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0, 0<6<2n, 0 < p < n . A zero value reflects a degeneration of the transformation on the polar axis p = kn, k = 0,1,....
1.1.3.3 Changes of reference frame Let us consider two distinct spaces in relative motion with respect to the other. For instance (E2) is assumed to be moving and ( E 1 ) is assumed to be at rest. Hence, the transformation of coordinates q'j, as defined in a frame at rest in (E 1 ), and qj as defined in a frame at rest in (E2), depends explicitly upon time:
Thus, the velocity components are transformed as:
Or, in matrix notation:
\df I dt\ is the so called transport velocity. It can be easily identified with the velocity of the moving frame, with respect to the frame that is assumed to remain at rest. The second component is the relative velocity, which is easily identified with the velocity of the system as defined in the moving frame. Clearly, relation [1.4] reduces to relation [1.3] when the transport velocity becomes zero.
8
Discrete systems
EXAMPLE. - A pendulum in a wagon Returning to Figure 1.1, in the Cartesian frame tied to (E^ one obtains:
Where X 0 ,Z 0 are the initial coordinates of the fixed point of the pendulum. Incidentally, such a result indicates that a change of reference frame does not modify the number of degrees of freedom of a system, except if the transport motion is not prescribed, as illustrated later in Chapter 4, example 3 of subsection 4.3.2.
1.1.4 Kinematical constraints In many mechanical systems, some generalized coordinates are connected to others through conditions which restrict the possible motions. In modelling real material systems, one is often led to formulate constraints of many kinds. To begin with, a distinction is made between two broad classes of conditions, namely those of holonomic constraints and those of nonholonomic constraints. Holonomic constraints reduce the number of degrees of freedom of the unconstrained system, while the nonholonomic constraints leave it unchanged. Though basically correct, this statement will be revised in Chapter 4, when formulation of constraints using Lagrange 's multipliers will be introduced.
1.1.4.1 Holonomic constraints To grasp what a holonomic constraint means, the simplest way is to start with a specific example. Consider a particle which is constrained to lay on the surface of a sphere of radius R, the origin of the frame being located at the centre of the sphere. This condition implies that the Cartesian coordinates of the point X, Y, Z obey the relationship:
Equation [1.5] reduces to two the number of degrees of freedom, as may be immediately checked by using spherical coordinates. The position of the point is then determined by the two independent variables (p and 0. Though ND is clearly independent of the coordinate system used, it is also worth studying the problem when it is formulated in terms of Cartesian coordinates. By using equation [1.5] to calculate Z as a function of X and Y, an ambiguity arises concerning the sign of Z. From a geometrical point of view, this reflects symmetry in the problem. Fortunately, ambiguity disappears as soon as the geometrical aspect of the problem is completed by the kinematical one. The latter implies that Z(t) is continuous and therefore the following initial condition must be verified:
Mechanical systems and equilibrium of forces
9
Since the configuration of the system is assumed to be known at t = 0, the initial condition fixes the relevant sign of Z(t). Thus, the present example suffices to emphasize the kinematical nature of the concept of degrees of freedom. Generalizing this example, constraints are said to be holonomic (from the Greek oXov entire, as a whole) if they can be expressed as relations between the coordinates of a mechanical system, which take the following form:
*0?m^...;0 = o
[1.6]
The coordinates that are connected through [1.6] may refer to a single or to several distinct particles. Whichever the case actually is, the major point remains that equation [1.6] can be used to calculate (at least implicitly) one coordinate in terms of the others, with the condition that kinematical continuity removes any ambiguity due to a possible non-uniqueness of the roots of [1.6]. Hence, the ND-DOP unconstrained system becomes a (ND - 1)-DOF system, when constrained. On the other hand, amongst the holonomic constraints, it is convenient to draw a distinction between those that change continuously with time, and those that do not. The first are called rheonomic constraints (the radical rheo comes from the ancient Greek verb peco which means to pour, to run) and the second ones are called scleronomic constraints (the radical sclero comes from the ancient Greek adjective (TKArjpov which means stiff, set). For instance, depending upon whether R is time dependent or not, relation [1.5] is a rheonomic, or a scleronomic condition. In both cases, the particle becomes a 3 - 1 = 2-DOF system. Such a result can be generalized to the case of motions constrained by several holonomic conditions according to the following rule: A set of N particles moving in a 3D-space constitutes a (3N - L)-DOF system if motion is constrained by L holonomic conditions which are mutually independent. Mutual independence of the constraint conditions implies that there is no redundancy in the set of equations which are used to formulate the constraints applied to the system. L linear equations are mutually independent, provided that no one of them can be obtained as a linear superposition of the others (see Appendix 1). In contrast, relations such as q n =0 and (q n ) = 0 clearly form a redundant set of conditions. EXAMPLE. - Degrees of freedom of a rigid body Two particles Pi, Pj are said to be rigidly connected to each other if their relative distance R^ remains constant. This scleronomic constraint may be formulated as the following (9^;.) condition:
10
Discrete systems
ri is the radius vector used to specify the position of the i-th particle. A possible set of generalized coordinates to describe this 5-DOF system may be defined as follows:
The first three coordinates set the position of Pi and the two remaining that of Pj, which lies on a sphere with centre Pi and radius Rij . It is of interest to go further by considering an arbitrarily large number of particles, thus modelling a rigid body. Let us start with a system of three particles. Five degrees of freedom are associated with two of them and the remaining particle is necessarily located on the circle defined by the intersection of two spheres of known centres P1, P2 and radii R13, R23. In this case, only one additional coordinate, for instance a curvilinear abscissa along the circle, is sufficient to determine unambiguously the position of this 6-DOF system. In the case of four particles, generally forming a tetrahedron, one is led to the same result. Indeed, six degrees of freedom are still associated with three particles and the remaining particle is located at the intersection of three known spheres. The geometrical problem yields two possible positions. One is eliminated by applying the condition of kinematical continuity. In the case of five particles, four radius vectors are known and three additional (5\)relations are associated with the fifth particle. They are formulated in terms of the following scalar products:
The known coefficients a 1 ,a 2 ,a3are the coordinates of P5 in the oblique frame (/]-^),(/j-r 3 ),(/j-/;). So, it can be easily verified that a further relation would merely be a linear superposition of the preceding relations. For instance, it is not difficult to check that r5.(r3 -r2) = a4 = al -a2, a relation which takes care of the compatibility of the redundant set of constraint relations. Generally, when formulating the rigidity of N particles systems, it could be thought necessary to write down C# = N(N -l)/2 (9\) relations. Fortunately, as soon as N > 4 this becomes unnecessary, because the number of mutually independent relations is reduced to L = 3(N-2). Such a reasoning can be extended to the case of a rigid body by considering N as being arbitrarily large. Hence, a rigid body can be described as a discrete system having at most 6 DOF. Of course, by using geometry, we could have derived such a result much more directly. Indeed, the displacement transformation of a geometrical figure can always be reduced to the
Mechanical systems and equilibrium of forces
11
product of a translation and a rotation. In a 3D-space, each of these elementary transformations is described by three independent parameters. As a final comment on the subject, the 5-DOF system of two particles is merely a particular case, arising because the particles having no dimension, any rotation about the P1P2 axis leaves the system unchanged.
1.1.4.2 Nonholonomic constraints The degrees of freedom of mechanical systems may be constrained by conditions which differ from the generic type [1.6]. Such conditions are called nonholonomic. As a first example, let us consider the case of a particle that is constrained to move inside a sphere, or at the surface of it. The Cartesian coordinates of the particle are thus governed by the condition:
Clearly, the condition [1.7] drastically restrains the space in which the motion can take place, but does not restrict the number of degrees of freedom of the particle. Relations of the type [1.7] are known as conditions of unilateral contact. During the contact, they reduce to equalities, and thus to holonomic conditions. Amongst the various kinds of nonholonomic constraints, the time-differential conditions are particularly worthy of mention because they are often used in rolling contact problems. They are expressed in terms of displacements and velocities of some points of the mechanical system:
It is also worth noting that [1.8] reduces to a holonomic condition when the differential equation can be integrated. An illustrative example is discussed in the next subsection. 1.1.4.3 Example: a constrained rigid wheel Let us consider a wheel modelled as a rigid circular disk of radius R, rolling on a horizontal floor, as shown in Figure 1.3. Oxyz is a Cartesian frame tied to the floor (Oxy-plane), with unit vectors i ,j,k . We are interested in determining the number of degrees of freedom and in defining convenient generalized displacements to describe the system, when subjected to various constraint conditions.
12
Discrete systems
1.
Unconstrained wheel
Figure 1.3. Rigid wheel rolling on a horizontal floor
Since the wheel is assumed to be rigid, ND = 6. A natural idea for defining convenient generalized displacements is to make use of the displacement of the centre G of the wheel, which yields the parameters of translation XG,YG,ZG, and to choose three angular parameters to describe the rotation of the wheel about G.
Figure 1.4. Angular displacements of the wheel
Figure 1.4 shows a possible choice where cp is the angle of rotation of the wheel about its axle, taken as the GX-axis. 0 is the angle of GX with the vertical Oz-axis of the reference frame. Finally, ip is the angle between the Ox-axis and the intersection of the wheel plane (P) with the floor plane.
Mechanical systems and equilibrium of forces
13
2. The wheel is constrained to keep in contact with the floor The contact point of the wheel with the floor is denoted C. The constraint takes the form of the holonomic condition ZG = Rsin0; then ND = 5, see Figure 1.5.
Figure 1.5. Wheel in contact with the floor 3. The wheel keeps in contact with the floor and its axle keeps parallel to the floor Since the plane of the wheel remains perpendicular to the floor, the additional holonomic relation 6 = n/2 holds and ND = 4. Motion can be described by using the horizontal translation of G, the rotation (p about the wheel axle and finally the angle i// between the O.x-axis and the tangent to the trajectory of C, or even better, the direct angle 0 = n - ip , as shown in Figures 1.6 and 1.7. The displacement of any point B at the disk periphery, which makes an angle (p with the vertical direction, counted starting from point C, is given by:
where
Figure 1.6. Vertical wheel in contact with the floor
14
Discrete systems
Figure 1.7. Condition of rolling without sliding
4.
Additional constraint of rolling without sliding
As shown in Figure 1.6, it is convenient to consider the direct Cartesian frame GXYZ with unit vectors J,J,K. One possible way to formulate the constraint condition is to balance the distances covered, during the same time, by the contact point along the disk periphery and along the path in the floor plane. Rolling without sliding implies that the length of arc CB is the same as the length of arc CC', see Figure 1.7. Accordingly, we have:
This yields the nonholonomic conditions: The second approach makes use of the concept of instantaneous centre of rotation, according to which the contact point is split conceptually into two distinct points, namely c1 = C is tied to the wheel and C2=C is tied to the plane. The velocities of the unfolded points are:
However, rolling without sliding implies that C1 and C2 have the same velocity, which thus turns out to be zero. This yields:
In the general case, the above equations cannot be integrated. Therefore, they are expressing nonholonomic conditions and the number of degrees of freedom remains unchanged.
Mechanical systems and equilibrium of forces
15
NOTE. - Parallel parking Any set of values Xc,Yc,
Figure 1.8. Parallel parking: the trajectory indicated is not always feasible! 5.
The preceding conditions hold and the path of the contact point is prescribed
Let the path of the contact point C be given, for instance by the explicit equation Yc = f (Xc ). It is expected that the wheel is restricted to a single degree of freedom system (in short, a SDOF system), which can be conveniently described by using either the variable (p, or Xc. Indeed, starting from equation of the trajectory, the following calculation is performed:
Since equivalent to each other:
the two following conditions are found to be
Moreover, they can be integrated. This yields for instance:
where, at this step, the subscript C has been dropped out tD simplify notation. The simplest path that can be conceived is a straight line. Then,X = R(p (Figure 1.9).
16
Discrete systems
Figure 1.9. Wheel rolling without sliding on a straight line
1.1.5 Forces formulated explicitly as material laws Various kinds of forces and moments may arise in a mechanical system. As it will be emphasized in subsection 1.2.2, it is convenient to begin by making a clear distinction between external and internal forces, or moments. By definition, the first do not depend upon the dynamical state of the system, while the second are related to it. The laws of mechanical behaviour, widely termed material laws, or constitutive laws, formulate the internal forces which are induced in the system (moving or not) as explicit analytical expressions. One is led first to make a distinction between static laws which involve only generalized displacements, (or their spatial derivatives in the case of continuous systems, cf. Volume 2) and dynamic laws which also involve time derivatives, or even primitives, of displacements. On the other hand, it is also useful to classify the laws, according to the linear or nonlinear nature of their analytical expression. EXAMPLE 1. - Linear spring
Figure 1.10. Linear spring acting in translation
Mechanical systems and equilibrium of forces
17
Figure 1.10 represents a spring that acts in translation along the Ox-axis. X1 and X2 denoting the displacements of the end points P1 and P2, the spring exerts the restoring forces:
where the positive quantity K is the stiffness coefficient of the spring, expressed in Newtons per meter (N/m) using the S.I. units, which are used in this book with few exceptions. Expression [1.9] is a particular case of the law of linear elasticity, which is the simplest that can be conceived. EXAMPLE 2. - Springs provided with gaps
Figure 1.11. Model of an elastic ball bouncing between two fixed walls
Figure 1.1la shows a simple system presenting nonlinear elastic behaviour. It consists of an elastic spherical ball, bouncing back and forth between two parallel rigid walls, which are separated from each other by the distance 2L. The ball of radius R, is slightly deformed elastically when in contact with one of the walls. As a first approximation, the following linear elastic contact law can be assumed:
where Kc is the equivalent stiffness of the ball when deformed in the radial direction by an amount &R/R of its surface at the contact point (here, the rather unrealistic assumption is tacitly made that contact is restricted to a single point of the deformed boundary of the body). Furthermore, the contact force Fc is assumed to be in the radial direction. The global behaviour of the system is then governed by the nonlinear stiffness coefficient:
18
Discrete systems
with where X is the displacement of the ball centre, with respect to the middle point at distance L from each wall. The gap J = L-R of each contact spring is equal to half the total length of "free flight". Thus, the system reduces to a particle whose one-dimensional motion is restricted by two springs of stiffness coefficient Kc, which are actuated as soon as the displacement becomes greater than the gap, see Figure 1.1 1b. The corresponding material law is expressed as:
otherwise NOTE. - Nonlinear contact spring Actually, the ball is not deformed according to a pin-point contact with the rigid wall, but according to a surface whose area varies during impact. This can be taken into account by using the more sophisticated Hertzian model of elastic contact, which results in the nonlinear elastic law Ft. =x(SR)
, where K depends upon the
elasticity constants of the materials and on the geometry of the contacting bodies, see for instance [TIM 70], [LAN 86], [YOU 01]. EXAMPLE 3. - Viscous damping
Figure 1.12. Conventional representation of a viscous damper
According to the considerations which shall be developed in Chapter 2 concerning mechanical energy, any real system interacts with others in such a way that it can either lose or gain mechanical energy. Many kinds of physical processes can be responsible for such interactions, for instance those which are described in
Mechanical systems and equilibrium of forces
19
Volumes 3 and 4, devoted to fluid-structure coupled systems. From the mathematical viewpoint, viscous damping is the simplest model that can be conceived for taking into account such irreversible exchanges of energy. Figure 1.12 is a conventional representation of a viscous damper, which acts in translation along the Ox-axis. It induces the dynamic forces:
C is the viscous damping coefficient, expressed in Newton seconds per meter (Ns/rri). Energy is dissipated, or gained, according to the positive or negative sign of C, as discussed further in Chapters 2 and 5.
1.1.6 Forces formulated as constraint conditions From the considerations discussed in the two last subsections, it is clear that the motion of a mechanical system can be restrained either by prescribing some constraint conditions, or by formulating explicitly some internal forces. For instance, in the example of the ball impacting against a wall, we could have opted for a model of unilateral constraint instead of the nonlinear elastic model (spring provided with gaps). In Chapter 5 subsection 5.3.2.2, this problem will be studied by using the two models successively. As an interesting result, it will be shown that discrepancies between the contact force and the unilateral condition models arise during the stages of contact, which concern the detailed description of the shocks (or impacts). Furthermore, the differences are found to vanish progressively as the impact stiffness coefficient Kc is increased. As a matter of fact, kinematical constraints formulate implicitly internal forcing terms, which can be identified precisely with the forces, or moments, that are necessary to enforce the prescribed conditions. Such forces and moments are termed constraint reactions. In Chapter 4, we shall see how to calculate conveniently such reactions. EXAMPLE. - Particle tied to a rigid string
Figure 1.13. Particle tied to a rigid string and rupture of the string
20
Discrete systems
Let a rigid body be connected to a fixed point O by a non-extensible string. When the body is rotating, it is the centripetal reaction induced by the condition of nonextensibility that maintains the body at a constant distance from O. To be fully convinced of that, cut the string and observe the motion of the particle, which becomes rectilinear, as shown in Figure 1.13. When modelling internal forces arising in a mechanical system, it is largely a matter of convenience to choose between an implicit model (constraint conditions) and an explicit model (material laws). For instance, in terms of degrees of freedom, it is far more convenient to model a solid, of which deformation is negligible, as a rigid body, instead of using a model of internal forces which would take care of the approximate rigidity of the body on the atomic scale. On the other hand, the distinction between constraints and material laws is of major importance to the manner in which a problem in mechanics may be formulated, as will be further developed in Chapters 2 to 4. Based on considerations about work, it will be found suitable to model only a specific class of internal forces as constraint conditions, namely that of the so called perfect constraints, which, by definition, do not perform any work.
1.2. Basic principles of Newtonian mechanics 1.2.1 Newton's laws In order to predict the changes in the configuration of a mechanical system over time, it is necessary to establish the equations of motion (also called the equations of dynamical equilibrium) which govern the time evolution of the coordinates (or displacements), of the system. The formulation of these equations rests on the three laws postulated by Newton (1687), which are taken as the foundation of Newtonian mechanics. It is worthwhile, for clarity at least, to review them briefly and to point out some consequences of major interest. As a preliminary, it is also worth recalling that Newtonian laws introduce force as the primordial physical concept of mechanics. Since force is a vector quantity, the Newtonian approach results in a vector formulation of the equations of motion. 1.2.1.1 Law of inertia A particle not experiencing any force either remains at rest, or is in uniform rectilinear motion. The law of inertia is restating in terms of forces the older inertial principle of Galileo, according to which an isolated and undisturbed body keeps a constant velocity vector. As a corollary, the reference frames, which are in uniform rectilinear motion in relation to each other, are equivalent from the point of view of dynamics. This is simply because the forces are the same in all such reference frames, which are therefore called Galilean or inertial frames.
Mechanical systems and equilibrium of forces
21
According to formula [1.4], position and velocity of a particle are transformed from one Galilean frame to another one, as follows:
r and V are the vectors of position and velocity in the unprimed frame, f' and V denote the transformed vectors in the primed frame. Vt is the transport velocity of the unprimed frame with respect to the primed frame, which is assumed to be fixed. Thus the inertial law of Galileo postulates the invariance of the dynamic equations with respect to the Galilean transformations [1.11]. Accordingly, it can be interpreted as a principle of relativity, which implies that in any inertial frame: 1. Time is the same. 2.
Mass of a particle is the same.
3.
Forces exerted on a particle are the same.
Another important consequence of such a principle is the non-existence of a unique frame of reference that would be an absolute frame which should be adopted to express the equations of motion in a privileged form. Moreover, even the idea of fixed frame has to be interpreted in a relative meaning. 1.2.1.2 Law of motion (basic principle of dynamics) A particle experiencing a force is prompted in an accelerated motion, so that the acceleration multiplied by the mass of the particle is equal to the force. Hence, the second law generalizes the first one, as it connects the acceleration to the force exerted on the particle, giving thus the following equation of motion:
X ( t ) is the displacement (or position) vector of the particle expressed in an inertial frame provided with a Cartesian coordinate system. F(t) designates the force, or the sum vector (the resultant) of the individual forces exerted on the particle. Such forces are generally time dependent. Equation [1.12] may also be rewritten in the following more general form:
Equation [1.13] introduces the linear momentum p of a particle as a new physical quantity which is defined as the product of the mass times the vector velocity of the particle. Incidentally, it may be noted that form [1.13] holds, even if m is time dependent.
22
Discrete systems
Finally, as we shall see on several occasions in this book, it is also of major interest to define the action of a physical quantity over a time interval t1 to t2 , as the integral of it over this interval. As a first example, from equation [1.13] it can be stated that p measures the mechanical action of the resultant of the individual forces exerted on the particle. Indeed, it follows immediately from equation [1.13] that:
The preceding considerations can be extended to systems of several particles. However, some care is required to deal in a suitable manner with the mechanical interactions between the particles. First let Fj be the resultant of forces exerted on the 7-th particle. The corresponding equation of motion is:
Now, Fj is generally the sum of internal and external forces. Furthermore, some internal forces, termed interaction forces, depend upon variables (coordinates, velocities etc.) which refer to distinct particles. Interaction forces are governed by a third law referred to as the law of action and reaction. 1.2.1.3 Law of action and reaction
Figure 1.14. Strong law of action and reaction in a system of particles The forces two particles exert on each other are equal and opposite to each other. According to this principle, when a particle Pi is acting on another particle Pj by exerting a force Fij then Pj is reacting on Pi by exerting the force Fji = -F ij .
Mechanical systems and equilibrium of forces
23
Furthermore, when dealing with purely mechanical applications, as it will be always the case in this book, a more restrictive form of this law, broadly referred to as the strong law of action and reaction, is adopted by adding to it the following postulate: The action and reaction forces are central; i.e. they lie along the line joining the particles. Thus, in a mechanical system comprising N interacting particles, for each pair one can define forces of mutual interaction which are central and exactly opposed to each other, as illustrated in Figure 1.14. EXAMPLE 1. - Two bodies interacting by gravitation Let P1 and P2 be a pair of particles with masses m1 and m2, which are attracted to each other by gravity. Newton's law of gravitation gives the interaction force:
G designates the constant of universal gravitation (G = 6.67 10 -11 N.m2 I kg 2) and r is the relative distance of the two particles (r = P1P2). A particular case of interest is that of a body with mass m orbiting round the Earth, which is modelled as a spherical rigid body with radius R = 6370km and mass M = 5.98 xlO24 kg. However, in so far as r remains sufficiently near to R and m remains sufficiently less than M, the force of interaction can be drastically simplified, being reduced as a first approximation to:
Fg is the gravity force and g designates the acceleration of Earth's gravity (g = 9.81ms-2 ). k is the unit vector along the local vertical, which is oriented from the orbiting body toward the Earth's centre. The important point worth emphasising is that according to the above approximation the actual interaction force is replaced by an external force exerted on the orbiting body, while the motion of Earth is implicitly assumed to remain unaffected by the motion of the body. EXAMPLE 2. - Motion of the centre-of-mass of a system of particles Let be a set of N interacting particles, subjected also to external forces F; . Equation [1.15] can be written as:
24
Discrete systems
Now, by summing the above equations over j, one is led to the following result, of remarkable simplicity, which is a direct consequence of the law of action and reaction (either in the weak or in the strong form):
It is then convenient to reduce the left-hand side of equation [1.16] by defining the centre-of-mass of the system of particles. The position of G is given by:
Using this definition, equation [1.16] becomes:
Hence, whatever the motion of the individual particles may be, the centre-ofmass behaves like a single particle with the total mass M of the system, which is subjected to the sum vector of the external forces. It is also convenient to define the resulting linear momentum P as:
Pj is the linear momentum of they-th particle. When F^e> = 0, P becomes constant and the motion of G is rectilinear and uniform.
1.2.2 D'Alembert's principle of dynamical equilibrium The concept of force of inertia - hereafter denoted F (l) - which was introduced by d'Alembert (1743), allows us to unify the formulation of statics and dynamics. Force of inertia is clearly an internal force because it depends upon the dynamic state of the system. By making use of F ( I ) , equation [1.12] takes the form of an equilibrated balance of forces:
Equation [1.20] extends to dynamic systems the same balance of forces as that utilized in statics. This balance of forces is contained in Newton's first law as a condition for mechanical equilibrium. Finally, it may be worth emphasizing that
Mechanical systems and equilibrium of forces
25
statics is only an idealized asymptotic case of dynamics in which motion is so slow that the time derivatives of displacement, and even the time evolution of forces, become negligible. In this book, a unified presentation of dynamic and static systems will be used, starting from the dynamical models. On the other hand, most systems are subjected to both internal and external forces at the same time. So, for sake of clarity, it is convenient to collect the internal forces on the left-hand side and the external forces on the right-hand side of the dynamic equations. Accordingly, equilibrium is written as the following balance of forces:
Hereafter, this convention will be systematically respected when writing down the equations in their final form. EXAMPLE. - The linear damped mass-spring system The equation of motion of the oscillator, shown schematically in Figure 1.15, results from the force balance written as:
Figure 1.15. Linear damped mass-spring system
-KX
is the stiffness (elastic) force, -CX is the viscous damping force and (e) - MX is the inertia force. Finally, Fv ' (?) denotes the external force, which excites the oscillator. Stress is laid on the point that the displacement X ( t ) is counted starting from the position O of the static equilibrium of the unloaded system
(e\
(e)
Fv ' = 0 . The equation corresponding to the static problem is KX = Fv '. Finally, (e) when the time evolution of Fv ' ( t ) is sufficiently slow, the inertia and damping forces remain negligible with respect to the stiffness force. Accordingly, the problem is governed by the equation KX(t] = F^e'(t], which is said to be quasistatic. The range of validity of such an approximation will be further discussed in Chapter 9,
26
Discrete systems
subsection 9.2.2, in relation to the frequency content of the excitation and response signals.
1.2.3 Equations of motion in terms of moments 1.2.3.1 Moment of a force and angular momentum
Figure 1.16. Moment of a force and angular momentum of a particle
One is often led to express equations of motion in terms of angular variables. In these cases, it is generally found convenient to formulate Newton's second law in terms of moments (or torques). It is recalled first that the moment of a force about a given point O is defined as the cross product:
As shown in Figure 1.16, r is the radius vector from the point O to the particle P, which is subjected to the force F. It is also useful to define the moment about a given axis of unit vector k , as the mixed product:
Substitution of equation [1.12] into equation [1.23], yields:
At this stage, it is useful to define the following vector quantity:
LPIO is the angular momentum of the particle P about the point O. Now, when O is a fixed point, it is found that:
Mechanical systems and equilibrium of forces
27
In this case, equation [1.25] takes the remarkably simple form:
Equation [1.28] formulates the dynamical equilibrium of the moments about a fixed point and indicates also that the angular momentum measures the action of a moment. If a component of the latter is zero, the corresponding component of the angular momentum is an invariant of motion. EXAMPLE. - Kepler's second law of planetary motion Even though the scope of the present book does not consider celestial mechanics, it would be a pity not to mention here one of the most famous occasions where astronomy provided the starting impulse to the development of Newtonian mechanics, which may be considered a direct application of equation [1.28]. Based on the careful measurements by Tycho Brahe, Kepler found by trial and error his well-known law about planetary motion, according to which the radius vector from the Sun to a planet sweeps out equal areas in equal times. It was a major achievement of Newton to be able to demonstrate that such a statement simply proceeds from the fact that the force of gravitation attracting two particles to each other is directed towards the centre-of-mass of the system (the centre of the Sun as a first approximation). Indeed, on one hand it is realized that the area swept per unit time is given precisely b y A = rxr
. On the other hand, as the force is always directed
towards a fixed point, the angular momentum is constant and so is A, since mass is constant too. As pointed out in the true enlightening lecture by R.P. Feynman on the relation of Mathematics to Physics [FEY 65], the proof given by Newton goes along the lines just outlined, but by using a quite ingenious and simple geometrical reasoning, instead of the vector calculus.
Figure 1.17. Angular momentum about the centre-of-mass of a set of N particles
More generally, equation [1.28] can be used to uncouple from each other the motion of the centre-of-mass G of a system and the motion of the system about G. To achieve this, O has to be taken at the centre-of-mass of the mechanical system, see
28
Discrete systemsms
Figure 1.17. Indeed, the total angular momentum of a system of N particles about a fixed point can be written as the sum:
Since G is assumed to be the centre-of-mass of the system, one obtains:
And so, by developing the right-hand side of [1.29], the remarkably simple result is established:
LG/0 is the angular momentum about O of the material point G where the mass of the system M = \
m; is concentrated.
Such a simplification is very convenient particularly for analysing the motion of rigid bodies. Indeed, in this case, motion of the system about G reduces to a 3Drotation, as further discussed in Chapter 2, subsection 2.2.2.
1.2.3.2 Plane rotation of a particle
Figure 1.18. Polar coordinate system
Mechanical systems and equilibrium of forces
29
Polar coordinates (A%#) are often used to describe motion taking place in a plane; see Figure 1.18. The following results are immediately established:
where u and u1 are the unit vectors in the radial and in the tangential directions, respectively. / = mr2 is the moment of inertia of the particle about the axis Oz, perpendicular to the plane Oxy. On substituting expression [1.31] of L in equation [1.28], it can be seen that if r is constant, 6 is governed by the following equation:
Now, comparison of equations [1.12], or [1.13], with equations [1.32], or [1.28], shows that Newton's equation of motion keeps the same form, whether a linear or an angular displacement is used. Such a result will be extended in Chapters 2 and 3 to the case of generalized forces and displacements of any kind. On the other hand, the second equation [1.31] shows that if the particle is rotating about the Oz-axis with the angular velocity Q = Qk , the linear velocity can be written as:
Since equation [1.33] is expressed in terms of vectors, its validity does not depend upon the direction of Q . 1.2.3.3 Centrifugal and Coriolis forces Referring back to Figure 1.18, let us consider now a particle sliding freely along the radial axis of unit vector u at constant velocity V = r. Furthermore, the radial axis is assumed to rotate at the constant angular velocity Qk about the fixed point O. The inertia force impressed to the particle is readily found to be:
The expression [1.34] makes apparent two new kinds of inertial forces which are not proportional to an acceleration, in contrast to that arising in Newton's second law of motion [1.12] or [1.32]. The first term on the right-hand side of [1.34] is known as a Coriolis force, which acts in the tangential direction, and the second term is known as a centrifugal force which acts radially in the outward direction from the centre of rotation. Such forces will be met again in subsection 1.2.4, in relation with transformations of reference frames.
30
Discrete systems
1.2.3.4 Applications to a few basic systems EXAMPLE 1. - The simple pendulum
Figure 1.19. Simple pendulum
The simple pendulum is shown schematically in Figure 1.19. It consists of a rigid body tied to a fixed point O by a non-extensible string, which is supposed to move in a vertical plane. The dimensions of the massive body are assumed to be so small with respect to the length R of the string that it can be modelled as a particle P with mass M. The most convenient way of deriving the dynamic equation of this SDOF system is to take the angle 0 as the variable of displacement and to work out the balance of moments about O. This gives:
Indeed, making use of the moments instead of the forces allows one to eliminate directly the - a priori unknown - reaction of the constraint condition OP = R, which identifies with the tension T of the string. 6 being counted from the lowest position on the vertical axis, it is readily shown that:
which gives the equation of motion:
Equation [1.35] sums two distinct terms. The first one is a linear dynamical term produced by the moment of the inertia force and the second is a nonlinear stiffness term produced by the moment of the weight. In terms of balance of forces, the formulation is a bit more heavy. Adopting the Cartesian coordinates ( which depend upon 0 ), we obtain:
Mechanical systems and equilibrium of forces
31
The internal forces comprise the force of inertia and the tension of the string:
~~(e\ The external force is the weight Fe ' = Mg. The horizontal and vertical balancing of forces give the two following equations:
In this elementary example, the string tension is very easily eliminated to obtain a unique equation in terms of the single independent parameter 6. As expected, this produces the same equation as that obtained by the balance of moments. Moreover, as an interesting by-product, the balance of forces method provides us with the reaction of constraint, which is found to be:
T comprises a nonlinear stiffness force induced by the weight and a centrifugal force, which is induced by the rotation of P around O. To conclude on this basic example, we must emphasize that elimination of the reactions of constraint in the case of systems having several degrees of freedom is generally much less simple than in the case of SDOF systems. We shall describe in Chapter 4 the very clever method devised by Lagrange to deal with such constrained systems in order to determine at the same time the equations of motion and the reactions of constraints. EXAMPLE 2. - Wheel connected to a linear spring Let us consider the rigid wheel already described in subsection 1.1.4.3. The track of it on the floor is now assumed to be a straight line, taken as the ox-axis. The Oxzplane is vertical and g designates the acceleration of gravity. The wheel centre is connected to a fixed and rigid wall by a spring (stiffness coefficient K) acting in translation, see Figure 1.20. M is the wheel mass, supposed to be uniformly distributed over the circular disk (radius K). The system is then moved away from its position of static equilibrium by imparting the horizontal displacement X0 to G. The exercise consists in formulating the equation of motion successively in the two following contrasted cases.
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Discrete systems
Figure 1.20. Wheel connected to a wall by a spring
1.
The wheel slides perfectly on the floor
The system is described by using the two independent variables ^Tand p. In the presence of gravity, the condition of contact with the floor Z = R implies a reaction from the floor T2 = Mgk that balances the weight of the wheel. Clearly, this reaction does not enter into the equations of motion. X is governed by the equation of equilibrium of the horizontal forces KX + MX'= 0. On the other hand,
(0) 2n V M about the wheel axle. However, in the context of this book, mathematical techniques used to solve this kind of differential equations fully deserve an extensive presentation, which is postponed until Chapter 5.
Mechanical systems and equilibrium of forces 2.
33
The wheel is rolling without sliding on the floor
As already shown, the wheel is now a SDOF system. Furthermore, the condition X= R(p implies the existence of a horizontal reaction of the floor Tx = Txi. The equations of equilibrium become:
Elimination of Tx yields:
A single equation of motion is thus produced, expressed here in terms of X:
NOTE. - Inertia of the rolling wheel As indicated by the last result, when the wheel is rolling without sliding, it oscillates at a lower frequency than when it slides without rolling. The reason for this is that rolling motion adds inertia to the system while stiffness remains unchanged. Accordingly, it can be advantageous to minimize the mass of rotating mechanical components. For instance, during acceleration, the legs of a cyclist are solicited "twice" by the mass of tires. Indeed, provided most of the mass is at the periphery of the wheel, it is found that J = MR2, hence MeX = 2MX . EXAMPLE 3. - Changing direction of a spinning wheel
Figure 1.21. Changing the direction of the axle of a spinning wheel
34
Discrete systems
Let us consider the simple experiment sketched in Figure 1.21, which consists of holding a spinning wheel by its axle and changing the direction of the axle. The spinning angular velocity Q, is assumed to be much larger than the angular velocity 9 imparted to the axle by the experimenter. If so, the angular momentum of the system remains practically collinear to the axis of spin. The mass M of the wheel is assumed to be essentially uniformly distributed at the periphery and so J = MR2. We want to know the force, or torque, which has to be exerted on the axle to change its direction from horizontal to vertical.
Figure 1.22. Rotation of the spin axis
As indicated in Figure 1.22, 6 is the angle of the spin axis counted from the initial configuration, which is horizontal. Assuming that Q>>0, for sake of simplicity, the following results hold, which are broadly known as the gyroscopic approximation:
Where MF = 2lI x F is the torque imparted by the experimenter:
Identifying component by component the rate of change of the angular 7/2/3
momentum with the external torque, it is found that Fxx = Fzz = 0; F}y = +
. As U shown in subsection 1.2.3.3, the force thus defined can be identified with a Coriolis force. Here, 6 is taken as negative, because the rotation considered in Figure 1.22 is from i to k , i.e. in the direction opposite to that of a direct frame. Therefore, the
Mechanical systems and equilibrium of forces
35
experimenter must exert a vertical moment by pulling with the right hand and pushing with the left hand on the wheel axle. The law of action and reaction as applied to the two connected subsystems (the wheel on one side and the experimenter on the other), implies that the experimenter is subjected to a torque equal and opposite to L. Hence, if he stands on a swivel chair, he will turn to the left. Application of such a result is very familiar in cycling. Provided the bicycle runs forwards, Q is negative. When the cyclist imparts a horizontal push with the right hand, the bike turns on the left and get tilted towards the inside of the bend. Conversely, by imparting a tilt to the bike, the cyclist can turn without activating the handlebar. However, such a way of driving is not advisable at low speed. Indeed, when the direction of L significantly departs from that of the axle of the wheels, it becomes harder to keep control of the motion!
1.2.4 Inertia forces in an accelerated reference frame It is not always convenient to formulate the equations of motion of a system by using inertial coordinates. Therefore, it is of interest to specify how the acceleration of a mass-point is transformed when noninertial coordinates are used. Here, for sake of simplicity, the problem is restricted to the case of motions of a particle taking place in a plane. The general case of 3-D motion is discussed in Chapter 2. Finally, a general formula can be written, going along the same line as that used in subsection 1.1.3.3 to derive the general formula [1.4] for the transformation of velocity.
Figure 1.23. Change of Cartesian reference frames
As shown in Figure 1.23, we define an inertial Cartesian frame ((^) (axes ox,oy and unit vectors i ,j ) and an accelerated Cartesian frame (C2) (axes OX, OY and unit vectors i, j). The vector position of a particle is then given by:
36
Discrete systems
Since ((^) can be transformed into (C2) by the product of a translation and a plane rotation, the coordinates are transformed as follows:
Incidentally, in accordance with the formula [1.4] the transformation [1.37] may also be expressed as:
Or, by using the notations of formula [1.4]:
By differentiating [1.37] with respect to time, the velocity of the particle is expressed as:
The last term of expression [1.39] is the relative velocity in the accelerated frame (C 2 ), as expressed in the inertial frame (C 1 ). The relative velocity as expressed in the frame (C 2 ) is obviously:
On the other hand, the sum of the first two terms in [1.39] is identified with the transport velocity which has a component of translation and a component of rotation. The last, as expressed in the frame (C2) is in the tangential direction. In vector notation, it takes the same form as relation [1.33]:
Mechanical systems and equilibrium of forces
37
where Now, by differentiating again the relation [1.39] with respect to time, the acceleration of the particle is expressed as:
The last term stands for the relative acceleration. As expressed in (C 2 ), this component is radial:
In the penultimate term we recognize a Coriolis acceleration, in accordance with [1.34]. As expressed in (C2) this component becomes:
The sum of the remaining terms can be identified with the transport acceleration. It is of interest to express the two components of rotation in (C2). The first rotation component is identified with the centripetal acceleration:
The second rotation component, called the Euler acceleration, is tangential:
Thus, it is found again that when expressed in terms of variables defined in a rotating frame, the acceleration of a particle may consist of three distinct components which are not second time derivatives of displacement variables. The corresponding inertia forces are the centrifugal force +m&2R, the Coriolis force -2mQk xR and the Euler force -mQkxR . Finally, for sake of completeness it is worth to quote the general formula for transforming the acceleration from one frame to another. Time differentiation of [1.4] produces the following formula:
38
Discrete systems
1.2.5 Concluding comments Basically, Newton's laws of motion introduce two vector quantities which stand for the fundamental concepts of Newtonian mechanics. In the case of linear displacements (i.e. translations), the primordial quantity is force, and linear momentum measures the action of the force. In the case of angular displacements, the primordial quantity is the moment (or torque), and angular momentum measures the action of the moment. Nevertheless, as already pointed out, Newton's equation of motion keeps the same form, whether a linear or an angular displacement is used. On the other hand, the traditional presentation adopted here, which starts from Newton's three laws, could have been replaced by a more contemporary and axiomatic one, which is based on only two founding principles, that of the existence of inertial frames and that of conservation of momentum. However, the author believes that by doing so little would have been gained for applications in the field of mechanical engineering. The reader interested in such more formal aspects of classical dynamics may be referred to [JOS 02]. On the other hand, in the vectorial approach to mechanics, Newton's law of motion is directly applied to produce the dynamic equations of a given system. As already emphasized, this method is traditionally adopted in elementary mechanics, for reasons of clarity and mathematical simplicity. However, it requires one to write down a detailed balance in which all the forces exerted on the system are formulated explicitly, including the reactions induced by the constraints. As illustrated in the next chapter, even for simple systems such as a double pendulum, the vectorial method becomes intricate and unduly tedious. Fortunately, it is possible to avoid this difficulty by adopting the so called analytical or variational approach, which introduces the work of the forces as the primordial quantity. By doing so, the reactions of constraints are automatically eliminated, provided they do no work. Thus it is this second approach that shall be largely favoured hereafter in this book. However, this does not mean that the vectorial approach has to be rejected. On the contrary, the vectorial method remains an efficient tool, at least in order to understand better the physical meaning of the equations of motion established by the analytical method. Indeed, to most of us the concept of force remains more familiar and is more intuitively understood than the concept of work, or energy. Finally, coming back to the comment concerning linear and angular variables, it will be shown that the analytical approach makes systematic use of generalized quantities, which can be defined independently from their physical nature. Accordingly, linear and angular displacements may be understood as two particular kinds of generalized displacements, while forces and moments are two particular kinds of generalized forces.
Chapter 2
Principle of virtual work and Lagrange's equations
To quote from Lanczos [LAN 70], "by founding the analytical mechanics, Lagrange added nothing fundamentally different to Newton's laws, but provided an immensely powerful weapon to solve any mechanical problem on the basis of pure calculation" and his book "Mecanique Analytique", first published in 1788 "may be considered rightly as an extraordinary achievement, which opened an entirely new world to mathematical modelling in mechanics". Analytical mechanics is entirely formulated in terms of generalized coordinates, providing thus mathematical expressions which hold independently of the specific coordinate system chosen. It introduces, as prime concepts, scalar quantities such as kinetic and potential energies, instead of force. Then, a variational principle concerning energy is used to derive the equations of motion in a purely analytical way. Variational principles and methods are very appealing in various branches of theoretical physics, as they bring out the symmetrical nature of the fundamental laws of physics. Following the approach of many textbooks devoted to mechanical engineering, the formalism of Lagrange's equations will be introduced here starting from the principle of virtual work. Nevertheless, it will be restated in the next chapter, starting from the principle of least action, as formulated by Hamilton, which offers richer perspectives both for a deeper understanding of physics and for the development of mathematical methods to solve practical problems.
