (−3ξ3 − ξ21 ) − 3

3 ω˜ 2 ξ3 )3 (ξ1 −ξ2 +ξ3 )3 , (14) + − 6εω02 χ˜ (−ξ2 + 3ξ3 )3 + 60 (ξ1 +ξ3!2 +ξ3 ) − 2 (ξ1 −2 3! 3! 2 + ω2 + ω 2 + 3ω 2 + ω ˜ 02 , ωs2 = 3ω0T ˜ 02 with three eigenfrequencies ω12 = ω˜ 02 , ω22 = ω0T 0 0 for the line’s linear system. For the double pendulum system governing nonlinear equations expressed by normal coordinates of the basic corresponding linear system have the following form: . ω˜ 2 ξ¨1 + ω˜ 02 ξ1 = 0 (ξ1 + ξ2 )3 + (ξ1 − ξ2)3 , 3!

ξ¨2 + ω˜ 2 ξ2 + 2ω 2ξ2 + 2ω 2 1 + γ T˜ (t) ξ2 + 4δ ξ˙2 = 0

0

0T

. (15) ω˜ 02 (ξ1 + ξ2)3 − (ξ1 − ξ2)3 , 3! 2 2 2 2 ∓ ω2 + ω2 with two eigenfrequencies ω1, 2 = ω˜ 0 + ω0 + ω0T 0 0T of the line’s linear system. For the double pendulum system, the first equation of the rheo-nonlinear system 15 in the linearized form represents partial pure harmonic oscillator, presented in Fig. 4 with frequency ω12 = ω˜ 02 = g of the free vibrations. In the non-linearized form the rheonomic members are not explicitly obtained. This linearized case is when both pendulums oscillate with same frequency, ω12 , as decoupled pendulums, as single mathematical pendulum, and then standard light thermo-visco-elastic element thermo-modified by temperature T (t) haven’t influence to this normal; coordinate composed by sum ξ1 = ϕ1 + ϕ2 . On this normal (main) coordinate oscillation in the linearized approximation are free without temperature influence. This is right for all cases of the multi pendulum systems presented in Fig. 3. For the double pendulum system the second equation of the rheo-nonlinear system (15), in the normal coordinate ξ2 = ϕ1 − ϕ2 and in the linearized form states the Mathieu-Hill type equation ( [16]) representing mathematical description of the thermo-rheological oscillator shown in Fig. 5 a with parallel coupled two light standard thermo-visco-elastic element thermo-modified by same temperature T (t) and one linear elastic spring with rigidity c0 = mg/ in the dynamic state. For this coordinate ξ2 = ϕ1 − ϕ2 , we can separate two main cases. For both cases, we take into consideration an asymptotic approximation of the amplitude and phase of the dynamic process on this coordinate ξ2 close around I∗ main 2 ˜ 23 + h0 T˜ (t) + 2ω02ε χξ = −2ω0T

Stochastic Dynamics of Hybrid Systems with Thermorheological Hereditary Elements T(t)

2P

0

295

x m

c0 = mg / l

Fig. 5 Thermo-rheological rheolinear system second partial oscillator 2

2 , and II∗ around parametric resoresonance, then Ω ≈ ω2 = ω˜ 02 + 2 ω02 + ω0T 2 . Next, we can conclude that along this nance if Ω ≈ 12 ω2 = 12 ω˜ 02 + 2 ω02 + ω0T coordinate it is possible to appear, under the corresponding kinetic parameters, some I∗ regimes closest to main resonance state, as well as one main resonance state and some II∗ regimes closest to parametric resonance state [17], as well as one resonance state under the thermo-viscoelastic temperature single frequency excitation.

3 Thermo-Rheological Double Pendulum System – System of the Averaged Equations For solving system of nonlinear differential equations 15 of the double pendulum system we take into account temperature excitation in the form: T˜ (t) = sin (Ωt + β ) with deterministic constant value frequency Ω and constant deterministic or stochastic phase β and we take into account the following form of the first asymptotic approximation of solution:

ξ1 (t) = C1 (t) cos Φ1 (t) and ξ2 (t) = C2 (t) cos Φ2 (t)

(16)

where amplitudes C1 (t) and C2 (t), and phases Φ1 (t) and Φ2 (t) are unknown function of time t. We obtain system of ordinary differential equations for these unknown amplitudes and phases, in the averaged form, along full phases Φ1 (t) and Φ2 (t), and for both cases we obtain [17] the following: a∗ Δdet(i) = ωi − Ω, Φi (t) = Ωt + φi , Ω = ωi − Δdet(i) main resonance . 3ω˜ 2 C˙1 (t) = 0, φ˙1 (t) = Δ(1) + 0 [C1 (t)]2 + 2 [C2 (t)]2 , Δdet(1) = ω1 − Ω1 8 ω1 h 0 cos (φ2 − β ) , Δdet(2) = ω2 − Ω2 C˙2 (t) = −2δ C2 (t) + ω2 . 3ω 2 ε χ˜ ω˜ 2 φ˙2 (t) = Δ(2) + 0 [C2 (t)]2 + 0 3 [C1 (t)]2 + [C2 (t)]2 4 ω2 12ω2 2 ω h0 sin (φ2 − β ) (17) − 0T ω2C2 (t)

296

K.R. Hedrih

Ω Ω Ω , Φi (t) = t + φi , = ωi − Δdet(i) parametric resonance 2 2 2 . 3ω˜ 2 Ω1 , C˙1 (t) = 0, φ˙1 (t) = Δ(1) − 0 [C1 (t)]2 + 2 [C2 (t)]2 , Δst(1) = ω1 − 8 ω1 2 ω 2 γ T˜0 Ω2 , C2 (t) cos (φ2 − β ) , Δst(1) = ω2 − C˙2 (t) = −2δ C2 (t) + 0T ω2 2 . 3ω 2 ε χ˜ ω˜ 2 φ˙2 (t) = Δ(2) + 0 [C2 (t)]2 + 0 3 [C1 (t)]2 + [C2 (t)]2 4 ω2 12ω2 2 γ T˜ ω0T 0 sin (φ2 − β ) (18) + ω2

b∗ Δst(1) = ω1 −

4 Stochastic Dynamics of the Thermo-Rheological Double Pendulum system – Parametric Resonance We intend to investigate role of the temperature T (t) for dynamic phenomena of the appearance of the parametric resonance regime for the case that bonded noise temperature excitation to the thermorheological connection between pendulums in the double pendulum system. For that reason we take into account that temperature excitation is random, bonded noise, taken in the following form T˜ (t) = sin Ω2 t + σ B (t) + β with deterministic constant value frequency Ω2 and that B(t) is the standard Wiener process, and β is a random uniformly distributed variable in interval [0, 2π ]. Then T˜ (t) is a stationary process having autocorrelation function and spectral density function [18]: σ2τ 1 R (τ ) = μ 2 e− 2 cosΩτ 2

and S (ω ) =

+∞ −∞

1 R (τ ) eiωτ d τ = μσ 2 2

ω 2 + Ω2 + σ4 .

2 2 ω 2 − Ω2 − σ4 + σ 2ω 2 2

(19)

/ / Stochastic process /T˜ (t)/ ≤ 1 is bounded for all values of time t. The change of coordinates phases and amplitudes in averaged system of equations 18 (using an idea of Ariaratnam [18]) in the following ways and in the following forms: ρi (t) = lnCi (t), ϑ (t) = φ (t) − 12 ψ , where ψ = σ B (t) + β and transforming the averaged system of differential equations 18 into the system of averaged stochastic differential equations of I to type with respect to the unknown amplitudes C1 (t) and C2 (t), and phases Φ1 (t) and Φ2 (t), results in the following forms:

Stochastic Dynamics of Hybrid Systems with Thermorheological Hereditary Elements

297

1 ˙ 1 Ci (t) dt and d ϑi (t) = φ˙i (t) dt − d ψ or Ci (t) 2 .5 2 3ω˜ 0 2ρ1 (t) 1 2ρ2 (t) e dt − σ dB (t), + 2e d ρ1 (t) = 0, d ϑ1 (t) = Δ(1) − 8 ω1 2 ω 2 γ T˜0 cos 2ϑ2 dt, d ρ2 (t) = −2δ dt + 0T ω2 .5 3ω˜ 02 - 2ρ1 (t) ω 2 γ T˜0 1 2ρ2 (t) d ϑ2 (t) = Δ(2) − 2e dt + 0T +e sin 2ϑ2 dt − σ dB (t). 8 ω1 ω2 2 (20) d ρi (t) =

The Lyapunov exponents λs of system mode processes (16) may be introduced by using the time mode – normal coordinates ξs , which by making use of the averaged equations have the following form: 5 1 1 1 2 2 λs = lim ln [ξs (t)] + 2 [ξs (t)] = lim ρs (t) . (21) t→∞ 2t t→∞ t ωs0 Now, we can use Lyapunov exponents λs as a measure of the average exponential growth of the amplitudes C1 (t) and C2 (t) of component processes. The Lyapunov exponents λs are the deterministic numbers with probability one for the system given by averaged equations. Solutions of the averaged differential equations depend on initial values. In order to calculate the expression and values for Lyapunov exponents λs , it is necessary to integrate both sides of the first two stochastic differential equations for the ρs of the system (20), so that, from expressions for Lyapunov exponents λs , we can write that ω 2 γ T˜0 (22) λ1 = 0 and λ2 = −2δ − 0T E cos 2ϑ22 . ω2 Following Ariaratnam [18] and Stratonovich [15] and also, our previous stochastic differential equations (21), the approximated values of the mathematical expectation E [cos 2ϑn ], for the process ϑ2 (t) = φ2 (t) − 12 ψ , and ψ = σ B (t) + γ are found to be in the form of expressions ⎛ 2 ˜ ⎞ ω0T γ T0 4Δ 1 I1+iq (z) I1−iq (z) (2)kor ⎠ ω + , (23) E [cos 2ϑ2 ] = F ⎝ 22 , , F (z, q) = σ σ2 2 Iiq (z) I−iq (z) where Δ(2)kor = Δ(2) − is taken into account of initial value.