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Discrete systems
2.1. Introduction Direct use of Newton's second law for establishing the dynamic equations of constrained multi degrees of freedom systems can often be a tedious and even an unfeasible task. Actually, the constraint reactions arise as supplementary unknowns, which have to be eliminated, because they cannot be considered as additional independent variables. In most instances the elimination process is not a straightforward task, even in the case of 2-DOF systems. This is illustrated in the following example, easily extended to an arbitrary number of DOF, to stress with the difficulty of the vectorial treatment of mechanics. EXAMPLE. - Double pendulum
Figure 2.1. Double pendulum Consider the system shown in Figure 2.1 where the two particles P1, and P2, with masses M1 and M2, are constrained to remain in the vertical plane Oxz. Since the connecting strings are assumed to be rigid, we have to deal with a 2-DOF system. Its motion is described by using the angular displacements 01 and 02. In the balance of forces (or moments), it is necessary to include the tensions T1, T2 as new unknown vectors, which have then to be eliminated to produce the final form of the dynamical equations, expressed in terms of independent variables only. However, the elimination process is much less obvious than in the case of the simple pendulum. On the other hand, it can be noted that T1 and T2 are normal to the path of the particles and hence do not perform any work. Moreover, this would remain true in the case of any kinematically admissible motion, i.e. a motion that is consistent with the kinematical constraints imposed on the system. Therefore, the idea is to establish the equations of motion, starting from a balance of work performed by the forces instead of forces themselves. This is the aim of analytical mechanics of which first notions are introduced in section 2.3, after having reviewed
Principle of virtual work and Lagrange's equations
41
the concepts of work and of mechanical energy and having discussed some general implications concerning the motion of discrete systems.
2.2. Mechanical energy and exchange of it Mechanical energy is a scalar and additive quantity, which depends upon space, but not upon the coordinate system used to describe the motion. A natural way to introduce this quantity is to perform a balance of the work done by the generalized forces exerted on the material system, between two arbitrarily given times t1 and t2.
2.2.1
Work and generalized forces
2.2.1.1 Work performed by a force Let F be a force, whose point of application is changed by an infinitesimal amount dX. It performs the infinitesimal work defined as the scalar product (also known as the inner product):
NOTE. - Notation of the scalar product In relation [2.1], the scalar product is defined in an Euclidean space with a dimension at most equal to three. Accordingly, the usual vectorial notation of Euclidean geometry is used. However, having in mind further extension to Ndimensional and functional vector spaces, the scalar product is also written by using either the matrix or the functional vector notation <, >. For instance:
Work done between t1, to t2, is given by the integral:
Such a result can be immediately extended to the case of any system of N particles, because it suffices to sum up the individual contributions of each particle forming the whole system. Let Fj be the force impressed on the j-th particle that is displaced by X j ( t ) . In Cartesian coordinates, the work produced by the whole system between t1 and t2 can be written as:
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Discrete systems
2.2.1.2 Generalized displacements and forces Since work is defined as a scalar product, it does not depend upon the coordinate system used to express the force and displacement vectors. For instance, shifting from Cartesian to generalized coordinates [q(t)], we obtain:
where [Q] denotes the vector of the generalized forces. When the scalar product [Q]T [dq] is written by using the index notation, the individual contribution of each pair of conjugate (dual) components of force Qk and displacement qk appear explicitly:
Now, using the transformation rules [1.3], we may shift from Cartesian to generalized components of forces as follows:
And so, identification of the force components gives:
where [J] is the Jacobian matrix already introduced in Chapter 1, subsection 1.1.3.1.
Principle of virtual work and Lagrange's equations
2.2.2
43
Work of inertial forces and kinetic energy
2.2.2.1 Linear motion (translation) in an inertial frame of reference Let X(t) be the displacement of a particle, as defined in a fixed Cartesian frame. The work of the inertia force is:
This result is now re-written as:
where £K is the kinetic energy of the particle, given by:
This scalar quantity is said to be positive definite as it becomes strictly positive as soon as the particle moves. Extension to a system of N particles proceeds simply by adding the individual contributions of each particle:
From the mathematical viewpoint, expression [2.11] is a quadratic form of the particle velocities, which is symmetrical, positive definite. It is written in matrix notation as:
[xl is the velocity vector of the system and [M] is the mass matrix. It takes here the particular form of a diagonal matrix, in which the mass coefficients related to the samey-th particle (velocities Xj, Yj, Zj) are all equal to mj. Since expression [2.12] remains invariant with respect to any change of coordinate system, [M] is actually a second-rank tensor. Moreover, as expression [2.12] is also invariant with respect to transposition, [M ] is symmetrical, a property which holds in any coordinate system.
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Discrete systems
2.2.2.2 Rigid body rotating in an inertial frame of reference Let us consider a rigid body free to rotate around its centre-of-mass G, which is assumed to be fixed. This 3-DOF system is described in a direct Cartesian frame with origin G. Q denotes the angular velocity vector, defined by:
where tffare the angles of rotation about the coordinate axes, as seen in Figure 2.2.
Figure 2.2. Direct Cartesian frame
The linear velocity of a current point of the body is then obtained by using relation [1.33]: V = ^2xr . Since d*P = Q(t)dt is an infinitesimal rotation, Q can also be thought as an instantaneous rotation vector. As shown in Appendix 2, the coordinate transformation of a point in the infinitesimal rotation is:
NOTE. - Infinitesimal andfiniterotations If a finite rotation W = Yxi j +tf/zk
is considered, the transformation
r' = Wxr does not correspond to a rotation anymore. Indeed, it can be easily checked that the Cartesian frame Oxyz is thus transformed into an oblique frame:
Principle of virtual work and Lagrange's equations
45
In order to describe finite rotations, one often makes use of the Euler angles, see Appendix 3. The reader is also referred to [GOL 80], [JOS 02] for a detailed discussion of finite rotations. The elementary angular momentum is (cf. formula [A2.7]):
where p is the mass per unit volume of the body and dV the elementary volume. Let —> x, y, z be the Cartesian coordinates of a current point. The components of dL are found to be:
Finally, by integrating [2.16] over the volume (V), the result is the matrix relation:
[I] is the matrix of inertia of the body, whose Cartesian components are:
[I] can be reduced to a diagonal form by using the coordinate frame of the principal axes of inertia of the body, which are given by the eigenvectors of [I]. Since the matrix is symmetrical, the eigenvalues and eigenvectors are real and the latter are orthogonal to each other (cf. subsection 2.2.2.5 and Appendix 4). Now, the kinetic energy of an elementary material volume with mass pdV is given by:
46
Discrete systems
Finally, integration over the whole body gives:
Here again, it is found that kinetic energy is a quadratic and symmetrical form of velocity, which is positive definite. Hence, shifting from a motion of translation to a motion of rotation, the linear velocity vector is replaced by the angular velocity vector [Q] and the mass matrix is replaced by the inertia matrix [I], both having the same basic mathematical properties in that, just like [M], [I] is a second-rank tensor. 2.2.2.3 Change of reference frame Let us consider a mechanical system set both in translation and rotation. Since kinetic energy is an additive scalar quantity, it can be expressed as a single quadratic positive definite form. However, it is often desirable to separate the translational and the rotational parts of motion. This is made possible by considering two frames of reference, one of them being an inertial frame and the other being tied to the centreof-mass G of the system. In most instances, G is in an accelerated motion with respect to the inertial frame. Then one is led to calculate the kinetic energy, referring to the inertial frame, but still using coordinates (or displacements) defined in the accelerated frame. More generally, let £K be the kinetic energy of a mechanical system in an inertial frame, as given by equation [2.12] in terms of the Cartesian velocity vector [X]. Let [q] be the displacement vector as defined in an accelerated frame. Relation [1.4] gives:
Substitution of this expression in equation [2.12] yields:
This form separates kinetic energy into three distinct components, namely: 1.
Transport energy
Clearly, this is the only nonvanishing term when the system is at rest in the accelerated frame.
Principle of virtual work and Lagrange's equations 2.
47
Relative energy
[M'] is the relative mass matrix, which is symmetrical and positive definite like [ M ] . It has to be noted that [J] and thus [M'] can depend upon the relative displacement vector [q], in contrast to[M]. Thus, relative energy is still a quadratic symmetrical and positive form of the relative velocity vector. However, it can also depend upon [q]. 3.
Mutual energy (or cross-energy)
Mutual energy Q gathers the last two terms present in the total kinetic energy £K. However, as each of them is related to the other by a transposition, they become identified to the same scalar. Hence, £r'
is re-written as:
Mutual energy couples linearly the relative and the transport velocity vectors. As a final comment, it can also be noted that generally, [J] and [M][7] are not necessarily symmetrical. EXAMPLE. - Particle rotating about a moving point
Figure 2.3. Particle rotating about a moving point
48
Discrete systems
Let X0i Y0j + Z0k be the position vector of a point 0, as defined in an inertial Cartesian frame with unit vectors i, j, k. Consider the Cartesian frame relative to O, with unit vectors I, J, K parallel to i, j, k. In the last reference frame, the position of the particle P, which rotates round O, can be described by the spherical coordinates R, 0,
The kinetic energy of P in the inertial frame is found to be:
The last step of the analysis expresses the result in terms of R,0,q> and in sorting out the relative component of energy, which is finally written as:
This yields a mass matrix [M'] function of the angular displacement (p. In a similar way, the mutual energy is given by:
This produces a non-symmetrical mass matrix, which depends on 0 and (p . 2.2.2.4 Generalized inertial forces in a rotating frame The generalized forces related to kinetic energy as expressed by using coordinates referring to an accelerated frame will be discussed later. The results obtained in Chapter 1 subsection 1.2.4. concerning a plane rotation can now be
Principle of virtual work and Lagrange's equations
49
extended to a 3-D rotation. Cartesian coordinates are transformed by a 3-D rotation as follows:
where [x] denotes the position vector of a particle referring to a Cartesian inertial frame and [X] denotes the position vector of the same particle referring to the rotated frame. The matrix of rotation [R] is formed with the direction cosines of the base vectors of the rotating frame, as expressed in the inertial frame:
It is easily shown that [R] is an orthonormal matrix depending upon three independent parameters:
As a consequence [R]-1 = [R]r. On the other hand, differentiation with respect to time yields the following results:
The instantaneous rotation of the particle is defined by the angular velocity vector Q - Qxi + Qyj + Qzk . Assuming here that the relative velocity [x] is zero, [x] reduces to the transport velocity vector, as expressed in coordinates of the inertial frame, whereas its expression [V] in terms of the rotated coordinates is given by the equation [1.33]. Thus, we have:
where [£2] is the instantaneous rotation matrix given by the formula [A2.3] of Appendix 2. From these two expressions of the transport velocity, it is readily found that:
50
Discrete systems
Therefore:
Finally, expressing the total acceleration which includes the relative terms
[x]
and [X], we obtain as a final result:
Thus the acceleration in the inertial frame, as expressed by using the coordinates of the rotating frame, is found to be the sum of the four following components: 1. Euler acceleration [Q] [X] 2. Centripetal acceleration [Q]2 [x] 3. Coriolis acceleration 2 [Q] [X] 4. Relative acceleration [x] in full agreement with the results of Chapter 1, subsection 1.2.4. 2.2.2.5 Properties of Hermitian matrices The Hermitian matrices, hereafter denoted [H], are built of complex elements and, as a definition, they verify the following condition:
where the star (*) marks the transformation by complex conjugation. The symmetrical matrices, hereafter denoted [S], which are built of real elements, constitute a subset of the Hermitian matrices and it is convenient to describe the properties of [5] as a particular case of those holding for [H].
Principle of virtual work and Lagrange's equations
51
A complex (NxN) matrix [A] can also be viewed as a linear algebraic operator in a complex vector space of dimension N, which transforms a vector X of that space into another vector Y of that space. In matrix notation, the transformation is written as:
In the terminology of linear operators, [A*]r is known as the adjoint matrix of [A] and matrices [H] or [S] are said to be self-adjoint. Going a little bit further, the vector space can be provided with a scalar product, which is written by using either the functional, or the matrix notation:
Incidentally, the second equality in [2.34] shows the invariance of scalar quantities with regard to transposition. As is the case in normal Euclidean 3D-space, the scalar product of real vectors is commutative and provides a real scalar. NOTE. - Hilbert's spaces More generally, a vector space which is complete (i.e. any Cauchy sequence converges to a vector within that space, for further details see Appendix 1 paragraph A1.4) and which is provided with a scalar product is termed a Hilbert's space. The scalar product is used to define the natural metrics of the space. Returning back to matrices operating on a Hilbert's space of finite dimension, it is recalled that the eigenvalues and the related eigenvectors of a matrix are the nontrivial solutions of the following homogeneous problem:
where [I] denotes the identity matrix I;t = 1, ifj = k and 0 otherwise . [pn] is the n-th eigenvector of [A] which is related to the n-th eigenvalue An. The natural norm of [#?„] is |[^n1| = y{^,,,^n). Unit eigenvectors are then produced by using the natural norm. Hermitian matrices have the properties which are listed below (for mathematical proofs, see Appendix 4): 1. All the eigenvalues are real numbers.
52
Discrete systems
2.
All the eigenvectors related to distinct eigenvalues are orthogonal to each others.
3. Starting from the whole set of eigenvectors, it is always possible to define an orthonormal basis of the Hilbert's space in which [H] is operating. This basis is characterized by the transformation matrix [O], of which columns are formed with a set of N orthonormal eigenvectors. Therefore [3>] is said to be orthonormal and it can be shown that its inverse is identical to its adjoint:
The matrix [<E>] can be used to transform [H] into a diagonal form as follows:
In the language of geometry, the transformation [2.37] is a similarity. Similarity maintains the parallelism of vectors, but not the direction. Two matrices related to each other by similarity are said to be similar. The eigenvalues of similar matrices are the same. 4. The orthonormal transformation [^] = [<^][^] can also be viewed as an orthonormal change of coordinates of the same vector from the initial basis of definition (coordinates qn) to the basis of the [p n ] (coordinates q'n). The transformation of coordinates is written as:
Any [H] matrix is thus transformed into a similar matrix noted [ H ] :
Going back now to the special case of symmetrical matrices [S] operating in a real vector space, we consider the quadratic form:
If the sign of a remains the same whichever [q] may be, except the null vector, [S] is said to be either positive definite, or negative definite, in accordance with the sign of a. Now it can be shown that the necessary and sufficient condition for [5] to be of a given sign is that all the eigenvalues are of the same sign, i.e. positive if [S] is positive. Obviously, two other cases are also possible. One is that all the
Principle of virtual work and Lagrange's equations
53
eigenvalues have the same sign, except some of them, which are found to be zero. Such a matrix is said to be positive, or negative, in accordance with the sign of the nonvanishing eigenvalues. A final occurrence is that all the eigenvalues do not have the same sign, in which case the sign of the matrix is undefined. EXAMPLE. - Principal axes of the tensor of inertia
Figure 2.4. Principal axes of inertia for a rigid body of revolution
The tensor of inertia [I] of a rigid body is symmetrical and positive definite. Generally, it is expressed as a full matrix, if referred to an orthonormal basis with no additional particularity. However, the eigenvectors of such a matrix define a principal frame of inertia (or basis) of the body in which the inertia matrix is diagonal. The coefficients are the eigenvalues identified thus with the principal coefficients of inertia. Now, if the body is symmetrical about some axis, taken as the Oz-axis, the latter is necessarily a principal axis of inertia corresponding to the eigenvalue Iz , called the polar coefficient of inertia and denoted J. The remaining two eigenvalues are identical to each other Ix = ly = J/2 and any pair of axes that are perpendicular to each other and to Oz, can be taken as the principal axes (i.e. orthogonal eigenvectors), see Figure 2.4.
2.2.3
Work performed by forces deriving from a potential
2.2.3.1 Potential energy The work done by a force that depends solely on the position of the particle can be written as:
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Discrete systems
where r denotes the position vector at time t, as defined in a Cartesian frame, dr is the infinitesimal displacement of the particle occurring over the infinitesimal time interval dt. The last expression of work in [2.41] introduces the concept of potential energy, denoted £ . Furthermore, letting t2 tend toward t1, the force can be expressed as the gradient of the potential, with the sign changed:
Here, the subscript i denotes the index of the Cartesian coordinates of the point. In accordance with Newton's second law, so far as the response of the mechanical system is concerned, knowledge of the gradient of the potential is sufficient. Thus, the potential can be defined save on an additive constant, which can be chosen arbitrarily, and in dynamics, only the fluctuating (i.e. time varying) part £p (t) of the potential enters into the problem. Moreover, it can be noted that the work done by a force derived from a potential does not depend on the time-history of motion, but only upon the change between the initial and the final configurations. In particular, it vanishes when r(t 2 ) = r(t 1 ). This is formulated as the following condition concerning the closed curvilinear integral:
dr is an infinitesimal arc, which is oriented along a closed trajectory. An interesting corollary of such a property is that potential energy is left unchanged by any frame transformation. Finally, it is found that £p(r} = £ p ( X } , where X denotes the displacement vector. 2.2.3.2 Generalized displacements and forces Let us consider a system of N particles. [X] denoting the vector of Cartesian displacements; the potential energy of the whole system is simply the sum of the potential energies of the individual particles:
where [Xj] is the displacement vector of the j-th particle. Now, if the same system is described by using an independent set of ND generalized displacements, the expression of the potential energy is transformed into:
Principle of virtual work and Lagrange's equations
55
Since the use of generalized displacements (or coordinates) leads us to specify the degrees of freedom, and not the particles, it becomes convenient to draw a distinction between the terms of £p [q] that involve a single DOF and those that depend upon several distinct DOF. The latter are recognized as interaction or coupling terms. Finally, the generalized displacements are related to the generalized forces by using the differential form of relation [2.5]:
EXAMPLE 1. - The simple pendulum Let us consider once more a simple pendulum of length L. If 9 denotes the angle with the vertical direction; the potential energy can be written as £p(0) = MgL(l -cos0), where the arbitrary constant has been chosen in such a way that £p vanishes if the particle is at the lowest position 6 = 0 + 2kn where k= 0,1,2,... The conjugate generalized force is Qe=-MgLsind. In physical terms, it is immediately recognized as the restoring moment that is induced by the weight of the particle about the fixed point O. Obviously, the force would be the same if the potential £p(0) = -MgLcos& would have been considered. EXAMPLE 2. - Coupling of two pendulums
Figure 2.5. A pair of identical pendulums, coupled by a linear spring The system to be studied is shown in Figure 2.5. The potential is found to be:
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Discrete systems
The generalized forces are:
They identify with the physical moments about the fixed points of the physical forces acting on the system. As expected, the moments induced by the restoring force of the spring comprise an interaction component that couples the two angular displacements 0, and 02.
2.2.4 Mechanical energy and the exchange of it with external systems 2.2.4.1 Conservative systems Let us consider a system, which is actuated by forces derived from a potential and by inertia forces solely. Starting from the preceding results, it is recognized that, at any time during the motion, the change of kinetic energy is exactly balanced by the change of potential energy:
And so:
Accordingly, the mechanical energy is defined by summing the fluctuating potential and kinetic energies:
£m is an invariant of motion. Therefore, a mechanical system which is actuated solely by forces derived from a potential and by inertia forces, is said to be conservative. By extension, the corresponding generalized forces are also said to be conservative. Clearly, the motion of a conservative system is perpetual in nature and governed at any time by a balanced exchange between kinetic and potential energies.
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EXAMPLE. - Oscillations of a simple pendulum Let us consider once more a simple pendulum of length L released from initial position 00 with zero initial velocity. The mechanical energy is immediately found
to be:
Now, this result can be used in order to prove that the motion is periodic and to determine the period. Indeed, the velocity can be expressed in terms of £m (00) and
Hence, it can be immediately deduced that |0| is upper-bounded by |0| and that the period is given by the following integral:
By carrying out the integration, it would be realized that T is an increasing function of 60. The pendulum is a classical example of nonlinear integrable systems, whose motion is rather simple. However, it has to be stressed that the dynamical behaviour of nonlinear systems is certainly not always simple, especially when they are forced, or when they have several degrees of freedom. The dynamical response of linear and nonlinear oscillators will be further discussed in Chapters 5, 7 and 9.
2.2.4.2 Nonconservative systems Experience shows that real material systems are always exchanging energy with other systems through various interaction mechanisms. Therefore, they are never conservative. Interaction may involve external or internal forces, as briefly outlined below. External forces: As already indicated in the first example of Chapter 1 subsection 1.2.1.3, external forces can be considered as an asymptotic case of interaction forces, such that one of the interacting systems is not significantly perturbed by the other. When a
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Discrete systems
conservative system is excited by external forces F^'(f) , the balance of energy becomes:
Depending upon the sign of this work, the mechanical energy of the system is an increasing, or to the contrary, a decreasing function of time. On the other hand, since the work of external forces depends upon time and variables of displacement only, it can be concluded that external forces are derived from potentials, which will be referred to as external potentials. Furthermore, both the excited and the exciting systems may be considered as two parts of a single system, which interact with each other through a potential. Clearly, the single system thus defined is conservative, as far as interaction between the two parts is potential in nature. Such a point of view will be further developed in Chapter 3, subsection 3.3.4. Internal nonconservative force: Experience shows that mechanical systems interact with thermo-dynamical systems through various and irreversible mechanisms. As a general and inevitable result, mechanical energy is finally dissipated into heat. A particularly famous consequence of omnipresent dissipation is the impossibility to build devices that would be in perpetual motion. Furthermore, any mathematical model based on conservative forces only, as useful as it may be, remains by nature defective as far as the energy balance is concerned. Fortunately, in numerous cases of practical interest, it is still possible to model such interactions by defining forces said nonconservative, work of which accounting for the exchange of energy. In this way, coupling with thermodynamics may be avoided, the problem being still modelled as a purely mechanical system. The viscous friction force, introduced in Chapter 1 subsection 1.1.5, is the simplest kind of dissipative force which can be conceived for that purpose. Major consequences of introducing dissipative forces in a dynamical model shall be discussed in several parts of this book. Suffice it to say here that a system initially provided with an amount of mechanical energy and subjected to dissipative forces is set in a motion which decays with time, so that the motion is damped out. However, the opposite case can also occur in which nonconservative forces supply the system with mechanical energy. A typical example is that of fluid-elastic forces which couple a moving solid and a flowing fluid. The work of such forces corresponds to an exchange of mechanical energy between the solid and the fluid, the direction of the transfer depending on the detailed features of the dynamically coupled fluidstructure system. Study of such systems is postponed to Volume 4 of this book. Besides their academic interest, such problems are also a real concern in many fields of mechanical engineering, owing to the risk of dramatic failures occurring. Indeed, when the solid is pumping energy from the flow, whether or not some external excitation is present, its motion is steadily amplified up to a certain limit, which is controlled by various nonlinear mechanisms that prevail as soon as the magnitude of the motion is large enough. Unfortunately, such limiting mechanisms result, in most
Principle of virtual work and Lagrange's equations
59
instances, in the destruction of the mechanical components, either by fatigue or by excessive wear.
2.2.5
Work performed by constraint reactions and perfect constraints
A priori, in the energy balance of a constrained system one has to include also the work of the reactions induced by the constraints. Since such reactions arise as additional unknown quantities of the problem, determination of their work is in most cases unfeasible. Consequently, to bypass such a difficulty the key is to model as constraint conditions only the workless constraints, which are said to be perfect. On the other hand, all the forces which do work are described through a material law. This aspect of force modelling is of major importance in the variational formulations of mechanical problems, as further discussed in the next section.
2.3. Virtual work and Lagrange's equations As emphasized for instance in [LAN 70], the idea of solving mechanical problems by considering not only the forces but also the velocities, or the displacements of their point of application is contained in the "Physics" of Aristotle (384-322 B.C.), in relation to the static equilibrium of the lever, see also [DUG 55]. Much later, in 1717, John Bernoulli gave the exact enunciation of the principle of virtual work, which was used first to solve problems in statics, and then introduced in dynamics as well, based on d'Alembert's principle.
2.3.1 Principle of virtual work Let us consider a system of N particles, referenced by a set of Cartesian coordinates. According to d'Alembert's principle, the mechanical equilibrium can be expressed by postulating that the resultant of the forces acting on each individual particle is zero:
In the presence of constraints, the natural idea is to make the distinction between the forces F^ that are expressed as explicit analytical laws and those R. that are defined implicitly as constraint reactions. Hence, the force resultant applied to the jth particle is written as:
Now, our purpose is to produce a new formulation of the mechanical equilibrium in such a way that the contribution of the reactions Rj is eliminated. The key point is to consider the work performed by the forces. As a preliminary mathematical tool, it
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Discrete systems
is convenient to define an infinitesimal field of virtual and admissible displacements 6 rj. Such a field is defined as an arbitrary variation of the coordinates of the system, considered at a fixed time t, which complies with the constraint conditions imposed on the system. It is termed virtual in order to mark a clear distinction from the real displacement that the system actually experiences over a time interval dt. Indeed, as time elapses, the constraint reactions like the other forces acting on the system may change, but the virtual displacements are entirely disconnected from any change in the forces, in contrast to the actual displacement of the system. As a trivial result, d'Alembert's principle leads to:
Relation [2.50] can be transformed into a very fruitful result, provided the study is restricted to systems involving solely explicit forces and perfect constraints. Indeed, as the work of perfect constraints is zero, the contribution of the constraint reactions to the balance of virtual work vanishes, and [2.50] reduces to:
This result, of paramount importance, is known as the principle of virtual work. It calls for the following comments: 1. In contrast with [2.50], equation [2.51] does not imply that all the coefficients of the virtual displacements Srn vanish. Indeed, since the constraints connect together some components of the displacement field, independence of the whole set of coordinates rn; n = 1...3N is lost. Therefore, prior to equating to zero the coefficients of the displacement variables, it is necessary to restate the problem in terms of an independent set of generalized displacements, which refer to the degrees of freedom of the constrained system. Such a change of variables yields an expression of the type:
[Q] is the vector of generalized forces and [Sq] is a vector of generalized virtual and admissible displacements, made of ND independent components. 2.
Equations [2.51] and [2.52] hold only if the constraints imposed on the system are perfect. The concept of perfect constraint, already introduced in subsection 2.2.5, can now be restated in a more suitable fashion: A necessary and sufficient condition for a constraint to be perfect is that the virtual work of the constraint reactions is zero.
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For instance, a constraint condition of rigid connection is perfect because any admissible displacement is orthogonal to the constraint reactions. A unilateral condition also is perfect, because in the absence of contact no reaction is induced, and during contact the reaction is normal to the surface of contact, while any admissible displacement is tangential to it. Another example of perfect constraint is given by the condition of rolling without sliding, which implies a fixed contact point (the instantaneous centre of rotation). In contrast, a contact condition including some sliding with friction could not be described as a perfect constraint. Hence the necessity to model it as an explicit material law. N
3.
The principle of virtual work can be restated as ^ F™ .6r. = 0, where Sir. is a ;=i
field of virtual and admissible velocities. It is worth noting that, historically, the principle was enunciated precisely in terms of velocities. Such a formulation in terms of work rate, or power, instead of work, may be found more convenient, in particular when dealing with nonlinear problems, which require a numerical procedure making use of small time increments to proceed step by step to the solution.
2.3.2
Lagrange's equations
Let us start from the principle of virtual work, written in the following dynamical form:
where Fj(a) is gathering together all the force components, except the inertia forces which are expressed in terms of momentum Pj. Now, the aim is to reformulate [2.53] in terms of the generalized variables qn and qn of the system. The Cartesian coordinates XrYrZ}\ j = !,... N of the position vectors rj are written as r k (q l ,q 2 ,...,q ND ;t'); k = 1,---3N , and the associated mass coefficients are denoted m;. Since 8rk are infinitesimal quantities they can be written as:
The non inertial terms are transformed according to equation [2.7]:
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Discrete systems
Incidentally, when Fj(a) is derived from a potential, the related generalized force is suitably expressed as Qn--d£pldqn. Transformation of the inertial terms is less straightforward. Consider first the kth component of momentum:
Here, E(x) denotes the integer part of x. Then, relation [2.56] may be developed as:
On the other hand, from [2.54], we have:
This expression has to be transformed further to let the kinetic energy of the system appear. The calculation is carried out as follows:
Once more, the velocity rk is written as:
Differentiation of relation [2.60] with respect to qn gives:
On the other hand, the following relation is readily found by permuting the order of the differentiations:
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63
Now, substituting relations [2.61] and [2.62] into [2.59], we obtain:
This result can be suitably expressed in terms of the kinetic energy of the system:
where £k(j) is the kinetic energy of they-th particle: with index k summed from Summation over all the particles of the system gives:
N
where £K - is the total kinetic energy of the system. system. ;=i Then, substituting relation [2.66] into [2.58], it becomes possible to express the virtual work of inertia forces in terms of the generalized displacements:
Finally, gathering together the partial results [2.55] and [2.67], the principle of virtual work can be written in a suitable form to derive the equations of motion:
Indeed, as the qn are independent variables, equation [2.68] is satisfied with the sole condition that the factor of every Sqn is zero. This provides ND equations of
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Discrete systems
motion, known as the Euler-Lagrange equations (often termed, in mechanics, after the last author's name only):
The calculation procedure described just above can be found essentially in the same form in many textbooks, for example [GOL 80], [GER 97], [JOS 02]. It can be noted that the way of introducing kinetic energy starting from the principle of virtual work is rather artificial. The reason is that we have to start from the virtual work of the inertia forces. As we will see in Chapter 3, the use of Hamilton's variational principle allows one to introduce the kinetic energy dependent terms of Lagrange's equations in a quite natural way, starting directly from the kinetic energy.
2.3.3 The Lagrange function (Lagrangian) Equations [2.69] are valid whatever the nature of the generalized forces. However, they comprise usually of at least one component derived from a potential 8p. Accordingly, Lagrange's equations are re-written as:
Qn denotes the forces arising neither from kinetic nor potential energies. £ is termed Lagrange's function, or the Lagrangian, which is given by:
On the other hand, as indicated by the notation used in the final form of equations [2.70], a suitable way for writing Lagrange's equations is to use the matrix formalism. This is particularly true in the linear domain, as we shall see in subsection 2.3.4. Finally, as already shown in subsection 2.2.4.2, external forces derive from an external potential, which is implicitly included in the Lagrangian [2.71]. However, it is also clearly possible to build an internal Lagrangian, which involves only the internal terms of energy. If such a choice is made, the system [2.70] takes the canonical form [1.21], expressed here as:
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65
2.3.4 Special form of Lagrange's equations in the linear case Provided the internal Lagrangian of the system may be modelled as a quadratic form of displacement and velocity, it is conveniently written in the following matrix form:
where the mass [M] and the stiffness [K] matrices have constant coefficients. The corresponding Lagrange's equations are then readily found to be:
where [Q] stands for the linear internal nonconservative force and
|Q
(t)|
stands for the external force. EXAMPLE. - Lagrange's equations of the double pendulum Adopting the notations of Figure 1.1, the kinetic and potential energies of the particle P1 are:
Then, the velocity of P2 is deduced from:
Whence:
And so, the kinetic and potential energies of P2 are found to be:
Now, from these results Lagrange's equations are obtained:
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Discrete systems
It is worth noticing that each step of the above calculation is straightforward. On the other hand, when the problem is restricted to the case of small motions about the state of static equilibrium Oei,0e2 = 0, it may be anticipated that it is sufficient to expand the above equations to the first order with respect to 01,,6?2. This results in the linear matrix system:
It is noted that the mass and stiffness matrices are symmetrical and that the mass matrix is full, coupling thus the two degrees of freedom of the system. The underlying theoretical background and physical interpretation of such results will be further described in Chapter 3. Moreover, Chapter 6 will present a mathematical formalism, which is especially well suited to analyse the small motions governed by this kind of linear systems. As we shall see, they correspond to small vibrations about the position of a state of equilibrium 9tl ,0e2 = 0, which is statically stable. To conclude on this example, external moments 9Jt^ (t},3ft^^(f) are applied to each pendulum. This loading may be considered as derived from the external potential:
The corresponding Lagrange equations are:
The last three chapters of this volume are devoted to the description of the mathematical tools and methods which are especially well suited to analyse the linear motions of such forced systems, in a systematic and straightforward way. In this respect, attention of the reader is drawn to the fact that such a mathematical background constitutes the backbone of the methodologies used by the engineer to perform the calculations required to analyse the design of a very large variety of important structures, even when calculation must be conducted beyond the linear domain of response.
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2.3.5 Lagrangian and Newtonian formulations Clearly, a formal equivalence must exist between the variational (Lagrange) and the vectorial (Newton) formulations of the equilibrium equations of any given mechanical system. The equivalence is readily established as follows. To begin with, the first inertial term of Lagrange's equations can be interpreted as the rate of the time variation of a generalized momentum, defined as follows:
The resultant of the other terms is identified with a generalized force Qn, that can involve three distinct components, arising from: 1.
A potential
£ p (q n ).
2.
The partial derivative of kinetic energy £K (qn, qn) with respect to qn.
3.
The remaining forces Qn, which are related neither to £p nor to £K. Accordingly, the general form of Newton's equations are found to be:
Since the vector equation [2.76] is expressed in terms of generalized quantities, it generalizes those already obtained in terms of physical displacements in Chapter 1, section 1.2. As a final note, let us consider a system which is described solely by an internal Lagrangian. If the latter is independent from a variable qn, the corresponding generalized momentum is obviously an invariant of motion.
2.3.6 Application to a building resting on elastic foundations Let us consider a building modelled as a homogeneous cuboid, with edge lengths a, b, H. r - xi + yj + zk is the position vector of a current point referenced in the Cartesian frame, the origin of which is at the centre-of-mass G and the axes are parallel to the edges of the cuboid. The corners of the base B1, B 2 ,B 3 , B4, are resting on elastic foundations, see Figure 2.6. Our purpose is restricted here to analysing the response of the building to a static external load applied laterally, assuming small displacements and neglecting gravity.
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Discrete systems
Figure 2.6. A cuboid resting on elastic foundations
2.3.6.1 Generalized displacements
Figure 2.7. Displacement of the base of the rigid cuboid
First, we adopt the six variables of physical displacement, namely the translations X1, Y1, Z1 of G about the position of static equilibrium in the unloaded configuration, and the rotations Vx,V/y,i/s2 about the axes of the fixed frame Gxyz. The external load is a force F0j applied to a point P0(0, + a/2, z 0 ). Foundations are modelled as a set of linear springs located at the corners of the base. The springs acting along the axial direction Gz have a stiffness coefficient denoted K. Those acting along the transverse directions Gx, Gy have a stiffness coefficient denoted k.
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Since the external load is a force acting along the direction Gy, the response can be restricted by the conditions Z1 = 0; X1 = 0 and i//y=i//z= 0. Hence, we consider only the two relevant variables Y1 and iffx. However, it is soon recognized that the elastic energy of foundations is more conveniently expressed in terms of the Cartesian displacements XBi,YBi,ZBi, / = l to 4 of the corners than in terms of yj and \ffx. The displacement of the base of the building is shown schematically in Figure 2.7., C being the middle point of segment B1 B2 (or B3 B4), to the first order in \i/x the displacements are found to be:
which are then re-written in terms of Y2 = y/xH 12 as:
where a = a /H denotes the reciprocal of the slenderness ratio of the solid. The small displacement field of the Gz-axis is Y(z) = Yl- zyfx.
2.3.6.2 Potential energy and stiffness matrix In so far as gravity is neglected, the potential energy of the system is entirely given by the elastic energy in the foundations. Summing the contributions of each spring, we obtain:
We recognize immediately a quadratic form, conveniently written as:
And so, the stiffness matrix is found to be:
where the stiffness ratio y = K/k factor.
is introduced as a physically pertinent scaling
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Discrete systems
By deriving -£e with respect to Y1, Y2, the generalized stiffness forces are produced according to the canonical form [Q(s)] = -[K][Y]. In the present case, they are:
2.3.6.3 Generalized external loading and solution of the forced problem The generalized forces relative to the external load are determined by calculating its work (or in an equivalent way its external potential) in terms of the generalized displacements Y1,Y2. This gives:
Therefore, the generalized forces are immediately found to be:
The static equilibrium of the forced system is governed by the matrix equation:
with
This linear system can be easily uncoupled by subtracting one row from the other. Doing so, the problem is immediately solved:
The displacement of the vertical Gz-axis is given by:
It is interesting to discuss this mathematical result as a function of the external load parameter z0 and of the internal parameters a, y. First, it is noted that the physical displacement field combines linearly a lateral translation and a rotation about Gx (rocking motion). As expected, the response is proportional to the external excitation and inversely proportional to the equivalent stiffness of the foundations.
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Then, if the force is applied to the lower base of the cuboid (z0 = -H/2), the latter is displaced according to a mode of pure translation Y(z) = Y0. Such a result is rather obvious since the rocking moment of the load vanishes for this particular location. Finally, it can be noticed that when z0 = -H/2, the response increases as a function of the slenderness ratio of the building (a —> 0). This is also easily understood because, for a given load, the restoring moment induced by the axial springs decreases as the reciprocal of the lever arm a/2. The factor a2 arises because the angular stiffness is itself proportional to a . The dependency on 1/y can be justified in a similar way. 2.3.6.4 Response to a distributed loading We deal here with another external loading, which is defined as the following one-dimensional force density:
where:
The lateral load is applied along the line parallel to Gz passing through P0. The results already obtained in subsection 2.4.2 are still available and the only remaining task for solving the new problem is to calculate the relevant generalized load. The work done by the external force field is now given by:
Hence, the generalized forces are found to be:
Physical interpretation of such a result presents no difficulty. Indeed, the lateral translation Y1 is excited by the resultant of the load, independently from the spatial distribution of the force density (governed here by the parameter T)). On the other hand, Y2 results from a rotation excited by the resulting moment, which depends upon the spatial distribution of the physical loading. In the present example, a law proportional to 77 has been assumed. The forced problem is then solved in the same way as previously:
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Discrete systems
Again, the response is a linear superposition of a translation and a rotation. Two particular cases of interest are 77 = -2 and n = -2(a2 y +1), which result in a pure translation and in a pure rotation, respectively. 2.3.6.5 Stiffness coefficients for distributed elastic foundations
Figure 2.8. Elastic foundations distributed over the base area
Let us consider the case of elastic foundations uniformly distributed over the whole area of the base. The corresponding stiffness coefficients (linear stiffness per unit area) are denoted KX, Ky, KZ , see Figure 2.8. If the displacement of the base is axial, the resultant of the elastic restoring force is:
This result may be used to define the equivalent stiffness of the foundations related to a pure axial displacement, which is found to be Kz1 = abKz. As we shall see, it is also worth noting that the same result could have been obtained by calculating first the work of the stiffness forces for an axial displacement Z1, (or the elastic potential):
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73
Kz1 = abK, is thus interpreted as the generalized stiffness coefficient related to the displacement Z1, which is twice the elastic energy for a unitary displacement (Z1 = 1). In a similar way, it is of interest to determine the generalized stiffness coefficients related to the lateral displacements Y1, Y2. It is recalled that the physical displacement in the lateral direction of any point of the lower base is Y c = Y 1 + Y 2 . Hence, the first method used in the preceding case becomes inoperative and we are led to adopt the second method, which consists in calculating the elastic energy induced by a lateral displacement. This yields:
However, Yc is coupled to the axial displacement field through the relation:
The corresponding elastic energy is:
Thus, the stiffness coefficient KY2Y2 is finally found to be:
For a displacement in the Gx direction, the calculation is identical. Results are expressed in terms of the displacement variables X1, X2=Hyy/2 and of the reciprocal of the slenderness ratio B = b/ H . The stiffness coefficients are directly reported in the global stiffness matrix written down in the next subsection. To build such a matrix, it is also necessary to determine the generalized stiffness coefficient related to the axial rotation iffz. For that purpose, let us consider a current point B ( x , y , - H / 2 ) which is moving along a circle centred at the middle point of the lower base, the radius of which is given by R^ = -yx 2 + y2. The amplitude of the small displacement induced by iyz is thus:
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Discrete systems
and the displacement vector is found to be:
The elementary elastic energy which is related to ysz is:
and so, by integrating over the base area:
Whence the generalized stiffness coefficient related to iffz:
2.3.6.6 Stiffness and mass matrices for any displacement field Gathering together the preceding results, the stiffness matrix related to any displacement field, function of the six variables X1, Y1, Z1, X2, Y2, i/sz is obtained by first adding together the partial energies calculated in subsection 2.4.5. This yields the following quadratic form:
From which the stiffness matrix for any displacement, hence for any external load distribution, is immediately obtained:
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[K] is symmetrical. The nondiagonal terms couple the components of the lateral displacements. On the other hand, the variables Z, and \f/^ remain uncoupled. Turning now to the kinetic energy of the system, it can be calculated as follows. The velocity of the centre-of-mass is VG = X 1 i + Y1 j + Z1 k and the angular velocity about G is Q = \jfxi + i/syj +i//zk . Hence, the kinetic energy of the building can be immediately written as:
Nevertheless, it may be of interest, at least as a training exercise, to check this result starting from the elementary energy of an infinitesimal volume in the solid. The velocity of a current point is:
The kinetic energy is thus given by the following integral:
The crossed term
vanishes. Indeed, as
the origin is taken at the centre-of-mass, r is necessarily an odd function of the coordinates. This result points out the basic property that a frame tied to the centreof-mass of a rigid body allows one to uncouple from each other the translational and the rotational components of the kinetic energy. The remaining non vanishing terms are:
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Discrete systems
The components of the tensor of inertia [I] are found to be:
As expected, the matrix [I], which refers to the principal axes of inertia, is diagonal. It is also of interest to discuss the coefficients Ixx and Iyy in relation to the geometry of the cuboid. For this purpose, it is convenient to express the coefficients in terms of slenderness ratios of the cuboid, or their reciprocals:
If a and B are sufficiently small, Ixx, Iyy can be simplified by neglecting the lateral dimensions of the cuboid and by concentrating on the Gz axis the mass per unit length of the cross-sections m = pab . This gives:
In Volume 2, this approximation will be made for modelling the inertial terms in the equilibrium equations of slender beams deformed in flexion. The mass matrix is:
Principle of virtual work and Lagrange's equations
where use is made of the variable transformation As expected, [M ] is diagonal.
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Chapter 3
Hamilton's principle and Lagrange's equations of unconstrained systems
One of the most famous fundamental principles of theoretical physics is certainly the law of least action enunciated as a universal principle for the first time by Maupertuis (1746): "when a change occurs in nature, the quantity of action necessary for the change is the least possible". Here, we shall introduce the exact formulation of this principle as made by Hamilton (1834), which states that the actual motion of a mechanical system between two arbitrarily fixed times t1, t2 makes stationary the time integral over t1, t2 of the extended Lagrangian of the system. Since this integral is suitably identified with the action of the Lagrangian, Hamilton's principle is thus a principle of stationary action, historically understood as a principle of least action. It can be either postulated as a first principle, or derived from the principle of virtual work. One interesting point for applications in mechanics is that it allows one to introduce kinetic energy in a quite natural way through the Lagrangian. Furthermore, starting from it, Lagrange's equations can be established by using a few mathematical procedures which are also of major interest in other problems in mechanics, which deal with continuous material systems provided with boundaries.
80Discrete Discrete systemss
3.1. Introduction In the preceding chapter, Lagrange's equations of discrete and unconstrained mechanical systems were formulated starting from the variational principle of virtual work. The latter may be considered as being differential in nature, because it deals with variations that are taken at a fixed time t.
Figure 3.1. Virtual variation of the actual configuration between two fixed times Here, we shall introduce another variational principle, where we consider the actual motion of the system between two arbitrarily fixed times t1, t2 and small virtual variations about it, see Figure 3.1. This is the principle of least action in its exact formulation given by Hamilton in 1834 and widely known as Hamilton's principle, which can be stated as follows: Amongst all the motions connecting in due time the given initial configuration C(t1) to the given final configuration C(t2), the actual one yields a stationary value of the time integral of the extended Lagrangian of the system, which is formed by adding the kinetic energy to the work of all the non inertial forces acting in the system. This variational principle is expressed analytically as follows:
Here, 6[ ] denotes the operator of variation. A is the action of the extended Lagrangian £, between times t1 and t2, defined by the integral:
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Hamilton's principle postulates that the action is stationary for any small virtual and admissible displacement. The necessary and sufficient condition for a virtual displacement to be admissible involves the fulfillment of the three distinct following requirements: 1. It complies with the constraint conditions ascribed to the material system. 2. It vanishes at initial and final times t1, t2: Sq(t 1 = Sq(t2) = 0. 3. It is reversible, i.e. if 8q is admissible, so is -Sq . To formulate Lagrange's equations starting from Hamilton's principle, a few basic concepts of variational calculus are needed. Introduced in mechanics by Bernoulli (1717), variational calculus was then developed by Euler and Lagrange. As shown by Euler, it allows one to express as a differential equation the condition that integrals depending on a single or several unknown functions be stationary. Such integrals are termed functionals. Stated briefly, a functional lets a number correspond to a function, like a function lets a number correspond to another number. The fields of application of variational calculus largely extend beyond that of mechanics and it is worthwhile to mention briefly its historical background, largely based on [DUG 55] and the outstanding book by Lanczos, [LAN 70], which was also of great help in the writing of this chapter and the next one. The idea to formulate the laws of physics starting from a principle of minimization is very old indeed. In the first century of our era, Heron of Alexandria made the statement that light travels according to the shortest possible path. This principle was restated much later by Fermat (1657,1662) in its correct form: light propagates from one point to another one by minimizing the elapsed time (in short: light follows the path of least time). In the same order of idea, Maupertuis (1746) postulated his principle of least action, based on theological arguments according to which "the very perfection of the universe demands a certain economy in nature and is opposed to wasting energy". As a specific application to material systems, he postulated that the motion minimizes a quantity he called action and he defined it as the product of momentum and distance, or as the product of energy and time, and this is basically correct. Lagrange produced the first mathematical formulation of such a principle (circa 1760), which was restated by Hamilton (1834) in its general form, applicable to dynamics as well as to statics. Finally, it may be noted that though Hamilton's principle postulates only that action is stationary, it is still referred to as a principle of least action, which extends to dynamical equilibrium the principle of potential minimization, which holds for static equilibrium, as seen in subsection 3.2.2. Actually, it has been proved [GEL 63] that action is truly a minimum in the case of the standard Lagrangian of conservative systems. However,
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the exact nature of the stationary value of action is of no concern in engineering mechanics. 3.2. The calculus of variations: first principles 3.2.1 Stationary and extremum values of a function Let F(u 1 , u2, . . . ,uN) be a differentiable function of N independent variables un, which may be interpreted as the Cartesian coordinates of a point P in a Ndimensional space. By plotting the value of F along the axis of an additional dimension, a surface in a (N + 1)-dimensional space is produced. The variation of F is defined as:
The way to determine if P(u 1 , u2, ...,uN) is a stationary point of F or not, is to explore the vicinity in all the possible directions and to check whether the variation of F vanishes to the first order in every direction, or not:
The expression [3.4] is called the first variation of F. It can be conveniently expressed by scaling the magnitude of Sun according to the following form:
where e is an infinitesimal scaling factor and the coefficients an are the director cosines of the vector Su . Equating to zero the rate of variation S ( 1 ) [F]/£ for any an implies that all the partial derivatives of F vanish also:
NOTE. - Geometrical interpretation of the stationary values of a function According to [3.6], the first variation of F is the scalar product of the gradient d F I du of the function and the vector 5u of virtual displacement, both of them being of dimension N. Restated in the language of geometry, it represents the projection of the gradient onto the varied displacement field. As Su is not a null vector, either the two vectors are orthogonal to each other, or the gradient must
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vanish. However, since the direction of Su is arbitrary, in a particular case it may be parallel to the gradient, and so the latter is necessarily a null vector. Going a step further, a stationary point can correspond to a local extremum (maximum or minimum), or to a saddle point, as sketched in Figure 3.2. In order to specify the nature of a stationary point, it is necessary to proceed with the analysis of variations to the second order. For that purpose, the concept of the second variation 8{2) [ ] is introduced. Assuming that F is a twice differentiable function, 8[F] can be developed up to the second order as:
At a stationary point and up to the second order, d[F] reduces to:
It can be noticed that the second variation S(2) [F] is a quadratic and symmetrical form, which is conveniently written in matrix notation:
Figure 3.2. Nature of a stationary point
The nature of the stationary point P is then discussed in relation to the sign of 8 [F] according to the following criteria: (2)
a is a local minimum. is a local maximum. 3° - otherwise P is not a local extremum. Now, let Aa , n = 1,2,..., Af, be the eigenvalues of the matrix [F (2) ]. According to the results already quoted in Chapter 2 and proved in Appendix 4 paragraph A4.4, it can be stated that: 1°- If
> 0 V n => P is a local minimum.
2°- If
< 0 V n =» P is a local maximum.
3°- If some An < 0, other > 0 => P is a saddle point. NOTE. - Stationary and extremum value When performing the preceding analysis, it has been tacitly assumed that P lies inside the space of configuration. This condition is necessary for the virtual displacements to be reversible. Indeed, the function could otherwise have an extremum value without being stationary. As a practical example, let us consider a ball rolling on a hilly ground. The gravity potential has local minimum at the bottom of the valleys. However, if the ball is restrained along the slope by a rigid wall at such a boundary, the potential will be minimum without being stationary and the virtual displacements will be irreversible, since starting from the wall, it is allowed to go up but not down the slope. On the other hand, [F(1)] is undetermined at such a boundary. EXAMPLE. - Stationary points of z = sinxcosy
The system dzldx = cosxcos y = 0. 3z / 3y = - sinx siny = 0 has two sets of roots x = n 12 + kn, y = nn and x = nn, y =• n 12 + kn. Here, k, n are either positive or negative integers. The nature of these stationary points is analyzed by using the eigenvalues equation: det
which yields: y). At the stationary points, we obtain: 1. For the first set of roots,
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If k and n have the same even/odd parity, \ = A^ - -1, the point is thus a maximum. If parity of k, n differs \ = ^ = +1, the point is thus a minimum. 2.
For the second set of roots,
Now it is found that /^ = -1; ^ = +1 if k, n are of the same even/odd parity, and that > ^ = l ; A 2 = - l if parity of k, n differs. Hence, any root of the second set corresponds to a saddle point. 3.2.2
Static stability
3.2.2.1 Criterion for stability The study of the first and second variations of a function may be rightly considered as the cornerstone of the analysis of static stability of mechanical systems. In statics, the Lagrangian reduces to a potential energy £p, with the sign changed. Since, the first derivatives of £ = -£p stand precisely for the forces acting in the system, a point P of the configuration space where £ (1) [ 0. 2.
P is unstable if at least in one direction repelling forces are induced that tend to remove further the system from P. This is the case either when P is a local maximum S(2) \£p (P)l < 0, or when it is a saddle point.
3.
Finally, the equilibrium point P is said to be indifferent if no force is induced, whichever the direction [a] may be. This is the case when
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It has to be stressed that the local analysis becomes insufficient to settle the actual behaviour of unstable or indifferent systems, since it deals with infinitesimal variations only. Actually, as soon as a real system is repelled sufficiently far from static equilibrium, its behaviour becomes governed by nonlinear forces. From all what precedes, it is possible to state the following mathematical criterion concerning the static stability of a mechanical system: A necessary and sufficient condition for a point of static equilibrium P to be stable is that all the eigenvalues of the matrix S(2) \£p (P) 1 are positive. 3.2.2.2 Static stability of a pair of upside-down and coupled pendulums
Figure 33. Coupled and upside-down pendulums Let us consider the system shown schematically in Figure 3.3. Each pendulum is maintained in the upward direction by a spring acting in rotation and the two pendulums are coupled to each other by a third spring. To simplify the algebra of the problem, the three springs are assumed to have the same stiffness coefficient K. The potential is written as:
Stability of the equilibrium configuration 0, = 92 = 0 is discussed in terms of the sign of the eigenvalues of the matrix:
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Thus, the system is stable only if K < 1. The similar problem referring to the non upside-down pendulums is deduced from the former one simply by changing the sign of K. As expected, this new configuration is always stable. It is also easy to verify that the matrix, whose elements are the second derivatives of the potential, can be identified with the stiffness matrix of the system, once linearized about the static equilibrium configuration of reference. Therefore, the stability of a configuration of static equilibrium is guaranteed, provided the related stiffness matrix is positive definite. Now, when the stiffness matrix depends upon a free parameter, such as K, the threshold of static instability occurs when the matrix becomes singular, passing from positive to negative. This kind of instability is termed divergence. However, in the language of engineering, it is broadly known as buckling. The linear analysis just performed provides the critical value of buckling, (KC = 1 in the present example), but gives no information concerning the direction in which the unstable pair of pendulums will fall down (toward the left or toward the right side?). This point will be further discussed in the next example. Finally, going back briefly to the case of the building resting on elastic foundations already studied in Chapter 2 section 2.4, it is found that when the gravity effect is taken into account, buckling occurs if the stiffness of the foundations are insufficient, or the slenderness ratio of the building is too large. Indeed, referring to the results of subsection 2.4.2, it is not difficult to show that the stiffness matrix including the gravity effect can be written as:
So, the building collapses as soon as a2y - K < 0 <=>
3.2.2.3 Buckling of a system of two articulated rigid bars We consider the system shown schematically in Figure 3.4. The bars (AC) and (CB) are of the same length L and assumed to be rigid. The possible motions are restricted to the plane of the figure. In addition, the end A can slide freely along the Ox-axis whereas B is fixed (position B0). However, the bar (CB ) can rotate freely
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about B0 . Such a constraint is termed articulation, or pinned support. Finally, the two bars are also articulated together at point C. The angle of the base of the isosceles triangle (AC B0) is denoted a. At A, a force F = FQi directed from A to B0 is applied. (A0 0C0) is the non deformed triangle, taken as the configuration of reference. Our purpose is to analyze the static stability of the system in relation to the base angle aQ.
Figure 3.4. Two articulated bars put in compression It is found convenient to divide the study into the following steps. /.
Kinematics of the system
Figure 3.4 shows a deformed configuration of the system. The displacement of point A is found to be:
Then the coordinates of point C are:
Now, the displacement of C is given by:
2.
The Lagrangian of the system In this problem, the Lagrangian reduces to the work of the external force:
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where the superfluous constant term was finally omitted. The external potential is £p = -WF. The generalized force conjugated to a is given by:
da e the moment of F0 about C, with sign changed, as implied by shifting from the right-hand side to the left-hand side in the equilibrium equation. 3.
Positions of static equilibrium and stability
Stable configuration
Unstable configuration
Figure 3.5. Configurations of static equilibrium
The positions of static equilibrium are deduced from the roots of the equation:
Two distinct sets of roots are found, which are denoted Op, a\ according to whether either the integer multiple of n is even, or not. The stability of the states of equilibrium is provided by:
The solutions ap = 2kn are thus found to be unstable. They correspond to a unique configuration in which the two bars are put in a line, with the bar extremities
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Discrete systems
ordered as A, C,B0. The equilibrium solutions a, = (2k + l);r are stable. Again, the bars are put in a line, but their extremities are now ordered as B0, C, A. Such results are immediately understood when looking at Figure 3.5. 4.
Stabilizing effect of a stiffened
articulation
Here, the articulation located at point C is provided with a linear spring acting on the angle (ACS0) = ft. Its coefficient of angular stiffness is denoted K. The restoring moment induced by the spring is M = -K (/? - /?0 ) = -2K (a - a0 ), in such a way that it is zero in the non deformed configuration, see Figure 3.6. The elastic energy is 4 = 2 K (a - a0) and the new Lagrangian becomes:
Figure 3.6. Angular stiffness of the articulation C
5.
Static equilibrium of the stiffened system: symmetrical case (X0 = 0 The equilibrium states ae are given by:
where 4> is the reduced load factor, which characterizes the relative magnitude of the destabilizing over the stabilizing moments. The second derivative of the potential is written in the following reduced form:
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It is realized that an equilibrium state ae *0 can arise only if 4>> 1. If <J>< 1. the only possible state of equilibrium is al = 0. Its stability is easily checked since
When <3E> > 1, two additional states of equilibrium occur, which are symmetrical to each other with respect to the Ox -axis: a2 = ae * 0; or3 = -ae. Thus it suffices to restrict the analysis within the interval [0,;r]. Owing to its definition, a2 has to verify the equilibrium equation: a2 -O sin
Figure 3.7. Potential of the symmetrical system (0^ = 0) Now, the stability of equilibrium can be checked by using the criterion:
Teh above inequality is valid since in the subinterval 0
n
than ae, whereas in the subinterval —
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Discrete systems
3.7 plotted in the domain [-#,+#] refer to five distinct values of the load parameter. The decisive change of shape when passing from a subcritical value of 3> to an overcritical one is obvious. 6.
Symmetrical equilibrium diagram
Figure 3.8. Equilibrium diagram in the symmetrical case
The reduced load factor as a function of the equilibrium angle ae is:
The diagram obtained by plotting this function in the domain [-#,+#] is displayed in Figure 3.8. As expected, the plot is symmetrical with respect to the a = 0 axis. The portion corresponding to <E><1 comprises a single rectilinear branch connecting the origin of the axes to the point [0,1]. In the domain > 1, in addition to the line ae = 0, two symmetrical branches ae*Q arise which increase monotonically when going away from ae = 0 (still remaining in the interval [-#,+#]). Such a diagram is interpreted as follows. Starting from the unloaded
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system, when the load is progressively increased, the system remains in the non deformed state of equilibrium ae = 0, in so far as the load is less than the critical value 3>f = 1. As soon as the load becomes larger than 4> f , the state of static equilibrium of the system evolves along one of the two symmetrical branches ±ae * 0. For this reason, at 3>c = 1, the system is said to bifurcate. However, it has to be stressed that because of perfect symmetry, there is no predisposition to let the bifurcating system evolve along one or other branch of the diagram. 7.
Static equilibrium of the stiffened system: case a0 * 0
Figure 3.9. Potential of the asymmetrical system G.Q ^ 0
Here, axial symmetry of the system is lost, as clearly evidenced by plotting the potential, see Figure 3.9. When a0 *0, the state of equilibrium is governed by the following condition:
Whatever the load may be, the above equation has a single root, which lies in the interval [0,;r] if aQ > 0, and in the interval [-^r,0] if a0 < 0. The smaller the value of O , the nearer to a0 is ae, whereas the larger the value of 4>, the nearer to ±n is ae. Properly speaking, there is no bifurcation anymore, because the diagram comprises a single branch only, which is a continuous line, see Figure 3.10. To conclude on the subject, it may be argued that real systems are never free from any geometrical defect, in such a way that a0 always differs from zero, even by a tiny value, which is still sufficient for determining which branch will be adopted by the
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system when evolving in the post-buckled state. The only remaining problem is thus to determine the sign of the angle a0 induced by the actual defects. Usually, this task is difficult, unless geometrical defects are designed on purpose.
Figure 3.10. Equilibrium plot, 3>(ae), Ofo * 0
3.2.3 Stationary value of a definite integral
Let us consider a functional of the following form:
where y(x) is an unknown function, assumed to be sufficiently regular to be differentiated twice. The interval [jt,, x2 ] is finite and y(x) satisfies the following boundary conditions:
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Figure 3.11. Admissible variation of y0 (x)
The problem is to find a function y0 (x) which produces a stationary value of F(y). To solve it, the starting idea is the same as that previously invoked to find the stationary values of a function. Indeed, we have to equate to zero the first variation £[F(v)] that is induced by a small virtual variation of y0 ( x ) , which is arbitrary but admissible, as illustrated in Figure 3.11. Let us define a function y(x) of the form:
rj (x) is an arbitrary function which can be differentiated at least once, in such a way that the integral [3.12] exists. Moreover, as the functions y0 (x) and y(x) have to satisfy the boundary conditions of [3.13], to be admissible rj(x) has to vanish at the boundaries of the definite integral: admissible
The scaling factor e in [3.14] is an arbitrarily small number. Now, we want to formulate the first variation £[F(y)] that is induced by the arbitrary variation Sy - £rj(x), Sy' - erj'(x), performed at a fixed value of x (S x = 0). It has to be emphasized that such virtual variations have not to be mistaken with real variations dy, dy', since the latter are induced by an infinitesimal variation dx * 0. Accordingly, it is also essential to make a clear distinction between the variation operator 8 [ ] and the differential operator d [ ]. In order to provide a suitable form of ^[F(y)], it is first necessary to show that S [ ] is permutable with the operators of differentiation and of integration:
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Discrete systems
the variation of fix) is defined by:
where / (x) is the varied function. Since fix) and 77 (jc) can be differentiated, one obtains:
Then, the variation of the derivative of fix) is:
Hence, the operators of virtual variation S [ ] and of differentiation d [ ]/dx can be permuted. In a similar way, variation of the integrated function gives:
Hence, S[ ] and J[ ]d!x are also permutable. In particular, we have:
Since variation is performed while keeping x fixed, it may be written as:
and so:
Now, substituting the last expression into equation [3.17], we obtain:
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NOTE. - Variational derivative It is also possible to establish the result [3.18] by introducing the concept of variational derivative, which provides the rate of variation of F (y) as a function of the scaling parameter £ as follows:
Accordingly, the variation 8 [F] is defined as being the variational derivative of F, calculated at e = 0, times e. This method could be preferred to the former one, as being more elegant and providing the desired result more directly. However, the first method has the advantage of making an explicit use of the variations. On the other hand, the expression [3.18] is not suitable as a final form, because rj(x) and rj'(x} are obviously related to each other. Fortunately, an integration by parts provides the opportunity to express d[F] in terms of the sole variation TJ :
Now, the key point is that the boundary term inside the brackets drops out owing to the conditions of admissibility [3.15]. Denoting £(x) the kernel of the integral, the condition for a stationary value of F (y) is thus found to be:
Furthermore, as 7 ( x ) is arbitrary, condition [3.19] must reduce to £(x) = 0, whatever the value of x may be within the interval of integration. Indeed, letting £e[x1,x2], we may choose a variation TJ(X) which vanishes outside a small interval ± y about £ and which is equal to Bly obtain:
within [-/,+/]. And so, we
The smaller the value of y , the less is the error. Moreover, as B is a nonzero finite number, the integral can vanish only if &(x) = 0. Thus, the function y0(x)
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which verifies the variational equation S[F( y0)] = 0 is solution of the differential equation, known as the Euler-Lagrange differential equation:
EXAMPLE. - Minimal surface of revolution
Figure 3.12. Surface stretched between two coaxial circles of same radius R We are interested in determining the surface of minimum area, which is stretched between two coaxial circles (Oz-axis) of same radius R, see Figure 3.12. The circles are located at heights -H and H. As the problem is clearly one of cylindrical symmetry, one is led to search first for one or several radial functions r(z), which comply with the conditions r(± H) = R and which provide a stationary value of the following integral:
Denoting the kernel of the integral by f ( z , r, /), it is found that:
The Euler-Lagrange equation is As this differential equation is nonlinear, there is no general mathematical method to solve it analytically. Nevertheless, its particular form suggests that a solution could be sought in terms of exponential functions. Moreover, owing to the symmetry of the problem with respect to the plane z = 0, one is led to try a
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hyperbolic cosine r(z) = acosh(Az). The surface so defined is called a catenoid. By substituting of this function into the differential equation, it can be verified that such a solution holds, provided that a = If A . The remaining constant A has to be adjusted to the boundary condition r(H) = R. This produces the transcendental equation cosh (/I//) = AR , which has either zero, a single, or two roots, depending on the value of the slenderness ratio H/R. It is easy to express H/R in terms of A :
Here, A, must be larger than one, whereas H/R must be less than one.
(a)
Equation for A
(b) H/R versus A
(c) Generatrix line of the stationary catenoids Figure 3.13. Surfaces of stationary area
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Discrete systems
Occurrence of two roots implies that of the two stationary values of the surface area, as illustrated in Figure 3.13, which one refers to H/R = 0.6. Figure 3.13a is a graphical representation of equation cosh (AH) = A R, where the presence of two distinct roots /i,,/^ is evidenced. Figure 3.13b is a plot of H/R versus A. Its maximum corresponds to the critical ratio (HI R\. =0.6627. Figure 3.13c shows the generator lines of the catenoid for A,,/^, which are the corresponding catenary curves of equation:
Hence, at each slenderness ratio H/R less than the critical value, we obtain two surfaces, which according to Figure 3.13 may be labeled as thick and thin waists respectively. NOTE. - Surfaces of minimal area In a similar fashion as in the case of the stationary values of a function, to determine the specific nature of the stationary solutions of the present problem it would be necessary to analyse the second variation of the area functional. However, this problem is far from simple and is clearly beyond the scope of the present book. Let us simply mention that there is an abundant literature on the subject, because, since the pioneering work by Lagrange, this field of research has continuously aroused a large interest in mathematics, physics and architecture. The interested reader may be referred in particular to [DAR 41], [NIT 89], [OSS 86]. NOTE. - Materialization of surfaces of minimal area Surface tension plays a major role in the mechanical equilibrium of thin liquid films. Experiment shows that work of surface tension is proportional to the film area (cf. Volume 3, Chapter 1). As a consequence, surfaces of local minimum area correspond to the states of statically stable equilibrium. Thus, the catenoids defined above can be conveniently materialized by using thin films of soaped water, supported by two coaxial circular rings, see for instance [ISE 92]. Clearly, the thickness of the film has to remain sufficiently small, in such a way that surface tension largely prevails on force of gravity, a condition which is easily fulfilled in practice. From such an experiment, it can be shown that the "thick waist solution" is stable and thus is found to stand for a minimum area solution, whereas the "thin waist solution" is not observed, and is thus found to stand for a saddle point, because it cannot be a maximum, for obvious reasons. By increasing progressively the distance between the rings, it may be observed that the film pops when H/R becomes larger than the critical value. Indeed, in this range the A equation has no root anymore. As soon as H/R is larger than the critical ratio, the physical solutions are two circular disks supported by the rings. Since, in the function [3.12] the second derivative of y(x) must exist, the variational method is clearly unable to find this kind of solution.
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3.3. Variational formulation of Lagrange's equations 3.3.1 Principle of virtual work and Hamilton's principle Let us consider a system of N particles Pj with masses mj, described by ND independent generalized displacements q\, qa,• • • q, which are defined in an inertial frame of reference. Let us assume that [q] is the solution of the dynamical problem (i.e. [q] provides the actual time histories of the particles). Now, let \6q\ be a virtual and admissible variation of [q], defined over the time interval [t1, t2]. Hence, it verifies the conditions:
The principle of virtual work (cf. Chapter 2, section 2.3) can be expressed in terms of generalized quantities, providing the matrix equation:
Hamilton's principle may then be conveniently introduced, by integrating equation [3.22] between f, and t2:
Now, as a definition of kinetic energy £K, integration of the inertial term has to produce its virtual variation between /, and t2. The key to establish this statement is to perform an integration by parts. Indeed, this allows one to express the variation in terms of the virtual variations of the velocity vector [Sq]. Taking into account the admissibility conditions [3.21], we obtain:
Gathering together results [3.23] and [3.24] yields Hamilton's principle, which takes the following variational form:
V
/
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Discrete systems
Expression [3.25] may be seen as the variation of a single functional A. As already pointed out in Chapter 1, subsection 1.2.1.2, the definite time integral of any physical quantity defines the action of this quantity over the interval of integration, hence A is a functional of action. Thus, according to Hamilton's principle, the generalized displacement vector which satisfies the matrix equation of motion leads to a stationary value of the functional of action. NOTE. - Nature of the stationary value of the functional of action Here again, to investigate the exact nature of the stationary values it would be necessary to consider the second variation of action. However, as already pointed out in the introduction (subsection 3.1), if the object is restricted to the formulation of the equations of motion, the nature of the stationary paths is of no practical concern. The interested reader is also referred to the quite enlightening lecture on the principle of least action by P. Feynmann [FEY 63]. 3.3.2 General form of the Lagrange's equations Going back to expression [3.25], in the general case the variation of kinetic energy can be written as:
The variation [3.26], can be expressed in terms of \Sq\ solely, by integrating by parts the second term. This yields:
In the above equation, the term within the brackets is zero, owing to the conditions of admissibility, and so the total variation of the action [3.25] becomes:
Since the 8 qk form a set of independent variables, while the time interval f,, t2 is arbitrary, the relation [3.27] is equivalent to the system of the second order differential equations in time:
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or, in matrix notation :
re here the Euler-Lagrange's equations, widely termed in mechanicsWe recognize here the Euler-Lagrange's equations, widely termed in mechanics WeWe recognize the Lagrange's equation of motion, which were already established in Chapter 2, section 2.3, starting from the principle of virtual work. A few particularly remarkable forms of such equations are discussed in the next subsections. 3.3.3 Free motions of conservative systems Let us consider a discrete mechanical system, actuated only by forces deriving from a potential £p (qk) and from kinetic energy. As already stated in Chapter 2, Lagrange's equations take the particularly elegant form [2.57,58]:
where £ (q It should also be noted that if the Lagrangian does not depend upon a variable qk, the corresponding generalized momentum pk = d Lid qk is a motion invariant. More generally, any variable qk which is not explicitly present in the Lagrangian is termed a cyclic variable. Existence of cyclic variables in a system greatly facilitates the study of motion, as will be outlined in the next example. On the other hand, use of the Lagrangian allows us to rewrite Hamilton's principle in a concise and elegant form:
Thus A coincides with the action of the Lagrangian of the system, as stated in the introduction, section 3.1. Another valuable property of the Lagrangian is that Lagrange's equations are unchanged when a total time derivative of any function of displacement and time is added to the Lagrangian. Indeed, let g(qk,t) be such a function. If dgldt is added to a Lagrangian, the action is changed as follows:
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Discrete systems
Since the virtual l displacements
are admissible, it is verified however that:
This remarkable result may be stated as follows: The Lagrangian of a system can be simplified by omitting or adding the total time derivative of any function of generalized displacements and time, judiciously defined to simplify its expression. On the other hand, it is also of interest to show that a system is conservative if its Lagrangian does not depend explicitly upon time, in agreement with the results already established in Chapter 2, subsection 2.2.4. Indeed, the time derivative of the Lagrangian may be written as:
Then, the second term of this expression can be transformed by using the corresponding Lagrange's equation to obtain:
and so:
where "C" stands for a constant, whatever its value may be. Let us discuss first the simplest and most usual case where the displacement vector [q] is referred to an inertial frame. Thus £k is a quadratic form of [q] and it can be easily checked that:
By substituting this result in the preceding expression, we are readily led to the expected conclusion that mechanical energy is a motion invariant:
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105 105
£KO and £po designate the kinetic and potential energies imparted initially to the system, in accordance with the initial conditions of motion [<7(0)J, [g(0)]. If now [q] refers to an accelerated frame, in most cases kinetic energy depends explicitly upon time, as may be anticipated from relations [2.22] and [2.23]. Accordingly, mechanical energy is not constant. However, it is worth mentioning that there are exceptions of practical interest. Indeed, if a mechanical system is set in rotation, at a constant angular speed, and is described by using a corotating frame, the transport and the mutual kinetic energies do not depend explicitly upon time and mechanical energy is conserved, as illustrated later in Chapter 4, subsection 4.3.2 and in Chapter 6, subsection 6.4.4. EXAMPLE. - The spherical conservative pendulum Let us consider a particle subject to a uniform gravity field, which is constrained to remain at a fixed distance R from a fixed point O in a 3-D space, see Figure 3.14. Motion is described by using the angular variables 6, (p of the spherical coordinates. The kinetic and potential energies are immediately found to be:
Figure 3.14. Spherical pendulum
It is convenient to formulate the problem in terms of dimensionless quantities, by using the following scaling factors:
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Discrete systems
where ( p 0 , P0, &0 are set by the initial conditions. The dimensionless Lagrangian is written as:
It can be noted that as 9 is a cyclic variable, Lagrange's equations are:
The second equation implies that Physically, the generalized momentum pe is identified with the angular momentum of the particle about the vertical Oz-axis. As it remains constant, we can conclude that if 6 is initially zero, motion is strictly restricted to the vertical #0 plane. Now, if 90 is not zero, 9 increases when the particle draws toward the poles (p=Q,tt . It may be anticipated that the latter are never reached, in such a way that motion is restricted inside a band limited by two parallels perpendicular to the vertical direction. It is worth mentioning that the equations of motion can be integrated because the problem has as many invariants (
In many cases, the motion \qB (f)] of the system (B) is practically independent of the motion of (A) and can be assumed to be known. For instance, this is the case of
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the satellite orbiting the Earth, described in Chapter 1, subsection 1.2.1. For such an approximation, £KB and £pB reduce to known time-functions and £pc is a function of [qA ] and time. The motion of (A) is thus governed by the simplified Lagrangian: [3.33] Thus, the terms £KB, £pB can be omitted, since they reduce to a total derivative of a time-function, cf. subsection 3.3.3. On the other hand, £pc is related to a fluctuating force field. As the Lagrangian [3.33] is explicitly time dependent, the mechanical energy of the system (A) is generally also time-dependent. The rate of variation of £A is found to be:
Transforming now the second term of the above expression by using Lagrange's equation, we get:
then, assuming that [qA ] refers to an inertial frame we obtain:
Finally, substituting the expression [3.33] of £A in the above relation, it is found that:
^mA = £KA +
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produce the potential of system (A), including the effect of the system (B), coupling being still neglected. Lagrange's equations are expressed in terms of the Lagrangian:
In the presence of an external force - induced by the fluctuating external potential linear in [qA ]- Lagrange's equations are written as:
where the subscript (
A
) has been dropped to simplify notations.
Incidentally, it has to be mentioned that a system can also be forced by prescribing the motion of some of its degrees of freedom. We will come back to this point in the next chapter for more detailed discussion. However, it is thought interesting to give here a simple example. EXAMPLE. - Seismic excitation of an SDOF mass-spring system
Figure 3.15. Seismic excitation of a mass-spring system
The mass-spring system of Figure 3.15 is rooted to the ground and is subjected to a seismic disturbance. As a result, a displacement D(t) is impressed to the rooted point in the axial direction of the spring. As in practice the mass of ground which is set in motion turns out to be much larger than the mass of the oscillator, D(t) may be assumed to be independent of the oscillator motion. Formulation of such a problem results in a forced equation of the following kind:
where the displacement Z of the particle is defined in the inertial frame.
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3.3.5 Nonconservative systems The change of mechanical energy in a nonconservative system is balanced by the work of nonconservative forces and of external forces. Let \Q ([q], [q], t)J be a nonconservative generalized force. Lagrange's equations are now written in the following form:
The force vector [Q] must be given in an explicit way, as a material law. For instance, viscous friction is modelled as a force vector of the kind:
where [C] is the matrix of viscous damping. NOTE. - Work of nonconservative forces As indicated in the introduction, it is possible, at least conceptually, to define an extended Lagrangian by adding to the kinetic energy the work of all the forces acting in the system except that of the inertial ones, cf. equation [3.2]. However, the task of building such an extended Lagrangian is often difficult and even intractable, since in many instances the analytical form of the nonconservative forces is not suited to perform the required integration. Fortunately, it is soon realized that in order to formulate the Lagrange equations, the only necessary task is to make explicit the variation of the work, which is immediate since SW = \Q\ \8q\. The same remark holds for any kind of forces. An elementary example is provided by the damped harmonic oscillator. The equation of motion can be easily established by starting from the variation of the extended Lagrangian, as follows:
Thus, it can be concluded that presence of nonconservative forces do not arise major difficulties for writing down the equations of motion of mechanical systems.
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Chapter 4
Constrained systems and Lagrange's undetermined multipliers
This chapter is devoted in its entirety to the description of the so called "Lagrange undetermined multipliers method". This very elegant and powerful mathematical procedure was devised by Lagrange to deal with constrained systems without having to first eliminate the superfluous variables. The availability of such a method magnifies clearly the advantages of using the analytical, instead of the vectorial, approach to establish the equations of motion of most material systems. Indeed, the task of eliminating dependent variables of a system is often tedious, or even inextricable. Hence, it is soon realized that in the absence of the Lagrange's multipliers method, the efficiency of analytical mechanics would be severely limited to a fairly restricted class of problem. Here, the mathematical aspects of the method are described and then the physical contents are illustrated by discussing several examples of practical interest, which involve holonomic constraints, first scleronomic and then rheonomic. Since the latter case is also classically treated by using a change of reference frame, this provides us with a good opportunity to discuss these two points of view for handling mechanical systems subjected to prescribed motion.
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4.1. Introduction In the preceding two chapters, Lagrange's formalism was introduced within the restricted field of unconstrained systems; that means mechanical systems described by independent coordinates, or displacements, in the same number as the degrees of freedom of the system. In this chapter, Lagrange's formalism will be extended to the case of discrete systems which comply with constraint conditions and are described by using a set of over-determined coordinates. As explained in Section 4.2, the key point is to use supplementary unknowns called Lagrange's undetermined multipliers, which are associated with the constraint conditions. Moreover, we shall see that such auxiliary unknown quantities can be interpreted physically in terms of generalized forces. As could be anticipated, these forces are identified with the reaction forces induced by the constraint conditions. This very elegant method may apply not only to holonomic constraints but also to some nonholonomic constraints, of the differential type in particular. Section 4.3 is devoted to the special case of systems with coordinates defined in an accelerated frame. The equations of motion of such systems may be established either by considering rheonomic constraints or more directly by expressing the kinetic energy in the inertial frame, in terms of the variables referred to the accelerated frame. As expected, Lagrange's equations obtained by one or other of these methods show the presence of additional inertia forces, which are induced by the acceleration of the "relative frame", namely the generalized transport forces, and the Coriolis forces, also termed gyroscopic forces, already identified in Chapters 1 and 2. 4.2. Constraints and Lagrange multipliers 4.2.1. Stationary value of a constrained function Let F(«p u2, ...,M y v )be a function of N variables, which is sufficiently regular. Let us suppose that the variables are related to each other by holonomic constraint conditions:
Therefore, it can be concluded that one variable, denoted Uj, is superfluous whereas the others form an independent subset. As already shown in Chapter 3, a stationary value of F implies:
However, such a condition does not imply here that all the coefficients of the variations Sun are zero, simply because one variable is not independent of the
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others. Our first thought to bypass such a difficulty would be to use the constraint condition (5H) in order to eliminate the superfluous variable, thus coming back to the problem already solved. Actually, this method is entirely justified and advisable in many cases as being the simplest one. However, in many other instances, it is difficult and even impossible to eliminate one variable by solving the (SR) equation. Indeed, this entirely depends upon the analytical expression of the specific relation one has to deal with. Moreover, in the absence of any objective criterion to identify which specific variable uj it is judicious to eliminate, the method becomes rather artificial and less advisable. This is typically the case when the constraint condition is symmetrical with regard to all variables. Let us consider for instance the condition of a rigid connection as written in Cartesian coordinates:
The choice of the independent variables is clearly arbitrary and the elimination of one variable, whichever it may be, results in the loss of the initial symmetry, unless spherical coordinates are used. The idea devised by Lagrange is much more elegant and, open to many applications of practical interest. At first, it is noted that any variation of the constraint condition is automatically zero:
Once more, since we deal with an over-determined set of variables, such an identity does not imply that the coefficient of each individual 8un vanishes. This having been recognized, it is possible to modify the condition to be verified by the constrained function for a stationary value, by combining linearly the relations [4.1] and [4.2]:
where an additional variable A, called an undetermined Lagrange multiplier, is judiciously introduced. Now, regrouping together the coefficients of each individual 6 un, we obtain:
At first sight it might be thought that little is gained by doing so, since the 8 un are still not independent of each other. Actually, on the contrary, the expression [4.3] is the key to the problem, and this specifically because the value of the a priori unknown A may be adjusted at will. Of course, the judicious choice is to set its
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value in such a way that the coefficient of a given 8 un vanishes, no matter which one. Thus, the value of A is taken as the root of the linear equation:
where the subscript j is solely complying with the condition that d^l d uj does not vanish at the stationary point P. Not only does equation [4.4] give the value taken by the unknown A, but also it allows one to eliminate the now superfluous term referring to q} in condition [4.3]. Assuming that the un are suitably numbered in such a way thaty = N, condition [4.3] reduces to:
Since condition [4.5] involves only N - 1 independent variables, it is now equivalent to the N - 1 equations:
Furthermore, starting from these results, the method may be restated more synthetically. Indeed, we have:
However SA = 0, since the value of A is fixed through the equation [4.4]. It is thus recognized that: In order to determine the stationary values of a function F constrained by a holonomic condition (91), it is sufficient to cancel the variation of F + ^(91), as if the N variables ofF were independent. Doing so, N equations of the form [4.6] are obtained and then one of them, chosen at will, is used to specify the value of Lagrange 's multiplier A. Extension of the method to the case of any number L < N of holonomic conditions (9^); i = 1,2,..., L is straightforward. Indeed, the variations of are equated to zero as if the N variables of F were independent, and L equations of the form [4.6] are used afterwards to specify the values of the L Lagrange multipliers. It has to be emphasized that, by this process, a system involving N - L independent variables is replaced by a system comprising N + L variables, namely
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the N variables of the free function and the L Lagrange multipliers. They are governed by N + L equations, namely the N equations of the type [4.6] and the L constraint conditions (SR,). Finally, there is still a possible way to go further into the synthetic formulation of the method, which is finally restated in the following fashion: The variables un and A.f may be considered as forming an independent set ofN + L variables and the variations of F + At IR, with regards to all the variables are equated to zero. Cancellation of the coefficients of the 6 un provides the N equations of the type [4.6] and cancellation of the coefficients of the 8kf provides the L constraint conditions. EXAMPLE. - Stationary points ofz = sinxcosy, constrained by x + y = cc Stationary points of this function were already studied in Chapter 3 subsection 3.2.1, however in the unconstrained case (free function) only. One has to realize that the occurrence of a holonomic constraint modifies the free function. In the present example this can be readily checked by using the following trigonometric identity:
Moreover, here the constraint condition allows one to define analytically the constrained function as a new free function, by eliminating the superfluous variable. Thus, the stationary points of the free form may be analysed according to the method already described in the last chapter. A single family of roots is found:
Nevertheless, in most instances the constraint condition does not allow one to define the new free function. If such is the case, the Lagrange multipliers method is the sole way to solve the problem. Here, we obtain:
A is then eliminated between the first two equations. This yields:
The constraint condition is then used to determine the values of x and y which provide stationary values of the constrained function. Obviously, the results are identical to those derived by applying the first method.