3ω˜ 02 2ρ10 2e + e2ρ20 8 ω1

(24)

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5 Conclusions We can conclude that the basic system (rheolinear-unperturbed system) corresponding to a rheolinear-thermo-rheological perturbed system has two main normal coordinates, and that for the thermo-rheological perturbed system our investigations results in one free partial and in the second mode, one forced and parametrically perturbed oscillator when the first mode becomes the unperturbed one. Non-linearities of pendulums explicitly appear in both phase equations and modes, and a nonlinearity of the spring appear explicitly only in the phase equation for second mode and for both cases of the main and parametric resonance. Also, on the basis of the analogy between models in Section 2 and these corresponding piezo-modifies rheological light elements, we can use all obtained models of the double pendulum system and all obtained approximation for amplitudes and phases of the nonlinear modes. By using previous expression for the Lyapunov exponent λ2 = 0 in the forms of expressions (23) with probability equal to 1 for evaluation of the stability or instability of the vibration process of the thermorheological double pendulum system stochastic stability, we must find the maximal value of the Lyapunov exponent and determine the kinetic parameters of the system such that this Lyapunov exponent takes negative values. We can also consider any cases when Δ(2) = ω2 − Ω/2 or Δ(2)kor = Δ(2) − 3ω˜ 02/(8ω1 )2e2ρ10 + e2ρ20 are equal to zero. Acknowledgment Parts of this research were supported by the Ministry of Sciences and Environmental Protection of Republic of Serbia through Mathematical Institute SANU Belgrade Grant ON144002 “Theoretical and Applied Mechanics of Rigid and Solid Body. Mechanics of Materials” and Faculty of Mechanical Engineering University of Niˇs.

References 1. Hedrih K (2007) Dynamics of Coupled Systems, Nonlinear Analysis: Hybrid Systems and Applications, Elsevier (to appear). 2. Hedrih K (2007) Energy analysis in the nonlinear hybrid system containing linear and nonlinear subsystem coupled by hereditary element, Nonlinear Dynamics 51(1), 127–140. 3. Hedrih K (2007) Transversal vibrations of the axially moving sandwich belts, Archive of Applied Mechanics, Springer, http://springerlink.com/content/?k =Hedrih. 4. Hedrih K (2006) The frequency equation theorems of small oscillations of a hybrid system containing coupled discrete and continuous subsystems, Facta Universitatis, Series Mechanics, Automatic Control and Robotics 5(1), 25–41, http://facta.junis.ni.ac.yu/facta/. 5. Hedrih K (2005) Partial fractional order differential equations of transversal vibrations of creep-connected double plate systems. In: Fractional Differentiation and Its Applications, A Le Mahaute, JA Tenreiro Machado, JC Trigeassou, J Sabatier (eds.), U-Book, 289–302. 6. Hedrih K (2007) Hybrid systems and hybrid dynamics: Theory and applications, 8th HSTAM International Congress on Mechanics, N Bazwos, DL Karabalis, D Polyzos, DE Beskos, JT Katsikadelis (eds.), I, 77–86. 7. Hedrih K (2004) Discrete continuum method, Computational Mechanics, WCCM VI, APCOM’04, Beijing, China, Tsinghua University Press/Springer, 1–11 (CD-ROM).

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8. Hedrih K (2006) Modes of the Homogeneous Chain Dynamics, Signal Processing, Elsevier, 86(2006), 2678–2702. 9. Goroˇsko OA, Hedrih K (2001) Analitiˇcka dinamika (mehanika) diskretnih naslednih sistema (Analytical Dynamics (Mechanics) of Discrete Hereditary Systems), University of Niˇs, Monograph, 426. 10. Hedrih K (1999) Thermorheological hereditary pendulum, Thermal Stresses 99, JJ Skrzypek, RB Hetnarski (eds.), Cracow, 199–202. 11. Hedrih K (2001) Differential equations of two mass particles constrained with a piezo-thermorheological hereditary element, Dynamics, Proceedings of 5th International Conference, PES, 77–80. 12. Hedrih K (2003) Discrete continuum’s models and thermo-rheological elements – basic idea and tensor equations, homogeneous linear chain and plane/space material nets, Proceedings of 6th International Conference Applied Electromagnetic, PES, Stability Pact for South Eastern Europe, DAAD, 127–130 and 131–134. 13. Goroˇsko OA, Hedrih K (2007) Construction of the Lagrange’s mechanics of the hereditary systems, APM Saint Petersburg, Minisymposium IDS – Opp. Lect., 133–156. 14. Goroshko OA, Puchko NP (1997) Lagrangian equations for the multibody hereditary systems, Facta Universitatis, Series Mechanics, Automatic Control and Robotics 2(7), 209–222. 15. Stratonovich RI (1967) Topics in the Theory of Random Noise II, Gordon and Breach, New York, 289 and 294–302. 16. Raˇskovi´c D (1965) Theory of Oscillations, Nauˇcna knjiga, Beograd, 504. 17. Mitropolyskiy YuA (1964) Problems of Asympthotic Theory of Non-stationary Vibrations, Nauka, Moskva, 431 (in Russian). 18. Ariaratnam ST (1996) Stochastic Stability of Viscoelastic Systems under bounded Noise Excitation, IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, Kluwer, Dordrecht, 11–18.

Tensegrity as a Structural Framework in Life Sciences and Bioengineering Simona-Mariana Cretu

1 Introduction In the fifth decade BC, Leucippus and Democritus had considered matter to be formed of indivisible particles, atoms, which were of all dimensions and forms. Pythagoreans presented the universe from the point of view of mathematics, asserting that everything is made up of numbers. Plato believes, like Empedocles, that matter is a combination of the four fundamental elements: fire, air, water and earth. In his book Timaeus (c. 360 BC), he makes known a new theory, equating the tetrahedron with the element fire, the octahedron with air, the icosahedron with water, the cube with earth and the dodecahedron with the stuff of which the constellations and heavens were made [30]. After Darwin the Platonic conception of forms was abandoned until the early 1970s, when researchers in protein chemistry suggested that the protein folds found in nature could be grouped into a finite number of distinct structural families. Denton put forward the hypothesis that they are made of Platonic forms [5]. Snelson [32] realized the first prestressed pin-jointed stable sculpture with struts and continuous tensions, which he named “Floating compression” [32]. Fuller coined the term tensegrity (a contraction of “tension” and “integrity”) to define this new family of structures. The first structures were made, being based on regular polyhedra. Tensegrities were applied to living forms, from micro to macro scale.

S.-M. Cretu University of Craiova, Faculty of Mechanics, 107 Calea Bucuresti street, 200512 Craiova, Romania, e-mail: [email protected]

J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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2 Regular Polyhedra and Their Applications to Life Sciences and Bioengineering For the first time the polyhedra with congruent regular polygonal faces were called regular polyhedra in Timaeus by Plato. Their vertices are on a sphere and the dihedral angles are equal. There are five regular polyhedra, also called Platonic solids or cosmic bodies: the tetrahedron (Fig. 1a), the octahedron (Fig. 1b), the icosahedron (Fig. 1c), the cube (Fig. 1d) and the dodecahedron (Fig. 1e). In Plato’s Timaeus these primary bodies are formed of triangles – the smallest part of Plato’s system. All triangles are made up of one right and two acute angles; they can be isosceles or scalene; the last-mentioned have their hypotenuses twice the lesser sides and they were used to compose the tetrahedron (Fig. 2, [8]), the octahedron and the icosahedron. Plato also used the isosceles right-angled triangles to compose the cube (Fig. 3, [8]).

a

b

Fig. 1 The regular polyhedra

Fig. 2 Regular tetrahedra: corpuscles of fire

Fig. 3 Cubes: corpuscles of earth

c

d

e

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Fig. 4 Regular jointed tetrahedra

In the case when there are joints at the vertices, only fully triangulated, regular polygon trusses are rigid. These are considered to be the most suitable structures for modelling anatomy. Figure 4a illustrates a jointed regular tetrahedron which is flexible (see also Fig. 4b) because the chain of one face has more than three elements and Fig. 4c illustrates a rigid jointed regular tetrahedron completely triangulated. Plato presented the transformation of fire, air and water into one another; earth could not be transformed into any other element because it is composed of one triangle different from the triangles of the rest of the regular polyhedra [30]. The “Jitterbug” is the name given by Fuller in [9] for the geometrical transformations of the flexible cuboctahedron. The cuboctahedron is the polyhedron obtained by segmenting the 12 edges of the cube into two and then by cutting off the 8 vertices. By pushing two opposite triangles of the cuboctahedron this is transformed into an icosahedron, after this into an octahedron (at this stage the model simulates chemical double bonding of atoms) and finally it may be collapsed and folded into the regular tetrahedron (quadrivalent tetrahedral structure described by organic chemists) [18]. Ingber proposed a tensegrity model of the cell undergoing “stress fiber” formation by using the “Jitterbug” [15]. It is constructed from plastic soda straws interconnected by a central filament of elastic thread. The theory of expandable mechanisms was developed in [17, 33, 34, 37–39]. Studying the mechanism of expansion of some viruses under the effect of a pH change, different expandable mechanical models, some of them regular polyhedra and the others irregular, were proposed by scientists in [1, 10, 19]. In nature, crystals are arranged as in the Platonic solids. The hydrocarbonats were also synthesized into tetrahedral, cubical and dodecahedral structures. In 1996 Curl, Kroto and Smalley 1996 Nobel Prize in Chemistry discovered the 60 atoms of carbon grouped together in the vertices of one truncated icosahedron, called the buckminsterfullerene structure [16].