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4.2.2. Nonholonomic differential constraints In the preceding subsection, we were concerned with the variation of holonomic constraints. It is noticed that they correspond to exact differentials, since the coefficients of the variations 8 un are the partial derivatives of a single function. Let us consider now the case of a particular family of nonholonomic conditions, whose variation takes the differential form:
Here, the coefficients \ are no more the derivatives of a same function, in contrast with [4.2]. It should be noted that differential conditions enter into this family of constraints:
Indeed, for any small and admissible variation S ql, Sq2, the following relation holds:
The Lagrange multipliers method applies to such constraints by considering the following variation:
which is immediately extended to the case of L conditions, in the following form:
where all the variables un may be considered to be independent. 4.2.3. Lagrange's equations of a constrained system In order to establish the equations which govern the stationary values of a constrained functional, the approach is the same as that already adopted with regard to constrained functions. Let a mechanical system be described by N generalized coordinates (or displacements) and by L holonomic constraint conditions:
N is equal to the DOF number of the unconstrained system (the so called free system) and L is assumed to be less than N. The Lagrangian of the free system is
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denoted ,£ and A designates its action. Cancellation of 6 [A] for the independent 6qn, provides Lagrange's equations of the free system. Turning now to the constrained system, its number of DOF is N - L, thus only N - L variations 6 qn are independent from each other. Incidentally, the over-determined case N > L is discarded here and will be briefly discussed later; see example 2 of this subsection. At any time, the variations of the constraint conditions are:
Accordingly, the action of the constrained system is postulated to be of the following form:
It must be pointed out that Lagrange multipliers At(t) are now as yet undetermined time-functions, instead of undetermined constants, as is the case of constrained functions. Indeed, the constraint conditions have to hold at any time (if not otherwise specified) during the motion of the mechanical system. Owing to equations [4.10], it is immediately recognized that the variation of the constrained system is identical to that of the free system:
where the convention of implicit summation on the repeated index t is used and will be used also hereafter to simplify notation. is then expressed in terms of the variations 8qn :
Regrouping now in 8 \A'] all the factors pertaining to a given 8qn , we have:
Once more, L multipliers Ae (t) are specified in a suitable way to cancel the factors of L coordinate variations 8qn, no matter which. Since, the constrained
a 8 Discrete systems
system has (N - L) DOF, the factors of the (N - L) remaining S qn also vanish. Therefore, the following N Lagrange equations governing the constrained system are obtained:
and these are completed by the L constraint conditions. Here again, the method consists in transforming a system of (N - L) equations involving (N — L) independent variables into another equivalent system of (N + L) equations involving (N + L) interrelated variables, namely the displacements qn (t) and the undetermined Lagrange multipliers A( ( t ) . Restated in another fashion, the Lagrangian of the constrained system is written in the following form:
A priori, the multipliers At should not be considered as additional degrees of freedom, since the calculation of the variation of A' is performed with regard to the qn only, which are considered as being the only variables of the problem. The undetermined multipliers Ae are calculated afterwards by using the constraint conditions. However, as in the case of constrained functions, we may perform again the variations as well with regard to the Af as with regard to the qn. Of course, the coefficients of SAf reduce to the constraint conditions. Whichever is the viewpoint adopted, one is led to the same system of equations:
The constrained Lagrangian £ can be written as the extended form:
The quantity Wcc, which arises as a direct consequence of the presence of constraint conditions, clearly stands for some kind of work. As it cannot be related to the work done by any force already acting in the free system, the only possibility left is to interpret Wcc as the work done by the constraint reactions. Furthermore, in so far as
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119
the conditions (91,) are referring to holonomic constraints, Wcc is a function of the generalized displacements and of time only. Hence, a quite natural idea is to let it correspond to a potential of constraints, which has to be added to the potential of the free system:
But now, as a natural consequence of the formalism involved in the method of the undetermined Lagrange multipliers, it becomes necessary to reconsider our first point of view concerning holonomic constraints, according to which they restrict the number of DOF of the system, as described in Chapter 1. In contrast, according to the present point of view it is now stated that: The holonomic constraint conditions do not restrict the number of degrees of freedom of the unconstrained system, but they modify its potential. Going a little bit further, by differentiating the potential of constraints with regard to the displacement variables qn, generalized constraint reactions R are produced. R is interpreted as being the resultant of the generalized reactions induced by all the individual constraint conditions, which are prescribed to the n-th degree of freedom of the free system. Indeed, differentiation yields:
It may be argued that, from the physical point of view, the present interpretation of the holonomic constraints is preferable to the mathematical definition given in Chapter 1, subsection 1.1.4.1. Actually, it is easily recognized that no realizable constraint can be made perfectly rigid. Consider for instance the mathematical condition qi - qj = 0. It can be conveniently modelled physically using an elastic spring connecting the displacements #, and qj. The potential of constraint is identified with that of the spring:
where Kcc designates the stiffness coefficient of the spring. The two constraint reactions acting on the constrained degrees of freedom are immediately found to be:
Since any part of the system must be in equilibrium, the force balance cannot depend upon the value of Kcc. In particular, the reactions must be independent
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of Kcc. On the other hand, the larger is Kcc the smaller is qi - qj . In other words, the elastic connection tends to an ideal rigid constraint as Kcc tends to infinity. To conclude, it may also be noted that the signs attributed to A and to the constraint conditions in the potential of constraint are arbitrary. This has no effect on the generalized and physical reactions, as illustrated in the next example. EXAMPLE 1. - Plane rotation of a particle about a fixed point
Figure 4.1. Circular motion of a particle
Let us consider a particle which is left at first to move freely in the plane (P). Polar coordinates r, 0 are used as generalized displacements. The Lagrangian of the free system reduces to:
Then, the particle is constrained to a circular path about O. Accordingly, the constrained Lagrangian is written as:
Lagrange's equations are found to be:
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Substituting now the condition r = R into the first equation, Lagrange's multiplier A(t) = —M R02 is found. A priori, we could wonder about the physical nature of the constraint work thus obtained, because in the present form A looks more like an inertia force (cf. Chapter 1 subsection 1.2.4) than as a force deriving from a potential. However, it may also be argued that the circular motion of the particle is uniform, since no external torque is acting on it. Indeed, the second equation provides the solution 0 = Q , where Q is constant. Hence, the reaction may still be interpreted as deriving from a potential of constraint, given here by:
As calculated according to the above procedure, Rcc is the generalized reaction exerted by the fixed point on the free system. Formally, Rcc and r are a pair of conjugated quantities. In physical terms, as r is a linear displacement, Rcc is a force. It is immediately identified with the centripetal force exerted physically by the fixed point to balance the centrifugal force caused by the acceleration of the particle. To conclude this elementary exercise, the following comments may be made: 1. Signs ofLagrange multipliers and constraint conditions It can be easily checked with the present example that the sign convention attributed to the multipliers or to the constraint conditions has no influence on the calculated reactions. Adopting Wcc =-/l(r-R), the first Lagrange equation becomes:
The two remaining equations are left unchanged, hence:
The constraint reaction is now given by:
2.
Nature of the constraint work and general relativity
The short discussion above on whether the constraint work is kinetic or potential in nature in the present example, points towards an equivalence between inertial and potential forces, indicating the presence of a link between accelerated motion and
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gravity. Indeed, even in everyday life, spinning motion is often used to mock up gravity. On the other hand, recognition of such a natural link was the starting "happiest thought" of Einstein in building up the theory of general relativity. As clearly discussed in layman's terms in [GRE 99], based on force measurements only, any accelerated motion can be described as a gravitational motion, provided suitable gravitational (or potential) forces are defined. As a consequence, "all observers, regardless of their state of motion, may proclaim they are stationary and the rest of the world is moving, so long as they include a suitable gravitational field in the description of their own surroundings. As a corollary, from the mechanical viewpoint all the vantage points are on equal footing". 3.
Physical modelling of the constraint condition
Turning back to the concrete problem of realizing the constraint condition involved in the present system, the particle will be connected to the fixed point O by using a slightly deformable string with an equivalent stiffness coefficient Kcc made as large as possible, but remaining finite in any case. To obtain radial equilibrium, the centrifugal force MQ2r has to be balanced by the stiffness force —Kcc (r-R), where r is the length of the extended string. Accordingly, ASr is identified with the virtual variation of the work of the spring and the following results are easily obtained:
Provided that Kcc is sufficiently large, the relation r = R holds in practice. EXAMPLE 2. - Static and hyperstatic equilibrium of a rigid rod Let us consider a rigid and uniform rod loaded by its own weight, which is resting on rigid knife edge supports, located at x - -L/2 and at x = Xj, as shown in Figure 4.2. The exercise consists in determining the support reactions as a function of x}, by using Lagrange multipliers. Considering first the unsupported rod, the analysis is restricted to small vertical displacements:
where x is the abscissa of a current point of the rod axis and where use is made of the two following independent variables: 1.
Z0 is the vertical displacement of the centre-of-mass O, located at the centre of the middle cross-section of the rod.
2.
6 is the rotation angle about 0, in the plane of the figure.
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Figure 4.2. Rigid rod resting on knife edge supports
In statics, the Lagrangian of the free rod subjected to gravity reduces to the term of potential energy of gravity:
a result which could be easily anticipated. Now, the rigid supports maintaining the rod in static equilibrium are equivalent to the constraint conditions:
Defining two undetermined Lagrange multipliers A 1 ,A 2 , the Lagrangian of the constrained system is written as:
from which the following Lagrange equations are derived:
whence:
Then, the generalized reactions exerted by the supports, which refer to the conjugate variables Z0 and 6, are given by:
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Using these generalized quantities, it is possible to return to the physical reactions which are applied to the rod by each support. Indeed, the work of reactions is unchanged in a transformation of coordinates (or displacements) and its variation as well. Hence we have:
Writing Z0 in terms of the "physical" displacements at the supports Zl and Z2, we obtain:
The following remarks are appropriate to check the relevance of the physical reactions thus obtained: 1. Clearly, the condition of global equilibrium of the rod implies that the resultant of the physical reactions must balance the weight of the rod, whereas the resultant moment about O must be zero. Obviously, these conditions could have been used to determine the support reactions, without having to resort to the method of Lagrange multipliers. 2.
As expected a priori, if x1 = L/2, the two reactions are equal, in agreement with the symmetry of the problem.
3.
If xl = 0, R = Mg and R = 0. Thus, the total weight is applied to the support located just below. This result could also be anticipated, since the centre-ofmass of the rod is located at mid-span.
4.
If x1 < 0, the sign of R2 is reversed and in the case of unilateral supports, the rod becomes unstable according to a rocking mode.
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Figure 4.3. Reaction torque tending to the distribution of a Dirac dipole
5.
If x1 — > - L / 2 , the two supports merge together and R1, R2 tend towards infinity with opposite signs. To interpret the physical meaning of such a result, we proceed as follows. Consider the support located at xl =-L/2 + £/2 . We have:
As shown schematically in Figure 4.3, the resulting moment of the reactions about the middle point x
Hence it is
noted that if e tends to zero, the reactive torque tends to MR = [MgL)/ 2, i.e. the finite value which allows one to balance exactly the external moment of the weight resultant; this result will be revisited later in terms of singular distributions, cf. Volume 2. Thus, provided the supports are bilateral, the rod can be maintained in stable equilibrium even if the lever arm is arbitrarily small, in principle at least. Such a support condition, known as a clamp, combines a zero displacement and a zero rotation condition. Finally, let us consider the cases where the rod is maintained by more than two supports. As easily understood, it becomes impossible to determine the support reactions if keeping in the framework of the mechanics of rigid bodies. Actually, with such support conditions, the number of unknown quantities entering in the Lagrange's equations of the constrained system becomes larger than the number of DOF of the free system. As a consequence, the system of equations to be solved is underdetermined. To illustrate this important point, let us consider the case of three supports located at x,,x2,x3, see Figure 4.4.
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Figure 4.4. Rigid rod with hyperstatic support conditions
The constrained Lagrangian is written as:
Lagrange's equations involving Lagrange multipliers are:
The first equation refers to the force balance. Once more, it teaches us that weight and resultant of support reactions have to be exactly balanced. The second equation refers to the moment balance. Obviously, it becomes mathematically impossible to determine the individual reactions induced by each support, as the number of equations is now less than the number of variables. More generally, any mechanical system constrained by a number of holonomic conditions which is larger than the number of degrees of freedom of the unconstrained system is known as a hyperstatic system. As a final comment, it is worth emphasizing that according to the definition given just above, the concept of hyperstatics is merely a by-product of modelling solids as perfectly rigid bodies. As already stressed, such a model is adopted because of its mathematical convenience. However, one has to be aware that hyperstatics does not fit within physical reality, so far as the number of DOF is concerned. Indeed, as real materials are always deformable, the degrees of freedom a material body cannot be enumerated. Moreover, since deformations occur at the supports, tiny as they may be, the reactions at each support can be unambiguously determined, whatever the number of supports is. Clearly, this kind of calculation has to be performed in the framework of the mechanics of continuous (deformable) solids, which is the object of Volume 2.
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4.3. Prescribed motions and transformation of reference frames In many instances, material systems are excited by prescribing the motions of some DOF. A typical example is provided by the case of a building subjected to a seismic shake. It is assumed that the basement accompanies the motion of the ground. Here, we will describe two distinct methods to deal with such dynamical problems, each one having its own interest. The first method consists of modelling the prescribed motions as rheonomic constraint conditions. This allows one to obtain the equations of motion in terms of generalized displacements referring to an inertial frame. As already shown, Lagrange multipliers are providing the constraint reactions. Here, they identify with the forces which induce the prescribed motions, when impressed to the unconstrained system. The second method consists in defining an accelerated reference frame tied to the prescribed motion. Then, the equations of motion are expressed in terms of the relative displacements, which refer to the accelerated frame. The generalized forces which are equivalent to the prescribed accelerations are now derived from the kinetic energy of the system, as determined in the inertial frame, but expressed in terms of the relative variables. 4.3.1. Prescribed displacements treated as rheonomic constraints A known displacement D} (t) impressed to a variable qj can be formulated as the rheonomic constraint condition:
Since rheonomic constraints belong to the class of holonomic constraints, the problem can be studied by using the method of Lagrange multipliers. Considering thus a system with a free Lagrangian of the type £ = £c - £p, we define the potential of constraint 8CC = - A j ( q j - D j ( t ) ) . system is:
Lagrange's equations can be written as:
The Lagrangian of the constrained
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The only new point with regard to the scleronomic case is that the nature of the generalized reactions may differ, depending whether the kinetic energy is an explicit function of q., or not. This is illustrated in the two following examples. EXAMPLE 1. — Seismic excitation of a mass-spring system Let us consider a mass-spring system in which the spring connects the mass M to a point O. The displacement of O, as defined in a given inertial frame, is a known function D(t) of time. Referring to this frame, X designates the displacement of M along the Ox-axis, counted from the equilibrium position at rest, and Y is the relative displacement as defined in the frame tied to O, see Figure 4.5. K is the stiffness coefficient of the spring and L is the length of the unloaded spring, which is specified here only to help in visualizing X and Y in Figure 4.5. Let X0 be the displacement of O in the unconstrained system. The constrained Lagrangian is written as:
Lagrange's equations are:
Figure 4.5. Seismic excitation of a mass-spring system
The constraint reaction is found to be A = K(X - D(t)] = KY(t). Hence the work of the constraint is found to be a potential and the constraint reaction is a stiffness force.
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EXAMPLE 2. - Rotating mass-spring system
Figure 4.6. Rotating mass-spring system
Consider a mass-spring system, which rotates at angular speed Q(t) about the Oz-axis. The mass slides without friction along the direction of the spring force, taken as the Ox-axis. L is the length of the unloaded spring, see Figure 4.6. The motion of the free system is described by using the rotation angle 6 of Ox about Oz and the elongation X of the spring. The Lagrangian of the constrained system is written as:
Lagrange's equations are found to be:
Hence, in contrast to all the former examples, it is found here that the Lagrange's multiplier is governed by the differential equation:
Physically A is the moment about Oz of the resultant of two distinct inertia forces, each one acting in the plane of rotation and perpendicularly to Ox, namely the Euler force -M Q(X + L), and the Coriolis force 2M QX . It can also be noted that A has the dimension of an angular momentum, which can be quite naturally interpreted as the action of the resulting moment of the Euler and Coriolis forces.
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Turning now to the second equation, it can be re-written as:
This form shows a negative stiffness term, which corresponds to the centrifugal force, as already discussed in example 1 of subsection 4.2.3. Clearly, this component is oriented in such a way as to move the mass away from the centre of rotation, whereas the spring restoring force is tending to bring it back. Thus the resulting stiffness coefficient (K -M Q2) can be either positive or negative, depending on the value of Q in relation to the critical spin velocity Qc =^K/M . Finally, assuming here that Q is constant, the term reported on the right-hand side of the equation is the permanent centrifugal force induced by the rotation of the mass, when located at the non-deformed state of the spring X = 0. As this component is merely a constant, it may be preferable to eliminate it by transforming the displacement variable:
X0 is the spring elongation which allows one to balance the permanent centrifugal force. Accordingly, the force equation is re-written as:
As written in terms of X, the equation describes the axial motion of the mass referenced to an unstressed initial state (£2 = 0), whereas the equation written in terms of £ describes the same motion as referenced to a permanent state ^ 2 ^ 0 , which is initially stressed, or "prestressed", by the permanent centrifugal force. Finally, as in example 1 of subsection 4.2.3, the centrifugal force may again be interpreted as deriving from a potential. 4.3.2. Prescribed motions and transformations of reference frame It is also worthwhile analysing the problem of prescribed motions by adopting another point of view to that of the rheonomic constraints. It is also possible, and even far more usual, to consider two distinct reference frames, namely: 1. The inertial frame ( R ' ) , in which the prescribed motion is defined. 2.
The accelerated frame (R) which accompanies the prescribed motion. Let us consider a system described by the displacement vector [q], as referred to
(R) .In (R'), the system is described by the displacement vector [q']. Our purpose
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is to formulate Lagrange's equations in terms of [q] and its derivatives. Once more, it is stressed that though such equations are written in terms of relative variables defined in the accelerated frame, they still govern the dynamic equilibrium in the inertial frame (R') . The key is to start from the Lagrangian referring to the inertial frame and written in terms of the inertial variables [q'],[q']• Then, a suitable transformation of coordinates of the type [q'] —» [q] is carried out to obtain the same Lagrangian, but written in terms of the relative variables [q],[q]- It is immediately recognized that only the inertial terms have to be transformed because the others can be written directly as a function of the relative variables [ q ] . Starting from the general form of Lagrange's equations:
Qn is the n-th component of the resultant of the forces other than inertia forces. Kinetic energy is the sum of three distinct components £$:', £$: and
Developing this expression and inverting the sign of it, the following expression of the generalized inertia force is obtained:
It is of interest to gather the various terms of this expression in such a way as to sort out the three following components, already identified several times in the context of rotating motions:
132 1.
Discrete systems Relative component
This component can be separated from the others by assuming transformation laws ffj which do not depend explicitly upon time, hence describing a change of coordinates and not a change of reference frame. 2.
Transport component
This component can be separated from the others by assuming that the system is at rest in the accelerated frame qn = 0, V n. 3.
Mutual component
This component gathers the remaining terms. It can be re-written in a skewsymmetrical form and it can immediately be checked that Whence:
are known as the gyroscopic coefficients. [G] is the gyroscopic matrix which is obviously skew-symmetrical. It has to be pointed out that gyroscopic forces are conservative since they do no work. Thus, we have:
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EXAMPLE 1. - Seismic excitation of a mass-spring system
Figure 4.7. Seismic excitation of a mass-spring system
The example already analysed in subsection 4.3.1 is revisited here by using the method of the reference frame transformation. Now the point O is subjected to a prescribed acceleration F(t) in the Ox direction, see Figure 4.7. Writing the equation of motion in terms of the variables Y and Y which refer to the accelerated frame, we obtain:
Thus, it is found that prescribing the acceleration F(t) to the relative frame is equivalent to forcing the oscillator by the external inertia force -M
r(t}.
EXAMPLE 2. - Rotating mass-spring systems
We consider once more the example 2 of subsection 4.3.1 where angular velocity 9 = Q is assumed to be constant. The Cartesian coordinates referred to the inertial frame are found to be:
Kinetic energy in the inertial frame can be calculated directly or by using the general transformation formulas [2.21, 2.22, 2.23]. Whichever method is used, the two following terms can be identified: Transport energy: Relative energy: Mutual energy is zero. The Lagrangian is:
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Of course, the axial equation of motion is the same as that already found. EXAMPLE 3. - Pendulum attached to a spring
Figure 4.8. Pendulum attached to a spring
Let us consider a pendulum in which mass is connected to the fixed point through a spring, as shown in Figure 4.8. L is the length of the unstrained spring. Restricting the study to motion taking place in a vertical plane, we have to deal with a 2-DOF system, which may be described by the angular variable 9 , defined in the inertial frame Ox, Oz and by the translation variable X, defined in the rotating frame, which goes along with the mass-spring system. Incidentally, it should be noted that in this problem we are making use of an accelerated frame, of which motion is not prescribed since 6 is unknown (cf. Chapter 1 subsection 1.1.3.3.). Potential and kinetic energies are found to be:
where transport and mutual energies are defined in the same way as though 6 (t) were a prescribed function. The Lagrangian can be written as:
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Thus the following equations of motion are obtained:
Once more, the permanent term in the right hand-side of the force equation could be eliminated. On the other hand, in the moment equation, the cross term is recognized as a Coriolis force, which couples nonlinearly the angular and the translation velocities of M. EXAMPLE 4. - Mass tied to a wheel through springs
Figure 4.9. Mass tied to a rotating wheel through springs As shown in Figure 4.9, the circular wheel rotates about its axle at constant angular speed Q . The point-mass M is connected to the wheel by linear springs K acting radially along two orthogonal directions defining a Cartesian frame in rotation with the wheel, of which the unit vectors are i, j, k . When the system is at rest, M coincides with the centre O of the circle and elongation of the springs vanishes. X, Y designate the coordinates of M referring to the rotating frame. We consider also the inertial frame Ox'y' and the X', Y' coordinates. (X', Y' ) and (X, Y) are related to each other through the rotation matrix:
where C = cos(i2?) ; S = sin (120
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Incidentally, it can be checked that the transformation matrix may be identified with the Jacobian matrix of the coordinate transformation. There is no difficulty in calculating the kinetic energy of M using the general expressions [2.21, 2.22, 2.23]. Indeed after some elementary algebra, it is found that:
However, it is still simpler to perform a direct calculation by determining the velocity of M in the inertial frame, in terms of the relative coordinates. This can be done either by using the vectorial relation:
or by making use of the coordinate transformation written in the matrix form, to obtain by differentiation:
In the inertial frame, the kinetic energy of M is thus found to be:
The potential energy is:
Finally, the two following dynamic equations are obtained:
As expected from [4.25], the gyroscopic forces couple the two equations in a skew-symmetrical way. On the other hand, the rotation induces a centrifugal force on each DOF. The dynamical behaviour of such rotating systems is rather intriguing and will be discussed later, in Chapter 6 subsection 6.4.4 of this volume and again in Volume 4. As a final remark it is also of interest to realize that if the equations of motion are expressed in terms of the Cartesian displacements X', Y' defined in the Ox'y' inertial frame, they reduce to those of two uncoupled harmonic oscillators:
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Hence this system may serve as nice example to illustrate that motion may be modified drastically according to the reference frame adopted to describe it.
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Chapter 5
Autonomous oscillators
In the previous four chapters of this book we have been concerned with the mathematical modelling of mechanical systems which results in a discrete set of time-differential equations. The object of the following five chapters will be to describe the mathematical procedures available to solve such analytical models and to develop the generic features of the dynamical responses, which are of major interest for the mechanical engineer. Because of the theoretical and practical importance of this field, broadly referred to as vibration theory, the literature on the subject is particularly abundant. Amongst the engineering oriented textbooks devoted mainly to linear vibrations, let us quote in particular [DEN 56], [ROC 71], [CLO 75], [MEI 67], [MEI70], [BIS 60], [GIB 88], [DEN 89], [WEA 90], [GER 97]. Most material systems, initially at rest in a given state of equilibrium, are likely to vibrate as soon as they depart from it. Vibrations are said to be free whenever they are triggered solely by delivering initially some amount of energy to at least one degree of freedom of the system. Such free vibrations reveal themselves as extremely useful for investigating the response properties of the material system to any kinds of excitations, as will be illustrated in specific examples in the following chapters of this book. This chapter is devoted to the analysis of the free vibrations of SDOF systems, starting with the so-called linear or harmonic oscillator which is the simplest vibrating system. It serves as a convenient introduction to several basic concepts, among them the concept of natural modes of vibration, which will be further extended in the next chapter to the case of MDOF systems. Furthermore, because of their relative simplicity, SDOF systems give us a good opportunity to make a first incursion into the nonlinear domain of dynamics. Indeed, even a limited study is enough to illustrate significant differences between the dynamical behaviour of linear and nonlinear systems.
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5.1. Linear oscillators 5.1.1 Mechanical oscillators A mechanical system is said to vibrate, or oscillate, when it is swinging back and forth about a permanent state of equilibrium, static or not. This state, which will be used as a reference to describe the vibration, is assumed to be stable. Indeed, vibration occurs as the result of the conflicting action of the inertial forces, which cause the resulting motion to overshoot the permanent state of equilibrium, and that of the restoring stiffness forces, which are tending to bring it back. Mechanical oscillators are the simplest systems of this kind. They correspond to devices which can be reduced to a mass-point and a spring. The back and forth motion takes place along a geometrical line which is prescribed a priori. Hence, according to the definition given in Chapter 1, a mechanical oscillator is a SDOF system. Clearly, such a concept arises as a mathematical idealization of a few real, but particularly simple, devices. Two archetypes, already introduced in the preceding chapters, are the mass-spring system and the simple pendulum. The basic difference between them is in the nature of the potential energy, and thus of the stiffness force, involved in the system. In masspring-like systems, restoring forces are induced by the elastic deformation of a body, while in pendulum-like systems, restoring forces arise as the consequence of a coupling between the motion and a permanent force field, the weight for instance. More generally, the permanent force field is said to prestress the system and the pendulum may be considered as a prestressed, or initially stressed oscillator. The linear, or harmonic, oscillator is the linearized version of such systems. Its motion is governed by a linear time-differential equation of second order and constant coefficients, which is written in the following canonical form:
It is recalled that K is the stiffness coefficient, C is the viscous damping coefficient and M is the mass, or inertial, coefficient of the oscillator. In this chapter, study is restricted to the case of autonomous oscillators. Therefore, there is no external excitation, except that motion is triggered by imparting initially some amount of mechanical energy to the system:
The motions of an autonomous system are said to be free in order to draw a clear distinction with the forced motions, which are induced by an external fluctuating (time-varying) excitation. Motions of the autonomous harmonic oscillator are thus governed by the homogeneous equation:
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This equation is provided with initial conditions of displacement and velocity:
Referring now to the notions developed in the first three chapters of this volume, it can be stated that solutions are oscillatory in nature, so far as the coefficients K, C, M of equation [5.4] comply with the following conditions: 1. Stiffness coefficient K> 0 The stiffness force -Kq is derived from a potential £p (g), which is written first in terms of a generalized coordinate denoted q. However, when the aim is to study the small motions about a position q0 of static equilibrium, one is naturally led to expand the potential in Taylor series with the point of expansion qQ:
Now, by definition (cf. Chapter 3, subsection 3.2.2) a state of static and stable equilibrium is such that:
As pointed out just above, the motion can remain confined to an arbitrary small neighbourhood of the static equilibrium q0 only if it is stable. Within such an infinitesimal domain, potential energy can be approximated by the quadratic and positive definite form:
2. Mass coefficient M > 0 Similar considerations hold concerning inertial forces. Kinetic energy can be written directly as a quadratic positive definite form:
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3.
Discrete systems
Viscous damping coefficient |C| < Cc
Viscous damping corresponds to a particular material law, which has been already introduced in Chapter 1, subsection 1.1.5. Moreover, as shown in Chapter 2 subsection 2.2.4.2 and again in Chapter 3, subsection 3.3.5, this force is nonconservative in nature, since its work describes the exchange of energy between the oscillator and its external environment. When C is positive, the viscous force dissipates the mechanical energy of the oscillator. It may be anticipated that if the damping coefficient is larger than a certain critical value, the mechanical energy given initially to the oscillator will be entirely dissipated before a single oscillation could be completed. The critical damping coefficient Cc marks thus the transition between the oscillating motions and the non-oscillating motions. Finally, if C is negative, the oscillator is pumping mechanical energy from the external surroundings. As a direct consequence, vibration amplitude is steadily increasing with time, in principle without any limit. However, it is clear that beyond a certain amplitude, the nature of the physical problem is drastically changed, since the real system becomes nonlinear. 5.1.2 Free vibration of conservative oscillators In this subsection C is assumed to be zero and equation [5.3] reduces to:
5.1.2.1 Time-histories of displacement Analytical solution of equation [5.9] is straightforward. It is recalled that the solution is searched in terms of the following trial functions:
By substituting expression [5.10] into [5.9], the following characteristic equation is obtained:
The two roots of equation [5.11] are:
The solution of equation [5.9] may thus be written in one or the other of the following equivalent forms:
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The two constants a + , a _ , or A,B are determined by specifying the initial conditions [5.4] of the motion and the following real time-history is found to be:
Another equivalent form, which is of interest for discussing the physical meaning of the result [5.14], is obtained by letting:
Formula [5.14] is then re-written as:
£Q is the mechanical energy imparted initially to the oscillator. Provided that K and M are both positive definite, the displacement [5.16] varies sinusoidally at the pulsation:
Pulsation, also called angular frequency, is expressed in radians per second. It may be emphasized that ft>, depends only upon the intrinsic properties of the oscillator, and not on the amplitude of motion. For this reason col is called eigenpulsation, or natural pulsation of the oscillator, depending to which, the mathematical or engineering flavour, one is inclined. It is also recalled that pulsation Q), frequency f and period T are interrelated as follows:
The frequency is expressed in Hertz (Hz), or in cycles per second (cps). The phase angle ^ of the displacement depends upon the initial conditions of motion. Physical interpretation of it will be made even clearer in Chapter 9, when studying forced oscillations in the frequency domain.
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Discrete systems
Finally, amplitude of the oscillation is found to be proportional to the square root of the mechanical energy given initially to the oscillator (which is a constant of the motion), and in the reciprocal ratio of the square root of the stiffness coefficient. Such a result is easily understood, as energy of the linear oscillator is a quadratic form of displacement and of velocity. NOTE. - Negative stiffness The resolution of the equation [5.9] made above can be extended without difficulty to the case of an "oscillator" of negative stiffness. Solution [5.14] becomes:
It is thus found that the system does not oscillate about the position of static equilibrium but moves away from it according to an exponential law. 5.1.2.2 Phase portrait A convenient way to study the dynamical behaviour of conservative oscillators, without embarking on the task of solving the equation of motion, consists in taking advantage of the invariance of energy by studying the level lines of energy:
They are plotted in the (q,q) plane, known as the plane of phase and the level lines are called phase trajectories. The set of such plots forms the phase portrait of the oscillator. It provides us with a picture that contains most of the interesting information about the dynamical behaviour of the oscillator, namely: 1. K and M being both positive definite, the phase portrait is a family of ellipses, which depend upon the energy parameter <^, see Figure 5.1. The ellipses are centred at origin, i.e. at the representative point of the state of static equilibrium. The midaxes are immediately found to be:
Such a portrait clearly characterizes back and forth motions, of which amplitude is proportional to the square root of the energy imparted initially to the oscillator. The trajectories in the phase plane are then in the homothetic ratio ^ .
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Figure 5.1. Phase portrait of a conservative linear oscillator
2. Since ellipses are closed lines, motion is a periodic oscillation, independent of the amount and the particular way energy is imparted to the system. As already pointed out in Chapter 2, subsection 2.2.4.1, the period of motion is given by:
3. The ellipses do not intersect each other so that only one phase trajectory corresponds to a fixed energy level £0. In other words, all possible physical motions are passing through the same dynamical states depicted by the £0 phase trajectory. The sole difference concerns the times at which a given state lying on the phase trajectory is reached. This depends upon the initial conditions which are prescribed to the oscillator. Of course, all these results are in agreement with those already established in subsection 5.1.2.1. However, the interesting point here is that they are now obtained without solving the equation of motion. Hence, the present method is well suited to deal with nonlinear conservative oscillators, as will be illustrated in section 5.2. Moreover, the phase portrait pictures conveniently the dynamical behaviour of the oscillator, discarding the less important details arising from the particular initial conditions adopted.
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5.1.2.3 Modal analysis Study of the linear conservative oscillator gives us a good opportunity to introduce the embryo of modal analysis, which is a particularly powerful method used to analyse a large number of mechanical problems, as we shall see step by step throughout the whole book. The cornerstone of modal analysis is to search for a solution of equation [5.9] in terms of the following complex trial function:
Clearly, expression [5.22] is very close to that already selected in subsection 5.1.2.1 as a trial solution of equation [5.9]. However here, we are no more interested in fitting the general solution to the initial conditions of a particular problem. Substituting expression [5.22] into equation [5.9] yields the modal equation:
Obviously the trivial solution (p = Q is of no interest to us. Existence of other solutions is thus conditioned by the roots of the characteristic equation:
As K and M are assumed here to be positive definite, the roots are a pair of real and opposite quantities. Going a little bit further, it is not difficult to understand that though the corresponding trail functions are linearly independent of each other, they still characterize the same vibration, save on the sign of the time. Now in the present problem, there is no difference between negative or positive time. Indeed, equation [5.9] is left unchanged when the sign of time is reversed and solutions [5.23] have not to comply with any initial conditions. In other words, as far as the conservative modal problem is concerned, time is reversible and has no origin. Therefore, we will retain only one of the two mathematical solutions of the characteristic equation [5.24], for instance the positive root, which identifies with expression [5.17]. To conclude on this point, it is realized that even if the base functions eia)l and e ia^ are linearly independent from each other, both of them stand for the same physical quantity, namely a harmonic vibration at pulsation ct){. On the other hand, magnitude (p of the vibration is clearly arbitrary. It is thus found convenient to normalise the modal vibration, by using for instance the additional condition:
More generally, it would be possible to normalize the modal motion by using any definite complex number, denoted (p^ . Again, this stresses that magnitude and particular phasing of the modal vibration are of no importance. The natural mode of
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vibration of the undamped harmonic oscillator is thus characterized by the natural frequency and the modal shape #?,, which in the case of a SDOF system reduces to an arbitrary complex number. A few consequences of such properties are worthy of further discussion. At first, returning to motion described in the real time domain, it is understood that any harmonic vibration defined by the complex amplitude:
allows one to describe a real time-history by retaining either the real or the imaginary part of it. Which particular choice is made is of no importance since the results thus obtained will differ only by a phase angle, which itself is arbitrary. Then, substitution of (p}e'(Wt'+''f) into the expressions of potential and kinetic energies would lead to complex quantities of the kind:
However, defining such quantities is not useful, since they cannot be interpreted in terms of real energies. A far more fruitful point of view is to consider the complex amplitude [5.26] as a complex vector. Accordingly, in the direct line with the results established already in Chapter 2 (cf. relation [2.71]), the quadratic forms defining kinetic and potential energies are interpreted as scalar products of complex vectors, cf. relation of definition [2.34]. In the particular case of a SDOF system, this yields:
Now substituting the complex amplitude [5.26] forX , the following functionals of energy are produced:
which reduce to quadratic forms of the magnitude of the modal vibration. Kl and M, are termed modal stiffness and modal mass respectively. They are also often referred to as generalized stiffness and generalized mass. The natural pulsation is given by the ratio of potential over kinetic energy functionals, a result known as the Rayleigh quotient:
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Such a ratio is clearly independent of the magnitude of vibration, which is arbitrary. The relations [5.29] may also be used to norm the eigenvector with respect to mass, in such a way that ^M^i = Mip? = 1, the modal stiffness being given by ipiKipi = Kq>l =0)^. Such a procedure, known as mass-normalisation, is found especially relevant in modal testing. Finally, it must be pointed out that the interest of introducing the modal quantities, just defined above, will become fully justified in Chapter 6, when discussing the case of multi degrees of freedom systems. 5.1.3
Free vibration of nonconservative linear oscillators
5.1.3.1 Time-histories of displacement Resolution of the equation of motion follows exactly the same steps as in the conservative case. As a preliminary, it is found convenient to rewrite equation [5.3] in the equivalent form:
where the dimensionless quantity £, termed reduced damping or damping ratio is defined as:
Considering first the dissipative case, for which £, is positive, the roots of the characteristic equation are found to be either complex or real according to whether £, is smaller than one, or not. Hereafter, the roots are written in the complex form directly applicable in the case of small damping:
Now, in the case ^ < 1, the time-history is found to be:
The pulsation of the sinusoid is given by:
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Figure 5.2. Free vibration of a lightly damped oscillator (subcritical damping) Figure 5.2 displays the typical profile of such a time-history, which is a sinusoid modulated by an exponentially decaying function. In most instances, serious vibration risks may occur when the systems are lightly damped, (values of £, typically less than a few per cent). In such cases, the formula [5.34] can be replaced by the simpler result:
The time-history [5.36] differs from that of the conservative case only by the exponential decay of the magnitude of displacement. Moreover, expression [5.36] may be re-written in a more suitable form to help physical interpretation:
Ta is the time-scale which characterizes the exponential decay of the vibration magnitude and y is the phase angle of the sinusoid. On the other hand, mechanical energy of the oscillator reduces to the simple expression:
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The relative decay of energy from one oscillation to the next one is:
Turning now to the case of large damping gl > 1, the time-history is found to be:
Figure 5.3 displays the typical profile of such a time-history. As expected, the mass-point is no longer oscillating about the static state of equilibrium, but is tending exponentially toward it. Such an oscillator is said to be overdamped and gc = 1 is the critical damping ratio which marks the boundary of two quite contrasting dynamical behaviours of the oscillator. Accordingly, subcritical damping refers to the domain £, < 1 and overcritical damping refers to the domain £, > 1.
Figure 5.3. Free vibration of an overdamped oscillator (overcritical damping)
Incidentally, the time-history related to critical damping is obtained by using the trial solution:
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Substitution of the function [5.41] into equation [5.31], leads one to:
Solution is produced by cancelling out every term of the above expression:
Hence, it is found that the free motion of a critically damped oscillator is of the same non-oscillating nature as that of the overcritically damped oscillator. Turning now to the case of negative damping, the analytical results established above are still valid, provided the sign of £, is changed. Clearly, the oscillator is dynamically unstable since its mechanical energy is steadily increasing with time, in an exponential way. The motion ceases to be an oscillation as soon as ^ < -1. Considering for instance the critical negative damping g c =-1, the motion is still given by the law [5.43], provided the sign of A = *y, has been changed:
It is of interest to note that the most important feature arising from the presence of nonconservative forces in a mechanical system is the loss of symmetry with regards to time which is thus introduced. In any real free motion, time elapses irreversibly, starting from a given origin, defined as the time at which energy is imparted to the system. In rather picturesque words, it may be said that time is now provided with an arrow. In Volumes 2 and 4, we will revisit the consequences of a lack of symmetry concerning the properties of the mathematical operators of mechanical systems. 5.1.3.2 Phase portrait
Figure 5.4. Phase portraits of a subcritical oscillator
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Figure 5.5. Phase portrait of an overcritical linear oscillator
Unlike the case of conservative oscillators, the phase portrait of a damped oscillator cannot be obtained directly, but only as a by-product of the time-histories. In particular, the phase trajectories become dependent on the initial conditions, in contrast to the undamped case. Nevertheless, for a given oscillator, all the phase trajectories are featured the same way. In the subcritical range £ i | < l , they are shaped as spirals which are either converging to, or diverging from, the origin (i.e. the state of static equilibrium), depending on the sign of £,, see Figure 5.4. Indeed, if damping is positive, the trajectories run from the representative point of initial conditions towards the representative point of static equilibrium. When damping is negative, they run in the reverse direction, even if the representative point of the initial conditions is "almost" coincident with the origin. Then, the origin of the phase plane is said to be either an attracting or a repulsing point, according to whether damping is positive or negative. Figure 5.5 displays the phase portrait of the overcritical oscillator. Of course, in this domain also, the origin of the phase plane is either an attracting or a repulsing point, according to the sign of damping coefficient. 5.1.3.3 Modal analysis We proceed along the same steps as in the conservative case. The characteristic equation now produces two distinct complex roots:
As in the conservative case, it is sufficient to retain only the root of which the real part is positive, defining thus the natural pulsation of the vibratory component of
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motion, also called natural pseudo-pulsation of the oscillation. The imaginary part of the complex pulsation [5.45] characterizes either the exponential decay, or increase in the magnitude of the oscillation, (depending on the sign of £j):
A more detailed discussion of complex modes is postponed to Volumes 3 and 4 of this book, where a few mechanisms of exchange of energy between a fluid and a vibrating structure are described. 5.1.4 Static instability (divergence or buckling) As could be expected, if the stiffness coefficient becomes negative, the oscillator becomes unstable. Indeed, the roots of the characteristic equation are now:
As a consequence, the mass-point is brought away from the rest position in a monotonical way, whatever the rate of dissipation due to damping may be. Damping only controls the speed of removal, see Figure 5.6. This may explain why buckling instability, which is static in nature, is also termed divergence.
Figure 5.6. Time-histories of a linear buckled oscillator (negative stiffness)
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5.2. Nonlinear oscillators Though the present book is essentially devoted to the study of linear mechanical systems, it is nevertheless worthwhile to make a few incursions into the nonlinear domain, so far as only limited theoretical developments and calculation procedures are required. Indeed, description of a few major features of the dynamical behaviour of nonlinear systems is sufficient to clearly mark the profound differences which exist between linear and nonlinear systems. This also gives us a good opportunity to emphasise that even if one has to deal with a nonlinear system, it is still advisable to start by analysing first its behaviour in the linear domain, before embarking on the analysis in the nonlinear domain. 5.2.1 Conservative oscillators The analysis of the dynamical response of autonomous conservative SDOF systems is straightforward, even in the nonlinear domain. As already pointed out in subsection 5.1.2.2, the problem can be solved analytically by using the first integral of energy. As in the linear case, the phase portrait is given by the level-lines of constant mechanical energy £m(q,q) = <^- As a general rule, according to whether they are closed or open, such lines characterize either periodic or non periodic motions. Moreover, the phase trajectories have no multiple points and do not cross each other. Consequently, under fixed initial conditions, there corresponds only one possible motion. As an exception worthy of note, a few specific trajectories, termed separatrices, may have multiple points. It has to be stressed now that occurrence of multiple points does not lead to any uncertainty concerning the motion of the system, since they can be reached only after an infinite amount of time. Thus, even if the separatrices are closed paths they do not stand for periodic motions. On the other hand, let us recall that the time-histories of motion, periodic or not, can be obtained implicitly in the form of the definite integral:
In most instances, potential energy £p (g) does not vary monotonically, thus at least one extremum value exists. Let qt be is a position of stable static equilibrium. By definition, qe is such that the following conditions hold:
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Thus if £Q lies within a certain neighbourhood of £p (qe), equation £p (q) = <^ has two roots denoted q^
and ^r raax , which bracket qe. In contrast with the linear
case, the corresponding interval of energy is finite instead of infinitesimal. Within such an interval, motion is necessarily periodic in nature, since the particle is caught inside a sink of potential. The period is given by the definite integral:
Since <^ = £p (fl^ ), the kernel of the integral has at least one singular point. As a consequence, before carrying out the numerical (or analytical) integration, it is suitable to make a variable transformation to remove the singularity. In contrast with the linear case, the period is a function of the magnitude of vibration. In addition it has to be noted that the integral [5.50] is not always convergent. This occurs in particular, along the separatrices. EXAMPLE 1. - Mass-point tied to a tensioned string
Figure 5.7. Mass-point tied to a tensioned string
The system studied here is schematized in Figure 5.7. The equivalent stiffness coefficient of the string is denoted K and its mass is assumed to be negligible in comparison with the mass M of the particle, which is located at mid-span of the string. Z designates its transverse displacement in the plane of the Figure. 2L0 is the length of the unstretched string. L(Z) is the actual half-length in the deformed state. It is assumed that L(0)> L0 in such a way that the string is prestressed in tension:
£Q is the string deformation associated with the initial stress.
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Making use of the dimensionless displacement £ = Z / Z>(0), we have:
The nonlinearity of the problem is thus geometrical in nature, arising as a direct consequence of Pythagoras' theorem. The linear approximation consists in neglecting any variation of the string length with £ Elastic potential energy may be written as:
where the constant is adjusted in such a way that £p (O;/?) = 0.
Figure 5.8. Elastic potential of the pre-tensioned string
This potential is plotted in Figure 5.8 for three values of the stretching coefficient fi>l. Provided £ is still sufficiently small, the potential can be expanded as a Taylor's series truncated to the fourth order:
where
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Hence, in the absence of permanent tension /?= 1, the elastic potential reduces to the quartic form:
Thus, it appears clearly that oscillations can be nearly linear provided the string is sufficiently tensioned and vibration amplitude is small enough, in such a way that the quadratic term of the potential dominates largely the terms of higher order. The quartic term is obviously responsible for the nonlinear behaviour of the oscillator. It dominates the quadratic term, as soon as vibration amplitude is larger than 2Ay£0 , a value which can be much less than one. Let us consider for instance a steel wire of section area 5 = 1 mm2 and length L^ = 1m. It can be shown that the equivalent stiffness of the wire in tension is K = ESIlQ = 210s Nm-1 ,where E = 2.1011 Pa is the Young's modulus of steel. The wire being tensioned at 1kN , it is found that e0 =510-3and 2^ = 0.14. NOTE. - Elastic potential and work of the tensioning (axial) force Instead of calculating the strain potential of the wire, it is also relevant to calculate the work done by the elastic forces in a transverse displacement Z. The tensioning force is:
The work resulting from the infinitesimal displacement dZ = j3L^d^ is:
Hence the work done to change from configuration £ = 0 to configuration £ = £, is found to be:
The potential, obtained by changing the sign of the work, is identical to the elastic potential, as expected a priori. However, the interest in performing the present calculation is not only to check the self consistency of the mathematical formalism, but also to clarify the origin of the quadratic term present in the potential. Indeed,
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even in the approximation L(£) = L(0), the tension T(0) is doing a certain amount of work when M is displaced by Z. This work is found to be:
Thus, it is found that the potential is the sum of two quite distinct terms, namely: 1. A prestress term, induced by the permanent tension of the wire. It reduces to a quadratic form, characterizing thus the linearized system. 2. An elastic term, induced by the change in length of the wire associated with the change of configuration. Approximated here as a quartic term, it governs the nonlinear behaviour of the oscillator.
Figure 5.9. Phase portrait of the mass-point on a tensioned string
As shown in Figure 5.8, the resulting potential £p has a single extremum, which is a minimum located at £ = 0. The phase portrait is made up of closed lines and all the possible motions are periodic in nature, see Figure 5.9. Referring to the Rayleigh quotient [5.30], it can be anticipated that the natural frequency of the oscillator increases with vibration amplitude £j, since £p (£,) is increasing faster than a parabola when departing from £ = 0. The Lagrangian of the system is written as:
Autonomous oscillators
Accordingly, the equation of motion, broadly known as Duffing's includes a cubic stiffness term:
159
equation,
The stiffness of the linearized oscillator is provided by stretching the string initially, hence it vanishes for the critical value /?c = 1. The corresponding natural frequency is:
The nonlinear frequency/^ is obtained by using relation [5.48], which takes here the form:
where £m is the magnitude of the oscillatory displacement and y=£/£m.
Figure 5.10. Nonlinear resonance frequency of Buffing's oscillator
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Then, the change of variable u = -Jl-y is used to avoid singularity of the kernel aty = 7. Accordingly, the integral is finally written as:
In Figure 5.10 the reduced frequency f = fNL/fL is plotted versus %m for several values of the tensioning parameter /?. The required integrals have been calculated numerically by making use of the software MATLAB. It can be verified that the increase of the natural frequency with vibration amplitude becomes barely noticeable, at least in the scale of the figure, as soon as £0 = ($ -1 becomes larger than 5%. This because if .f is sufficiently large, the prestress force largely prevails on the elastic force. At the opposite, if f0 is less than 0.05%, the frequency becomes very sensitive to the vibration amplitude. Incidentally, such results indicate that when playing a string instrument, it is advisable to provide the strings with a fairly high tension. Piano strings are made of high strength steel wire tensioned up to around 109Pa (ikN/mm2pleading to a typical value of £0 =0.5%. Finally, in the absence of initial stress (e0 = 0), the static equilibrium position £ = 0 is indifferent, from the linear viewpoint; the system is, however, stabilized by the geometrical nonlinearity. EXAMPLE 2. — Articulated bars, prestressed in compression
Figure 5.11. Mass-point linked to two rigid bars preset in compression
The system is schematized in Figure 5.11. The two rigid bars are linked together by an articulation whose mass M is much larger than their own mass. On the other
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hand, each bar is connected to a fixed point through a linear spring of stiffness coefficient K, acting in the axial direction of the bars. In the unstretched state, the total length of a bar including its connected spring is Z^. When the bars are set in coaxial configuration, the springs are compressed and the length becomes L
Figure 5.12. Potential of the compressed bars
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The potential is displayed in Figure 5.12, for a few values of /3. The presence of three extrema, hereafter denoted (A),(B),(C), is clearly conspicuous. (A) refers to the static equilibrium position %A = 0, which is unstable. (B) and (C) refer to the symmetric positions of static equilibrium, which are stable:
Figure 5.13 is the phase portrait of the system. It comprises two distinct families of closed trajectories, of which the physical meaning is rather obvious. Indeed, if the imparted energy <^ is less than the potential barrier £p (0), the particle remains trapped in one of the potential sinks which are centered at (B) and at (C). Hence, the phase trajectories are made of a pair of closed orbits, each one surrounding (B), or (C). The specific orbit selected by the particle depends upon the initial conditions of motion.
Figure 5.13. Phase portrait of the compressed bars
On the other hand, if <% is larger than £p (0), the pair of closed paths merge together to produce a single closed orbit surrounding the three positions of static equilibrium. Clearly the particle is now provided with a sufficient amount of energy to jump the potential barrier centered at (A). Finally, when £^ is exactly equal to £p (0), the corresponding trajectory has a double point at (A). This particular trajectory is the separatrix, which marks the boundary between the two families described just above. Calculation of the period of motion occurring along the separatrix, would result in an infinite value. As a matter of fact, the repulsing point (A) can never been reached in a finite time, whatever the actual initial conditions may be.
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EXAMPLE 3. - The plane pendulum
The potential can be written as £p = MgL(l-cos#). Accordingly, it vanishes at the lowest position (0 = 0). Provided the cyclic nature of the angular variable 9 is disregarded, it can be stated that the pendulum has the following positions of static equilibrium: [ stable equilibrium :0S = 2nn [unstable equilibrium :9, = (2n + \)n
Figure 5.14. Phase portrait of the plane pendulum
The phase portrait displayed in Figure 5.14 comprises two distinct families of paths. The family of the closed orbits is related to energy levels ^ less than 2LMg .They stand for periodic oscillations about the lowest position. Calculation would show that the natural frequency is a decreasing function of the oscillation amplitude. This is simply because |sin#| is less than \9\. Energy levels higher than 2LMg result in open paths, which stand for non-periodic motions, in which the pendulum is revolving around the fixed point. The separatrix has an infinity of double points which correspond to the angular positions of unstable static equilibrium. As a final remark on this system, it is noted that the cyclic nature of 9 can be accounted for simply by rolling the phase portrait of Figure 5.14 on a circular
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cylinder of unit radius. In this way, all the phase trajectories are suitably transformed into closed paths and the periodic nature of all the possible pendulum motions is recovered. 5.2.2
Damped oscillators
In contrast with the conservative case, no general methods of establishing exact analytical solutions for phase trajectories exist in the nonconservative case, since energy is no more invariant. In so far as nonlinearities remain sufficiently weak, it is still possible to obtain approximate solutions analytically by using suitable perturbation methods. However, description of such methods, which are well documented in particular in [NAY 73], [NAY 79], is beyond the scope of this book. On the other hand, nowadays, it is also possible to perform numerical investigations on the computer, by integrating step by step in time the nonlinear equations of motion. Two typical direct integration methods are further described in section 5.3, namely central difference and Newmark's algorithms. Applications provided in this book were obtained by implementing them in the MATLAB software. EXAMPLE. - Damped Duffing's
oscillator
Duffing's oscillator was already introduced in subsection 5.2.1. Here, the purpose is to investigate its behaviour in the presence of viscous damping. The equation of motion is thus written as:
It is recalled that the position of static equilibrium q = 0 is unstable. However, the cubic stiffness term prevents the system going to infinity. Without making any calculation, it can be anticipated that the stable equilibrium positions (B) and (C) are now attracting points, since viscous damping is steadily dissipating mechanical energy of the system. The presence of two distinct attractors provides the dynamical behaviour of the nonlinear system with a somewhat intriguing feature, which contrasts with the case of the linear oscillator, namely a high sensitivity to the initial conditions of motion, at least in a certain range of values. Indeed, provided the mechanical energy initially imparted to the system is not too large in comparison with the potential energy at point (A), tiny variations in initial displacement and velocity, performed at constant energy, can be sufficient to change the final destination of the particle, from (B) to (C) or the reverse, as illustrated by the plots in Figures 5.15 and 5.16.
Autonomous oscillators
Figure 5.15. Time-history and phase portrait of Duffing's oscillator ( m=l, a>= 0.2n) case 1: the panicle is attracted by (B)
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Figure 5.16. Time-history and phase portrait of Duffing's oscillator ( m=l, CD= 0.2n) case 2: the panicle is attracted by (C) Furthermore, when a quantity such as the initial displacement is parametrically varied, the mechanical energy remaining constant, the particle is alternatively attracted by the point (B), or by the point (C), in successive intervals of g(0) values, which can be fairly narrow. A striking analogy with the game of roulette is worthy of
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mention. Fundamentally, the pits in the rolling track of the game behave as attractors (sink of potential) and tiny variations in the initial conditions are sufficient to modify into which pit the ball will stop. More generally, the high sensitivity of nonlinear systems also occur in relation to changes in quantities other than the initial conditions. Since it is impossible in practice to control the exact value of any parameter of a real system, the sensitivity actually introduces the idea of chance or randomness into the behaviour of otherwise deterministic systems. Finally, when numerical investigations are carried out, one has to be conscious that, even if tiny, the errors which are unavoidably introduced in the computation process can have major consequences on the final result, as further discussed in section 5.3, in relation to the present example. These considerations are sufficient to point out that the task of analysing the response of dynamical systems is generally much more arduous in the nonlinear than in the linear domain. This will be further illustrated in Chapter 9, in connection with the forced motions of Duffing's oscillator, see subsection 9.4.2.
5.2.3 Self-sustaining oscillators A self-sustaining oscillator is an oscillator which is dynamically unstable from the linear standpoint, for instance because its viscous damping coefficient is negative, but which is stabilized by the presence of nonlinear dissipative forces. The resulting oscillations, of which the magnitude is stabilized to a finite value by nonlinear forces, are then termed steady self-sustained oscillations, or vibrations. An archetype of such systems is the Van der Pol oscillator, which is governed by the equation:
This equation is characterized by the presence of a nonconservative force comprising a linear and a nonlinear component, both of them being proportional to the velocity q(t\ As a consequence of nonlinearity, the nonconservative force is found to dissipate, or alternatively, to produce mechanical energy depending upon the magnitude of the oscillation with respect to the critical value L(f )l = 1. We are interested here in discussing the case of positive values of the parameter a. Indeed, it is easily verified that this condition is necessary for obtaining steady self-sustained oscillations: 1. If motion is initiated with |g0|«l, the oscillator starts behaving as a linear oscillator, which is dynamically unstable (negative damping). Accordingly, |g(f)j is found to increase from a pseudo period to the next one. However, the efficiency of the nonlinear component to dissipate the energy produced by the linear component of the nonconservative force steadily increases with the vibrational amplitude. Thus, it is conceivable that a periodic motion can be
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sustained which will correspond to a global conservation of energy at the timescale of the period. 2.
The case |<70|>1 can be discussed qualitatively in the same way. Now the oscillator starts by losing energy. However, as soon as the amplitude of the vibration becomes sufficiently small, the same regime of steady sustained oscillations as above takes place, in which the energy balance over a cycle of vibration is exactly zero.
Figure 5.17. Phase portrait of the Van der Pol oscillator The steady oscillation qualitatively inferred from the above considerations defines a closed limit cycle in the phase plane. This cycle is immediately recognised as an attractor, on which all the possible phase trajectories are finally converging. The numerical integration of the Van der Pol equation confirms such a behaviour, see Figure 5.17. Furthermore, according to the computed time-histories it is found that provided the mechanical energy £^ of the limit cycle is large enough, within each period the motion may be split into two well contrasted stages of relatively slow and relatively fast motion, which occur alternately, see Figure 5.18. Clearly, such a pattern corresponds to the distinct stages of motion, within the periodic cycle, during which energy is gained and then lost by the self-excited oscillator. Such sustained oscillations are often termed relaxation oscillations.
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Figure 5.18. Time-history of a relaxation oscillation (displacement and velocity)
5.3. Numerical integration of the equation of motion Methods for producing approximate solutions of the equations of motion have been researched for a long time, A plethora of papers and many books have been devoted to the subject. The approach here is restricted to two basic algorithms which are widely used in structural dynamics. The reader interested in the topic is referred to [BAT 76], [BEL 83], [PRE 89], [ARG 91], [GER 97]. The starting point of any numerical integration scheme is to discretize the time derivatives of any quantity, replacing it by a suitable finite difference approximation:
Therefore, q(t) is computed only at a discrete and finite sequence of successive time-steps. The various integration schemes available can be distinguished from each others by the two major following features: 1.
The value of the discretized time-step at which the equilibrium equations are verified.
2.
The order of the truncated Taylor series which are used to discretize the derivatives.
The first point is of paramount importance since it leads one to the existence of two quite distinct classes of algorithms, namely the explicit algorithm and the implicit algorithm. Hereafter, an example of each of these is described. As we shall see best in Chapter 7 in relation to MDOF systems, the general advantages of the explicit algorithms are that they are easier to programme and are, in many cases, more efficient than the implicit algorithms, especially for solving nonlinear problems. Nevertheless,
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they are also marked by conditional stability. Fundamentally, the time-step, hereafter denoted h, must be less than a certain critical value hc, otherwise the magnitude of motion is found to increase exponentially with time, independently of the physical model to be treated. For certain applications, in either the linear or the nonlinear domain, conditional stability becomes a major drawback because computational efficiency of explicit algorithms may be compromised by the necessity to select a time-step much smaller than the physical time-scales of the problem. On the other hand, it is possible to establish implicit algorithms which are unconditionally stable, at least in the linear domain, for which computational efficiency can be optimised by selecting the value of h in accordance with the physical time-scales of the problem. However, in nonlinear applications, the computational efficiency decreases as a consequence of the need to balance the system at each time-step by using an iterative process. Such preliminary remarks serve to emphasize that, the choice of a specific algorithm has to be made in accordance with the particularities of the problem to be solved.
5.3.1 Explicit scheme of central differences of second order 5.3.1.1 Recursive process Let us start with the Taylor series, truncated to the second order:
where n =1,2,... is the n-th time-step, of duration h. From relation [5.52], it is immediately found that:
Now, let us consider the forced equation of a damped harmonic oscillator:
The reason for introducing an external force, already here, will become apparent in subsection 5.3.2 where the treatment of nonlinearities will be described. Then, the force balance is written at the n-th time-step:
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Substitution of the discretized derivatives [5.53] into equation [5.55] gives:
which can be written as:
where the dimensionless frequency G7, = hco{ is used. Finally, relation [5.57] is written as a recursive sequence, producing the value of qn+l expressly in terms of quantities defined at the foregoing time-steps n and n-1:
where
5.3.1.2 Initialisation of the algorithm The actual motion of the oscillator is analysed starting from an initial dynamical state determined by the physical displacement X0 and velocity V0. However, in order to initialise the recursive sequence [5.58], it is necessary to define two fictitious displacements at time-steps -h and -2/i, the actual motion starting at time t = 0. One is thus led to express the physical initial conditions in terms of the recursive sequence:
Using the expressions [5.59], the displacement values required to initialise the sequence are easily obtained. The following intermediate results are produced:
substituting [5.60] into [5.59], the desired results are readily obtained:
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5.3.1.3 Critical value of the time-step for stability It must be realized that the fact of writing the force balance at the n-th time-step (see equations [5.55] and [5.58]) leads one to compute the configuration of the system at the next time-step. Basically, the configuration at the step n is assumed to be already known and the sequence is used to extrapolate the configuration at the step n + 1. We may thus anticipate that if too large a time-step is chosen, such an extrapolation is likely to become unrealistic, inducing eventually a numerical instability, marked by the divergence of the sequence [5.58] whatever the external excitation may be. This can be checked analytically in the linear domain of response. The proof is worth considering, since a striking analogy exists between the behaviour of the recursive sequence and the dynamical behaviour of a linear oscillator. In the absence of any external excitation, the process [5.58] can be written in matrix notation, starting from n = 1:
Now, the process [5.63] converges only if the modulus of the eigenvalues of the recursion matrix [L] is less than, or equal to one. Let us assume that [L] is a regular matrix with two distinct eigenvalues, it is then possible to transform it into a diagonal matrix by using the eigenvector matrix, hereafter denoted [<X>]. In geometrical terms, this is interpreted as a transformation of similarity defined by:
which leaves the eigenvalues and the angle between two vectors unchanged. Now, in the frame of the eigenvectors, the recursion process is written as:
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and the mapping [5.63] is transformed into:
It is immediately realized that convergence of the process requires that the moduli of \ and ^ be less or equal to one. After some elementary algebra the following results are obtained:
The simplest case to investigate is the conservative case (^ = 0) where it is found that:
Therefore, the following critical value of the time-step for stability of the recursive scheme is derived:
It can also be verified that damping does not modify this stability threshold, provided its value is positive, as shown in Figure 5.19. As could be expected, in the case of negative damping, the modulus of the two eigenvalues are found to be larger than one, even if the time-step is arbitrarily small. This is clearly a direct consequence of the instability of the physical system.
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Figure 5.19. Modulus of the eigenvalues of matrix [L] versus reduced damping
5.3.1.4 Accuracy of the algorithm
Figure 5.20. Computed time-history of the undamped mass-spring system
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Figure 5.21. Error signal It is first emphasized that stability and accuracy of a dynamical algorithm are two distinct properties which must not to be confused with each other. After having discussed the problem of stability, it is appropriate to investigate the errors introduced in relation to the time-step of the algorithm. In this respect, it turns out that computing the free vibration of an undamped linear oscillator is sufficient to provide us with the essential information about accuracy. As a typical example, consider the case of a mass-spring system, mass 4 kg and natural frequency 75 Hz, which is set in motion by the initial displacement X0 = 1 m and the initial velocity V0 = Om/s. The computations have been performed using MATLAB. The timehistory of Figure 5.20 indicates that the central difference scheme provides the correct value of the vibration amplitude, without inducing any numerical attenuation or damping. Nevertheless, a systematic error is still made as shown in Figure 5.21, which is a plot of the error signal obtained by subtracting the analytical solution from the numerical one. The error signal can be described as a sinusoidal component at the natural frequency of the mass-spring system which is "modulated" by a linearly increasing envelope. The slope of the envelope signal is an increasing function of h. Actually, the error is caused by a small shift in time in the computed solution with respect to the theoretical solution. As a general result, it can be stated that no algorithm can be free from any error either in amplitude or in time. However, the errors can be made as small as desired, by diminishing the value of the time-step. Nevertheless, even tiny round-off errors can be of importance when dealing with nonlinear systems. For instance, in the case of the damped Duffing's oscillator, numerical simulations performed by using the central difference scheme with the
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Discrete systems
same initial conditions but with distinct time-steps may result in distinct trajectories ending either at (B) or at (C), as illustrated in Figure 5.22.
Figure 5.22. Sensitivity of the numerical solution of the damped Duffing's equation to the time-step of the central difference algorithm
5.3.2 Application to a parametrically excited linear oscillator Linear oscillators are said to be parametric when at least one of their coefficients K,M, or C is time-dependent. As an example, we consider a simple pendulum vibrating in a fixed plane with a small angle 9 about its support point O'. The latter is moving in the vertical direction about the fixed point O, according to the given oscillation Z 0 sin(fty), where the displacement Z0 is assumed to be very small in comparison with the length R of the pendulum, see Figure 5.23. The Lagrangian of the system is:
Whence the following equation of motion:
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where
Figure 5.23. Pendulum with a moving support According to this equation, known as Mathieu 's equation, the pendulum behaves as if the support would be fixed and the mass would be subjected to a gravity field oscillating sinusoidally at the prescribed angular frequency co0. Solutions of more general parametric equations, known as Hill's equations, are extensively discussed from an analytical point of view in many textbooks, see in particular [WHI 44], [ANG 61], [MAG 66], [BER 84], [NAY 79]. As a major result, the presence of parametric instabilities (also termed parametric resonances) is shown to occur if 0)0 lies inside intervals centered at the resonance frequencies coQ/ct)l=2/n, n = 1,2... The width of such intervals is an increasing function of K . Without entering into a detailed discussion of such systems, it can be said in a qualitative way that the physical mechanism responsible for the parametric instability is clearly an energy transfer from the prescribed motion toward the pendulum. As would be expected, such a transfer can be globally positive (i.e. when averaged over a cycle of oscillation) only when the ratio of the response frequency of the pendulum to that of the prescribed oscillation of the support is sufficiently close to particular values. Width of the intervals where the so called parametric resonant responses can take place is expected to increase with the mechanical energy involved in the prescribed motion.
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Figure 5.24. Two distinct time-histories of the parametric pendulum
The integration algorithms of the explicit type allow one to solve numerically the equation of the parametric pendulum in a quite convenient way. It is found suitable to deal with the parametrically varying stiffness force as an external force. In accordance with the recursive scheme [5.58], this force component is known explicitly at the required n-th time-step. Figure 5.24 displays two samples of such simulated time-histories; one refers to a non-resonant (stable) response and the other refers to a resonant (unstable) response. In this kind of application, it is found that a time-step of the order of one tenth of the critical value for stability provides satisfactory results. In the non-resonant response, the difference between the pendulum and the excitation frequency results in a beating phenomenon marked by a low frequency modulation of the amplitude of the response. The beating phenomenon will be further discussed in Chapter 7.
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5.3.3 Application to an oscillator impacting against an elastic stop 5.3.3.1 Impact force model
Figure 5.25. Mass-spring system impacting against an elastic stop
The explicit algorithms may be used to deal with nonlinear problems in the same way as in the case of the parametric pendulum, since the nonlinear forces can be treated as external forces. However, a difficulty arises if the nonlinear terms depend upon velocity or acceleration, because these quantities are not yet known when writing down the force balance at the n-th time-step. Hence, one is constrained to formulate some suitable approximations to express the forces, which are distinct from those already adopted to model the linear terms. As a consequence, the accuracy and stability of the algorithm deteriorate. The practical importance of such problems is highly dependent upon the specific nonlinearities to be treated. As a further example, worth being discussed in some detail because of its physical interest, let us consider again the case of a harmonic oscillator. But now, the motion is limited on one side by an elastic stop. The stiffness coefficient Kc of the impacted support is assumed to be much larger than that of the oscillator (case of so called stiff impacts). Starting from a state of equilibrium at rest, the oscillator is set in motion by imparting an initial velocity. With the notations detailed in Figure 5.25, the system is governed by the following set of equations: Forced oscillator: Initial conditions : Nonlinear forcing function :
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Discrete systems
An analytical solution is easily obtained in the case of stiff impacts. The motion before the first impact is governed by the following linear equation:
Referring to formula [5.14], the time-history is:
Hence the dynamical state at the starting time t0 of the first contact with the obstacle is:
The next stage is a contact or shock-stage, which is governed by the following linear equation:
Since we assumed that k << Kc, the kX term in the equation of motion can be neglected:
The variable transformation q = X - L leads one to the autonomous system:
Thus, motion of the oscillator during the shock is described by the following time-history:
Duration TC of contact between the oscillator and the obstacle, also termed the shock duration, is inferred from the contact condition X (t0 + Tc) = L. Whence we deduce that:
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where Tc is the natural period of the "shock-oscillator". Incidentally, it may be noted that shock duration does not depend upon the energy imparted initially to the oscillator, and thus is also independent of the strength of the impact. Moreover, it is also easy to check that, in agreement with the principle of conservation of energy during an elastic impact, the velocity of the oscillator at the end of the shock is precisely -X (t0). On the other hand, the impact force is identified with the shockspring reaction, which is thus given by:
The interesting point, worth emphasizing and easily checked, is that the action of the impact force identifies with the change of linear momentum of the oscillator from start to end of the shock:
Figure 5.26. Potential and phase portrait of the impacting oscillator
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Discrete systems
Finally, the period of the nonlinear oscillation is calculated by using the invariance of energy. The procedure is readily understood by referring to Figure 5.26, which sketches the potential and the phase portrait of the system. The potential curve is made up of two distinct parabolic segments, one of them refers to the "free-flight" stage of motion and the second, much stiffer than the first one, refers to the shockstage. Therefore, the phase portrait is made of two distinct elliptic arcs. They correspond to the phase portrait of two linear oscillators, distinct from each other by the value of the stiffness coefficient, namely k for the "free-flight" oscillator and K1 = k + Kc = Kc for the shock-oscillator. Duration of a cycle of "free flight" is given by the definite integrals of the type [5.21], calculated between the following boundaries:
So, in reduced quantities: The period is thus given by:
In the case of violent impact, the above result takes the simpler form:
As expected, the period of the impacting oscillator is found to be less than the period T. of the non impacting oscillator. Moreover, if the shock stiffness tends to infinity, the shock duration tends to zero. However, it may be verified that important aspect of the motion such as the "free flight" stage and the action of the impact force remain unchanged. This interesting aspect of the problem will be discussed further in the next subsection. On the other hand, from the analysis made above, it may be readily understood that numerical simulations using explicit algorithms require a time-step the value of which is related to the shortest time-scale of the problem, i.e. TC and not T1. More specifically, when using the central differences scheme, in formula [5.69], fi>, has to be replaced by coc. As a consequence, in the presence of very stiff impacts, computational time can become excessive. Hence, from a practical point of view, it is advisable to determine first the minimum time-scale which is needed to interpret
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usefully the results of the numerical simulation and then to adjust the value of the impact stiffness accordingly. Let us consider for instance the crash of a car against a rigid wall. Such an event lasts typically about 100 ms. Crash tests clearly indicate that such a transient may be satisfactorily described by using a time-step of about one millisecond. Hence, to perform numerical simulations of the crash, a suitable value of Kc such that w c / 2 n = 1kHz is appropriate.
Figure 5.27. Displacement of the impacting oscillator
Figures 5.27 to 5.30 illustrate the use of the central differences method to treat the present problem. The simulation refers to the case k / K c =0.01 .The nonlinearity of the response and the elastic compression of the impacted spring during shocks are clearly detectable in Figure 5.27, which shows the time-history of displacement. The velocity plot of Figure 5.28 makes the nonlinearity even more conspicuous, because velocity reverses abruptly during the shocks. As the system is conservative, maximum velocity remains exactly equal to the value imparted initially to the oscillator ( X0 = 54 km/h). The time-history of the impact force is plotted in Figure 5.29. It is shaped as a negative half-sine curve repeated periodically at the frequency of the oscillation. Force amplitude depends upon the stiffness coefficient of the shock-spring but the action over a shock is precisely equal to the change of linear momentum 2M1X0. Anticipating the mathematical formalism, it can be said that in the limit of infinite stiffness, impact forces tend to a periodic series of Dimc & distributions (cf. Chapter 7 and Appendix 6), weighted by 2MjX0. Moreover, in the
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limit of infinite impact-stiffness, the shock-spring model could be advantageously replaced by a nonholonomic constrained model, from the analytical viewpoint at least, as discussed in the next subsection. However, so far as the dynamical response of the oscillator is concerned, not much is gained by letting the impact time be vanishingly small. Indeed, as soon as Kc is sufficiently large, the truly relevant quantity is the action of the impact force and not the particular force value at a given time.
Figure 5.28. Velocity of the impacting oscillator
Figure 5.29. Impact forces
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Figure 5.30. Phase portrait
Finally, the phase portrait shown in Figure 5.30 is found to be in satisfactory agreement with the theoretical sketch of Figure 5.26. When plotted over a large number of cycles, it provides a good test to check the periodicity of the computed response. Indeed, any error with respect to periodicity results unavoidably in trajectories which superpose imperfectly from one cycle to the next. 5.3.3.2 Constrained model Starting from the last model, it is of interest to further investigate the asymptotic case Kc —> °° analytically. For this purpose, impacts are now modelled according to the following conservative unilateral constraint condition:
Thus, T_ is the time "just before" an impact and T+ is the time "just after" the same impact, occurring at time T . Before the first impact, the motion triggered by the initial velocity X0 is:
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The impact constraint implies that:
with the auxiliary condition for impact: X0 > Lo)l The first impact occurs at the incident velocity:
Thus, the impact constraint provides us with the necessary initial conditions to describe the "free-flight" motion after the first impact. It is found that:
where Now, periodicity of the motion implies that:
where Tc is the still unknown period of the impacting system. Therefore, the condition of periodicity can be suitably written as the following mapping:
The above system can be solved directly, or as an eigenvalue problem. Let us outline the direct method first. From the first equation, it is found that:
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The second equation provides another relation which turns out to be compatible with the first one:
After a few trigonometric calculations, the above result may be re-written as:
which is a form suitable for comparison with the result provided by the impact force model:
Substituting this expression in the result of the constrained model, it is found that:
which is the correct result, since
as already shown.
Now solving the problem as an eigenvalue problem, the mapping is first changed into the equivalent form:
According to this new formulation, the required solution must correspond to the eigenvalue A = +1 and the related eigenvector may be interpreted as an invariant or fixed point of the mapping. Actual resolution of the problem presents no difficulty. The eigenvalue equation is:
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The eigenvectors are conveniently written as:
providing thus the same compatible equations as those obtained by using the direct method. As a final remark, the reader can check that the mapping:
obtained by discarding the finite jump condition of velocity at the impact, leads one to the linear period Tc = — .
5.3.4 Newmark's implicit algorithm As seen in the last subsection, explicit algorithms use the equilibrium equation at the rc-th time-step in order to determine the dynamical state of the system at the next (n+1)-th time-step. It may be noted, that in this procedure, the force balance written down refers to a known configuration and is used to extrapolate the configuration at the next time-step. As already pointed out, the drawback of such an explicit scheme is its conditional stability, which is governed by the smallest time-scale of the numerical model. In contrast with this, the implicit schemes use the force balance written down at the (n+1)-th time-step to determine the state of the system at the same (n+1)-th time-step. Such a procedure allows one to build unconditionally stable algorithms. Though there exists a whole family of Newmark's algorithms, it is sufficient for our purpose to describe here the most broadly used version of them, which is implicit and free of numerical damping. Let us start from the Taylor's formulae:
where the residue is given by:
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Velocity and displacement are then written as:
The particular scheme treated here makes use of the following approximation, known as the mean value of acceleration:
Substituting the approximation [5.72] into the formulas [5.71], it is found that:
Writing down the equilibrium equation at the (n+1)-th time-step, we have:
Now, by using the expressions [5.73], the force balance [5.74] becomes:
The force balance written in the form [5.75] allows one to compute the acceleration at the time (n + 1)h. Then, use is made of relations [5.73] to compute the velocity and the displacement at the same time. It can be shown that this algorithm is unconditionally stable and free from numerical damping. For that purpose, the recursive scheme is written in matrix notation and the eigenvalues of the matrix are determined. It turns out that their modulus is independent of h and precisely equal to one. Examples of application are postponed to Chapter 7, when dealing with the response of multi degrees of freedom systems.
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Chapter 6
Natural modes of vibration of multi degree of freedom systems
As seen in the last chapter, modes of vibration arise as a natural concept in the study of the free vibrations of linearized systems. This concept may be considered as the central tool for studying the dynamical behaviour of most MDOF systems in the linear domain. Although, they are basically a linear concept, they can nevertheless be of great value also for modelling and solving many nonlinear problems, as further illustrated in Volume 2. The first objective is to show that the natural modes of vibration are directly connected to the problem of uncoupling the coordinates of a MDOF system. Indeed, from a theoretical point of view, they arise as the solutions of an eigenvalue problem, which is restricted here to conservative models. Owing to the symmetrical nature of the matrix operators involved, the eigenvectors, or mode shapes, can be used to define a vector basis of the space of configuration, which is real and orthogonal. These eigenvectors or mode shapes can be used to transform the equations of motion into a set of uncoupled differential equations, described by diagonal matrices of stiffness and mass which operate on the so-called natural, or modal, coordinates of the material system. From a more physical viewpoint, there exists a close connection between natural modes of vibration and standing waves, as will be shown using the example of a chain of coupled oscillators. Finally, a few extensions of the modal concept are presented, within the domain of conservative systems.
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6.1. Introduction Restricting the analysis to the linear domain, equilibrium of discrete mechanical systems having N>l degrees of freedom, (also called multi degrees of freedom systems, or briefly MDOF systems) is governed by a system of N linear equations, algebraic in the case of statics and differential with respect to time in the case of dynamics. In most instances, such equations couple together several generalized displacements. Therefore, it is extremely useful to find a systematic procedure allowing one to uncouple such systems of equations, in statics as well as in dynamics. Indeed, if suitably uncoupled, the system is reduced to a set of N oscillators independent of each other, which is very convenient for further analysis. The problem of uncoupling will thus serve as a guideline for most of the considerations which shall be made in this chapter. In the first instance, section 6.2 is concerned with the task of linearizing Lagrange's equations about a static state of stable equilibrium. In the conservative case, after linearization, the system is thus characterized by a stiffness matrix [K] and a mass matrix [M]. Both of them are symmetrical, [M] is positive definite and [K] is positive. Section 6.3 constitutes the core of the present chapter. It deals with autonomous conservative and linear systems which vibrate freely about a static state of stable equilibrium. We shall show that a judicious transformation of displacement variables allows one to uncouple the equations of motion. Moreover, the same uncoupling procedure applies also in the case of statics. The column vectors of the transformation matrix are formed by N linearly independent mode shapes of vibration of the system. As expressed on this modal basis the dynamic equations of the mechanical system reduce to a set of N uncoupled linear oscillators. The natural frequencies of such oscillators are the modal frequencies, of the system, whereas the masses and stiffness coefficients are called the modal mass and stiffness coefficients (or the generalized mass and stiffness coefficients). Clearly, the coupling between the variables on the physical basis (or in any non modal basis) is accounted for by the mode shapes which interrelate the displacement of each oscillator. Section 6.4 extends the concept of vibration modes to systems other than those already considered in section 6.2. Namely, the following items are discussed: 1. Natural modes of vibration of constrained systems. After linearization of the constraint conditions, the Lagrangian of a constrained system gives rise to a set of linear differential equations, mixing the variables of displacements and constraint reactions. In the conservative case, such systems can be characterized by a [K] and a [M] matrix, which are still symmetrical. However, as the variables of the problem now mix components of displacement and of forces, the physical meaning of stiffness and mass matrices [K] and [M] does not
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hold anymore. Nevertheless, modal analysis of such systems can proceed in the same way as in the unconstrained case. 2.
The free modes of rigid bodies, of which the natural frequency is zero. They correspond to motion allowing the potential energy of the system to remain unchanged.
3.
The modes of elastic buckling, which may occur in many prestressed systems. Such modes are analysed successively from the static and dynamical point of views.
4.
The whirling modes, which take place in rotating systems as a consequence of gyroscopic coupling (cf. Chapter 4, subsection 4.3.2).
On the other hand, extension to nonconservative systems will be described in Volumes 3 and 4, in relation to fluid-structure coupled systems. Indeed, at this stage, we will be able to bring the physical meaning of such modes to light by discussing the behaviour of a few examples.
6.2. Vibratory equations of conservative systems 6.2.1. Linearization of the equations of motion Let us consider a conservative W-DOF mechanical system, of which P is a static state of stable equilibrium. Motions around P are described by using N generalized displacements independent from each other, which are the components of the displacement vector [q\. In accordance with this definition, [q] is zero at P. The system is governed by Lagrange's equations:
Now, we restrict our interest to motions of small magnitude, in such a way that the equations can be linearized about the configuration P, which is taken as a reference. As already indicated in Chapter 2, subsection 2.3.4, the system [6.1] takes the canonical form:
Here, it is of interest to start by relating the stiffness matrix [K] and the mass matrix [M] to the quadratic approximation of the Lagrangian. With this aim in mind, it is appropriate to carry out an expansion of £ in Taylor's series, limited to the
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quadratic terms in [q], or [q]. Referring to the forms [2.12] and [2.20] of Chapter 2, kinetic energy is written as:
where only the constant terms are retained in [M ]. Clearly, such a mass matrix is symmetrical and positive definite, i.e. all the eigenvalues are positive, as already discussed in Chapter 3 subsection 3.2.1. On the other hand, it is recalled that the state of static equilibrium is defined by the conditions:
[K] is the stiffness matrix, which is thus found to be symmetrical. Moreover, provided the equilibrium state is stable, [K] is also positive definite, becoming eventually simply positive, if the case of indifferent equilibrium is included as a possibility. Of course, potential energy takes the quadratic form:
To conclude this subsection, it is noted for further reference that the equations governing the linear system, when excited by an external force vector, take the canonical form:
6.2.2. Solution of forced problems in statics 1.
Resolution by inversion of the [K] matrix
Adopting the static, or quasi-static approximation, the forced problem reduces to the following linear algebraic system:
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Provided the state of equilibrium taken as a reference is not indifferent, [K] is a regular matrix so the system [6.7] can be solved by using the inverse matrix [K] , at least formally:
However, it has to be stressed that such a solution is physically meaningful only if [K] is positive definite. Otherwise, the matrix would be singular or the material system would be unstable and its mechanical response to any external excitation would be time dependent and nonlinear in nature. EXAMPLE. - Coupled pendulums Going back to the pair of coupled pendulums already considered in Chapter 3, subsection 3.2.2, the forced problem takes the form:
The roots of the determinant A = (2 - k)2 -1 are k1 = 1 and K2 = 3 . KJ corresponds precisely to the threshold of static instability of the system. Hence, it is rather obvious that the displacements which may be calculated by using the above formula are physically meaningful in the domain K < K1 = 1 only. Indeed, when the system becomes statically unstable, the pendulums are attracted toward the position of stable equilibrium 0es = n. As is well known in practice, it follows that large oscillations about 9es occur instead of the solution given by (a). 2.
Inversion and triangular decomposition of a matrix
In the process of solving large systems of linear algebraic equations numerically, calculation of the inverse matrix is not advisable because is a costly and weakly accurate operation. It is much more preferable to perform a triangular decomposition, using Choleski's decomposition described in Appendix 5. Doing so, the transformed matrix takes the form:
where [T] is an upper (or right) triangular matrix, i.e. a matrix in which all the elements below (or to the left of) the main diagonal are zero. [Tj
is the lower (or
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Discrete systems
left) triangular matrix obtained by transposing [T]. Moreover if [K] is regular, [T] is regular too. Substituting [6.8] into [6.7], the system to be solved takes the convenient form:
Then, we can proceed easily to the solution of the two triangular systems involved in [6.9], without having to perform any matrix inversion, provided of course that no diagonal term of [T] vanishes, that is to say provided [K] is regular.