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3 Tensegrity and Polyhedra The tensegrity structure is a prestressed (internal stress prior to application of external force) stable closed structural system, composed of compression struts within a network of tensile elements. In the definition of a class I tensegrity structure the struts do not touch one another but in a class II tensegrity structure some struts may have a common vertex. The loads can be applied on them in any direction, but the elements remain only under tension or compressed. Local increased tension is reflected throughout the structure. This kind of structure was first explored by Snelson, and has been the subject of many studies since. The early tensegrities of Snelson, Fuller and Emmerich used mainly regular polyhedra as the basis for finding new shapes. But the shape of the tensegrity corresponding to a particular polyhedron is not identical to that of the polyhedron. The T-tetrahedron, called the ziz-zag tetrahedron or truncated tetrahedron, was first exhibited by Francesco della Sala at the University of Michigan (1952). Its tensegrity structure has six struts and four tendon triangles. Snelson also modeled a version of the truncated tetrahedron with six struts [32]. To realise a tensegrity model “V expander” Raducanu and Motro utilised a tetrahedron with two bending struts [37]. It can be obtained by transforming the tetrahedron model from Fig. 4c; the angles between two opposite apothems will be modified from 60◦ to 90◦ and the lengths of the elements will be adjusted, as well. Burkhardt and Hunter present the calculus and the construction respectively of a tensegrity tetrahedron [3, 11]. They used two bending struts and seven tendons with different lengths, one of them being the central tendon. The tetrahedral radii constitute the struts of the structure. The units for Fuller’s mast at the Museum of Modern Art exhibition (1959) were constructed by Sadao and Price in the same manner [9]. A version of the two-stage X – a module column model with some changed tendon lengths – was computed by Burkhardt in [3]; its vertices match a regular tetrahedron and one end of the struts touches the face of the tetrahedron [3]. Some classes of tensegrities are chiral; they come in left and right-handed versions of what look like rotated prisms.

4 Applications of Tensegrity Concepts to Living Forms In living forms, the fully triangulated tensegrity systems adapted to allow movement have elastic tension nets that are taut enough to maintain the shape of the model, yet have enough residual elasticity to change the lengths and to increase or decrease the mobility of the systems. When the size and weight of a tensegrity model increases, so does its prestress.

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Ingber proposed in [14] the tensegrity model for the cell, which provides the best explanation for its movement and behavior [12–15]. The cytoskeleton, the extracellular matrix (ECM) and cell adhesion are of major importance for the study of morphogenesis at cellular level. The cytoskeleton is a complex network of three different types of molecular protein polymers, known especially as: intermediate filaments, microtubules, and microfilaments. The intermediate filaments form a meshwork along the inner membrane of the nuclear envelope. The microtubules establish the positions of different organelles (i.e. for the eukaryotic animal cell: the nucleus, the endoplasmic reticulum, the Golgi apparatus, the mitochondria and the lysosomes). The microfilaments determine the shape of the cell [6]. In the cellular tensegrity model the microfilaments and the intermediate filaments bear the tensional forces and the microtubules bear the compression forces. The cell is embedded in the ECM. The intracellular cytoskeleton interconnects with the ECM and with neighboring cells through focal adhesion complexes and specialized junctional complexes respectively. On these are preferentially focused mechanical forces. Ingber presented nucleated tensegrity models for the cell using tensegrity icosahedrons. Between the smaller tensegrity icosahedron – the nucleus, and the larger one – the cell, he stretched elastic strings to mimic cytoskeletal connections. The model of a six-strut tensegrity system was utilized by Mudana to simulate the red blood cells and the rod cells (photoreceptor cells found in the retina of the eye) [26]. Scientists came to the conclusion that the central nervous system behaves like a tensegrity system, too [16]. Levin coined the term biotensegrity. He introduced the tensegrity structure to model the torso, the spine, the shoulder, the scapula, the arm and the pelvis [20–24]. At each stage of development the evolving structure exist with the least amount of energy expenditure and the tensegrities are low energy requiring structures. Muscles and all other soft tissue elements in the human body are always under tension and bones are compressed. The tensegrity icosahedron is the most suitable for biological musculoskeletal modelling. Icosahedral tensegrity structures are hierarchical and evolutionary. Levin considers the shoulder part of an integrated mechanical system with the tensegrity icosahedron as its finite element. Flemons realized tensegrity models which are abstractions of the body designed for a close match in form and function: an expanded octahedron tensegrity which contracts and expands in the same way the torso does, some stellated tetrahedron tensegrities which model the spine, the tensegrity cushions that model the intervertebral discs which act as couplers, a modified expanded octahedron to model the pelvis and the knee saddle joint [7]. Flemons also proposed chiral tensegrities known as T-prisms which are single layer and can be clockwise or counterclockwise. They are threefold T-prisms or fourfold T-prisms and describe the forearm, or the weight transfer from the leg to the foot respectively.

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C A

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Fig. 5 Tensegrity T prism

5 Form-Finding Analysis An important step in the design of tensegrity structures is form-finding analysis, meaning: to determine a connectivity pattern and to determine the lengths of the elements. Some methods which determine the lengths of the struts and tensile elements use the non-linear programming, the relaxation process and the calculation of force density. Other methods for regular structures address both connectivity pattern and determining the lengths of the elements. The methods for form finding are classified into two categories: kinematic methods and static methods. A review of these methods can be found in [35]. Recently, evolutionary computation has been applied to evolve the connectivity pattern and parameter values of a tensegrity structure [27]. We utilise below the kinematic method for form-finding analysis [2] of a tensegrity threefold prism (Figs. 5b–d). It has three struts, each of them of length s and equal to 250 mm (the average length for the ulna bone). We use some elastic elements to tie the three struts together in this way: we connect one end of a strut to one end of the other strut and we obtain an equilateral triangle of elastic elements (DEF) with the length of the side l, equal to 47 mm (the distance between each pair of struts). The opposite ends of the struts are tied together in a similar manner and we obtain another equilateral triangle (ABC) with the same side, l. These two sets of tensile elements form the two bases of a triangular prism whose side edges are marked out by the struts: AD, BE, CF (Fig. 5a). One triangular base of the prism rests fixed (DEF) and the second base of the prism (ABC) is twisted relative to the other; the rectangular sides of the prism become non-planar quadrilaterals and each of them has two opposite obtuse angles and two opposite acute angles. We tie the vertices of each quadrilateral corresponding to the two obtuse angles using an elastic element (AE, BF, CD) which has the minimum length equal to the minimum diagonal (which changes its length when we twist the upper base relative to the lower one) and so we obtain a stable structure, a tensegrity T prism (Fig. 5b).

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r

2p/

Dh

3

a

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A h

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r O G

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Fig. 6 The rotation of the triangular prism

We’ll calculate the minimum-length tensile element, t, for the chosen tensegrity model, the T prism (Fig. 6). The radius of the circle circumscribed to the equilateral triangle of the base, r, is obtained as follows: l 47 r= = 27.167. (1) π = 2 sin 3 2 sin π3 From the right-angled triangle ADG: s2 = 2r2 − 2r2 cos α + h2,

(2)

where α is the angle of the twist of the upper base relative to the lower one. From the right-angled triangle AGE: t 2 = 2r2 − 2r2 cos(

2π − α ) + h2 . 3

(3)

From Eqs. (2) and (3) we obtain: t 2 = s2 + 2r2 cos α − 2 cos(

2π − α ). 3

(4)

In order to obtain the minimum-length elastic element, we derive with respect to α Eq. (4) and equating the result to 0 we find its solution for α = 5π/6. Substitution of the values 250 for s and 27.167 for r into the Eq. (4) yields: t = 244.8 mm. The T prism can be clockwise or counterclockwise, and can model the right or the left forearm respectively. The tensile elements must be taut enough to maintain the model in this shape and have enough residual elasticity to change their dimensions for the movement of the forearm-model.

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6 Transformations of the Tensegrity Systems A tensegrity system can be modified by reducing or increasing the lengths of some struts or tensile elements and by changing their relative positions. In the variable-geometry structures the lengths of the elements are changed continuously in order to pass from one configuration to another. Furuya, Hanaor, Oppenheim, Williams, Bouderbala, Motro, Sultan, Skelton, Tibert, Aldrich et al., Defossez, El Smaili et al., Fest et al., Schenk et al. analysed and designed tensegrity structures with these characteristics. A tensegrity structure can be actuated by three methods: strut-collocated (the actuators alter the strut lengths), cable-collocated (the effective rest length of the cables are changed) and noncollocated actuation (actuation is applied between two struts, two cables or a strut and a cable). Paul, Lipson and Cuevas simulated two tensegrity robots for locomotion (one based on a triangular tensegrity prism and the other one based on a quadrilateral tensegrity prism) by changing the length of the cables. The evolutionary optimization was used to obtain periodic gait controllers for forward locomotion of these robots [28, 29]. Micheletti and Williams propose an algorithm to modify a given stable tensegrity configuration by altering the length of a given edge. They obtain another one and calculate the change in length of the other edges [25]. Referring to living forms, Muddana’s simulations demonstrated how filopodia, erythrocytes and rod cells acquire their shapes [26]. These were modeled using the tensegrity architecture of the cell, i.e. a six-strut tensegrity system – icosahedron tensegrity, by increasing or reducing the lengths of some struts. In the six-strut tensegrity system the natural length of one strut was increased beyond that of other struts to form filopodia. To simulate red blood cells Muddana increases the natural length of a parallel pair of struts and so the six-strut tensegrity system is transformed into a biconcave disk. To obtain the ellipsoid shape he decreases the natural length of a parallel pair of struts in the initial system. The cylinder shape is obtained if both these changes occur simultaneously. The idea advanced in this study is the question whether tensegrity structures may be obtained from the primary bodies formed of triangles, presented in Pato’s Timaeus. At the beginning we analyze whether we can obtain a tensegrity prism model from a triangulated tetrahedron model. We’ll also try to obtain a tensegrity system that better approximates the human forearm by transforming the tensegrity prism into a tensegrity structure such as the letter X. We consider a triangulated tetrahedron such as in Fig. 4c. In this system we place three struts which do not touch one another: AE, BD and CF, the other segments being elastic elements (Fig. 7a). If we modify the lengths of the elastic element we transform the regular tetrahedron into a tensegrity triangular prism, with each base as a scalene triangle formed by elastic elements, ECD and BAF (Figs. 7b, c).

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F

F

A D D

B B

C

E

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E

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c

Fig. 7 A regular polyhedron and a tensegrity prism A

D

B E

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Fig. 8 A tensegrity system such as the letter X

However, in humans, the forearm contains two long bones, the radius and the ulna. The ulna bone is longer than the radius. The radius can rotate over the ulna, when the forearm twists. The extremities of the two bones are connected together by ligaments, and the bones are also connected by the oblique cord and the interosseous membrane. The fibers of oblique cord run in opposite direction to those of interosseous membrane extending downwards and sideways, from the lateral side of the tubercle of the ulna to the radius [39]. We diminish the length of one edge for each base of the system from Fig. 7b, EC and AF. So we obtain two segments instead of two bases, ED ≡ DC and BF ≡ BA, such as the oblique cord and the interosseous membrane respectively. We adjust the other two, BE and FD (such as the distance between the ulna and the radius bone at the proximal and the distal end of the forearm respectively). We obtain a tensegrity system such as the letter X (Figs. 8a, b), with two congruent struts (AE ≡ FC) and the third (BD) bigger than the other two (such as the radius bone and the ulna bone, one shorter than the other).

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7 Conclusions This paper underscores the influence of the tensegrity theory especially in the study of living forms and some tensegrities obtained from platonic polyhedra. It presents the calculus for the form-finding of a tensegrity system that approximates the human forearm – a tensegrity threefold prism – and one model that better approximates the forearm, being obtained by the transformation of a platonic tetrahedron. At the beginning we obtained a tensegrity prism model from a triangulated tetrahedron model and finally a tensegrity system such as the letter X. The new idea that I put forward in this study is the possibility that tensegrities may be obtained from the primary triangulated bodies presented by Plato in his book Timaeus. The examples analyzed so far support this idea. Acknowledgement I thank my brother, Augustin Cretu, for his substantial contribution to the translation of the papers concerning this subject.

References 1. Baker JE, Tarnai T (2004) On modelling an expandable virus. In: Huang T (ed) 11th World Congress in Mechanism and Machine Science, China Machine Press Tianjin 3, 1295–1299. 2. Burkhardt RW Jr (2008) A practical guide to tensegrity design, http://bobwb.Tripod.com. 3. Burkhardt B (2007) Tensegrity tetrahedron, http://members.tripod.com/bobwb/synergetics/ photos/datasheets.html#tetra 4spec. 4. UH RF, Kroto HW, Smalley RE Press Release: The 1996 Nobel Prize in chemistry, nobelprize.org/no bel prizes/chemistry/laureates/1996/press.html. 5. Denton M, Marshall Etkin B (2001) Laws of form revised. Nature 410, 417. 6. Farr G (2002) The human cell, http://www.becomehealthynow.com/ebookrint. php?id = 709. 7. Flemons TE (2006) The geometry of anatomy – the bones of tensegrity, http://www. intensiondesigns.com/itd-biotensegrity/biotensegrity/papers/geometry of anatomy.html. 8. Friedlander P (1958) Plato: An Introduction. Harper & Row, New York, http://faculty. washington.edu/smcohen/320/timaeus.htm. 9. Fuller B (1961) Tensegrity. Portofolio and Art News Annual 4, 112, 127, 144, 148, http://www. rwgrayprojects.com/rbfnotes/fpapers/tensegrity/teneg01.html 10. Guest SM, Kovacs F, Tarnai T, Fowler PW (2004) Construction of a mechanical model for the expansion of a virus, http://www.2.2ng.cam.ac.uk/-sdg/pre print/virusmodel.pdf. 11. Hunter S (2005) Posting by Spencer Hunter, bit.listserv.geodesic, http://members.tripod.com/ bobwb/synergetics/photos/spencer.html. 12. Ingber DE (2003) Tensegrity I. Journal of Cell Science 116, 1157–1173. 13. Ingber DE (2003) Tensegrity II. Journal of Cell Science 116, 1397–1408. 14. Ingber DE (1998) The architecture of the life, Scientific American Magazine 278, 48–57. 15. Ingber DE (1993) Cellular tensegrity: defining new rules of biological design that govern the cytoskeleton. Journal of Cell Science 104, 613–627. 16. Jauregui VG (2004) Tensegrity structures and their application to architecture. Thesis for M.Sc. in Arhitecture, School of Architecture, Queen’s University, Belfast, http://www. alumnos.unican.es/uc1279/Tensegrity Structures.html. 17. Jensen F, Pellegrino S (2005) Expandable “Bob” structures. Journal of the International Association for Shell and Spatial Structures 46(3), 151–159.

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18. Judge A (1980) Vector equilibrium and its transformation pathways, http://laetusinpraesens. org. 19. Kovacs BF, Tarnai T, Guest SD, Fowler PW (2004) Double-link expandohedra: a mechanical model for expansion of a virus, http://biophysics.asu.edu/banf files/guest/double.pdf. 20. Levin SM (2007) Hang in there! The statics and dynamics of pelvic mechanics. http://www. biotensegrity.com/. 21. Levin SM (2002) The tensegrity-truss as a model for spine mechanics: biotensegrity, http://www.biotensegrity.com/. 22. Levin SM (1997) Putting the shoulder to the wheel: a new biomechanical model for the shoulder girdle, http://www.biotensegrity.com/. 23. Levin SM (1995) The importance of soft tissues for structural support of the body, http://www. biotensegrity.com/. 24. Levin SM (1980) Continuous tension, discontinuous compression: a model for biomechanical support of the body, http://www.biotensegrity.com/. 25. Micheletti A, Williams W (2007) A marching procedure for form-finding for tensegrity structures. Journal of Mechanics of Materials and Structures 5, 857 882, http://www.math.cmu.edu/ users/wow/papers/marching.pdf. 26. Muddana HS (2006) Integrated biomechanical model of cells embedded in extracellular matrix. Thesis for Master of Science, Texas A&M University, http://research.cs.tamu.edu/bnl/ papers/muddana.thesis06.pdf. 27. Paul C, Lipson H, Valero-Cuevas FJ (2005) Evolutionary form-finding of tensegrity structures. In: The 2005 Conference on Genetic and Evolutionary Computation, http://ccsl.mae.cornell. edu/papers/GECCO05 Paul.pdf. 28. Paul C, Roberts J, Lipson H, Valero-Cuevas FJ (2005) Gait production in a tensegrity based robot. In: The 2005 International Conference on Advanced Robotics, http://ccsl.mae. cornell.edu/papers/ICAR05 Paul.pdf. 29. Paul C, Valero-Cuevas FJ, Lipson H (2006) Design and control of tensegrity robots for locomotion. IEEE Transactions on Robotics 22(5), 944–957. 30. Plato (360 BC) Timaeus, http://www.ellopos.net. 31. Raducanu V, Motro R (2001) Patent 2 823 287, Stable self-balancing system for building component. 32. Snelson K (1948) http://www.kennethsnelson. 33. Tibert AG (2002) Deployable tensegrity structures for space applications. Ph.D. dissertation, Royal Institute of Technology, Stockholm, Sweden. 34. Tibert AG, Pellegrino S (2003) Deployable tensegrity masts, http://www.2.mech.kth.se/gunnar/AIAA-2003-1978.pdf. 35. Tibert AG, Pellegrino S (2003) Review of form-finding methods for tensegrity structures. International Journal of Space Structures 18(4), 209–223. 36. Wesley N (1999) The anatomy lesson, http://home.comcast.net. 37. Wohlhart K (2007) Cupola linkages. In: Merlet JP, Dahan M (eds) 12th World Congress in Mechanism and Machine Science I, 319–324. 38. Wohlhart K (2005) Double pyramidal linkages. In SYROM I, 293–300. 39. Wohlhart K (2001) New regular polyhedral linkages. In SYROM II, 365–370.

Study of Nonlinear Dynamics in a Buck Converter Controlled by Lateral PWM and ZAD Fabiola Angulo, Jorge E. Burgos, and Gerard Olivar

1 Introduction Power Electronics is an area that stands out today for its growth and applicability in industry. This area regards all about conversion, shapes and voltage levels in the handling of systems that require high precision in the signals of energy transmission. To achieve this conversion efficiently, it is necessary to use semiconductors in a switching process together with other passive components as inductors and capacitors [1]. On the other hand, power converters show a variety of nonlinear phenomena like periodic orbits, bifurcations and chaos. The converter switching components and the use of control techniques such as the Pulse Width Modulation (PWM) make the system nonlinear and nonsmooth. A previous work about these topics can be found in the paper reported by Hamill [2], somewhere the first studies about the existence of bifurcations and chaotic behavior in buck converters controlled by PWM were done. We can also mention about works of Chakrabarty [3], di Bernardo [4], Fang [5] or Olivar [6]. Many control techniques have been designed which use a sliding surface defined as a linear combination of the error and its derivative. This combination is forced to have zero average for each sampling period. This technique is known as ZAD. We can mention the works of Angulo [7], Fossas [8] useful and Ramos [9] treating about this technique. Fabiola Angulo Universidad Nacional de Colombia, Sede Manizales, Campus La Nubia, Manizales, Colombia, e-mail: [email protected] Jorge E. Burgos Universidad Nacional de Colombia, Sede Manizales, Campus La Nubia, Manizales, Colombia, e-mail: [email protected] Gerard Olivar Universidad Nacional de Colombia, Sede Manizales, Campus La Nubia, Manizales, Colombia, e-mail: [email protected] J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

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In this paper some results related to the presence of nonlinear phenomena in a buck power converter controlled by a Lateral Pulse-Width Modulation (LPWM) and Zero Average Dynamics (ZAD) techniques are presented. The computation of the duty cycle by linear approximation of the sliding surface and the analysis of existing nonlinear phenomena in the converter are the main items of this work. Supplementary, we confirm the existence of periodic orbits, bifurcations and chaos in the analysed electric circuit. Section 2 include a buck converter modeling controlled by LPWM and ZAD techniques. In Section 3 the duty cycle for the control law is calculated. In Section 4 the nonlinear dynamics of the buck converter is analyzed with the control law {−1, 1}.

2 Modeling the DC-DC Buck Converter with LPWM and ZAD Fig. 1 shows a schematic diagram of the buck converter. For the state-space representation of the system, we consider the capacitor voltage v and the inductor current i as state variables. This system can work as a DC-DC or DC-AC converter depending on the reference signal. In our case, the reference value Vre f is constant, and thus the regulation problem will be studied. The state-space modeling of the system is described as follows: 1 1 0 v˙ − RC C v + E u, = (1) i i˙ − L1 0 L where the dot over the state variables means d/d τ and u is the control variable which regards to the switching action between positions 1 and 2 visible in Fig. 1. For a representation of the converter as a dimensionless system, it is necessary to perform the following change of variables and time: ! τ v 1 L x1 = , x2 = i, t = √ . (2) E E C LC

+ −

E 1 2

+ −

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Fig. 1 Scheme of the buck converter

L

i C

+ −

V

R

Study of Nonlinear Dynamics in a Buck Converter

Then, the new state-space modeling can be rewritten: x1 x˙1 −γ 1 0 u. = + 1 −1 0 x˙2 x2

315

(3)

The voltage v and current i variables are now related to x1 and x2 , respectively.