6.3. Modal analysis of linear and conservative systems Obviously, the task of solving any system of the kind [6.6] is made much easier if both [K] and [M] are diagonal. Indeed, in such a case, one has to deal only with a system of N forced harmonic oscillators, which are independent from each other. Solution of any forced problem in statics is thus immediate. Resolution of any forced dynamical problem is reduced to that of a system of N uncoupled ordinary differential equations. Moreover, such equations are linear and their coefficients are constant. The mathematical techniques for solving analytically such equations form the subject of the last three chapters of this volume. In the present section it will be shown that conservative and linear dynamical systems can always be transformed into a diagonal form. This can be achieved by using a judicious linear transformation of coordinates.
6.3.1. Coupling and uncoupling of the degrees of freedom In order to grasp better the problem of DOF coupling and uncoupling in a material system, it is found appropriate to analyse first the system of two harmonic oscillators, coupled together by a linear spring, as shown in Figure 6.1. The Lagrangian of the system is:
which gives the following system of dynamical equations, written in matrix form:
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This system is clearly coupled through the nondiagonal terms of the stiffness matrix. At first sight, such a result seems quite natural since the physical component connecting the two massive bodies is precisely a spring of stiffness coefficient Kc.
Figure 6.1. Two harmonic oscillators coupled together by a spring
Nevertheless, this point is worthy of further consideration. Indeed, from the mathematical viewpoint, it is clear that the basis of the eigenvectors of [K] allows one to transform [K] into a similar diagonal matrix. Therefore, it is realized that such a transformation lets the stiffness coupling disappear. However, there is no reason why, through the same basis transformation, the mass matrix [M] would be changed into another diagonal matrix. As a consequence, the transformed equations are no longer coupled through stiffness but they are likely to be coupled through inertia! This clearly indicates that: The inertial, or the elastic nature, of the coupling is not an intrinsic property of a given mechanical system but merely a consequence of the choice made in defining the generalized displacements which serve to describe it. The reasoning above leaves the possibility of the existence of a particular basis in which both the [K] and [M] matrices would take a diagonal form. Obviously, if it exists, such a basis is the key to the problem, as it will allow one to uncouple the system [6.6]. It can be easily shown that such a basis does exist for the system [6.10], provided the two oscillators are identical to each other. Indeed, in this case, any vector [a /3J is a possible eigenvector of [M]; thus, in particular, the eigenvectors of [K] can be selected. When normalized to a unit modulus, they take the transposed forms:
The physical interpretation of such eigenvectors is clear: The symmetrical shape [p] is said to be "in-phase": the two masses move in the same direction at any time, with the same amplitude. When the system
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Discrete systems
vibrates in accordance with the in-phase eigenmode, the connecting spring does no work. •
[02] is said to be "out-of-phase": the two masses, move in the opposite direction at any time, again with the same amplitude. When the system vibrates in accordance with the out-of-phase eigenmode, the connecting spring does work.
The transformation of displacement variables related to the uncoupling is of the kind:
where the scaling factor a is arbitrary. Such a result is rather intuitive and could indeed be obtained directly by the elementary procedure of variable elimination. Adopting a = 1/2, the transformed system is:
Therefore, in terms of generalized quantities the "in-phase" oscillator is characterized by the mass coefficient M and by the stiffness coefficient K, whereas the "out-of-phase" oscillator is characterized by the mass coefficient M and the stiffness coefficient K + 2KC. Moreover, the only difference between the generalized and the physical oscillators is the existence of a fixed phase relationship between the two physical displacements, which does not hold in the case of really uncoupled oscillators. It is also of interest to define the following functionals of potential and kinetic energy:
Such functionals are independent from the coordinate system, provided the basis vectors are normed in the same manner, for instance by the condition:
These very important results, which are introduced in this subsection by taking a particularly simple case, can be extended to any system of the type [6.6], by introducing the concept of natural modes of vibration. This is the object of the next subsections.
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6.3.2. Natural modes of vibration 6.3.2.1. Basic principle of the modal analysis Here the embryo of modal analysis introduced in Chapter 5 subsection 5.1.2.3, is extended to the case of Af-DOF systems. With this object in mind, we search for non trivial solutions of the autonomous version of system [6.6] which are of the harmonic form [q]elQit, independently from any particular initial conditions of motion. We are thus led to formulate the problem as the following linear and algebraic system:
Since such a system is homogeneous, it is clear that non trivial solutions, denoted [O n ], can be obtained only if w2 takes the discrete values w2n, which are the roots of the characteristic equation:
It is immediately noticed that [6.12] is a polynomial equation of degree N with respect to the variable T = w2. Consequently, it has N A roots, some of them being eventually multiple. The natural modes of vibration are then defined as harmonic vibrations characterized by the following complex amplitude:
wn
is the natural pulsation of the mode indexed by the integer n (n = 1,...N),
[tpn ] is the corresponding mode shape. 6.3.2.2. Basic properties of the natural modes of vibration Starting from the properties of symmetry and positive sign of the [AT] and [M] matrices, which are inherent to any conservative and stable mechanical system, the following properties of paramount importance may be inferred: 1.
The modal pulsations are positive and the mode shapes are real. At first, we show that the An are necessarily positive, if [K] and [M] are
positive definite. Indeed, letting [on] be a vector solution related to a given Tn, we can state that:
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Discrete systems
where Relations [6.14] are obtained by performing the scalar product by [#>„] of the terms of the modal equations. In the language of geometry, such a scalar product is interpreted as the projection of the modal equations onto a mode shape vector related to An. Now, since [K] and [M ] are assumed to be positive definite, Kn and Mn and thus An are also positive definite. This basic result is conveniently written in the form of a Rayleigh quotient, which extends to N-DOF systems the quotient already introduced in Chapter 5, (cf. relation [5.30]):
According to the Rayleigh quotient, the modal (or generalized) stiffness can be identified with the functional of potential energy <Epn> and the modal mass with the functional of kinetic energy <E K n > . The modal pulsation wn = +Tn is also taken as positive. Its physical meaning is clear, since it is the angular frequency of the harmonic oscillation of the real system, vibrating freely in the n-th mode. Incidentally, it also happens that one has to deal with a null stiffness matrix. This is in particular the case of free (i.e. not provided with any support) bodies which have six modes of rigid body at zero frequency. In subsection 6.4.1 we shall come back to this particular case, which is of practical importance in many instances. On the other hand, since An and the elements of the [K] and [M ] matrices are real quantities, the mode shapes are necessarily real too. The vectors [On] provide the relative algebraic amplitude of the displacements of the various degrees of freedom of the system. The term of "relative amplitude" is highly appropriate since [pn] is defined except for an arbitrary multiplicative constant. This is also precisely the reason why [pn] is called a mode "shape". Usually a specific value of the constant is used to define a norm for [pn]- Amongst the most widely used, let us quote the norm of the unit modulus: |P n | = l, that of the largest component of generalized
displacement max(Ui,n) = 1,i = l,2,...,N
and finally
normalisation already mentioned in Chapter 5, according to which:
the mass-
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201
Finally, it may be also useful to mention that when the generalized displacements of two distinct degrees of freedom have the same sign they are said to be in-phase, whereas if they are of opposite sign they are said to be out-of-phase. 2.
The mode shapes are orthogonal to each other with respect to [K] and [M ]
By orthogonality of any pair of mode shapes with respect to the stiffness and the mass matrices is meant that the vectors [Oj] and [pk] comply with the following conditions:
The most direct way to prove that the conditions [6.16] hold is to transform the modal system [6.11] in such a way as to turn it back to a standard eigenvalue and eigenvector problem of a symmetrical matrix. Indeed, it is recalled that the eigenvalues of a symmetrical matrix are real and the related eigenvectors are orthogonal with each other (for mathematical proof, see Appendix 4). The close analogy between the modal calculation presented just above and the standard eigenvalue problem of a matrix is rather obvious. Indeed, since [M] is always regular, it is possible to multiply equation [6.11] by the inverse [M] , to obtain the standard eigenvalue equation:
However, the range of this first idea remains too limited, since in most instances [A] is not symmetrical, even if both [K] and [M] are symmetrical matrices. Therefore, it is necessary to have recourse to a more subtle transformation, which preserves the symmetry of the modal equation. The key is to transform the massmatrix by using Choleski's decomposition, already invoked in subsection 6.2.2. Doing so, the transformed matrix takes the form:
Since [M] is regular, [T] is regular too. Equation [6.11] is thus re-written as:
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Then, we introduce the transformed modal vector:
The transformed system is:
Once again, we obtain a standard eigenvalue problem. However, this time, the eigenvalue equation involves a symmetrical matrix. Whence, it may be concluded that the eigenvectors [yrn ] related to distinct eigenvalues An are orthogonal with each other. Now, returning to the mode shapes [#>„], it is easily shown that the orthogonality of the vectors [Wn ] leads necessarily to the orthogonality of the vectors [pn] with respect to [M] and [K]. Indeed, we have:
On the other hand, since the matrix equation [6.11] is verified, the orthogonality with respect to [M ] leads necessarily to that with respect to [K] and the proof of the relations [6.16] is thus completed. 3.
Orthonormal modal basis
As detailed in Appendix 4, even if the multiplicity m of the eigenvalues An is larger than one, an Orthonormal basis of N eigenvectors can always be associated with the Af-DOF system [6-21], and also to the original system [6.11]. Such a basis of the natural modes of vibration is conveniently described by the modal matrix [O] which is built by assembling the column vectors [pn] of a set of N Orthonormal mode shapes. Indeed, according to such a definition, it follows immediately that:
Accordingly, any coupled system of linear dynamical equations, forced or not, becomes uncoupled when written in the modal basis. In Chapters 7 and 9, we will take advantage of this basic result to study the forced vibrations of N-DOF systems.
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203
Keeping here to the case of free vibrations the equations [6.6], as written in the modal basis, take the particularly simple form:
where [p] is rightly termed the natural coordinate vector of the material system. Of course, the displacement vector [q] as expressed in the "physical" coordinates and the displacement vector [p] as expressed in the natural coordinates are related to each other by the transformation rule [q] =[p][o]• The physical meaning of equation [6.24] may be stated as follows: The transformation of any linear and coupled N-DOF system into the modal basis provides an equivalent system involving N harmonic oscillators which are uncoupled from each other. Stiffness and mass coefficients of these oscillators are the generalized stiffness and mass of the corresponding mode. So, the N-DOF linear model of the material system is reduced by the modal transformation to a set of N independent mass-spring systems. 6.3.2.3 Modal analysis of 2-DOF systems EXAMPLE 1. - Two coupled oscillators of distinct masses We go back to the system [6.10] for the case K1 = K2 = K , but with M1 * M2. For further discussion of the results, it is found convenient to write the modal system in the reduced form:
where the following dimensionless quantities are introduced:
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The characteristic equation of the system is found to be:
The discriminant is A = (1 + k)2(1 + u) - 4X1 + 2k) and the natural frequencies are given by:
Figure 6.2. Reduced natural frequency vs. mass ratio: u= M1 /M2
The mode shapes, as normalized by the condition X2 = 1, are found to be:
The scalar product of the two modal vectors is different from zero, except in the special case u = l. Indeed, an elementary calculation produces the following simple result:
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205
On the other hand, it may be easily checked that they are orthogonal to each other, with respect to the mass and the stiffness matrices.
Figure 6.3. Mode shape (pn (1)/pn(2) vs. mass ratio: u = M1 /M2
The natural frequency of the out-of-phase mode decreases with u, tending asymptotically to l + K". The natural frequency of the in-phase mode also decreases with //, tending asymptotically to zero. Such a behaviour displayed in Figure 6.2 can be better understood by inspecting how the mode shapes are changing with u, see Figure 6.3. It is found that for the in-phase mode the preponderant motion is that of the more massive oscillator. Therefore, its frequency tends to zero as u —» °° . In contrast, for the out-of-phase mode the preponderant motion is that of the less massive oscillator. Therefore, when u —»°°, its frequency tends to that of a single oscillator with mass M2 and stiffness K + Kc = K 1+K . Of course, such results are fully corroborated by the evolution with ju of the modal masses u1,u2 , as shown in the plots of Figure 6.4, where the value of M2 is again adopted as a reference. The representative curves of the in-phase (u1) and out-of-phase ( u 2 ) modal masses are symmetrical to each other with respect to the line u = 1. As expected, the
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Discrete systems
crossing point is u = 1, //j = U2 = 2 since in this case the two masses vibrate with the same amplitude. As expected too, in the asymptotic case U—>°°, u1 = u and u2=l-
Figure 6.4. Modal mass vs. mass ratio: u = M1 /M2
EXAMPLE 2. - Rigid rod supported by springs The system considered here is shown in Figure 6.5. It is made of a cylindrical and homogeneous rod (radius R and length L), which is supported at the ends A and B by linear springs of stiffness coefficients KA and KB, acting along the transverse Oz direction. The rod is assumed to be much stiffer than the supports in such a way that it may be modelled as a rigid body, at least for analysing the first low frequency modes. A priori, we have to deal with a 6-DOF system, which can be conveniently described by using the linear displacement X,Y,Z of the centre-of-mass O and the three angular displacement variables which describe the rotations about O. Since the problem is restricted to the case of small motions, we could use the angles V^Y ,V about the Cartesian axes Ox,Oy,Oz (cf. Chapter 2, subsection 2.2.2.1).
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Figure 6.5. Rigid rod supported by linear springs
However, the present problem is further simplified by restricting our interest to small motions in the Oz direction only (i.e. motions transverse to the rod's axis and in the plane of the Figure 6.5). Therefore, we now have to deal with a 2-DOF system only, described first by using the variables Z and Wy = 9 . As the rod is assumed to be homogeneous, the centre-of-mass is at the middle point and kinetic energy can be directly written as:
Thus, shifting to the cylindrical coordinate system r,o,x:
n = L/R is the slenderness ratio of the rod. The same remark as that already made in Chapter 2 subsection 2.4.6 can be repeated here. If rj is sufficiently large, the (R/ L) term may be neglected when calculating the coefficient of inertia /
. As an immediate corollary, the kinetic
energy related to the rotation of the cross-sections can be neglected too. Therefore, it is found that:
The last result could be obtained by concentrating the mass of the rod uniformly along its axis. Indeed this gives:
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where m m designates the equivalent mass per unit length of the rod. Concerning the potential energy, let us start with the particular case of identical springs KA = KB = K. Elastic energy may be written as:
At this step, it is found convenient to use the new displacement variables given by:
The Lagrangian is:
Defining col =2K/M , as a suitable scaling factor for angular frequency, the following uncoupled modal system is obtained:
Whence, the two following modes: 1.
Pure translation mode:
2.
Pure rotation (rocking) mode:
In this particular case, they form an orthogonal pair. This is a direct consequence of the uncoupling of the displacement variables Z, and Z2. Turning now to the general case: KA = K; KB = K(l + 2k), elastic energy is written as:
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209
The modal system is found to be:
This time, the displacement variables are coupled together and though the analytical solution of the system presents no difficulty, expression of the final results is rather cumbersome and is omitted. The major point is that two mode shapes are found, which combine linearly the two mode shapes of the uncoupled system. Expressed in mathematical language, the mode shapes of the coupled system [On], n = 1,2 may be obtained as a linear superposition of the mode shapes of the uncoupled system:
The case of a rod with mass per unit length m(x) depending on x would be analysed in a similar way. For instance, in the case of a rod with a linearly increasing radius (conical frustum), we have:
Here, coupling between the variables of translation and rotation is introduced through the mass matrix. Once more, two modes are found, which combine a translation and a rotation. Mode shapes form an orthogonal pair of vectors, but with respect to the mass and stiffness matrices only. 6.3.2.4 Natural modes of vibration as standing waves The material system to be discussed here is shown in Figure 6.6. It is made of a chain of identical mass-spring systems (coefficients of mass M and of stiffness K). connected to each other by a spring. Moreover, the stiffness coefficient of the coupling springs is also equal to K. The physical displacement of the n-th particle is Xn;n = l,...N.
Figure 6.6. Chain ofN identical and coupled oscillators
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Discrete systems
The modal equations are written as the NxN
where w2 =K/M
matrix system:
.
Most often, as soon as N becomes greater than 2, resolution of this kind of system is carried out using numerical methods. Several types of algorithms are available, the efficiency of which varies according to the size of the system to be solved. In Appendix 8, we describe a method, which is based on the iterative solution of a forced system. It is well suited to the treatment of large systems, such as those arising in the finite element models of complex structures (for a description of the basic principles of the finite elements method, cf. Volume 2, Chapter 3). The numerical results presented here were obtained by using the MATLAB software . However, in the particular case of the present system, an analytical solution is available, which illustrates clearly the connection between the vibration modes and waves. As described for instance in [COR 77], [JOS 02], the key point is to take advantage of the common structure of the current rows of the system, nonzero terms of which are simply shifted to the right by one column, when proceeding from one row to the next, except for the first and the last rows. Accordingly, it is assumed that the modal displacement of the current degree of freedom is related to the next simply by a fixed phase shift, in such a way that in complex notation:
It is of special interest to focus first on the physical meaning of this type of solution. The starting point is the relation between the complex amplitude of the harmonic vibration of a current particle to that of the preceding one, and finally, through a cascade, to that of the first one:
Now, the vibration in the real domain may be obtained by taking either the real or the imaginary part of the complex amplitude, as already explained in Chapter 5, subsection 5.1.2.3. So we can assume a time history, for instance of the kind:
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211
where As expected, according to the type of solution which has been assumed, the harmonic vibration of the (k+1)-th mass at time t + kr is the very same as that of the first mass, at former time /. This kind of motion is known as a travelling wave which propagates here to the right, if referring to Figure 6.6. Further, if one assumes that in the physical system the distance from one particle to the next is also constant, it is possible to define the phase speed of the wave as:
Here the wave is said to be longitudinal because the oscillatory motion of the particles is parallel to the direction of the wave propagation, see Figure 6.7.
Figure 6.7. Oscillation of the particles in a longitudinal wave
As further illustrated in Chapters 7 and 8, it is important to know whether c varies with frequency, or not. If not, the wave is said to be "non-dispersive", otherwise the wave is "dispersive". To decide on the dispersive nature of a wave, it is clearly necessary to relate the phase shift to the frequency. Such an equation is known as a "dispersion equation". In the present system the dispersion equation is readily obtained by substituting the trial solution into the current row of the system. The following equation is found:
The equation which thus provides the dispersion curve is:
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Discrete systems
According to its analytical form, it may be concluded that waves which travel along the chain of oscillators become dispersive, as soon as w- and thus also frequency - is sufficiently large. Returning to the problem of the finite chain, we can analyse the system further by specifying some conditions at the ends of the chain. Using the modal pattern evidenced in the case of N = 2, a first idea is to extend it to the case of a larger number of particles, by making a distinction between two families of mode shapes, namely those which are symmetrical with respect to the "middle particle" of order N/2, and those which are antisymmetrical. Of course "middle particle" is either an actual or a virtual particle, depending whether N is an odd or an even number. The condition to be verified for the first family clearly is that the first and the last masses must vibrate "in-phase". Whence, the permitted phase shifts between the first and the last DOF:
The condition for the antisymmetrical family is:
Based on these infinite sequences of permitted values, it is possible to define the N natural modes of vibration of the finite chain. The modal frequencies are:
Finally, including the modal phase angle into the complex solution, we get:
Recalling that the mode shapes are necessarily real, because the system is conservative, they may be obtained as a linear superposition of the real and imaginary parts of the complex solution:
Natural modes of vibration
The coefficients arb-
213
are real numbers which can be determined - save an
arbitrary multiplicative constant - by setting suitable conditions on the displacement of the particles at the ends of the chain. For instance, if the first and the last particles were not attached to a fixed wall through a spring, and so completely free to move, we would have necessarily aj * 0, bj = 0 . In the case they are fixed to the wall, and so completely prevented from moving, we have a. = 0, bj,# 0 . Finally, the case depicted in Figure 6.6 may be immediately identified with the last one. Indeed, nothing prevents us from assuming that there exist particles of mass M at the anchoring points of the first and last springs, labelled j - 0 and j = N+l, respectively. Such a (N+2)-DOF free chain is then constrained by the conditions
The complex amplitude of the modal vibration is given by:
This form may be interpreted as a standing wave, since at any time all the particles vibrate either in phase, or out of phase with the other, depending on the sign of the mode shape. A first interesting point is that whereas the frequency of an harmonic wave travelling along an infinite chain can be varied continuously, the standing waves exist only at discrete frequencies, given here by the first relation [6.37]. The physical reason for such a quantization is that in a finite chain the displacements of the particles have to comply with the end conditions. A second interesting point, which is also helpful for the better understanding of the quantization of the modal properties, is that the modal vibration can be interpreted as the linear superposition of two waves which travel in opposite directions, at the same speed. Indeed, the complex amplitude [6.38] may also be written as:
where the mode shape [6.37] was first multiplied by -2i.
214
Discrete systems
The first exponential stands for a wave which travels to the right and the second for a wave which travels to the left, in agreement with the result [6.27]. The wave [6.39] is merely a particular case of the following linear superposition:
To comply with the end conditions q> (0) = (p (N +1) = 0, it is necessary that:
The first condition connects the relative amplitude of the two travelling waves, in such a way that they are reflected with a change of sign at both ends of the chain. The second condition states that nontrivial solutions occur solely at particular values of the phase of the waves, which are the integral multiples of an elementary phase angle. Physically, except for the specific values of the modal phase angles wj, given by [6.41], the incident and reflected travelling waves interfere destructively. Turning now to the results produced by solving numerically the modal system, we adopted the following numerical values: K = 4 000 N/m, M = 4 kg, N = 60. Figure 6.8 is a plot of the natural frequencies /„ versus the modal order n. Figure 6.9 displays a few mode shapes. The relative displacement of every particle (or oscillator) is symbolised by an asterisk (*). Superposed on such a plot, is also represented, as a full line, the continuous function sin(n^), in the interval 0 < £ < 1.
Figure 6.8. Natural frequencies versus modal order
Natural modes of vibration
215
Figure 6.9. Low order mode shapes
As we shall see in the next two chapters, this chain of identical oscillators is a very convenient system for discussing the propagation of non harmonic waves, in the dispersive and then in the non dispersive case. 6.4. A few extensions of the modal concept 6.4.1. Natural modes of vibration of constrained systems Let us start with an autonomous system described by a vector \q\ of generalized displacements comprising N components, which is constrained by an independent set of / < N scleronomic conditions, written as a vector denoted |9l ([
where the elements of the matrix of linear constraints [L] are constant. The constrained system is governed by the constrained Lagrangian:
216
Discrete systems
£> is the Lagrangian of the unconstrained system, which is characterized by the matrices [K] and [M]. The set of the undetermined Lagrange's multipliers related to the constraint conditions form a vector denoted [A]. Since scleronomic constraints are modelled as a potential, the constrained system remains conservative and we may proceed to the modal analysis of it. Lagrange's equations of the constrained and linearized system take the form:
Since [A] and [q] are the unknown of the problem, it is suitable to gather them into a unique vector. Such a procedure allows one to transform the system [6.44] to the symmetrical matrix form:
The system [6.45] is said to be of mixed type, because the vector of unknowns comprises both generalized displacements and forces. As a consequence, the global matrices do no longer have the physical meaning of stiffness and mass matrices. Incidentally, it is noted that the second matrix is singular. Such particularities do not prevent in any way the modal analysis of the constrained system. The modal vectors thus found gather the physical information about the modal displacements and the modal reactions of constraints. EXAMPLE. - Two engaged gears on flexible supports
Figure 6.10. Simplified model of two gear-wheels on flexible shafts
Natural modes of vibration
217
We want to determine the natural mode of vibration of the system sketched in Figure 6.10, which is a very simplified model of two parallel rotating shafts coupled to each other by two gear-wheels (radii R1 and R2). The angular displacements are related to each other by the condition of rolling without slipping, which is written as:
where 0, and 02 are positive in the anticlockwise direction. On the other hand, the shafts consist of straight beams of revolution which are deformed in torsion (cf. Volume 2, Chapters 3 and 4). We adopt here a drastically simplified model in which elasticity of the shafts is taken into account by two angular springs of stiffness coefficients K1 and K2, whereas their inertia is neglected. The wheels are assumed to be rigid with polar inertia Jl and J2. Mathematical formulation of the problem is quite straightforward. The Lagrangian is written as:
Hence dynamic equations are found to be:
The modal equations are written in the matrix form:
The singularity of the "mass matrix" does not generate any difficulty. The quotation marks are used to stress again that we have not to deal here with the physical mass matrix, but only with a matrix which plays the same role from the mathematical point of view. Of course, in the present example a solution can be easily be obtained by eliminating the superfluous variable. The following natural pulsation is obtained:
The mixed mode shape, normalized to a unit displacement #2 is found to be:
218
Discrete systems
whence the physical constraint reactions are deduced. The torque exerted on the first wheel is
and that exerted on the second wheel is
It is noted that they are found to be precisely in the ratio of the wheel's radii. This could be expected, as the reaction forces exerted on each wheel have to be exactly opposite to each other, in agreement with the principle of action and reaction. Therefore, the following relation is necessarily verified:
As a final remark on the subject, it may be noted that if Kv - K2 = 0, the wheels are left to spin freely at constant angular speed. Such a free SDOF system complies with the condition of rolling without slipping with a zero contact force and its natural frequency is also zero.
6.4.2. Free modes of rigid body As already discussed in Chapter 3 subsection 3.2.2, and indicated again above, a null eigenvalue in the [K] matrix indicates the presence of a state of indifferent static equilibrium. This also implies the presence of a mode with a generalized stiffness and natural frequency equal to zero, (cf. last remark of the last example). Such particular modes, which involve zero potential energy, are termed free modes of a rigid body. Indeed, they are encountered in material systems provided with no supports. EXAMPLE. - A pair of particles coupled by a spring Let us particularize the system shown in Figure 6.1, by assuming K1 = K2 = 0, M, * M2. The interested reader will have no difficulty in finding the presence of an in-phase mode [^,]r =[l
1]; #>, =0, together with that of an out-of-phase mode,
Natural modes of vibration
219
It may also easily be checked that such modes are orthogonal to each other with respect to [K] and [Af]. As a further example, we could have chosen also the chain of N >2 identical oscillators studied in subsection 6.3.2.3 with the end masses left free. In the same way a solid body, left totally free, has six free modes of rigid body. 6.4.3. Prestressed systems and buckling modes In most cases, the small oscillations we are interested in take place about a state of static equilibrium which is already in a stressed state, resulting from the initial application of a certain static load. Moreover, in many instances, the dynamical equations are found to depend upon the prestressed state of static equilibrium. A familiar example of such systems is the simple pendulum, which is subjected to the gravity field. If such is the case, it is pertinent to separate the potential energy into two distinct parts, namely the so called elastic component, which is independent from the static load and the prestress component which is related to it. Accordingly, linearization of Lagrange's equations produces two distinct stiffness matrices, namely the prestress matrix [KQ] and the elastic matrix [Ke]. Like [AT, ], [£0] is symmetrical, however its sign depends on the stressed state of equilibrium which is taken as a reference. As expected, [K0] is negative if the state of equilibrium is unstable in the absence of elastic restoring forces, and [K0] is positive if it is stable. It turns out that in many systems the initial stresses have a destabilizing effect, stability of the system being recovered by providing the necessary elastic forces. We have already discussed such a system in Chapter 3, subsection 3.2.2.3. It is found convenient to characterize the magnitude of the initial stresses which governs the state of static equilibrium by a dimensionless parameter /I, called the load factor. This allows one to write the modal equations of the system as:
As a consequence, the natural modes of vibration are found to depend upon the load factor. In particular, they become statically unstable when the modal stiffness Kn (/I) passes from a positive to a negative value. The threshold of modal buckling instability occurs at the critical value Ac „ of the load factor, such that Kn (Tc n ) = 0:
Actually, the mechanical system taken as a whole buckles according to the mode which has the smallest critical value of load factor. On the other hand, it is also clear
220
Discrete systems
that provided A is still less than T c n , the natural frequency of the n-th mode remains positive. It decreases as the load is enhanced to vanish at the critical load of modal buckling. Beyond this threshold, the positive branch wn (A) bifurcates into a pair of imaginary conjugate branches. Incidentally, this kind of bifurcation is known as a pitchfork-like Hopf bifurcation. As already shown in Chapter 5 section 5.2, with regard to the system of the articulated bars put in compression, to study the behaviour of the system in the post-buckled state it is necessary to have recourse to a nonlinear model. On the other hand, it is worth noting that the study of buckling instability can be performed by making a static analysis. Indeed, the necessary and sufficient condition for buckling instability to occur can be expressed as the following static equation:
Now, a striking analogy between the static equation [6.48] and the modal equation [6.11] is obvious. Indeed, it suffices to change co2 into A and [M ] into [KO] . The nontrivial solutions of matrix equation [6.48] are known as the modes of elastic buckling. EXAMPLE 1. - Upside-down and coupled pendulums Figure 6.11 shows schematically a system made of two rigid masts erected vertically. They are supported by a free articulation at the bottom and by linear springs at the upper end. A mass-point is located at a height 0 < £ < L on each mast. M denotes the mass supported by the first mast and aM that which is supported by the second mast. The mass of the masts is assumed to be negligible.
Figure 6.11. Pair of upside-down and coupled pendulums
a
1.
Static analysis of buckling
Figure 6.12. Eigenvalues of the global stiffness matrix versus the load factor
We start by analysing the static stability of the system in relation to the load parameter (.. Potential energy of the system for small angular displacements 01, 02 is written as:
Whence the following stiffness matrices:
Buckling modes are obtained by solving the modal equation:
The modal critical loads for elastic buckling are the roots of the characteristic equation:
222
Discrete systems
The mode shapes are then deduced from the equation:
A particularly simple case is that of two identical mass-points (a= 1), which gives the following results: In-phase buckling mode: Out-of-phase buckling mode As expected, the lowest threshold for buckling instability is provided by the inphase mode which is independent from the coupling stiffness and less stiff than the out-of-phase mode. Therefore, in practice, when the system becomes unstable, the masts crash down in the same direction. Such results are illustrated in Figure 6.12, where the eigenvalues of the modal matrix of elastic buckling are plotted versus the load factor A. If a * 1, the results are qualitatively the same, except that the angular displacement of the in-phase buckling mode of the heaviest mass is larger than that of the other mass. For this reason the threshold of instability depends not only upon Kg but also upon Kc. 2.
Dynamical analysis of the system
Figure 6.13. Natural frequencies versus load factor
Natural modes of vibration
Kinetic energy of the system is
223
Thus the mass matrix
is similar to the matrix of initial stress:
The modal problem is written in terms of reduced variables as:
Figure 6.13 illustrates the manner in which the natural frequencies of vibration of the system change when the load factor is increased. Only the real values are displayed in this graph. As expected, the frequencies are found to decrease monotonically as A increases and to vanish at the critical load. It is also worth mentioning that in the vicinity of Ac n the decrease of the corresponding modal frequency is fairly fast. As a consequence, it is advisable to adopt a suitable safety margin for designing this kind of structures. EXAMPLE 2. - Two articulated bars set in compression
Figure 6.14. System of two articulated bars with axial load
224
Discrete systems
The 2-DOF system to be analysed here is a natural extension of the SDOF system studied in Chapter 3 subsection 3.2.2.3. As shown in Figure 6.14, it is made up of three bars (ACj); (C{C2); (C 2B ) of same lengths L, which are assumed to be rigid. In the non deformed configuration, the bars are set in line along the Ox axis. Possible motions of the system are restricted to those taking place in the plane of the figure. Moreover, the end A slides freely along the Ox -axis, and B is fixed (position 50). The bar (CB) is articulated at B0. The bars are also articulated to each other at C, and C2, however the connections are also provided with an angular stiffness K and a mass M, assumed to be much larger than the mass of the bars, to simplify the algebra. Finally, the system is prestressed by applying to A an axial force F0. #,,# 2 ,0 3 standing for the angles made by the bars with the Ox-axis, the transverse displacements of Q and C2 are found to verify the following relations:
As a consequence, it is found that 6?,,02,03 are bound to each other by a holonomic condition. The system vibrates about the state of static equilibrium defined by: 0, =02 = 0; 93 -n. It could be easily checked that the displacements in the axial direction are of the second order in 6. Therefore, only the transverse displacements are considered here. Performing the variable transformation if/ = n - 03, it is found that:
Neglecting the mass of the bars, kinetic energy is: the mass matrix:
Elastic energy is:
Whence
Natural modes of vibration
whence the elastic matrix
225
K
Prestress energy is deduced from the work done by the load F0 when A slides along the Ox-axis. It is found that:
As the aim here is restricted to formulating the linear equations of motion, the above expression can be simplified by performing a Taylor's expansion about the position of static equilibrium, limited to the second order. This gives the quadratic form of potential energy:
and the prestress matrix:
The modal system is thus written as:
where The natural pulsations are the roots of the characteristic equation:
Here again the mode shapes are found to be: [$?, ]T = [l l]; [q 2 ] r = [l related configurations of the bars is shown in Figure 6.15.
-l]. The
226
Discrete systems
Now, if F0 > 0, the system is compressed, which turns out to have a destabilizing effect. The system buckles according to the in-phase mode at the critical load parameter: Acl = (OQ ==> Fc, = 2 K / L . The out-of-phase mode is destabilized at the higher critical load parameter: T2 = V3A fl . The last result is not surprising since, for the same amount of destabilizing work H£ o , the stabilizing springs do more work according to the out-of-phase mode than according to the in-phase mode.
Figure 6.15. Mode shapes of the system of two articulated bars
Finally, if F 0 < 0 , the system is tensioned, which turns out to enhance the stiffness of the system. Therefore, the natural frequencies of the two modes increase with the magnitude |F0| of tension. EXAMPLE 3. - Stretched chain of coupled mass-spring systems
Figure 6.16. Stretched oscillator chain and transverse waves
Let us consider once more the chain of N oscillators described in subsection 6.3.2.4. But here we assume that the chain has been stretched in such a way that the state of static equilibrium is now stressed. The length of the stretched chain is L = (N+ l)€, instead of L^ =(N + l)l 0 which refers to the relaxed state. Further,
Natural modes of vibration
227
we are now interested in the small vibrations which take place in the plane Oxz of the Figure 6.16. The tension in the stretched chain is readily found to be T0 = K(f-(0). Denoting Zn the small transverse displacement of the n-th particle, the elastic energy of the first oscillator is calculated as follows:
The problem being restricted to small linear oscillations, it is left as an exercise for the reader to verify that the following approximation suffices, which is identified with the prestress potential:
Applying the same procedure to the other oscillators, the total prestress potential is found to be:
On the other hand, it is also easy to show that in the linear approximation the longitudinal oscillations are independent from the stressed state. Thus, it turns out that the transverse oscillations of the system are governed by the same equations as the longitudinal oscillations, provided the elastic coefficient K is replaced by the prestress coefficient:
The above result shows that the ratio of the natural frequencies of the transverse to the longitudinal modes is simply the square root of the stretching deformation, which is usually a small parameter.
6.4.4. Rotating systems and whirling modes of vibration Let us consider a mechanical system vibrating about a state of permanent rotation, characterized by the spin velocity Q . [q] denotes the vector of generalized displacements defined in the rotating frame. In agreement with the results established in Chapter 4 subsection 4.3.2, the kinetic energy of vibration is the sum of the three following components, which are suitably expanded in Taylor's series up to the second order:
228 1.
Discrete systems Relative energy is reduced to a quadratic form in \q\, involving constant coefficients of mass. The latter constitute the elements of the mass matrix [M ], which is symmetrical and positive definite.
2.
Transport energy is reduced to a quadratic form in [q], involving constant coefficients of stiffness. The latter constitute the elements of the stiffness-matrix [KO] of centrifugal prestress, which is usually symmetrical and negative definite.
3.
Mutual energy is reduced to a bilinear form in [g] and [q~\, the coefficients of which constitute the elements of the antisymmetrical gyroscopic matrix [G]. Therefore, the system of modal equations takes the following canonical form:
In this modal system, Q is thus acting as a load factor. On the other hand, though system [6.49] involves imaginary terms, it still remains conservative as already shown in Chapter 4, relation [4.26]. Consequently, the natural frequencies of the system remain real and positive, in so far as the threshold for buckling instability is not yet crossed. On the other hand, the gyroscopic coupling induces mode shapes which are complex. It is worth examining further such a particularity by considering two distinct examples. 6.4.4.1 Particle tied to a rotating wheel through springs
Figure 6.17. Mass-point tied to a rotating wheel through linear springs
Natural modes of vibration
229
Returning to the system shown in Figure 6.17 which was already considered in Chapter 4, subsection 4.3.2, we recall that the equations of motion written by using the displacements X,Y defined in the rotating frame were found to be:
As a first step, it is of interest to isolate the gyroscopic coupling effect. We start thus by discarding the centrifugal terms. Accordingly, the system of modal equations reduces to:
The characteristic equation is:
which has the two following positive roots:
The reason for using the subscripts 'F' and '£' will be made clear below by investigating the mode shapes related to the modal pulsations [6.52]. Substituting caF,coB in the system [6.50], the following complex mode shapes are produced:
Such shapes are conventionally normalized here by the condition XF = XB = 1. On the other hand, the real vibration is obtained by retaining either the real part or the imaginary part of the complex amplitudes:
Figure 6.18. Forward mode: (a) Re[Af], (b) lm[AF]
230
1.
Discrete systems
Forward mode The real part of A
gives
The imaginary part of AF = [i -ij ]eiwFt gives:
As shown in Figure 6.18, plotting the motion of the mass-point given either by [6.55] or [6.56], results in a circular motion at angular speed COF , which is oriented in the same counter clockwise direction as the permanent rotation. For this reason, such a mode is known as a forward (or direct) mode. In the inertial frame, the angular speed is found to be w F + Q = w2 + Q1 . 2.
Backward mode
Figure 6.19. Backward mode: (a) Re[A R ], (b) Im[A R ]
The real part of A
The imaginary part of A
gives;
gives;
Natural modes of vibration
231
As shown in Figure 6.19, circular motions governed by [6.57] or [6.58] are oriented in the clockwise direction, opposite to that of the permanent rotation. For this reason, such a mode is known as a backward (or retrograde) mode. In the inertial frame, the angular speed is found to be opposite to that of the forward mode:
A straightforward calculation shows that the forward and the backward modes are orthogonal to each other:
Figure 6.20. Campbell's diagram of the system in the presence of gyroscopic coupling and without centrifugal stiffness, case of symmetrical supports K = Kz
In Figure 6.20, the natural frequencies of the system are plotted versus the speed of permanent rotation. Such a plot, broadly known as a Campbell diagram, is presented here in a dimensionless form, by using w0 as a pertinent scaling factor. It is noted that the backward frequency tends to infinity with O / w 0 , whereas the forward frequency tends asymptotically to zero. In this particular system, the
232
Discrete systems
gyroscopic coupling is found to increase the frequency of the backward mode and to diminish that of the forward mode. However, we shall see later that the trend is inverted in many other systems. On the other hand, even if either the backward or the forward natural frequency tends to zero when Q tends to infinity, the gyroscopic effect does not induce any buckling instability of the rotating system. Taking now into account both the gyroscopic and the centrifugal effects, the modal system is written as:
The characteristic equation is:
which has the four roots:
Figure 6.21. Campbell's diagram of the system in the presence of gyroscopic and centrifugal effects, case of symmetrical supports Ky = Kz
Restricting the analysis to non negative frequencies, it is found that the forward and the backward modes have the natural pulsations:
Natural modes of vibration
233
In the range Q<(OQ, there is a forward and a backward mode and it is found that (OB increases linearly with Q, whereas COF decreases linearly with Q. The frequency of the forward mode vanishes as Q = a)0. In the range Q > fW 0 , it is found that two backward modes are coexisting, their frequency increasing linearly with Q, see Figure 6.21. Because of the presence of gyroscopic coupling, analysis of stability is less straightforward than in the case of systems vibrating about a static state of equilibrium. However, it may be noted that the four roots [6.62] of the characteristic equation are real at any spin velocity. Therefore, in contrast to the systems analysed in subsection 6.4.3, there is no bifurcation, marking a change of sign of stiffness and consequently, no buckling. However, the analysis of the same system, performed in the case of asymmetrical supports (Ky * Kz), evidences the existence of a finite interval of angular speeds into which the system is buckled. This interval is found to be precisely equal to the gap between the natural frequencies of the uncoupled system at .(2 = 0, see Figure 6.22.
Figure 6.22. Campbell's diagram of the system in the presence of gyroscopic and centrifugal effects, case of asymmetrical supports Ky ^ Kz
234
Discrete systems
6.4.4.2 Fly-wheel on flexible supports The system to be discussed here is sketched in Figure 6.23. It is made of a rigid disk locked at mid-span to a rigid axle, the ends of which are supported by linear springs acting into two transverse directions (fixed unit vectors j,k ), orthogonal to each other. It is found convenient to define the inertial Cartesian frame Oxyz, of unit vectors i ,j,k . Origin O coincides with the centre-of-mass of the disc.
Figure 6.23. Fly-wheel on flexible supports
The system spins about the axle at the angular speed Ql, where / is the unit vector along the axle in its actual configuration. It can be defined as the transform of i through the small rotations Vy,Vz about the axis j,k which describe the rocking modes of vibration of the system about O. We are interested here in investigating how such modes are coupled by the permanent spin. Equations of small vibrations can be obtained either as a direct application of the theorem of angular momentum (cf. Chapter 1, subsection 1.2.3.2, example 3) or by using the formalism of Lagrange equations. In the present problem, the first method is the easiest one. Indeed, as we shall see a little later, calculation of kinetic energy is not so straightforward here, because the rate of rotation involves the large component Ql . 1.