The matrix representation (3) is called Normalized Modeling, where γ = R1 CL , and in compact form X˙ = AX + Bu. √ The sampling period T = Tc / LC, where Tc = 50μ s. It is desired to control this system by means of a lateral pulse width modulator (LPWM) which at each interval of length T must guarantee a zero average value for a function s(x) defined in Eq. 15. The control signal u can take the discrete two values −1 or +1 being defined by: −1 if kT ≤ t ≤ kT + dk , u= , (4) 1 if kT + dk ≤ t ≤ (k + 1)T where (the duty cycle) dk varies between cycles. We define s(x) (known as sliding surface) [7, 9] by s(x) = (x1 − x1re f ) + ks (x˙1 − x˙1re f ),

(5)

where x1 is the controlled variable, x1re f is the reference signal (x1re f = Vre f /E where Vre f is the desired output voltage in the dimensional model) and ks is the surface time constant. Also, x˙1re f = 0 since we study the regulation problem. For a linear and time invariant system subject to any unitary pulses, the general solution of the system can be expressed by the Poincar´e map, which relates one sample state variable at time (k + 1)T to the previous value at time kT . After some algebra, and taking into account Eq. (4), we get

x((k + 1)T ) = eAT x(kT ) − eAT − 2eA(T −dk ) + I A−1 B. (6) One computes the duty cycle (the time while the control variable is −1) by a procedure associated with the design of the control strategy applied to the system. The objective is to guarantee the existence of a duty cycle that assures a zero average dymanics in the sliding surface s(x) at each sampling period. This restriction is known as the ZAD technique allowing calculation of the duty cycle from the following equation [7]: (k+1)T

s(x(t))dt = 0.

(7)

kT

3 Computation of the Duty Cycle The duty cycle can be found from Eq. (7) where a zero average of the sliding surface for each sampling period is obtained. This allows the controlled variable to follow an established reference value. The exact solution of this equation is an inconvenience

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if a physical implementation must be realised. Then, it is necessary to use other simpler techniques for the duty cycle calculation. One of the efficient approaches takes into consideration a piecewise-linear approximation of the sliding surface. In this case, the following is assigned [7]: 1. The sliding surface behaves like a piecewise-linear function. 2. The sliding surface slopes are given by the slopes calculated at the beginning of each sampling interval. According to the above, we obtain the following expression for computing the integral of the sliding surface for each sampling period (0, T ), T

s(x(t))dt ∼ =

0

d 0

(s0 + t s˙2 ) dt +

T

((s0 + s˙2 d) + (t − d)s˙1 ) dt

(8)

d

where s0 is the value of the surface at the sampling instant, s˙1 is the surface slope when the signal control is 1 and s˙2 is the slope when the signal control is −1. Considering (3) and (5), the values of s0 , s˙1 and s˙2 , based on some parameters of the system, can be computed as follows: s0 = (1 − ksγ )x1 (0) + ks x2 (0) − x1re f

(9)

s˙1 = (1 − ksγ )(−γ x1 (0) + x2(0)) + ks (−x1 (0) + 1) s˙2 = (1 − ksγ )(−γ x1 (0) + x2(0)) + ks (−x1 (0) − 1) Solving Eq. (7) with the approximation in Eq. (8), we obtain the duty cycle d : ⎛ ⎞ s ˙ + 2s /T 2 0 ⎠T d = ⎝1 − (10) s˙2 − s˙1 To guarantee a duty cycle between 0 and T, it is necessary to take into account the following considerations: 1. Root existence. 2. The root must be less than or equal to 1 to avoid negative duty cycles. With these considerations, Table 1 describes the possible duty cycle values and the conditions under which they are obtained. Table 1 Duty cycle Duty cycle Conditions 0 /T 1 − s˙2 s+2s T s0 + T2 s˙2 ≤ 0 and s0 + T2 s˙1 ≥ 0 ˙2 −s˙1 0 T

s0 + T2 s˙2 ≤ 0 and s0 + T2 s˙1 < 0 s0 + T2 s˙2 > 0

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4 Bifurcational Analysis In this section the nonlinear dynamics of the converter, when the control law is u = {−1, 1} is studied analytically and numerically as well. The complementary case {1, −1} can be analysed also with the same tools, but since the obtained results are radically different, they will be shown in another paper. Phenomena like periodic orbits, bifurcations and chaos are present. The stability of the system will be also analyzed by computing the characteristic multipliers. Bifurcations occur when a change of the system dynamics is present due to the variation of a parameter (for example, the input voltage or the load). An appropriate way to perform the study of the bifurcations is by the use of bifurcation diagrams. Figure 2 shows the bifurcation diagrams for the voltage, current and duty cycle respectively. The reference value is 0.8. For obtaining these diagrams, the following values were assigned to the converter parameters: E = 40V , L = 2mH, C = 40μ F, and Tc = 50 μ s, which yield γ = 0.3536 and T = 0.1767 [7]. In our case, the bifurcation parameter is the surface time constant described in Eq. (5). From an approximate value ks = 0.183, the system shows a 1-Periodic orbit. Within the range ks < 0.183, we can see a period-doubling bifurcation, and at the instant when the saturation of one of the duty cycles occurs through a discontinuityinduced bifurcation, the system behaves in a chaotic way. Based on these results, we analyze in detail the existence of bifurcations in the system. Accordingly to Fig. 2, we can observe the effectiveness of this technique for the regulation problem. The system shows a good regulation for values ks > 0.183. For example, for ks = 0.2, the voltage is 0.7989, (and thus the regulation error is 0.1603%).

Bifurcation Diagram - Reference = 0.8

Bifurcations Diagram - Reference = 0.8

0.82 0.8

Bifurcation Diagram - Reference = 0.8

0.45

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0.1 0.08

0.02

0.05 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 Bifurcation Parameter- ks

0 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 Bifurcation Parameter- ks

Fig. 2 Bifurcation diagrams. Reference = 0.8. Voltage, current and duty cycle vs ks , respectively

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4.1 Period-Doubling Bifurcation In this case, for a value of the bifurcation parameter, there is observed a change from one- to two-periodic stable orbit. Next we analyze the existence and stability of these two kinds of orbits.

4.1.1 1-periodic Orbits A 1-periodic orbit exists when x(T ) = x(0), that means, the state conditions at the end of the first sampling period are the same conditions at the beginning of that period. To obtain an expression that allows to evaluate the existence of these orbits, we start from the solution of the system expressed by Eq. (6). The following mathematical procedure consists on a matrix representation with which it is possible to obtain the initial conditions when the 1-periodic orbit exists . We have the following variable assignment [6]: ! γ 2 −1 −1 A B= = −C, ω = 1− . −γ 2 Considering the value of ω from Eq. (11), the state transition matrix can be expressed as: γ 1 sin ω t At − 2γ t cos ω t + 2ω sin ω t ω . (11) e =e cos ω t − 2γω sin ω t − ω1 sin ω t Splitting the sine and cosine components from equation (11) we have: γ 1 γ γ At −2t 2 ω ω e =e I cos ω t + sin ω t = e− 2 t [I cos ω t + M sin ω t] , − ω1 − 2γω

(12)

where I is the identity matrix and M is the matrix of constant terms of the sine function component. Now we proceed with the normalization of the duty cycle from 0 to 1 (previously the duty cycle was evaluated within the range [0, T ]). The normalized duty cycle is assigned with the variable d.ˆ Its value is given by dˆ = d/T , with ˆ ). Also, replacing α = t/T in Eq. (12), we obtain the d ∈ (0, T ), (and thus d = dT matrix N(α ), which represents the state transition matrix based on the normalized time α : γ eAt = N(α ) = e− 2 T α [I cos ω T α + M sin ω T α ] . (13) Therefore, the system solution for the cases u = −1 and u = +1 based on N(α ) is defined as follows: • Control signal u = −1: • Control signal u = 1:

then x(α ) = N(α )(x(0)) − [I − N(α )]C, then x(α ) = N(α )(x(0)) + [I − N(α )]C.

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Now with the knowledge of these expressions we find the one that indicates the system solution when the first sampling period x(1) occurs:

ˆ = N(d)(x(0)) ˆ ˆ C, • Interval (u = −1): then x(d) − I − N(d)

ˆ d) ˆ + I − N(1 − d) ˆ C. • Interval (u = 1): then x(1) = N(1 − d)x( ˆ we obtain: Replacing the value of x(d)

ˆ N(d)(x(0)) ˆ ˆ C + I − N(1 − d) ˆ C. x(1) = N(1 − d) − I − N(d) According to the property N(α )N(β ) = N(α + β ), this expression is reduced to

ˆ + N(1) C. x(1) = N(1)(x(0)) + I − 2N(1 − d) (14) The normalized duty cycle dˆ is equal to: dˆ = 1 −

s˙2 + 2s0 /T s˙2 − s˙1

(15)

and based on the initial conditions of the system, it can be expressed as dˆ = 1 − c1 x1 (0) + c2 x2 (0) + c3 , where c1 , c2 and c3 are: 1 2 2 −γ + ks γ − ks + (1 − ksγ ) , (16) c1 = − 2ks T 1 2 1 − k s γ + ks , c2 = − 2ks T 1 2 c3 = − −ks − x1re f . 2ks T Considering the system solution described by (14) and knowing that a 1-periodic orbit exists when x(1) = x(0), we have the following expression:

ˆ + N(1) C x(0) = [I − N(1)]−1 I − 2N(1 − d) (17) With Eq. (17) and since the duty cycle dˆ was computed according to Eq. (15), it is possible to obtain the initial conditions of the state variables (the voltage and current) when a 1-periodic orbit exists. Fig. 3 shows the evolution of the 1-periodic orbit for ks = 0.184. From point A to B, there is a duty cycle dˆ = 0.10083, and the signal control is −1. At point B the switching occurs and now the signal control is 1 and then the orbit goes again to point A. The initial conditions of voltage and current are 0.7982 and 0.2983, respectively. Now we analyze the stability of this orbit by using the characteristic multipliers method. The main idea is to determine the range of values of the bifurcation parameter where the orbit is stable. The characteristic multipliers method consists of evaluating the jacobian of the Poincar´e map (discrete representation of the system) on the equilibrium point and then the eigenvalues (characteristic multipliers)

320

F. Angulo et al. 1-Periodic orbit, ks = 0.184 0.305 0.3

C1=(0.79872 V.0.29828|

A 0.295

Current

0.29 0.285 0.28 0.275 0.27 B 0.265 0.798 0.7981 0.7982 0.7983 0.7984 0.7985 0.7986 0.7987 0.7988 0.7989 Voltage

Fig. 3 1-periodic orbit. ks =0.184

are calculated [6, 7, 10]. We have ˆ

∂ x(1) ˆ C ∂d , = N(1) + 2N (1 − d) ∂ x(0) ∂ x(0)

(18)

ˆ ∂ x(0) is the partial derivative of the duty cycle with regards to the initial where ∂ d/ conditions. These partial derivatives are expressed as: 2 3 2 3 ∂ dˆ ∂ dˆ ∂ dˆ 1 = , = (−c1 , −c2 ) (19) ∂ x(0) ∂ x1 (0) ∂ x2 (0) 2 c1 x1 (0) + c2x2 (0) + c3 and the state transition matrix derivative N (α ) is: γ

N (α ) = e− 2 T α [M1 cos ω T α + M2 sin ω T α ] ,

(20)

where M1 = −(γ /2)T I + M ω T and M2 = −(γ /2)T M + ω T I. Since we have a second order system the number of characteristic multipliers to analyze is two. The criterion used to find the value of the bifurcation parameter where the 1-periodic orbit becomes unstable is when one of the eigenvalues passes through −1 [6,10,11]. Thus, it must be fulfilled that 1 + trace(J(x)) + det(J(x)) = 0.

(21)

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Table 2 1-periodic orbit characteristic multipliers ks 0.18332384176 0.18332384177 0.18332384178 0.18332384179

α

m1

m2

0.100831 0.100831 0.100831 0.100831

0.48535188798 0.48535188799 0.48535188800 0.48535188801

−1,00000000010 −1,00000000003 −0,99999999996 −0,99999999989

Table 2 shows that for ks = 0.18332384177, the characteristic multiplier m2 passes through −1. Therefore, a period-doubling bifurcation occurs (or Flip Bifurcation). In this case, the 1-periodic orbit still exists but it is now unstable and a 2-periodic orbit with two stable fixed points appears. Fig. 4(a) shows a bifurcation diagram in which the existence of the perioddoubling bifurcation is visible in details.

4.1.2 2-periodic Orbits The last results confirm the existence of a period-doubling bifurcation. That indicates the occurrence of a 2-periodic stable orbit, with two non-saturated duty cycles dˆ1 and dˆ2 . The analysis related to the existence and stability is similar to the one done for the case of 1-periodic orbits. This time we have two duty cycles dˆ1 and dˆ2 whose values are dˆ1 = 1 ˆ − c1 x1 (0) + c2x2 (0) + c3, d2 = 1 − c1 x1 (1) + c2x2 (1) + c3, where x1 (0), x2 (0) are the initial conditions when the first sampling period occurs and x1 (1), x2 (1) are the initial conditions for the second sampling period. Therefore, a 2-periodic orbit exists when x(2) = x(0). Now we obtain a mathematical expression that allows the calculation of the initial conditions when the orbit with two non-saturated cycles exists. For that, it is necessary to evaluate the state variables at every switching. In this case, those instants are x(dˆ1 ), x(1), x(dˆ2 ) and x(2):

• first subinterval: x(dˆ1 ) = N(dˆ1 )(x0 ) − I − N(dˆ1 ) C;

• second subinterval: x(1) = N(1 − dˆ1)x(dˆ1 ) + I − N(1 − dˆ1) C;

• third subinterval: x(dˆ2 ) = N(dˆ2 )x(1) − I − N(dˆ2 ) C;

• fourth subinterval: x(2) = N(1 − dˆ2)x(dˆ2 ) + I − N(1 − dˆ2) C. Replacing x(dˆ2 ), x(1) and x(dˆ1 ) in the above, we obtain the following expression which represents the solution of the system after two sampling periods: x(2) = N(2)x(0) + [I + 2N(1) + N(2) − 2N(2 − dˆ1) − 2N(1 − dˆ2)]C,

(22)

A 2-periodic orbit occurs when x(2) = x(0); then the initial conditions x(0) for this orbit can be determined as: x(0) = [I − N(2)]−1 [I + 2N(1) + N(2) − 2N(2 − dˆ1) − 2N(1 − dˆ2)]C.

(23)

322

F. Angulo et al. Bifurcation diagram - Doubling period 0.04 0.035 0.03

Duty cycle

0.025 0.02 0.015 0.01 0.005

a

0 0.18320.18330.18330.18330.18330.18330.18330.18330.18330.18330.1833 Bifurcation parameter - ks 2-Period orbit, ks = 0.183325 0.305 C12=(0.79898 V,0.30181)

0.3

C

0.295 A

C12=(0.79847 V.0.294611)

Current

0.29 0.285 0.28 0.275 0.27 B 0.265 D

b

0.26 0.798

0.7982

0.7984

0.7986

0.7988

0.799

0.7992

Voltage

Fig. 4 (a) Bifurcation diagram. Duty cycle vs ks . Existence of doubling period bifurcation. (b) 2-periodic orbit with non saturated duty cycles. ks = 0.183325

Study of Nonlinear Dynamics in a Buck Converter

323

Studying the stability we evaluate the Jacobian of the Poincar´e map, which consists on the partial derivative of the system solution x(2), that was obtained in Eq. (22) with regards to the initial conditions x(0): ∂ x(2)/∂ x(0). Therefore,

∂ dˆ1 ∂ dˆ2 ∂ x(2) = N(2) + 2N (2 − dˆ1) C + 2N (2 − dˆ1) C , ∂ x(0) ∂ x(0) ∂ x(0) where ∂ dˆ1 /∂ x(0) is calculated as before, while ∂ dˆ2 /∂ x(0) is equal to: 2 3 ∂ dˆ2 ∂ x1 (1) ∂ x2 (1) 1 =− + c2 , c1 ∂ x(0) ∂ x(0) ∂ x(0) 2 c1 x1 (1) + c2x2 (1) + c3

(24)

(25)

where x(1) is given by Eq. (14) and its derivative is expressed by Eq. (18). As we can see in Fig. 4(a), we evaluate ks until one of the duty cycles is saturated (in this case, the saturation is to zero). For ks between 0.183252254 and 0.183323842 the 2-periodic orbit with non-saturated cycles exists. Fig. 4(b) shows the evolution of the 2-periodic orbit for ks = 0.183325. The starting point is indicated by A and the sequence of the orbit evolution by A-B-C (first sampling period) and C-D-A (second sampling period).

4.2 Border Collision Bifurcation and Chaos For ks = 0.183252254 the duty cycle dˆ2 is saturated to zero. Having precised the 2-periodic orbit for this value, it is necessary to perform the previous analysis for with one non-saturated cycle and one saturated cycle. The value of the duty cycle is equal to dˆ = 1 − c1 x1 (0) + c2x2 (0) + c3 and the system is evaluated at the state ˆ x(1) and x(2) : variables x(d),

ˆ = N(d)(x ˆ 0 ) − I − N(d) ˆ C; • first subinterval: x(d)

ˆ d) ˆ + I − N(1 − d) ˆ C; • second subinterval: x(1) = N(1 − d)x( • second interval: x(2) = N(1)x(1) + [I − N(1)]C. ˆ in these expressions, we have: Replacing x(1) and x(d)

ˆ + N(2) C, x(2) = N(2)x(0) + I − 2N(2 − d)

(26)

and the initial conditions x(0) are expressed as follows:

ˆ + N(2) C. x(0) = [I − N(2)]−1 + I − 2N(2 − d)

(27)

Regarding the stability, we evaluate the Jacobian of the Poincar´e map, which is the partial derivative of the system solution x(2) with regard to the initial conditions x(0): ˆ

∂ x(2) ˆ C ∂d . = N(2) + 2N (2 − d) (28) ∂ x(0) ∂ x(0)

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F. Angulo et al.

Table 3 Characteristic multipliers for a 2-periodic orbit with a saturated duty cycle ks 0.183263672 0.183252254 0.183252253 0.183252252

m1

m2

0.237938 0.238395 0.261107 0.261107

0.998350 0.998034 −2.020980 −2.020980

Table 3 presents values of the characteristic multipliers for some values of ks at which the 2-periodic orbit with one saturated duty cycle exists. The results also confirm that the characteristic multipliers m1 and m2 have a discontinuous jump in their values indicating the presence of a discontinuity-induced or border collision bifurcation due to the existence of a saturated duty cycle [10,12,13]. Besides, the characteristic multiplier m2 becomes unstable. This means that the 2periodic orbit losses its stability when the saturation occurs. At this point, the system starts its chaotic behavior.

5 Conclusions An expression for the duty cycle was obtained when the converter is controlled by the LPWM technique and the ZAD strategy. The conditions for the existence and stability for 1- and 2-periodic orbits, with and without saturation were obtained. It was possible to check the existence of a 1-periodic orbit for a wide range of the parameter ks . As the value of ks decreases, a period-doubling bifurcation occurs. A stable 1-periodic orbit becomes unstable and a stable 2-periodic orbit is born. The range of parameter values for the stability of the 2-periodic orbits is very narrow, and this means that probably, in the experiments, this behaviour will not be observed. When the saturation of one of the cycles occurs, there is observed a border collision bifurcation and the converter starts its chaotic behavior.

References 1. Banerjee S, Verghesse G (Ed) (2002) Nonlinear Phenomena in Power Electronics, IEEE Press. 2. Hamill D.C, Jefferies D.J (1988) Subharmonics and chaos in a controlled switched-mode power converter, IEEE Transaction on Circuits and Systems, Vol. 35, No. 8, pp. 1059–1061. 3. Chakrabarty K, Poddar G, Banerjee S (1996) Bifurcation behavior of the Buck converter, IEEE Transactions on Power Electronics, Vol. 11, No. 3, pp. 439–447. 4. di Bernardo M, Garofalo F, Glielmo L, Vasca F (1998) Switchings, bifurcations, and Chaos in DC/DC converters, IEEE Transactions on Circuits and Systems 1: Fundamental Theory and Applications, Vol. 45, No. 2, pp. 133–141.

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5. Fang C, Abed E.H (1998) Analysis and Control of Period Doubling Bifurcation in Buck Converters Using Harmonic Balance, Technical Research Report, Department of Electrical Engineering and the Institute for Systems Research, University of Maryland. 6. Olivar G (1997) Chaos in the Buck Converter, Ph.D. Thesis, Universidad Polit´ecnica de Catalu˜na (available at http://www.tdx.cesca.es/TDX-0921104-170716/). 7. Angulo F (2004) An´alisis de la Din´amica de Convertidores Electr´onicos de Potencia usando PWM Basado en Promediado Cero de la Din´amica de Error, Instituto de Organizaci´on y Control de Sistemas Industriales (IOC), Ph.D. Thesis, Universidad Polit´ecnica de Catalu˜na, (in Spanish) (available at http://www.tdx.cesca.es/TDX-0727104-095928/). 8. Fossas E, Biel D, Ramos R, Sudria A (2001) Programmable logic device applied to the quasisliding control implementation based on zero averaged dynamics. Proceedings of the IEEE Conference on Decision and Control, Orlando, USA, pp 1825–1830. 9. Ramos R, Biel D, Fossas E, Guinjoan F (2003) A Fixed-Frequency Quasi-Sliding Control Algorithm: Application to Power Inverters Design by Means of FPGA Implementation. IEEE Transactions on Power Electronics, Vol. 18, No. 1. 10. Zhusubaliyev Z, Mosekilde E (2003) Bifurcations and Chaos in Piecewise-smooth Dynamical Systems. World Scientific Series in Nonlinear Science, Series A, Vol. 44. 11. Wiggins S (2003) Introduction to Applied Nonlinear Dynamical Systems and Chaos. SpringerVerlag, New York. 12. Banerjee S, Jain P (2003) Border-Collision Bifurcations in One-Dimensional Discontinuous maps. International Journal of Bifurcation and Chaos, Vol. 13, No. 11, pp. 3341–3351. 13. Yuan G, Banerjee S, Ott E, Yorke J.A (1998) Border-Collision bifurcations in the Buck Converter. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 45, No. 7, pp. 707–716.

Stability of the Periodic Solutions for Quasi-linear Controlled Systems in Critical Case Mihai Popescu and Alexandru Dumitrache

1 Introduction The analysis of some dynamic phenomena leads to mathematical models represented by controlled differential system. From the command system, the major interest is represented by the controllable systems. In this hypothesis, the construction of a control, given by a linear from of state variables, might modify the eigenvalues of the linearized system [1, 2]. For all negative real parts of eigenvalues the controlled system has an asymptotic stable trivial solution. If some of the eigenvalues are pure imaginary and the others have the real part negative, the system constitutes the critical stability case [3–5]. An equivalent system will be found by application of a serial successive transformation and its solutions stability will be determined as well. The transformed system is a nonlinear one having a non-homogenous form of state variables. It will be analyzed with respect to conditions of stability of its periodical solution or the periodic solutions orbital stability [8, 9].

2 Quasi-linear Controlled Systems We will consider the controlled nonlinear system [6, 7]: ∞ dz = Az + bu + ∑ z(m) , dt m=2

(1)

M. Popescu and A. Dumitrache Romanian Academy of Science, Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie no. 13, sect. 5, 050911 Bucharest, Romania, e-mail: ima popescu@ yahoo.com, [email protected]

J. Awrejcewicz (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems. c Springer Science + Business Media B.V. 2009

327

328

M. Popescu, A. Dumitrache

where A, b are constant matrices, z is the n – dimensional state vector, z(m) are homogenous forms of the variables z1 , . . . , zn and u1 , . . . , ur , u is the command vector. The coefficients of z(m) are time-independent. Assuming that the linearized system (1) is controllable, one can construct the following control law: u = c∗ z. (2) so that the resulting matrix A¯ will not have any eigenvalues with the positive real part, which would mean the instability of the trivial solution (z = 0) of system (1), where A¯ = A + bc∗. If the characteristic equation of the matrix A¯ has k purely imaginary roots λs (s = 1, . . . , k) and the other (n − 2k) roots have negative real parts, then system (1) is transformed into the form: dxs = −λs ys + Xs (x, y, z) , dt dys = −λs xs + Ys (x, y, z) , s = 1, . . . , k dt dxs = −Pz + Z (x, y, z) , dt

(3)

where Xs and Ys have the following properties: Xs (x, y, z)|x=y=0 ≡ 0, Ys (x, y, z)|x=y=0 = 0. Solution of the system (3) is sought in the form: ∞

∑

xs = cos αs cs +

(m) τs (α1 , . . . αk ,

c 1 , . . . ck ) ,

m=2

∞

∑

ys = sin αs cs +

(m) τs (α1 , . . . αk ,

(4)

c 1 , . . . ck ) ,

(5)

m=2

z j = cos αs

∞

(m)

∑ Zj

(α1 , . . . αk , c1 , . . . ck ).

m=2 (m)

(m)

Functions τs and Zs are of rank m homogenous forms in c1 , . . . , ck with periodical coefficients related to the variables α1 , . . . , αk . The functions α1 , . . . , αk are satisfies the following set of equations: d αs = λ s + Δs . dt

(6)

that has been obtained from (3) by introduction of polar coordinates: xs = τs cos αs , ys = τs sin αs .

(7)

Stability of the Periodic Solutions for Quasi-linear Controlled Systems in Critical Case

329

The trivial solution stability of the system (1) is based on theorem 1. Theorem 1. If the system (3) is admitting a bounded solution family (5), than the trivial solution of the controlled system (1) is stable. The integral curves defined by (5) are periodical. The study of stability for the periodical solution system (1), has as a consequence, the determination of the orbital stability conditions.

3 Orbital Stability The transformed system (1) might be written as the form of the autonomous system: dx = X (x) . dt

(8)

We presume that this system admits the periodical solution x = u (t) with the period ω . Based on the unicity theorem we have X (u (t)) = 0 for all t. Denoting X (u(t)) (9) X (t) = X (u(t)) where X meets the local Lipschitz condition, the curve x = X (t) will not cover the entire unit sphere, thus there is a unit vector e1 so that X (t) + e1 = 0 for each t. Starting from the constant vector e1 there will be constructed an orthogonal and normalized system of vectors (e1 , e2 , . . . , en ). Let X (t) , ei cos θi = = X (t) , ei (10) xei and consider the vector

ξν = c ν −

cos θν e1 + X 1 + cos θν

(11)

which are periodical functions of period ω and have the same regularity property as X (t). It is easy to demonstrate that the vectors X (t) , ξ1 , ξ2 , . . . , ξn are an orthogonal normalized system. In this way, an orthogonal system of normalized vectors on X is attached to the periodical solution u (t). The orthogonal system consists of unitary vectors, which are tangent to the curve x = u (t). Using this orthogonal and normalized system the variable change x = u (θ ) + S (θ ) y is performed, where:

(12)

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M. Popescu, A. Dumitrache

S (θ ) = (ξ2 (θ ) , ξ3 (θ ) , . . . , ξn (θ )) = (Siy (θ )) , i = 1, . . . , n ∗ y = y1 , y2 , . . . , yn−1 , j = 1, . . . , n − 1 ∗ u [θ ] = u1 (θ ) , u2 (θ ) , . . . , un (θ ) .

(13)

Using this transformation and the orthonormality of S (θ ), one obtains a system of differential equations equivalent to the initial one. Therefore, the stability of the trivial solution of the transformed system determinates the stability of the periodic solution for the analyzed system., and / / / / det (S (θ ) , X [u (θ )]) = /X [u (θ )]/ det S (θ ) , X [u (θ )] = / / (14) / / = /X [u (θ )] = 0/, since det(S(θ ), X [u(θ )]) = 1 for an orthogonal and normalized system (ξ2 , . . . , ξn , X). Using this change of variables the curve x = u (t) becomes y = 0, θ = t and the system (8) is transformed into: X [u (θ )]

d θ d θ dS (θ ) dy + = X (u (θ ) + S (θ ) y) . y + S (θ ) dt dt d θ dt

(15)

By multiplying (15) toX ∗ [u (θ )], and due to the orthogonality of the columns of S (θ ) and X ∗ , the system (8) becomes: dy = Y (θ , y), dt dθ = Θ(θ , y), dt

(16)

after the transformation (12) is used, where the values of Θ(θ , y) and Y (θ , y) have the properties Y (θ , 0) = ξμ∗ (θ ) , X [u (θ )] = 0, (17) X ∗ [u (θ ) X [u (θ )]] Θ (θ , 0) = = 1. 2 X [u (θ )] The resulting system is: 1 dy Y (θ , y) , = dθ Θ (θ , y)

(18)

Y (θ ,y) where function Θ( θ ,y) is periodic in θ with the period ω and has (for a sufficient low |y|) the same regularity properties as X because Θ (θ , y) = 0. Expanding X [u (θ ) + S (θ ) y] in a Taylor series around y = 0 we have:

H [u (θ ) + S (θ ) y] = X [u (θ )] +

∂ X [u (θ )] S (θ ) y + O |y|2 . ∂x

The stability matrix of the variations system is given by:

Stability of the Periodic Solutions for Quasi-linear Controlled Systems in Critical Case

A (θ ) =

∂ X [u (θ )] . ∂x

331

(19)

By an analog Taylor expansion, one can obtain: 1 = 1 + O(|y|), Θ (θ , y)

(20)

where O (|y|) represents the terms converging to zero together with y. The system (19) becomes: dy = B (θ ) y + O(|y|), dθ where the elements b μν of the matrix B are given by: d ξν . b μ ν (θ ) = ξμ∗ (θ ) A (θ ) ξν (θ ) − dθ

(21)

(22)

The elements bμν (θ ) are periodical functions of period ω . Neglecting the term O (|y|), which is approaching zero like y does, the normal variations system (21), becomes: dy = B (θ ) y. dθ

(23)

The variations system corresponding to the periodic solution u (θ ) is: dv = A (θ ) v. dθ

(24)

Each element of the above solution space is a linear combination of the base ξν of the orthogonal normalized space: v (θ ) = p (θ ) X [u (θ )] +

n

∑ pν (θ ) ξν (θ ).