Vector formulation: rate of change of angular momentum In the absence of vibration, the angular momentum of the disk is: [6.64] In the presence of a vibration according to the rocking modes, it becomes:
Natural modes of vibration
235
The equations of motion are thus readily obtained as a direct consequence of the balance of moments (cf. equation [1.27]), which takes here the form: L = restoring moment exerted by the springs Therefore, the following equations are obtained:
Substituting the result [6.65] in [6.66], the equations take the form:
whence the modal system:
The system [6.68] is quite similar to [6.50], the only difference being the sign inversion in the gyroscopic coupling. Accordingly, the modal pulsations are:
The sign inversion of the coupling implies that of the evolution with Q of the forward and backward modes is also inverted, see Figure 6.24. The physical displacement of the end A of the axle is given by:
As expected, the vibration takes place in a plane passing through Ox, which rotates in the forward direction Q at the angular speed (OF . It is worth emphasizing that, in contrast with the former example, in the present case rotation of the modes is relative to the inertial frame.
236
Discrete systems
Figure 6.24. Campbell's diagram of the disk on flexible supports
2.
Lagrange 's equations
The crucial point in formulating Lagrange's equations of this system is to produce a correct expression for the kinetic energy of rotation. Use is made of the noninertial Cartesian frame, Ox'y'i , which is defined in such a way as to keep the same notations as in Appendix 3. Accordingly, Oz is the spin axis and Ox\Oy' are an orthogonal pair of principal axes of inertia of the disk. Therefore, kinetic energy of rotation can be written as:
where Q' is the vector of rotation rate as defined in the frame Ox'y'z
which is co-
rotative with the disc. The problem now is to determine the components of Q' in terms of the angular velocities as defined in the inertial frame. Thus, it may be noted that the frame transformation required here is the reciprocal of that used in the first example. As a large rotation rate Q( is present in the problem, it is no longer possible to linearize the frame transformation. It may be worth noting that such a linear calculation would lead to wrong results, as spurious centrifugal forces which have no physical meaning would appear, since the system rotates about the centre-ofmass. In order to carry out the pertinent calculation use is made of Euler's angles. The procedure is detailed in Appendix 3.
Natural modes of vibration
237
In the frame used to define the Euler angles, the Lagrangian can be written as:
Lagrange's equations are:
Of course, the equations [6.73] differ from [6.67] only by the different labelling of the coordinate axes. The following permutations: z—> x, x—> y, y—> z allow one to return to the definition used in [6.67]. Dynamical behaviour and stability of rotating systems will be further investigated in Volume 4, Chapter 2, including damping, see also for instance [GEN 95] and specialized books such as [LAL 90], [CHI 93], [KRA 93], [ADA 01].
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Chapter 7
Forced vibrations: response to transient excitations
The study of the forced responses of material systems to various kinds of external excitation is usually the central object of modelling. Determination, or at least some global description, of such responses is clearly of major concern in many fields of physics and of mechanical engineering. In contrast to the vibration modes, which remain an abstract concept, forced responses are amenable to measurement. Furthermore, they control operating and mechanical integrity conditions of most material devices and structures. The problem of solving the equations of motion is often extremely difficult, requiring generally the use of advanced computational methods. This is mainly because of the presence of nonlinearities. However, linear Lagrange's equations may usually be solved analytically, provided that the coefficients are constant and that the excitation has a suitable analytical form. Here, we describe a general method for solving this kind of problems based on the Laplace transformation, which allows one to replace the time dependent differential equations by algebraic equations, expressed in terms of the Laplace variable. As will be shown, this method presents several advantages over the direct method of solving the differential equations in the time domain.
240
Discrete systems
7.1. Introduction Let us consider a discrete and linear mechanical system, which is excited by an external force vector, applied starting from time t = 0. As already seen in the preceding chapters, the motion is governed by a set of ordinary differential equations with constant coefficients and of second order with respect to time. The problem is fully determined if the equations are provided with a suitable set of initial conditions, which specify the dynamical state of the system at t = 0. Equations of motion are thus written as:
It has to be emphasized that the vector of external force \Q^ (t )| is assumed to be identically zero at any time in the range t < 0, i.e. before motion is studied. In this chapter, a general method is described for analysing the system [7.1]. Furthermore, by applying the method to a few simple cases for which analytical solutions are available in closed form, it is possible to highlight the major features of the dynamical responses to a broad class of excitation signals, known as transients. As a preliminary, section 7.2 describes the mathematical properties of the deterministic transients, which are physically relevant as excitation signals in mechanics. Section 7.3 deals with the forced responses of SDOF linear systems. Analysis of such problems is performed by using the Laplace transformation. For this mathematical tool, a brief review of the definition and the few properties that are necessary in our applications are presented. The reader, who would like to deepen or refresh his/her, theoretical background on the subject, is referred to standard textbooks such as [DETT 84], [ZEM 65]. In the domain of the Laplace variable (the so called image domain), response is conveniently expressed as the product of two distinct functions. One of them, known as the image of the external loading, characterizes the excitation signal and the initial conditions, whereas the other one, known as a transfer Junction, characterizes solely the response properties of the excited system. Thus, in terms of Laplace transforms, a very clear distinction is made between the intrinsic properties of the excited system and those of the external excitation, in such a way that the influence of each of these quantities in the response can be investigated separately. The inverse Laplace transformation is then used to obtain the response in the time domain. At this step, mathematical difficulties may arise, because the inverse Laplace transform is available analytically, or not, depending on the analytical form of excitation signal. Fortunately, in practice the study of a few typical transients leading to simple analytical solutions is sufficient to illustrate the main features of the response signals which are of major interest to the engineer.
Forced vibrations
241
In section 7.4, the harmonic oscillator is used to introduce the concept of Green's Junction - also called unit impulsive response in the language of engineering - which is defined as the inverse Laplace transform of a transfer function of the system considered, or in more physical terms, as a response to an impulse of unit magnitude, applied to a single degree of freedom. When extended to MDOF and continuous systems, Green's functions provide us with a powerful tool for analysing various dynamical problems, e.g. waves travelling in continuous media, as further discussed in Volumes 2 and 3. These are also often encountered in modal testing procedures, since excitation of the structures by an impact is an experimental technique which is widely used. In section 7.5, the formalism is extended to the case of MDOF linear systems. Their response properties are now entirely characterized by a transfer function matrix, which is generally full. Fortunately, it can be uncoupled by projecting it on a modal basis, provided a suitable model for damping can be adopted. Then, Newmark's implicit algorithm for numerical integration of the equations of motion, which was introduced in chapter 5 in the case of SDOF systems, is extended to MDOF systems. Indeed, implicit algorithms present a few interesting particularities worthy of mention when passing from SDOF to MDOF systems, in particular, the solution procedure becomes truly implicit. Application is made to a heavily damped system in modelling the shock absorber in a car suspension. 7.2. Deterministic transient excitation signals 7.2.1 Locally integrable functions and regular distributions Let us start with deterministic and real functions of time/(f). By definition, such functions connect to each other the real numbers t1 and / (^ ), where /, lies within the range of definition of/(f). Moreover, the mapping ?„ — > / ( f j ) is deterministic, which means free of any uncertainty, or randomness. Restricting study to the physical context of mechanics, /(?) is assumed to be a continuous or a piecewise continuous function of finite magnitude:
Bounded functions with a finite number of jumps are locally integrable, which means that: dt does exist, V the finite interval T
Such functions generate regular distributions defined by the integrals:
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where use is made of the functional vector notation <, > of the scalar product. Accordingly, f ( t ] and \j/(i) are interpreted here as functional vectors, as detailed in Appendix 1. On the other hand, J/(f) designates an auxiliary function, termed test function that complies with the following very restrictive conditions: •
i/f(t) is identically zero, outside a finite interval i.
•
iff (?) can be differentiated up to any desired order.
NOTE. - Theory of distributions The concept of distributions, introduced first by Dirac, and then formalized mathematically by Schwartz, is a convenient tool for analysing mechanical systems. Appendix 6 may be used as a mathematical refresher concerning the few definitions and theorems which are necessary for our applications. To the reader interested in the subject, the book by Stakgold [STA 70], in which there is a clear and pragmatic presentation of the mathematical formalism, is recommended. Amongst many other references of interest, let us quote [SCH 50], [BRE 65], [ZEM 65]. 7.2.2 Signals suited to describe transient excitations For mathematical convenience a signal suited to describe a transient excitation is first defined as a locally integrable function that complies further with the following conditions: 1.
f(t)*0,onlyifO<
t
2. f ( t ) is upper-bounded. As a corollary, such transients are integrable in any time interval:
According to this definition, p is the action of the excitation signal over its total duration T. As already stated in Chapter 1, if f(t) stands for a force, p is a linear momentum, and if f(t) stands for a moment, p is an angular momentum. On the other hand, when writing the integral of action in the functional form (i.e. as in the last expression of [7.5]), it is considered that the unit function can be considered to be a particular test function, since the integral is restricted here to the finite interval T.
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3. Furthermore it is also required that /(t) be square integrable:
Considering f(t] as a functional vector, 8 stands for its natural norm, as made explicit by the last expression of [7.6] that identifies 8 with the scalar product of /(r) by itself. 8 is known as the energy of the signal f(t). The reason for this terminology is clear, since in many physical applications energy can be formulated as a quadratic form of a given function, e.g. in mechanics. On the other hand, when dealing with physical problems, it seems rather natural to be interested in finite energies only, as stated by the condition [7.6], though impulsive excitations are a notable exception, as we shall see in subsection 7.2.3 and further in Chapter 8. NOTE. - Lebesgue's integrability and functions QR'(0,T) From a formal viewpoint, we should specify that formula [7.6] refers to a Lebesgue integral, since the space of the Riemann-integrable functions is not extensive enough to accommodate the natural norm [7.6]. Furthermore, in the direct line of vector algebra, it may be said that f(t] is a vector of Hubert's space denoted /£ (0,r), where the suffix (R) indicates that f ( t ) is a space of real valued functions only, and where the subscript 2 indicates that Lebesgue's integral of the squared functions exists over any interval, finite or not. Finally, the values within the parentheses specify the domain in which the function is not identically zero. However, such formal aspects are quoted here for reference to the mathematically oriented literature. More details on Lebesgue's integral are far beyond the scope of this book and would not add to a better physical understanding of its content. A convenient way to define analytically a transient signal is to make use of the Heaviside step function, which shall be advantageously interpreted as a distribution, just below. If considered as a function, the Heaviside step is written as:
It is worth noting that the particular value assumed to hold at t = 0 is selected in accordance with the expansion of !/(t)as a Fourier series over a finite interval, as detailed in Chapter 8, section 8.2. However, anticipating the results established in section 7.3, it is worth stressing here that, from the physical point of view, the actual value of &(t) taken at t = 0 is unimportant. Actually, it is much more fruitful to consider the Heaviside step as a distribution than as a function. Accordingly,
i/(t)
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is characterized by its action on functions g ( t ) , which are not necessarily transients. By definition, the action is:
Hence, it is found that the action of &(t) on g(t) coincides with the quantity already termed action of g(t), as defined in Chapter 1. Further, !/(t) is found suitable to truncate a function g(t) - initially defined over an infinite domain - over a finite interval [0, T] , producing thus the transient f(t):
7.2.3 Impulsive excitations: Dirac delta distribution External impulses are defined as peculiar transients which have the characteristic property of having a nonzero finite action, whereas their duration is arbitrarily short. An impulse of amplitude p applied at time t0 is written analytically as the so called Dirac pulse pd (t - t 0 ), where use is made of the Dirac delta distribution, noted S( ). S(t) is a singular distribution that is identically zero if t*Q and has an arbitrary value at t = 0. The operating characterization of S( ) is made through the integral of action. Indeed, by definition, the action of £(ty)on a test function ^(?) is:
The action of S ( t - t 0 ) on a function g ( t ) , which is subjected to the sole condition of existence at time t0, can be defined over any arbitrary time-interval [t Pt2 ]as:
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NOTE. - Two false problems: numerical value of £(0) and energy of 8 (t) ?
Figure 7.1. Sequence of functions converging to
§(t)
For sake of clarity at least, it is useful to insist on the fact that £(t) cannot be interpreted as an ordinary function. Indeed, as already indicated, its value at t = 0 is unimportant, the sole quantity that is relevant is its action on other functions. In particular, it may be shown that by defining £(t) as the limit of a sequence of suitably defined functions Sn (t), we can assign what value we want to (0), for instance +°° in example (a), and -«> in example (b) of Figure 7.1. Moreover, the reader should have no difficulty in building other suitable sequences for which lim (S (0)) is a real number, arbitrarily selected. For a more detailed discussion on n—>~
v
f
this point, see in particular [STA 70]. The practical importance of such a remark can be further emphasized by asking about the energy of a Dirac pulse. According to the general definition [7.6], energy would be given by the scalar product (S(t],S(t)). However, such an expression is meaningless since the action of S(t) on itself, as defined by [7.11], would be £(0), which is undetermined. Stated informally, the purpose assigned to S(t) is to fix the time of occurrence of an event, therefore squaring such an assignment is clearly meaningless! More generally, and for the same basic reason, it would also be meaningless to define a function by using a Dirac distribution as an argument. On the other hand, we shall see in the next chapter that energy contained in a Dirac distribution can be considered to be infinite, based on the spectral point of view. It is necessary to interpret properly the dynamical equation of a harmonic oscillator, which is forced by a Dirac pulse:
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Here again, it is suitable to start from the relation [7.11]. According to it, equation [7.12] has to be interpreted in terms of action. With this object in mind, we proceed to an integration of [7.12] over a time interval of arbitrary duration £, bracketing t0. This results in:
Now, when £ —» 0 the result [7.13] tends to:
Indeed, since the displacement q(t) is a continuous function of time, the terms arising from stiffness and from damping vanish when e -»0. The only non vanishing term arises from inertia, producing a finite jump of velocity occurring at t0. The symbolic equation [7.12] is thus equivalent to the operative system:
As a particular case, the system [7.15] furnishes the physical interpretation of a Dirac pulse 8 (t - tQ ), which stands for an impulsive load of unit amplitude, applied at t0. Another particular case of practical interest is that of an impulsive load applied at t0 = 0 to an oscillator, which was at rest before. Indeed, it is found that:
Put in words, loading a system by an initial impulse is equivalent to prescribing a nonzero initial velocity.
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7.2.4 Excitations of infinite duration and finite energy It is also possible to extend the class of the transient excitations to the case of functions f ( t ) which are not identically zero beyond a certain time, but which decay sufficiently rapidly when t—>°o, so that /(f) and (/(O) are integrable over the range t > 0. The action and energy of such functions are thus finite. As a simple example, let us consider the force signal:
7.3. Forced response and Laplace transformation The system [7.1] reduced to a single equation and provided with given initial conditions is written as:
According to the general theory of linear differential equations, solution of the problem [7.17] can be obtained by adding a special solution of the inhomogeneous differential equation to the general solution of the corresponding homogeneous one. Constants of integration are calculated afterwards in order to fit the initial conditions of the specific problem to be treated. However, for a systematic study of the solutions, where the exciting signal is varied at will, it is found preferable to make use of the Laplace transform technique. As we will see in the next subsection, obtaining the Laplace transform of the solution is an easy task, producing quite valuable information on the physical problem. Then, in order to shift from the image to the time domain, it is necessary to carry out an inverse Laplace transformation. In many instances, the calculation can be performed analytically without major difficulties, producing the time-history of the response.
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Discrete systems
7.3.1 Laplace and inverse Laplace transformations Let X (f) be a function, or a distribution, with the following property:
where a is a real and finite constant. Such a function has a Laplace transform, which is noted either LT^X (f)], or more concisely X (s). The Laplace transformation is defined by the integral:
where the Laplace variable s is usually complex. Writing the Laplace variable as s = a + ico and assuming that the condition [7.18] holds, it can be concluded that the integral [7.19] exists for any value of a larger than a, i.e. on the right side of the line x = a in the complex Oxy plane. Now, in order to obtain a one-to-one correspondence between the original function X (f) and its image X (s) through a Laplace transformation, the additional condition must be satisfied:
Properties of the Laplace transformation which are needed for the applications in this book are given in Appendix 7. Most of these may be easily inferred from [7.19]. The major advantage of using the Laplace transformation to solve ordinary differential equations, which are linear and having constant coefficients, is that they are thus transformed into linear algebraic equations in the image domain. Obtaining the Laplace transform of the solution is thus immediate. The original solution in the time domain is then calculated by using the inverse Laplace transformation, noted LT~l[x (s)J, which is defined by the following integral, to be carried out in the complex plane:
where the line joining the points a - i°° and a + i<*> in the complex plane leaves on the left hand side all the possible singularities arising in the kernel of the integral [7.21].
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Whether the calculation can be carried out analytically, or not depends upon the nature of the singularities. Calculation is especially straightforward when the method of residues can be used, as detailed in Appendix 7, paragraph A7.9.
7.3.2 Transfer functions of the harmonic oscillator The Laplace transformation of equation [7.17] gives:
This result is established by using the theorem of differentiation [A7.3]. Now, it is of particular interest to express equation [7.22] according to the following canonical form:
Figure 7.2. Transfer box of a linear system with single input and output
The form [7.23] is quite remarkable not only because of its simplicity but also because of its physical content. Indeed, it shows that in the domain of the Laplace variable (image domain), the response of the system is obtained as the product of two distinct functions: 1. H(s) characterizes entirely the response properties of the system considered. H(s) is known as a transfer function. According to this terminology, the system is viewed as transferring an input signal (the excitation), to an output signal (the response), as depicted in the logical scheme of Figure 7.2. On the other hand, depending upon the physical nature of the input and output signals, distinct transfer functions can be defined for a same system. For instance, in the case of a harmonic oscillator, the transfer function relative to the pair displacement/force of output/input signals is:
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Discrete systems
Transfer functions for the pair's velocity/force and acceleration/force are obtained by multiplying [7.24] by s and by s2 , respectively. The transfer function [7.24] has two poles. In the most usual case of subcritical damping, they form a pair of complex conjugate quantities:
2. The function Q(s) is given by:
It characterizes, of course, the external excitation applied to the oscillator starting from t = 0, which is thus found to comprise two distinct components. One of them is the image of the external force and the other arises from the initial conditions. 7.3.3 External loads equivalent to nonzero initial conditions As already mentioned just above, relation [7.26] clearly shows that non vanishing (inhomogeneous) initial conditions are equivalent to some kind of external timedependent (fluctuating) loads. This point is worthy of further discussion. 7.3.3.1 Initial velocity and impulsive loading The response of a linear oscillator initially at rest, to an impulsion applied at t = 0, is governed by a set of equations which can be written into the two following equivalent forms, (see subsection 7.2.3):
Such an equivalence suffices to prove that initial velocity can be modelled as an impulsive external load. However, it is also of interest to see how the equivalence is translated in the image domain. The remarkably simple result that LT^S(*)] = ! follows immediately from [7.11] and [7.19]. Accordingly, the image of the impulsive response is found to be:
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Thus, from [7.28] it can be stated that: The transfer function of the linear oscillator may be identified with the Laplace transform of the response of the oscillator, initially at rest, to an impulse of unit magnitude, applied at t = 0. 7.3.3.2 Initial displacement and relaxation of a step load
Figure 7.3. Loading equivalent to an initial displacement
The case of an initial displacement is more subtle than that of an initial velocity. Assuming that at times t < 0, the oscillator is prestressed by a static load Q0, it is thus displaced by q0 = Q01K about the position of the unloaded static equilibrium. At time t = 0, the load is released as shown in Figure 7.3. From the dynamical point of view, this release is equivalent to the step load:
where ll(t] is the Heaviside step function, already defined in subsection 7.2.2. Therefore, the two following systems of equations are found to be equivalent to each other:
where q\-q-q^ is the displacement of the oscillator about the position of the loaded static equilibrium. In the image domain, we have:
which reduces to the expected result:
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Discrete systems
It may be noted that the multiplying factor 1/s arises directly from the definition of the Laplace transformation, which implies that q0 vanishes identically in the range t < 0. Therefore, the variable transformation, expressed in the static form as qi=q-q0, is suitably expressed in dynamics as qi^(t} = (q-q0}-^(t}- A non vanishing initial displacement is thus equivalent to the sudden release, or relaxation, of a prestressed state. 7.3.4 Time-history of the response to a transient excitation The inverse Laplace transformation is used to shift from the image domain to the time domain. In many instances the singularities of q(s) are poles; if such is the case, q(t) can be calculated easily by using the residue theorem (see Appendix 7, paragraph A7.9). When applying such an analytical procedure, it is suitable to make a clear distinction between the poles arising from the excitation signal Q (s) and those arising from the transfer function H ( S ) . The poles belonging to Q (s) characterize the forced motion, which may be identified with the special solution of the inhomogeneous differential equation. On the other hand, the poles [7.25], which belong to #(•$), characterize a free vibration at the natural frequency of the oscillator. It may be identified with the general solution of the homogeneous differential equation, where the integration constants are suitably chosen so as to fit the initial conditions of the problem. It is also appropriate to stress that a free vibration is present even if the oscillator is initially at rest. In such a case, it characterizes the starting effect of the transient excitation. In accordance with the above considerations, it is appropriate to separate the response to a transient load into two stages occurring successively. The forced stage takes place during the transient and, of course, depends upon the time-history of the transient. However, it also includes a free vibration, superimposed linearly on the forced motion. The free stage starts at the end of the exciting transient. Then, the oscillator vibrates freely according to a time-history which fits the initial conditions of this second stage. Eventually, the oscillator is at rest at the end of the forced stage; if so, the oscillation of the free stage vanishes completely. On the other hand, in the presence of damping, the magnitude of the free vibration decays as an exponential with time (cf. Chapter 5, subsection 5.1.3), and this during the forced stage as well as during the free stage. Thus, if excited by a transient, the system finally comes back at rest, in an asymptotical manner.
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It is quite instructive to analyse the specific response of a harmonic oscillator to a few typical transients. From such an analysis, a few results which are of practical interest to the engineer can be illustrated. As a preliminary, the system [7.17] is conveniently re-written in terms of dimensionless quantities, by using the following scaling factors: 1. The natural period of the conservative oscillator: T 2.
The magnitude (or maximum amplitude) of loading: Q
3.
The magnitude of displacement due to static loading: q The dimensionless variables are then defined as follows:
and the system [7.17] takes the reduced form:
The Laplace transformation of [7.34] gives:
7.3.4.1 Response to a rectangular pulse The transient is written as:
The Laplace transform is:
The oscillator is assumed to be initially at rest. Substituting [7.37] into [7.35], it follows that:
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Discrete systems
Figure 7.4. Unit rectangular pulse Junction
Since the exponential factor implies a delay in the time domain (cf. Appendix 7, A7.5), the associated term has to be taken into account for T > 0 only, that is during the free stage of the response. Therefore, during the forced stage, the time-history of motion is given by the integral:
are the residues belonging to the three poles:
The following intermediate results are then easily found:
Finally, the time-history of response is written as:
where
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If damping is sufficiently small, the last term of [7.41] can be neglected:
Incidentally, the result [7.42] shows that the response of the oscillator to a step load is the sum of the response to the static load defined by the magnitude of the step and of a harmonic oscillation at the natural frequency of the oscillator, induced by the initial jump in the load. Provided damping is sufficiently small (typically gl <0.7), the conservative approximation is sufficient to assess a reasonable upper bound to the maximum amplitude of the response in relation to pulse duration, which is information of major interest to the mechanical design engineer. Therefore, in what follows we assume that £, = 0 . Motion during the free stage starting at T = 6 , can be deduced from the previous one simply by using the delay theorem. From the simplified result [7.42] it is found that:
1. Pulse of long duration: 6 > 0.5 The time-history of displacement is plotted in Figure 7.5. During the forced stage, the response is a harmonic vibration of unit amplitude, at the natural frequency of the oscillator, which is centred about £ = 1. During the free stage, a similar response is observed, except that the free oscillation is now centred about £ = 0. The magnitude of displacement is always less, or equal to twice the static value. The first occurrence of a maximum (or peak value) is at time rlm =0.5, i.e. the half-period of the free oscillation. The first maximum of the free motion is defined by:
Its value is thus less or equal to that already reached during the forced stage. Incidentally, it is noted that if the duration of the pulse is an exact multiple of the natural period of the oscillator, the free stage oscillation vanishes. Indeed, the oscillator is exactly put at rest at the end of the pulse and it will remain at rest for ever, if not excited again.
a 56 Discrete systems
Figure 7.5. Response to a pulse of long duration (dashed line: excitation, full line: response)
2. Pulse of short duration: 0 <0.5 Figure 7.6 shows a typical response to a short pulse. Its amplitude increases steadily during the forced phase up to:
Maximum displacement occurs during the free stage:
Figure 7.6. Response to a short pulse (dashed line: excitation, full line: response)
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Figure 7.7. Maximum displacement as a function of the step duration
Finally, it is of interest to note that when 9 tends to zero, rm tends to 0.25 and £m = Ind. Accordingly, if the pulse is suitably normalized to a unit area, it can be replaced by a Dirac pulse, since the actual response of the oscillator is practically the same as the impulse response:
This result could be anticipated, since we knew already that £(f)can be defined as the limit of the pulses displayed in Figure 7.1 (a). Figure 7.7 is a plot of the maximum displacement versus the step duration 6 . On such a plot a clear distinction between short and long pulses is conspicuous. NOTE. — Derivative of the Heaviside step distribution The considerations discussed just above can also be used to show that il(f) is differentiable in the sense of distributions, the result being dii(t)/dt = S(t). Indeed, let us calculate the following quantity:
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Discrete systems
one obtains:
The above result can be interpreted as:
which can also be found using the Laplace transformation as follows:
7.3.4.2 Response to a trapezoidal transient Of course, a step signal does not occur in practice, because in reality neither the rise nor decay times of any transient can be zero. It is thus of interest to investigate the effect of a finite rate of change in the amplitude of the excitation. This can be performed conveniently by considering the trapezoid signal of Figure 7.8, where again 6 is the total duration of the transient and © stands for the rise time of it. Calculation similar to that already detailed in the last example shows that, provided Q is sufficiently long (d > 0.5), the maximum displacement is now given by:
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Figure 7.8. Trapezoidal transient with a nonzero rise time
Figure 7.9. Maximum displacement versus the rise time of the pulse Figure 7.9 is a plot of the function [7.48]. It shows in particular that the discrepancy between the static and dynamic magnitude of the response is less than 10%, as soon as © becomes greater than about four. From the foregoing analyses, the two following conclusions of practical interest arise, which hold for every transient presenting a single peak: 1.
2.
The response amplitude of an oscillator excited by a single peaked transient is at most twice the response to the static load of peak value. The longer the duration of the transient (9 > 0.5), the shorter the rise (or decay) time, the closer to this upper bound is the actual peak value of response. A damping ratio up to a few percent does not modify the above results very significantly.
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Discrete systems
The second point stems from the fact that, as already shown in Chapter 5, the characteristic time-scale for damping is:
Therefore, if £ is sufficiently small, Ta » 0.5 . Nevertheless, the above statements must be drastically modified if the transient excitation presents several peaks, acquiring thus an oscillatory nature. This point is evidenced in the next subsection, based on a simple analytical example. 7.3.4.3 Response to a truncated sine Junction
Figure 7.10. Sine Junction truncated to six periods
Using the same scaling factors as those already defined in subsection 7.3.4, the analytical expression of the signal displayed in Figure 7.10 is:
where a is the reduced frequency of the signal. The oscillator is assumed to be conservative and initially at rest. In the image domain, the response during the forced stage is found to be:
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Figure 7.11. Response of the undamped oscillator to a truncated sine
Figure 7.12. Response of the undamped oscillator to a truncated sine Let us consider first the nonresonant case a * 1. The inverse Laplace transform
is:
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Discrete systems
This result shows that the total response is the sum of a free oscillation at the natural frequency of the oscillator, which is caused by the starting effect of excitation, and of a forced oscillation at the frequency of the excitation signal.
Figure 7.13. Response of the damped oscillator to a truncated sine
Figures 7.11 and 7.12 show two samples of time-histories, referring to a = 1.5 . In the first case, the oscillator remains at rest during the free stage, since duration of the transient excitation is an exact multiple of the natural period of the oscillator. This peculiarity may be used to check errors induced in algorithms of numerical integration of the equation of motion. In the second case 0 = 3.66... so the free stage oscillation is clearly present. Figure 7.13 refers to a similar calculation for a damped oscillator (g = 2%). The magnitude of the response during the forced stage is still very close to that calculated in the conservative case, and a damped oscillation is taking place during the free stage. Finally, Figure 7.14 refers to the same case as Figure 7.13, except that damping is greater and that duration of excitation is practically infinite (equal or longer than the duration of the computed response). As evidenced in Figure 7.14, motion during the forced stage tends progressively to a steady harmonic oscillation at the frequency of excitation. The underlying reason for such a result is the exponential decay of the free vibration.
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Figure 7.14. Response of the damped oscillator to a truncated sine, case 6 —> °°
Therefore it is useful, in practice, to separate the forced stage into two distinct regimes. The first one is a transient regime during which the presence of the free oscillation, induced by the starting effect of excitation, is still significant. The second regime is a steady vibration resulting from the sole persisting response component, namely the response forced by the sine. Of course, the larger the damping, the shorter is the transient regime. In terms of energy, the existence of steady vibration indicates that, when averaged over a cycle, the work produced by the excitation is exactly balanced by the energy dissipated by viscous damping. In this respect, the present situation is similar to that of the limit cycle of the autonomous Van der Pol oscillator (cf. Chapter 5, subsection 5.2.3). It is then of particular interest to discuss the nearly resonant case a ^ 1, as it provides us with the opportunity to introduce the beating phenomenon, which is encountered in many fields of physics. Basically, beats result from the linear superposition of two harmonic signals of similar amplitude and of similar but still distinct frequencies:
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Discrete systems
Figure 7.15. Response to nearly resonant harmonic excitation
The result [7.53] shows that the response resulting from the interference between two harmonic signals of close frequencies cov,0)2 may be interpreted as an harmonic signal occurring at the central frequency (Oc, which is modulated in amplitude at the much lower frequency Aco. Accordingly, the response involves two quite distinct time-scales 7j =;r/ft> c ,and T2 = nl Aco.
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Coming back to the present problem, the general result [7.52] can be re-written as:
Owing to the factor lie, the magnitude of the nearly resonant response is much larger than in the nonresonant cases, at least if the transient is long enough (9 > 7j). Figure 7.15 shows two typical responses to a nearly resonant excitation. The first plot corresponds to a relatively short duration transient T{ < d < T2 and the second corresponds to a long lasting transient 0 > T2. In terms of energy, it can be noted that the mechanical energy of the oscillation, during the forced stage, varies according to the slow time-scale T2. Indeed, it is easy to show that:
Thus, the work done by the nearly resonant excitation changes sign at the low modulating frequency. Finally, we consider the resonant case a = 1. The response of the undamped oscillator, in the image domain, is found to be:
Multiplicity of the poles s = ±2in equals two and the method of residues provides the following solution in the time-domain:
This result clearly shows a drastic change in the motion, when shifting from a non resonant or nearly resonant excitation to an exactly resonant one. The second term of solution [7.57] includes a linear envelope which modulates the amplitude of the harmonic oscillation. Accordingly, from one cycle at the natural frequency to the next, the magnitude of motion is found to increase proportionally to the elapsed time. In terms of energy, the work produced by the resonant force is now always positive, when averaged over the natural period, in contrast to the non resonant case, see Figure 7.16.
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Discrete systems
Figure 7.16. Response to a long lasting resonant sine excitation
Figure 7.17. Resonant response of the dissipative harmonic oscillator
The presence of dissipation modifies this unrealistic result. Indeed, in the presence of damping it is found that, provided d is sufficiently long, the response tends to a steady oscillatory regime in which energy produced by the external force is dissipated by the viscous damping force, see Figure 7.17. The response in the steady regime is a cosine, the magnitude of which is proportional to the reciprocal of damping:
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The major conclusion of practical interest arising from the result [7.58] may be put in words as follows: A resonant excitation applied to a harmonic oscillator during a time larger than the time-scale of damping induces a steady oscillation, the magnitude of which is inversely proportional to the damping ratio. Thus, when the system is lightly damped, as it is often the case, the magnitude of the dynamical response is much larger than that of static response. This will be emphasized again in Chapter 9 by adopting a spectral standpoint. By itself, such a result suffices to motivate the dimensioning of structures against vibrational problems, when they are intended to withstand dynamical loads. 7.4. Impulsive response and Green's function 7.4.1 Green's function of a harmonic oscillator The response in the time domain of an oscillator to initial conditions is easily calculated by using the residue theorem. In the case of subcritical damping, which is of major interest for structural dimensioning against vibration problems, it is found that:
which identifies of course with the result already established in Chapter 5 by solving directly the differential equation of motion (cf. subsection 5.1.3.1). We consider now the response to a unit impulse occurring at time t0. This response is known as a Green's function of the oscillator. It is given by:
a result which can also be written as:
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Discrete systems
G(t) is thus the inverse Laplace transform of the transfer function. This is not surprising since we have:
7.4.2 Green's function and forced response to any transient
Figure 7.18. Sequence of pulses converging to 8(t - T) as n —> °°
Figure 7.19. Transient signal decomposed as a sequence of successive pulses
The Green's function allows us to express the response of a harmonic oscillator to any transient excitation Q (f) as a convolution product. This is an immediate consequence of the convolution theorem (formula [A7.8] of Appendix 7) as applied to relation [7.23], which gives:
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Physical understanding of such a result can be gained by splitting up Q^*' (?) into a sequence of impulses. Indeed, it may be noted that:
On the other hand, S(t-r^ can be obtained as the limit of a sequence of the rectangular pulses displayed in Figure 7.18. Accordingly, we have:
The series [7.64], where time rk is within the interval \tk , tk +hln\, is suitable to state the Riemann's integrability of Q (*)• On the other hand, the same series can also be interpreted as splitting up Q (f) into a sequence of successive pulses, the number of which tends to infinity and width of which tends to zero as n tends to infinity, see Figure 7.19. Since the response of the oscillator to the impulse Q^e\r)S(t - T) is Q^e\T)G(t - T), the result [7.62] appears as a mere consequence of the superposition principle, which holds for any linear system. According to this principle, the response can be built by summing the individual responses to partial excitations. In the present problem, the latter are made up of the successive impulses, which result in <2 when superposed. Before leaving the subject, it may be noted that the formulation [7.62] is more interesting as a nice theoretical result than as a convenient formula to calculate the response of the oscillator. Actually, in most instances the convolution integral is not easily calculated analytically, and its numerical computation is cumbersome. Nevertheless, a very simple application of [7.62] will serve us in Chapter 8 to introduce the spectral analysis of linear systems. 7.5. Response of MDOF linear systems 7.5.1 Transfer function matrix of a conservative system The Laplace transformation of the matrix system [7.1], without damping, results in:
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The relation [7.26] becomes:
In principle, the system [7.65] can be solved by inversion. One is thus led to define a transfer junction matrix \H(s}\, in such a way that:
The matrix equation [7.67] is the natural extension to MDOF systems of the formalism already introduced in subsection 7.3.2, in connection with the harmonic oscillator. The transfer function matrix enables us to relate the response, a displacement in this case, of any individual degree of freedom to the excitations acting on the whole set of the degrees of freedom of the mechanical system. Clearly, the concept of the transfer function matrix can be extended to any linear system, mechanical or not.
Figure 7.20. Transfer box of a linear system with N inputs and N outputs
The transfer box shown in Figure 7.20 extends that of Figure 7.2 to linear systems provided with N inputs and N outputs. Every element //,-,- of the transfer function matrix can be interpreted as a transfer function connecting the input of index i to the output of index j. In the case of multiple inputs (excitations), the output of index j is the superposition of the partial responses of the y'-th DOF to the individual inputs.
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On the other hand, in the case of conservative mechanical systems, the matrices [AT| and [M] are symmetrical. Thus, the transfer function matrix is also symmetrical. A corollary of such a symmetry is known as Maxwell's theorem of reciprocity according to which: The response ofthej-th DOF to a unit load applied to the i-th DOF is the same as the response of the i-th DOF to a unit load applied to thej-th DOF. Finally, as an immediate extension of SDOF systems, H^s) is the Laplace transform of the response of the y'-th DOF, to a unit impulse applied to the i-th DOF at time t = 0:
where Gi} (t) is the Green's function relative to the i-th input and they-th output.
7.5.2 Uncoupling by projection on the modal basis 7.5.2.1 Principle of the method Using directly formula [7.67] to calculate the response of a Af-DOF system would be very clumsy, at least as soon as N > 2. Of course, it is advisable to use the symmetry properties of the operators [K] and [M] of the conservative mechanics by projecting first the equations of motion on a modal basis, (cf. Chapter 6, subsection 6.3.2.2). Let us thus consider a mechanical system described by using physical coordinates (or displacements), which is written in the image domain as:
As already discussed in Chapter 6, the basis of the natural modes of vibrations is complete and the mode shapes \
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qk (5) are the natural coordinates of the vector [X (s)]. Moreover, since [
where 0)k is the natural pulsation and Mk is the generalized mass of the k-th mode. Qk(s) is the generalized force applied to the fc-th mode, which is defined as the projection of the physical load [F (s)] on the k-th mode shape:
Of course, the physical load can also be expanded as a modal series, since it is a vector pertaining to the same vector space as the response vectors. Hence, we have:
Accordingly, the generalized forces Qk (s) are simply the coordinates of [F(s)] on the [<£] basis. Thus, one is naturally led to the following statement: The concept of natural modes of vibration enables us to transform a forced and coupled system, of size NxN, into a forced system ofN harmonic oscillators, which are uncoupled from each other. As we shall see on many occasions hereafter, such a result is of paramount importance, from the viewpoints both of physical modelling and computation. 7.5.2.2 Modal expansion of the transfer and Green's functions Let us consider a unit impulse applied to the /-th DOF of a Af-DOF material system, which is assumed to be initially at rest. In the time-domain the (transposed) load vector is written as:
where S(t) is located at the a j'-th column. The Laplace transform of the load vector is:
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Projection on the modal basis produces the following vector of generalized load:
where
It is convenient to put the indices i and j as arguments of the transfer functions and mode shapes for future extension of the formalism to continuous systems. Indeed, it will suffice to replace the indices by the coordinates of a pair of points in a Euclidean space, namely the loaded point and the point at which response is calculated, or measured (see Volume 2). The inverse Laplace transformation of [7.77] gives the expansion in modal series of the impulsive response (or Green's function), which is immediately found to be:
According to either relation [7.77] or [7.78], an impulse applied to the z-th DOF is able to excite all the natural modes of the physical system, except those which are orthogonal to the impulse, i.e. the modes such that 0?t (i) = 0. Furthermore, in order to get a nonzero contribution of mode [
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EXAMPLE 1. - Impulsive responses of two identical oscillators in the presence of weak coupling The system shown in Figure 7.21 comprises two identical mass-spring systems connected together by a spring with a stiffness coefficient Kc much smaller than that of the oscillators. One is interested in determining the displacement response induced on the second oscillator by an impulsive excitation of the first one. By definition, the response in the time domain is the Green's function G(l,2;f), times the amplitude P0 = MV0 of the exciting impulse. The modal properties of the system are easily obtained by adapting to the present problem the results already established in Chapter 6, subsection 6.3.1. The modal system is written as:
whence the following modal properties:
Figure 7.21. Identical oscillators weakly coupled by a spring
Substituting these results into [7.78], one obtains:
Now, if K is sufficiently small, the preceding result can be written as:
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This result can be written in the remarkable form, already encountered in subsection 7.3.4.3, formula [7.53]:
The plots shown in Figure 7.22 are the responses of the two oscillators, as computed with the following numerical values:
Plotted displacements are in cm.
Figure 7.22. Response of the two oscillators
Here the beats occur as a consequence of the superposition of the two modal contributions (partial responses) to the global response, which results in either constructive or destructive interferences, depending upon the relative phasing of the partial responses. Furthermore, it is worth noting that the response of the non excited oscillator lags behind the response of the excited oscillator by a quarter period of the envelope signal. The physical reason for such a delay clearly lies in the inertia of the
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non excited oscillator. As a consequence, the mechanical energy stored initially in the first oscillator is exchanged between the two oscillators at the beating period. EXAMPLE 2. — Impulsive responses in a chain of identical oscillators Let us consider a chain of oscillators comprising N identical mass-spring systems, with stiffness coefficient NxK and mass coefficient MIN, which are coupled together by springs with stiffness coefficient also equal to N x K . Thus, the global stiffness of the spring series is K and the global mass of the particles is M. The calculations to be qualitatively discussed here refer to a chain of N = 40 identical undamped oscillators (M = 10 kg, K = 4 000 N/m ). Figure 7.23 is a plot of the natural frequencies versus the modal order, which is quite similar to that of Figure 6.6, except that the frequency range expands now from about 60 Hz, to 1600 Hz.
Figure 7.23. Natural frequencies versus modal order
Two Green's functions are plotted in Figures 7.24 and 7.25. They show the two following characteristics: 1.
The time-histories are far from simple and even by scrutinizing a much longer lasting plot than those displayed here it would be difficult to ascertain whether motion presents any periodic feature, or not. This point will be further discussed in Chapter 8, which is devoted to spectral techniques to analyse the frequency contents of time-signals.
2.
Early, the response of the excited particle looks like a rather short transient, which is marked by one peak of fairly large amplitude followed by a few oscillations of much lower magnitude, see Figure 7.24. Afterwards, the particle remains practically at rest during a rather long time interval, then it oscillates
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again. Response of the other particles is delayed, the lag being proportional to the rank of the particle in the chain. The motion is a rather intricate oscillation combining the natural frequencies of the system in a complicated way.