(25)

ν =2

By derivation of the expression (25) one shall obtain: n n dp d pν ν d ξν ξν + p = ∑ p ν A ( θ ) ξν , X [u (θ )] + ∑ dθ dθ ν =2 d θ ν =2 and by multiplying the Eq. (26) with ξ ∗ we shall have: n n d ξν (θ ) ν d pμ p = ∑ bμν (θ )pν . = ξμ∗ ∑ A (θ ) ξν (θ ) − dθ d (θ ) ν =2 ν =2

(26)

(27)

Proposition 1. The normal components pν of the variation ν , are verifying the normal variations system (24).

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M. Popescu, A. Dumitrache

Since: d du (θ ) ∂ X [u (θ )] du (θ ) d dX [u (θ )] du = = (X [u (θ )]) = = A (θ ) , dt dθ dt dθ ∂ u (θ ) dθ dθ (28) it follows: θ) Proposition 2. du( d(θ ) = X [u (θ )] is a solution of the system of variation (24). Let Φ (θ ) be the matrix of fundamental solutions for the variations system with the first column represented by the solution X [u (θ )]. Thus it can be written: Φ (θ ) = U (θ )C. (29)

where U (θ ) is the solutions matrix with U (θ ) = E. We have: Φ (θ + ω ) = U (θ + ω )C = U (θ )U (ω )C = U (θ )CC−1U (ω )C = Φ (θ ) K, (30) where K = C−1U (ω )C. For matrix K the following proposition is formulated: Proposition 3. The matrix K has also the monodromy matrix U and, thus, its eigenvalues are multipliers of variations system. Let (v2 v3 . . . vn ) be a different solutions of X [u (θ )] that generates matrix Φ, where: ∗ vν = v1ν v2ν . . . vnν , ν = (2, . . . , n) X = (X1 X2 . . . Xn )∗ . The columns of K can be written as: ∗ j = 1, . . . , n. k j = k1j k2j . . . knj The solution (25) of the variations system becomes: vν = pν X [u (θ )] +

n

μ

∑ pν ξμ ν = 2, . . . , n.

(32)

μ =2

Expanding the relation (30) we obtain: X [u (θ + ω )] = X [u (θ )] k11 + vν (θ + ω ) = X [u (θ )] kν1 +

n

μ

∑ vμ (θ ) k1 ,

μ =2 n

μ

∑ vμ (θ ) kν .

μ =2

(33a) (33b)

Stability of the Periodic Solutions for Quasi-linear Controlled Systems in Critical Case

333

We will prove the following: μ

Proposition 4. The functions pν (θ ) , ν = 2, . . . , n represent the terms of the fundamental matrix for the normal variations system (23). The functions νv (θ ) are solutions of the variations system. Let P(θ ) be the matrix of the fundamental solutions for the normal variations system, and let μ K2 = kν , ν = 2, . . . , n; μ = 2, . . . , n, (34) μ

where k11 = 1, k1 = 0, μ = 2, . . . , n. μ

n

μ

μ

Relation pν (ω ) = ∑ pν (0) kν can be written as follows: μ =2

μ

μ

μ

pν (ω ) = ∑ pν (0) kν ,

(35)

P (ω ) = P (0) K2 .

(36)

K1 = kν1 , ν = 2, . . . , n

(37)

which provides: If then matrix K is given by:

K=

1 K1 0 K2

.

(38)

From (36) and (38) we have: Proposition 5. The normal variations system has the eigenvalues of K2 as multipliers; these values are the eigenvalues of K excepting the multiplier equal to 1 corresponding to the periodical solution x[u(θ )]. From the Proposition 3 and Proposition 5 it can be concluded that: Proposition 6. The normal variations system multipliers are obtained from the system multipliers eliminating the multiplier equal to 1. The notion of orbital stability – to be defined below – is connected to the stability of the periodic solutions. Let Γ be the integral curve corresponding to system (8) in the phase space. The distance between the point x and Γ is given by:

ρ (x, Γ) = inf x − z. z∈Γ

(39)

Definition 1. A periodical solution x = u(t) of the system (1) has orbital stability if for each x0 with ρ (x0 , Γ) < 0 one has ρ (x (t, x0 ) , Γ) < ε for any t. Using this definition and the above results the following conclusion arise: Theorem 2. If (n−1) multipliers of the variations system corresponding to the periodical solution x = u(t) are inside the unit circle than the periodical solutions x = u(t) is orbitally stable.

334

M. Popescu, A. Dumitrache

In the sense of the orbital stability, the statement of theorem is equivalent to: Assuming that x1 (t) is an other solution of system (8) for which x1 (t0 ) is sufficiently close to the curve x = u(t) than, there exists c so that: lim [x1 (t) − u (t + c)] = 0.

(40)

t→∞

Proof. If the (n−1) multipliers of the variation system are inside the unit circle, from the previous presentation it will follow that the normal-variation-system multipliers are located inside the unit circle and, thus, the trivial solution of the normal variations system: dy 1 Y (θ , y) , = (41) dθ Θ (θ , y) is uniformly asymptotic stable, which implies that there exists δ0 > 0, so that for |y(0)| < δ0 we have: lim y (θ ) = 0. (42) θ →∞

Using substitution (12), the relation (49) becomes: lim {x [t (θ )] − u (θ )} = 0.

(43)

θ →∞

Consider 0 < θ < θ . From (16) and (20) we get:

t θ − θ − t θ − θ = =

θ

θ

θ θ dt 1 − 1 d θ = O (|y|) d θ . − 1 dθ = dθ Θ (θ , y)

θ

(44)

θ

Since the trivial solution of the normal variations system is uniformly asymptotic, it follows that it is exponentially stable, and – thus – there exist the constants h, α > 0 so that: |y (θ )| < he−αθ . (45) Relation (32) implies: O (|y|) < Me−αθ , M = const. > 0.

(46)

From (51) and (53) we have: θ

θ

M t θ − θ − t θ − θ ≤ M e−αθ d θ < M e−αθ d θ = e−αθ . α θ

(47)

θ

and for a sufficiently great θ , the relation (47) becomes: lim [t (θ ) − θ ] = θ0 .

θ →∞

(48)

Stability of the Periodic Solutions for Quasi-linear Controlled Systems in Critical Case

335

Also: t( θ )

x (t (θ )) − x (θ + θ0 ) =

X [x (α )] d α = [t (θ ) − (θ + θ0 )] X [x (α )],

(49)

θ +θ0

such that: |x [t (θ )] − x (θ + θ0 )| ≤ |t (θ ) − (θ + θ0 )| sup |X [x (α )]| .

(50)

Since the trivial solution of the system (41) is asymptotically stable for each ε > 0 there exists δ > 0 for which |y (θ )| < 0 once |y (θ )| < δ . For ε = 1, the relation (12) becomes: |x (u (θ ))| < |u (θ )| + |S (θ )| .

(51)

and since u (θ ) and S (θ ) are periodical, i.e. bounded, X [u (θ )] is also bounded. Using relation (48), we have: lim [t (θ ) − (θ + θ0 )] = lim [(t (θ ) − θ ) − θ0 ] = 0.

θ →∞

θ →∞

(52)

In conclusion, the boundedness of X [u (θ )] together with relations (50), (52) leads to: lim {x [t (θ )] − x (θ + θ0 )} = 0. (53) θ →∞

From (43) and (53), by deduction, it will result: lim {x (θ + θ0 ) − u (θ )} = 0.

(54)

θ + θ0 = t, −θ0 = c,

(55)

lim {x (t) − u (t + c)} = 0.

(56)

θ →∞

By denoting the relation (54) becomes: t→∞

Thus, the theorem is proved.

4 Conclusions The research performed until now [1] has touched the problem of critical cases by means of theory of Lyapunov. Thus the construction of a Lyapunov function is providing an answer regarding the asymptotic stability or the instability of the trivial solution of the case under consideration, but it cannot state if the system will be stable, and if the stability is asymptotic.

336

M. Popescu, A. Dumitrache

This paper provides a method to analyze the stability of the periodical solutions for the system obtained by the substitution of the control. A theorem to prove the existence of the orbital stability is also presented. The stability of the trivial solution of the nonlinear system is determined by studying the stability of the periodical solutions for the transformed system – i.e. orbital stability. Acknowledgement This work is partially financed from the government support of scientific research during 2006–2008, under grant CNCSIS, no. 11.

References 1. Malkin I (1952) Theory of Loading Stability, Edition Mir, Moscow. 2. Zoubov V (1978) Th´eorie de la Commande, Edition Mir, Moscow. 3. Popescu M (1995) Stability of motion on three-dimensional halo orbits, Journal of Guidance, Control and Dynamics 18, 1119–1126. 4. Popescu M (1998) Existence and orbital stability of periodic solutions for small parameters, Nonlinear Analysis 33, 773–784. 5. Popescu M (1999) Study of the orbital stability of nonlinear controlled systems, Continuous Discrete and Impulsive Systems 5, 451–463. 6. Berkovitz LD (1974) Optimal Control Theory, Springer, New York. 7. Popescu M (2002) Singular Optimal Control for Dynamical Systems, Romanian Academy Publishing House, Bucharest. 8. Popescu M (2003) Periodic solutions for nonlinear differential systems of equations with a small parameters, Nonlinear Analysis 52, 535–544. 9. Reithmaier E (1991) Periodic Solutions of Nonlinear Dynamical Systems, Lecture Notes in Mathematics 1483, Springer, New York.