Figure 7.24. Displacement of the first (excited) particle
Figure 7.25. Displacement of the last particle
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Such features may be qualitatively interpreted as follows. Initially, the impulse provides the system with an amount of energy, which is first delivered to the excited oscillator, whereas the other particles are still at rest, because of inertia. Then, energy is transferred from one particle to the next, with a finite velocity of transport. After a while, the energy is spread out in an intricate and fluctuating way amongst all the particles of the system. Such a behaviour may be suitably confirmed by plotting in the same figure the displacement of the chain of particles for a few fixed times, see Figure 7.26, where the progression of the peak of response from one particle to the next one can be clearly discerned.
Figure 7.26. Displacement of the chain of particles at a few fixed times Referring back to the study already performed on a similar chain in Chapter 6 subsection 6.3.2.4, the observed features of these impulsive responses indicate clearly that the impulse triggers a transient which travels along the chain as a longitudinal wave and which is reflected at the fixed ends. Propagation is not easy to follow because in this discrete medium the waves are dispersive, and the transient is far from being a simple harmonic wave. In the next chapter we will see that propagation of the same transient becomes far more simple, provided a non dispersive model is adopted.
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EXAMPLE 3. - Standing waves in a chain of identical oscillators Finally, forced standing waves can also be excited, as shown here. As already pointed out in Chapter 6, the natural vibration modes of the system are either symmetrical or antisymmetrical with regard to the median particle of rank N/2. Therefore, if the external force is distributed symmetrically, only the symmetrical modes are excited, and if the force is distributed antisymmetrically, only the antisymmetrical modes are excited. This is illustrated in Figures 7.27 and 7.28, which plot the displacements, for a few fixed times, of a chain of 20 oscillators, excited by imparting an initial velocity to each particle of the chain. Figure 7.27 refers to the symmetrical case: +1 m/s is imparted to each particle and Figure 7.28 refers to the antisymmetrical case: +7 m/s to the first 10 particles and -7 m/s to the others. An additional feature of the responses worthy of note is the progressive change in the shape of the displaced chain as time runs. Such behaviour indicates that the relative contribution of the various modes to the global response is also timedependent, since each of them is varying in its own time-scale. Nevertheless, in both cases, the response is clearly shaped as a standing wave.
Figure 7.27. Displaced chain of particles (symmetrical impulsive loading)
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Figure 7.28. Displaced chain of particles (antisymmetrical impulsive loading)
7.5.3 Viscous damping We have already emphasized the paramount importance of dissipation for limiting the magnitude of the response of the harmonic oscillator at resonance (cf. subsection 7.3.4.3). This is of course also true in the case of MDOF systems. In order to include dissipation into a linear model with constant coefficients, it is necessary to adopt the viscous damping model, which is now formulated as a matrix [C] operating on the velocity vector. The elements of [C] must be constants, otherwise ordinary products such as Ctj (t )qj (0 would be transformed into products of convolution in the image domain of the Laplace transforms. Actually, linear systems provided with time varying coefficients are much more involved than those analysed here and the related physics is also very different as already illustrated in Chapter 5 on the parametric oscillator studied in subsection 5.3.2.1. In the presence of viscous damping, equation [7.65] becomes:
The load vector is changed into:
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Finally, the transfer function matrix connecting
However, the trouble is that there is no reason for [C] to be transformed into a diagonal matrix, when expressed on the modal basis of the related undamped system. Therefore, if [C] is not assumed to have suitable mathematical properties, the advantage of the modal uncoupling procedure and the validity of the modal expansions is finally lost, which is rather unfortunate! 7.5.3.1 Model of viscous and proportional damping In many cases, one has to deal with lightly damped systems in which the detailed damping mechanism is rather unimportant, provided the rate of energy dissipation, as averaged over a cycle of vibration, is correctly described. In such cases, it is of course very convenient to model dissipation by using a damping matrix [C] which becomes diagonal when projected on the modal basis of the related conservative system. Although such a simplification is a mathematical trick without any physical background, it is still reasonable to neglect coupling by the nondiagonal terms of [<J>]~ [C][O], provided their magnitude is small enough. Adopting such a model, the modal projection of [7.80] results in a diagonal system. As expected, the n-th row of it takes the canonical form of a damped and forced harmonic oscillator:
The modal damping ratios gn are given by:
In practice, modelling of dissipation based on the physical mechanisms encountered in real structures remains intractable in most cases. Nevertheless, reasonable values of modal damping ratios can be assessed, based either on vibration measurements (for modal testing techniques, see for instance [EWI00]), or on values prescribed in regulatory guides used in structural design, see for example [LAL 99, Vol. 5]. The rules and norms found in such guides provide the current state of the art in a specific domain of engineering; the degree of uncertainty concerning realistic values of some physical quantities, such as modal damping, is accounted for by introducing a suitable safety margin.
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Discrete systems
A model which is widely used to obtain modal damping of the type [7.84], is the so called proportional model, according to which [C] is assumed to be a linear form of the [K] and [M] matrices:
The damping ratios are found to be:
The free parameters a and (3 of the model can be- adjusted by prescribing specific values qi and g} to the resonances co^CDj, in such a way that every damping ratio lying inside the interval f^y,, (O- J is less than the highest value of g( and q.j, but still remaining reasonably large, as illustrated in Figure 7.29. Suitable values of a and fi are found to be:
Figure 7.29. Adjusted proportional damping
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7.5.3.2 Non proportional viscous damping Actually, there are also many cases of practical importance which cannot be reasonably fitted by using a proportional viscous damping model. As a first consequence, the vibration equations cannot be uncoupled by using the modal basis of the related conservative system. The response properties of such systems may be described in terms of complex modes of vibration. This aspect shall be further developed in Volumes 3 and 4 of this book. However, complex modes are used mainly to study the dynamical stability of nonconservative systems and far less often to study forced motions. Indeed, if the number of degrees of freedom is small (typically two or three) it may be found more convenient to produce an explicit formulation of the transfer functions which are required, by solving analytically the coupled system in the image domain of the Laplace transformation. In the other cases, it may be preferred to carry out numerical simulations using an implicit integration algorithm, as further illustrated in subsection 7.5.3.3. EXAMPLE. - Quarter-car model of a car suspension
Figure 7.30. Quarter-car model of a car suspension
This example is taken from [GEN 95]. The quarter-car model shown schematically in Figure 7.30 is the simplest that can be conceived to analyse the dynamical behaviour of a car suspension. It consists of a 2-DOF system where K stands for the equivalent stiffness of the tire and m for the equivalent mass of the wheel. M is the quarter of the mass of the car body, k is the stiffness coefficient of the suspension and C is the viscous damping of it. Z1 is the vertical displacement of the wheel and Z2 that of the car body, which are both defined in the inertial frame tied to the road. Z0 (x) stands for the vertical profile of the road which is uneven. Irregularities induce a prescribed vertical displacement Z0(W) of the contact point between the wheel and the road, where V designates the cruising speed of the car. Vibration equations of the model are written in the image domain as the following matrix system:
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Discrete systems
The practical aim is to avoid an excessive vertical acceleration of the body, say higher than 1 g, and that the magnitudes of the displacements |Z2| and |Z,| remain less than 75 cm. The following numerical values are adopted: K = 10s Nm~' ;m = 25kg;k= 2,5104 Nm'1; M = 250kg; A = 2.15103 Nsm'1 Modal analysis of the conservative system (C - 0), gives: /; = 1.42 Hz; M, = 251 kg ;K,=2104 N/m; [(p, ]T = [0.203 f2 = 11.28 Hz; M2 = 25.1 kg ; K2 = 1.2610sN/m; [#>2]r = [l
l] -0.0203]
The first mode is in-phase, with a vibrational amplitude of the body about five times larger than that of the wheel. The second mode is out-of-phase, with a vibrational amplitude of the body about 2% of that of the wheel. The separation of modal frequencies is easy to understand. The frequency of the first mode is convenient so far as the comfort of the passengers is concerned, since its value is near to that of human walk. As pointed out in particular in [ROC 71], at frequencies less than 7 Hz, one feels sea-sick, whereas at frequencies higher than 2 Hz, one suffers backache. Modal projection of the damping matrix gives:
The nondiagonal terms cannot be neglected since they are of the same order of magnitude as the diagonal terms. Moreover, even if such a gross approximation is made, large values of reduced modal damping are obtained: £ j = 30%; $2 =63% emphasizing again the importance of retaining all the terms in [C]. From the transfer function matrix, it is straightforward to deduce the following transfer functions which are of practical interest:
In principle, they can be used to calculate the forced response of the system in the time domain, provided that a suitable analytical form of the prescribed displacement Z0(j) is available. However, the calculation is rather tedious and, as we shall see in Chapters 8 and 9, the most efficient way to use transfer functions is to shift from the
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domain of the Laplace variable to the frequency domain, with the aim of describing the spectral properties of steady state vibration. On the other hand, if we are interested in determining the time-history of the motion induced by the short lived transient connected to an individual bump or pothole, this approach could be preferred, especially in engineering applications, to perform numerical simulations based on a step-by-step integration technique. In the presence of large viscous damping it is found advantageous to use an implicit algorithm which produces the velocity vector at the current time-step. We will use here Newmark's scheme which is extended first to MDOF systems in the next subsection. 7.5.3.3 Implicit Newmark algorithm Extension of the implicit Newmark algorithm described in Chapter 5 subsection 5.3.3, in the case of a single damped linear oscillator, is straightforward. It is recalled that the dynamical equilibrium is written at the time-step n+1 to produce the acceleration and then the velocity and displacement vectors at the same instant of time. In the case of a linear MDOF system, the algorithm takes the following matrix form:
Substituting the truncated Taylor's series into the equilibrium equation leads to the following implicit equation:
The first equation [7.89] is said to be implicit because the equivalent of a matrix inversion is required to solve it (see below for numerical method used):
where the following matrices have to be determined:
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Discrete systems
Calculation of the matrices [7.91] is immediate if [AT],[C],[M]are diagonal. Otherwise, [E] is determined using Grout's algorithm, as described in Appendix 5. Another feature of major interest of this Newmark algorithm is its unconditional stability. Indeed, for industrial applications, one is often used to build mechanical models, arising from a discretization in finite elements of complex structures (cf. Volume 2), which unavoidably include a fairly large number of degrees of freedom. As a consequence, such models are likely to involve implicitly several modes of vibration with natural periods substantially less than the smallest time-scale one is interested in. Now, as seen in Chapter 5 subsection 5.3.1, the use of an explicit algorithm would require the choice of a time-step less than the smallest natural period which is embedded in the system to be integrated (see formula [5.69]). The use of an implicit algorithm is then preferable, since it makes it possible to select the value of the time-step which is appropriate to the physical time-scales of the problem, independently of the modal time-scales embedded in the discrete model. EXAMPLE. - Car suspension at a bump overcrossing We come back to the quarter-car model described in subsection 7.5.3.2. Here, the dynamical problem we are interested in is the car response at the crossing over a speed limiting band, assimilated to a half sine bump, with height ZQ=10cm and width LQ = 30 cm. The aim is to determine the maximum cruising speed at which the criteria of subsection 7.5.3.2 are not yet exceeded. Simulations have been performed by adopting for [C] successively a diagonal, then a proportional and finally the full form of the damping matrix. 1.
Diagonal form of[C] Here, we adopt the simplistic model:
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Figure 7.31. Diagonal [C]: time histories of acceleration at car speeds 1 and 4 kms
The main results are displayed in Figures 7.31 and 7.32. The efficiency of the shock absorber is particularly conspicuous on the plots of acceleration. Indeed the magnitude of the acceleration peak is found to be divided by a factor of about five, when passing from the wheel to the body. In principle, the wheel resonates according to the mode at 11 Hz; however the corresponding period is much shorter than the crossing time over the bump and modal damping is fairly high. Accordingly, two shortly lived peaks of acceleration are observed at the beginning and at the end of the
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Discrete systems
obstacle. The body response is essentially according to the mode at 1.4 Hz. Consequently, acceleration remains perceptible during the whole crossing time, and even later, though the fairly high damping again ensures a fast decay of the response after the crossing of the bump. This is better seen in the plot relative to the highest car cruising speed 4 km/h. It is also noted that, at 4 km/h, the criterion of admissible acceleration is not yet exceeded.
Figure 7.32. Diagonal [C]: time histories of displacements at car speeds 1 and 4 ferns'
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On the other hand, the plots of Figure 7.32 indicate that displacements of the wheel and of the body are similar, though distinct from each other, differences increasing with car speed. At 1 km/h the response of the wheel is essentially quasistatic: Zt(0 = Z0(V/), as expected. Indeed, the time-scale of excitation is large in comparison with the period of the responding mode. This is not the case as far as the car body is concerned and it is found that displacement of the body is larger than that of the wheel and delayed to some extent. However at 4 km/h, a dynamical effect can be detected on both the wheel and the body response signals and displacement of the body is less than that of the wheel. The maximum displacement of the wheel indicates a compression of the tire (spring K) of about 16 mm.
Figure 733. Proportional [CJ: time histories of acceleration at 1 and 4 kms~
290 2.
Discrete systems Proportional damping matrix
Figure 734. Proportional [C]: time histories of displacements at 1 and 4 kms~
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The model [7.85] is adopted where the coefficients a and J3 are fitted to the modal damping values g} = 30% and g2 = 63%. It is found that:
The acceleration plots of Figure 7.33 are similar to those of Figure 7.31. The peak of acceleration of the body is slightly larger. The displacement plots of Figure 7.34 are also similar, but not identical to those of Figure 7.32. In particular, peak displacement of the body is higher according to the proportional model than according to the diagonal model. 3.
Non proportional damping matrix Finally, the exact form of the original matrix is used:
Qualitatively, the results shown in Figures 7.35 and 7.36 are similar to the preceding ones; however, a few quantitative differences are worth emphasising. In particular it is noted that the criterion concerning the magnitude of the acceleration peak is now exceeded at the car speed 4 km/h. It may be also noted that the wheel takes off from the road by about / cm. Such results indicate that caution has to be taken when using simplified models such as the proportional damping model. This is not astonishing since heavy damping induces large dynamical forces, hence the nondiagonal terms of [C] reinforce the coupling between the wheel and the car body. In a certain manner, coupling by damping contradicts the weak coupling by stiffness (K/k - 0.1). Before leaving the subject, it is of interest to outline some additional results, which are obtained by letting a few key parameters of the model vary.
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Figure 7.35. Exact [C]: time histories of acceleration at 1 and 4 kms~
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Figure 7.36. Exact [C]: time histories of displacement 1 and 4 kms'
1.
Characteristics of excitation
As soon as the maximum admissible cruising speed of the car is exceeded, acceleration of the body reaches peak values of about 2 g. However, this order of magnitude remains practically constant and even is found to decrease when speed is
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Discrete systems
further increased. In the same way, it is found that the magnitude of the displacements substantially decrease with speed. The underlying reason for such results is the fact that the action of excitation is a decreasing function of crossing time. Nevertheless, great care has to be taken concerning the limits of validity of the present model, in which the horizontal efforts are entirely discarded. On the other hand, it is also clear that the geometry of the obstacle, not only the height and the width of it, but also the shape of its vertical profile, are important parameters of the problem. The stiffer the slope of the profile, the less is the speed at which it is suited to cross over the speed limiter. 2.
Characteristics of the car model
The influence of the stiffness coefficients and of their ratio is rather obvious, so far as the natural frequencies and the mode shapes are concerned. In particular, it is clear that the ratio k/K must be large enough in order to obtain sufficiently contrasted modal displacements of the body and of the wheel. Turning now to the damping of the shock absorber, zero or small damping leads to an acceleration of large magnitude of the wheel but not of the body. On the other hand, motion is almost periodic, which is a very bad point. On the other hand, when [C] is ten times higher than the nominal value of the present model, coupling between the wheel and the body becomes too strong and the motion of the two components are found to be very similar to each other. As a consequence, a transient acceleration signal of 4g at 4 km/h is observed in the car body, which is far beyond the accepted limits. Finally, if [C] is ten times less than the nominal value, the exact and the proportional damping matrices provide quite similar numerical results, which are not acceptable for the comfort of the passengers.
Chapter 8
Spectral analysis of deterministic time signals
The reason for devoting an entire chapter to the basic mathematical tools of spectral analysis and leaving the applications to mechanics to the next and last chapter of this volume, is the extraordinary importance of the subject in many fields of physics, including mechanics, and in signal processing, including signals of mechanical origin. Historically, spectral analysis stemmed from Newton's work on the decomposition of white light through a prism. At the outset, it is appropriate to be more specific about the concept of signal. As defined for instance in [HAR 98], a signal is a physical effect which propagates from a material object and can be described mathematically. In particular, a signal is deterministic if it can be defined as a single function, or distribution. Spectral analysis of time signals consists basically of decomposing them into a series of harmonic oscillations. An harmonic component is defined by three scalar quantities: frequency, phase and amplitude. Therefore, as a fundamental result of the spectral analysis, description of the signal is shifted from the time domain to the frequency domain, the phase and amplitude being generally frequency dependent. The mathematical tools necessary to perform the transformation are the Fourier series and the Fourier integral.
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8.1. Introduction Calculation of the response of a harmonic oscillator to a sinusoidal force, even if applied during a finite time interval only, emphasizes the significance of the ratio of the excitation frequency to the natural frequency of the oscillator, (cf. chapter 7, subsection 7.3.4.3). This point of major importance can be extended to other excitation signals, by expressing the response as the convolution product of the force and Green's function of the oscillator. Indeed, starting from the integral [7.62], in which the Green's function [7.59] is substituted, we obtain the following expression for the response:
where damping is discarded.
Figure 8.1. Signal chopped by a sine function It is often difficult, or even intractable, to express analytically the definite integrals involved in the formula [8.1]. However, the convolution product can still be used in order to bring out the importance of the ratio of the time-scale Te, at which Q^ (?) varies, over the natural period 7] of the oscillator. It is easy to see that the major contribution to the response q(t) originates from that part of the signal Q"e' (/) which varies at the resonant scale Te = T{. Such a conclusion results directly from the chopping property of the circular functions. The sinusoidal modulation implies, of course, the transformation of the original signal into a periodic alternating sequence of values, known as the chopped signal, see Figure 8.1. Therefore, partial
Spectral analysis of deterministic time signals
responses interfere destructively, except if the time variation of Q
297
(?) is able to
compensate for the sign alternation induced by the sine modulation. Thus, in order to get constructive interferences leading to a response of large amplitude, it is necessary that the sign of Q(e) (t) alternates at the same frequency as the modulating signal. The chopping effect may also be interpreted in terms of action. Indeed, as indicated again in Figure 8.1, the time integral of any signal, multiplied by a sine or cosine function of period T1 « Te, is almost zero. A similar result also holds if T1 » Te, since the respective parts of the modulated and modulating signals are, of course, interchangeable. As a consequence, we are led to the same conclusion that the closer to unity is the ratio Te 17}, the larger is the magnitude of the response. EXAMPLE. — Response to a resonant sine acting over an interval t > 0 The force signal is written as: Qe (t) = F0 (sin*y,f )M(t), where co^ = ^JKIM The convolution product is:
where qs - F0 / K which is re-written as:
and finally which identifies with the result [7.56]. In a real system, damping implies a progressive loss of memory of the initial conditions because the free vibration excited at the starting time is decaying in an exponential way. So, it is natural to be interested in truly periodic excitation and response signals, as defined in a time interval extending from -°° to + °°. The energy of a periodic signal is of course infinite, since it accumulates from one period to the next one. However, it is easily understood that it remains possible to define a new finite quantity, known as the mean power, given by the following integral:
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where T stands for the period of the signal and a is an arbitrary time. Spectral analysis is aimed at focusing the interest on the frequency content of the signals, and not to the peculiarities of their time histories. The basic mathematical tool used to shift from time to frequency domain is the Fourier transform. The introductory considerations at the beginning of this chapter suffice to anticipate the advantage of using such a technique to analyse the excitation and response signals of mechanical systems. In section 8.2, the notion of the spectral density of time signals is introduced, starting from a brief review of the Fourier series and the Fourier integral. As in the case of the Laplace transformation, the aim here is restricted to introduce the main definitions and theorems which are necessary for applications in mechanics, without going into the detailed mathematical subtleties and formal proofs. The reader interested in the theoretical aspects of the subject is referred to specialized books such as [LAN 56], [SAG 61], [ANG 61], [PAP 62], [BRA 78]. As an instructive application, propagation of nondispersive one-dimensional waves is investigated, based on the space-time representation of the waves in terms of Fourier series. Section 8.3 is concerned with the problem of loss of information which can occur when a time signal is discretized, since numerical simulations as well as digital processing of measured data produce discrete quantities. Digitizing signals has been fully proven to be the most convenient and accurate way to process and to store large amounts of data. In the present book, again discussion is necessarily restricted to the most fundamental and basic aspects of the problem, namely the Shannon sampling theorem and the definition of the discrete Fourier transform. A plethora of papers and many textbooks have been, and are still devoted, to digital signal processing, among them [BEN 58], [BEN 71], [BEN 80], [MAR 87], [BLA 91], [PRE 90], [MAX 96]. 8.2. Basic principles of spectral analysis 8.2.1. Fourier series Any periodic function X ( t ) , which is upper bounded and which comprises only a finite number of extreme values and discontinuities, can be expanded as a Fourier series. In other words, the functions of period T provided with the additional properties mentioned just above, can be expanded on the orthonormal set of the following circular functions:
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which clearly satisfy the following conditions:
Similarly, the function can be expanded on the basis of the complex exponentials:
which satisfy the following conditions:
The real version of the Fourier series is written as:
The complex version of it takes the following form, which has the advantage to be symmetrical with respect to time:
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Therefore, the Fourier series enables us to express any signal of period T as a series of harmonic components, frequencies of which are the multiples of the fundamental frequency 1/T. Incidentally, when X (f) has a finite discontinuity (finite jump), the series is found to converge towards the mean value, (see for example [ANG61],or[ZEM65]):
8.2.2. Hilbert space of the functional vectors of period T In writing down the relations [8.4] and [8.6], we used on purpose the functional notation of the scalar product. Based on the general definitions and results reviewed in Appendix 1, the reader can check that the set of all the time functions (real or complex valued) which have the same fundamental period T, may be viewed as the elements of a Hilbert space of infinite dimension (assuming completeness), which is provided with the scalar product, defined in agreement with [A1.15]:
where the asterisk * marks the complex conjugation and where the subscript T specifies the time interval of integration involved in the scalar product. The weighting factor 1/T is appropriate to recover a signal power. Furthermore, the circular functions, or the complex exponentials, present in the Fourier series [8,7], or [8,8] form an orthonormal basis of this Hilbert space. Thus, the expansion of X(t) as a Fourier series can be interpreted merely as the projection of a vector on an orthonormal basis and the coefficients of the Fourier series are identified with the components of the vector X(f). Orthogonality of the basis vectors enables us to calculate each component separately. Put in more concrete terms, the harmonic components of the periodic signal are uncoupled from each other. This aspect is extremely fruitful in the context of mechanics, as is similar to the methods of modal analysis of conservative systems, either discrete, or continuous as discussed in Volume 2.
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EXAMPLE. - Periodic sequence of rectangular pulses
t
Figure 8.2. Periodic sequence of rectangular pulses
Let X(t) be a periodic sequence of rectangular pulses of amplitude A and duration T= oft, where T is the period and a a coefficient selected between 0 and 1. X(t) may be written as:
Applying the relations [8.7], the following coefficients are found: ^ a0 = aA;an =—sin(^na) n = \,2,...;bn = 0 nn
a0 identifies with the mean value of the signal. The coefficients bn are zero because the signal is even, X (t) = X (-t). It is also worthy of interest to reconstitute the initial signal from the Fourier series, truncated up to a given order N. As an example, we consider a pulse A = 10, T=0.03 s, which is periodically repeated at the frequency 20 Hz. Computation of the series was performed numerically, using MATLAB. The consequences of the time discretization will be discussed when necessary. The Fourier series calculated by taking into account the first hundred harmonic components is shown over four periods, in Figure 8.3. It may be verified that truncation at order N = 100, provides accurate results, except in the vicinity of the discontinuities of the original signal. Figures 8.4a, and 8.4b show in more detail the reconstitution of a single rectangular pulse, using first one hundred and then five harmonics, for emphasising the errors induced by the truncation of the series. The most conspicuous feature, which is present in both figures, is the so called Gibbs oscillations, which of course involves frequencies proportional to N. If N = 100, the oscillations rapidly decay with the distance from the discontinuity, whereas the decay is barely detectable when N = 5. On the other hand, it could be checked that the rising time of the reconstituted pulse is proportional to 1/N. In terms of frequencies, as the Fourier series is truncated at
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the order N, all the components referring to frequencies higher than the cut-off frequency fc = NIT are discarded in the reconstituted signal. More generally, to truncate the Fourier series of any periodic signal up to order N is equivalent to filtering it by using a low-pass filter provided with a sharp cut-off frequency fc = NIT , where T denotes the period. As a consequence, any feature in the original signal is erased in the truncated Fourier series, whenever the time-scale of the variation is less than TIN.
Figure 8.3. Signal as represented by the truncated Fourier series
Furthermore, as the Fourier series converges, it can be integrated term by term to ft obtain the series expansion of the integrated signal Y(t) = I X(u)du . In the present case, the process leads to a monotonically increasing function, which looks like a staircase line made of equal steps repeated periodically, as shown in Figure 8.5. The basic step involves a linear increase of duration T, followed by a plateau of duration 6. Clearly, the loss of periodicity of the integrated signal is due to the presence of the so-called secular term a0t. It may be also noted that integration lengthens significantly the time-scales and diminishes correspondingly the spectral extent of the original signal. As a consequence, Gibbs oscillations are essentially absent on the scale of the figure.
Spectral analysis of deterministic time signals
Figure 8.4a. One pulse as given by the Fourier series, truncated to N = 100
Figure 8.4b. One pulse as given by the Fourier series, truncated to N = 5
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Figure 8.5. The integrated signal given by the integrated series
Now, if we start from a periodical sequence of centred pulses, the mean value of the signal vanishes, and so does the secular term of the integrated signal Y(t). The coefficients of the Fourier series of the centred signal are now:
Figure 8.6. The integrated signal given by the integrated series
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In the absence of a secular term, the integrated signal Y(t) is a periodic triangle function and the series integrated term by term is a Fourier series. As an illustration, Figure 8.6 shows the reconstitution of Y(t) for the pulse parameters T = 0 = T/2. Y(t) is a sequence of isosceles triangles, which could still be accurately described by using a truncation order substantially less than 100. Finally, differentiation of a periodical sequence of rectangular pulses (centred or not) produces a sequence of Dirac's pulses (see Appendix 6, relation [A6.21]):
By differentiating term by term the Fourier series:
the following series is produced:
It is of interest to discuss, at least briefly, the major features of the reconstitution of the sequence [8.12] by [8.13]. Indeed, as [8.12] is written in terms of singular distributions whereas [8.13] is written in terms of functions, it can be expected that some difficulties may arise. It can be easily checked that Z(t) is highly sensitive to the time-step h, which is used to compute numerically the truncated series. This is not surprising since S(t-t0) vanishes almost everywhere, a property which is likely to induce misleading results when Tjt) is sampled with a finite time-step. Figure 8.7 is a plot corresponding to N = 700 and h = 0.25 ms. According to the upper plot in full line, the Dirac pulses are represented as very short lived peaks, with constant magnitude, whereas according to the plot in dotted line, the pulses are reproduced as single nonzero values repeated periodically, see in particular the lower plot. Such a result can be easily checked analytically. Let Np denote the number of the sampled values of Z over a full period. Hence, the sampled series can be conveniently written as:
Now, for n, = N /4 , we obtain:
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a result which shows that the Fourier series Z(t) does not converge uniformly.
Figure 8.7. Truncated Fourier series of the differentiated signal Z(nh)
From a more practical point of view, as the series is truncated to the N first harmonics, it can be verified that:
Spectral analysis of deterministic time signals
In the same way, for HI = 3N /4, we obtain:
Figure 8.8. Truncated Fourier series of the differentiated
signal Z(nh)
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Discrete systems
Hence, provided Np /4 is an integer, the magnitude of the pulses is AfN in full agreement with the results of Figure 8.7: AfN = 10x20x100 = 2104. Figure 8.8 displays two other plots of Z(nh), as sampled at a much slower rate Np=S, and then at much higher rate Np = 2000 than in Figure 8.7.
Figure 8.9. Residual oscillations in Z(nh)
For n * n p n 2 the calculation of Zn is tedious; however it can be expected that the corresponding values are much less than the peak values, since destructive interferences between the harmonics occur as soon as A i ^ n p n 2 . By comparing Figures 8.9 and 8.7 it can be seen that such residual oscillations depend upon Np. Finally, if the sampling rate does not fulfil the condition that Np /4 be an integer, the restitution of the Dirac's pulses becomes very poor, as the magnitude of the pulses are found to be highly dependent upon the sampling rate. Moreover, they can be modulated from one pulse to the next, according to a low frequency signal as illustrated in Figure 8.10. Thus, we may conclude that, provided the truncated Fourier series [8.13] is suitably sampled, it provides an acceptable representation of the sequence [8.12], since the sifting property of the Dirac distribution is reproduced satisfactorily, except for the scaling factor. This may be easily restored by dividing Z(nh) by /TV, where it is recalled that / is the frequency of the rectangular pulses signal X(t) and N the maximum order of the harmonics retained in the truncated series.
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Figure 8.10. Signal Z(nh), as unsuitably sampled, N 14 is a fractional number
8.2.3 Application: propagation of nondispersive 1-D waves We return to the chain of N identical and coupled oscillators described first in Chapter 6, subsection 6.3.2.4 from the modal viewpoint and then in Chapter 7, subsection 7.5.2.3 in which propagation of the dispersive travelling waves was discussed, based on a few numerical calculations. From the analytical results [6.28] and [6.33] it may be inferred that if the number of particles is increased while the global characteristics of the chain are maintained constant (total length L, stiffness K and mass M) the number of natural modes of vibration which are nearly in harmonic
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Discrete systems
sequence is increasing too. Extrapolating such a model to the continuous case (N —»oo ), one is naturally led to adopt the following modal basis:
X
where £ = — is the abscissa along the chain LJ
As will be shown in Volume 2, such a model can be used as a first approximation to describe the axial waves in homogeneous beams of uniform cross-section, and in Volume 3, it will be shown that it models even more satisfactorily the plane acoustic waves in a homogeneous fluid. The response to an impulse P0 = MV0 impressed initially on the 7-D continuous medium at £=1/2 is obtained by substituting [8.14] into [7.78]. This gives the following Fourier series:
where r=colt This result is quite noteworthy as we recognize here the Fourier series of a periodic sequence of centred rectangular pulses. Furthermore, the same series holds if considered from time, or from space points of view. Figure 8.11 displays the time response at £ = 0.5and £ = 0.75. The length of an individual pulse is found to be:
Figure 8.12 displays a sequence of superposed snapshots which show the progression of the wave front. Physical interpretation of such results is clear. If the natural frequencies are simply proportional to the rank of the natural mode of vibration, the phase speed of the waves becomes independent of frequency, and thus the waves are no longer dispersive. As a consequence, the propagation of travelling waves is drastically simplified in comparison to the dispersive case investigated in Chapter 7.
Spectral analysis of deterministic time signals
Figure 8.11. Oscillatory displacement of a particle versus time
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Figure 8.12. Superposed snapshots: progression of the wave front
Quantitative interpretation is left as an exercise for the reader. In particular, it is of interest to relate the phase speed to the material characteristics of the chain and to explain the features of the pulses in term of wave reflection at the ends of the chain. Furthermore, it is also of interest to study the propagation of the elastic force signal:
JY(n;f) is the elastic force exerted by the particle of rank n+1 on the particle of rank n. As we shall see in Volume 2 the continuous version of it will be defined as the normal stress in a straight beam, or bar. The reader will easily check that JV(n;f) is shaped as a periodic series of rectangular pulses occurring at the wave fronts. Their action identifies with that of the exciting impulse. Of course, as the number of particles in the chain tends to infinity, the pulses tend to Dirac's impulses, in agreement with the derivation of series [8.15]. Finally, by completing this exercise it clearly appears that the excitation signal triggers a stress-wave along the chain. Provided propagation is not dispersive, the stress-wave travels through the chain at constant speed and without any distortion, but the sign is changed at each reflection at the fixed ends of the chain. On the other hand, in this 7-D system, the displacement wave [8.15] is essentially the integral of this stress-wave.
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8.2.4. Fourier transformation 8.2.4.1. Definitions Let X(0 be a function which can be nonzero, in a time interval of arbitrary extent. The Fourier transformation of X(0 is denoted 7F[x(r)] = X (/)= X (co). Frequency/ and thus pulsation a), are defined as real quantities:
Of course, for such a definition to be meaningful, it is necessary that the integrals be convergent. This is obviously the case, provided \X(t)\ is integrable. It is also possible to prove that it is also the case if X(i) is square integrable (i.e. if energy of X(t) is finite). This last result is known as the Plancherel theorem, whose formal proof is rather involved. On the other hand, it is of interest to note that if both the Laplace and Fourier transforms of X(t) exist, one can shift from one to the other, simply by performing the following variable transformation:
where it is recalled that 5 is a complex and (O a real variable. The inverse Fourier transformation is defined by the following integral:
8.2.4.2. Properties of Fourier transforms Many properties of Laplace and Fourier transforms are quite similar. In particular, most of the formulas collected in Appendix 7 can be adapted to Fourier transforms by using the rule of correspondence [8.18]. In particular, the convolution theorem [A7.8] can be extended to Fourier transforms, provided the lower limit of integration is properly shifted from 0 to -«>. Indeed, a major difference between the two transforms is that in the Fourier integral, time origin does not play any particular role, in contrast to the case of the Laplace integral. As a further consequence of particular interest in dynamics, the terms connected to the initial values in the differentiation theorem [A7.3], have to be dropped when adapting the formula to the case of the Fourier transforms. In consequence, motion calculated by using a Fourier transformation of the dynamical equations automatically discards the free oscillations induced by nonzero initial conditions of the external excitation. This is a suitable property when interest is restricted to the study of the steady regime of forced responses.
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8.2.4.3. Plancherel-Parseval theorem (product theorem) The Plancherel-Parseval theorem belongs to the Fourier transform. Although, its proof is far from obvious, the result is easily grasped, as it simply states that the crossed (or mutual) energy of two signals is independent of the domain of description (time or frequency). It is formulated as follows:
Of course, [8.20] holds also in the case of a single signal Y (t) = X(t). As will be seen in subsection 8.2.3, this theorem is of major importance for applications, as it forms the cornerstone of spectral analysis. 8.2.4.4. Fourier transform in the sense of distributions and Fourier series When applying the definition [8.19] of the inverse Fourier transform to the case of a Dirac impulse, one realises that it becomes possible to define a Fourier transform of periodical functions in the sense of distributions. Indeed, it is found that:
From the formulas [8.21] and the Euler relations concerning circular functions, it is easily inferred that:
In accordance with the results [8.22], the Fourier series can be written as the Fourier transform in the sense of distributions:
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8.2.5. Spectral content of time signals 8.2.5.1. Spectral density of energy of a transient signal Let X(t) be a deterministic signal with finite energy. The product theorem [8.20] implies that:
where Sxx (/) stands for a new quantity naturally called the spectral density of energy, in short notation ESD. Now, when X(i) is real, as it is usually the case, then X (-/) = X* (/). As an immediate consequence, Sxx (/) is an even function. It is thus found convenient to restrict it in the domain of non negative frequencies:
The physical meaning of the ESD can be made even clearer by considering the signal energy contained in the infinitesimal band of frequencies [/,/ + df] which is of course:
Figure 8.13. Energy spectrum of a rectangular pulse of energy A 1
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Discrete systems
This is illustrated schematically in Figure 8.13, taking the example of the energy spectrum of a rectangular pulse of length r, measured in seconds. If U stands for the physical unit of the amplitude A, the spectral density of energy is expressed in (U I Hz) . Now, it is also interesting to calculate the spectrum of a rectangular pulse of variable length but of constant action (amplitude AI T ):
Most of the energy is contained in the frequency range [0,1/rj. fc = I11 is the cut-off frequency of the spectrum, see Figure 8.14. Incidentally, the product theorem implies that:
in agreement with the mathematical formula
Figure 8.14. Energy spectrum of a rectangular pulse in semi logarithmic scale
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317
When T — » 0 , the pulse of action A approaches the impulse AS(t)whose spectrum is flat, with the constant density 2A 2 . A flat spectrum is known as a white spectrum, using an expression borrowed from optics. Incidentally, it is also found that though energy of a Dirac impulse cannot be defined in the time domain (cf. NOTE in subsection 7.2.3), it would be found infinite if calculated in the spectral domain. In any case, the importance of such peculiarities is more formal than practical, as they merely indicate that a Dirac impulse is not physically feasible, though it can be satisfactorily mocked up by suitable transients, such as a rectangular pulse, provided the proper time-scale is respected. As a general rule, the frequency range covered by the spectrum of a signal increases in proportion to the reciprocal of the time-scale T of the signal, more specifically the spectrum extends up to the cut-off frequency fc=l/t. Figure 8.14 is a typical example, as the rectangular pulse lasts 0.01s whereas the cut-off frequency is precisely found to be WOHz. In other words, the size of the time and frequency ranges filled by the signal, denoted A/ and A/respectively, is inverted when shifting from one domain of description to the other, in such a way that the following invariant holds:
The similarity of relation [8.27] with the Eisenberg principle of uncertainty met in quantum mechanics is rather striking. NOTE. - Shock or impact signal As outlined just above, the spectrum of a rectangular pulse can be used to represent a shock, or impact signal produced by a shock hammer (cf. Chapter 5). By adjusting properly the stiffness of the impactor, it is possible to adjust the frequency range over which the spectrum is essentially white. However, the contact force induced by the impact of a harmonic oscillator against a rigid wall is shaped as a half sine and not as a rectangular pulse, as detailed in subsection 5.3.3.1. Then, as an exercise, it is of interest to compare the spectra of such transients suitably scaled to last the same time 1 and have the same unit action. Accordingly the half-sine signal is written as:
The Fourier transform is given by:
After some straightforward algebra, we obtain:
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Discrete systems
where fr The energy spectrum follows immediately:
It may be noted that the spectrum has no singularity, even at the reduced frequency /r = l. As shown in Figure 8.15, the spectrum of the half-sine is found to be essentially the same as that of the rectangular pulse, provided the time-signals are properly scaled.
Figure 8.15. Spectra of equivalent rectangular pulse and half-sine transients
8.2.5.2. Power spectral density of periodical functions The theory of distributions enables us to extend the spectral concept to the case of periodic functions of finite power. Accordingly, the physical quantity of relevance here is the power spectral density, (PSD in short notation) expressed in U2/Hz, where again U stands for the physical unit of the signal, which is now periodic. From the Fourier series [8.7], it is clear that we have to deal with a discrete spectrum, in which the signal power is entirely concentrated at the frequencies of the harmonic
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319
contributions to the signal. Such contributions are naturally viewed as spectral lines of arbitrarily small width and finite power. Total power of the signal is thus:
It is found appropriate to write this line spectrum as a series of Dirac's impulses 6(f - n/j) where /, is the signal frequency (the so called fundamental frequency):
Such a result is consistent with the Fourier transform [8.23]:
EXAMPLE 1. - Power spectrum of a sinusoid The periodical signal which has the simplest spectrum is of course the sinusoid X ( t ) = Asin(#>,/ + (p}. Its power spectrum, when restricted to the domain / > 0 , comprises the single line:
A sinusoid is said to be monochromatic using again an expression borrowed from optics. EXAMPLE 2. - Power spectrum of a periodic sequence of rectangular pulses Let us start by expanding the signal shown in Figure 8.2 as a Fourier series in the complex domain:
For n * 0 it is found that:
whence, for n varying from 0 to + «>:
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Discrete systems
Thus the power can be expanded as:
whereas the power spectrum is written as:
It consists of a sequence of lines spaced from each other by /j. The envelope is precisely the continuous spectrum of a single rectangular pulse, see Figure 8.16 where the spectrum is plotted using linear scales, in contrast with the loglog plot of Figure 8.15. The longer the duration of the pulse (a—>l), the narrower the frequency range of the spectral lines with a significant amount of power. Besides this, it may be noted that to shift the signal by a certain delay, leads simply to add a phase angle to the spectral components of the Fourier transform, and of course, the spectrum remains unchanged.
Figure 8.16. Power spectrum of a periodic sequence of rectangular pulses
On the other hand, it is also noted that as P™ = a A2, we can infer that:
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321
One major interest of such a result is that it gives the relative amount of the total power which resides in the mean (static) and in the fluctuating parts of the signal. As would be expected, the power stored in the static component increases with the duration of the pulses. Finally, it is found that balanced sharing of power occurs when a = 0.5, and it can be verified that to take account of 98 % of the total power of the signal, it suffices to truncate the Fourier series at order 10. As a consequence, in most applications it would be unecessary to truncate the series at an order higher than 10. NOTE. - Nearly periodical functions From the Fourier series we know that any linear combination of sine and cosine functions, whose frequencies form a harmonic sequence, results in a periodic function which oscillates at the fundamental frequency of the sequence. This is also the case whenever the frequencies are in a rational ratio, since rational sequences can be transformed into harmonic sequences by selecting properly the fundamental frequency. On the other hand, if at least one frequency ratio is irrational, periodicity of the resulting function is irremediably lost, though its spectrum still consists of spectral lines only. Such functions are said to be nearly periodic to mark the distinction from functions which have a continuous spectrum. Nearly periodic signals are common in mechanics. A typical example may be the impulsive response of a MDOF system, in which the resonant frequencies are not in a rational ratio and the damping very small (for instance, the Green's functions displayed in Figures 7.24, 7.25). 8.2.5.3. Mutual or cross-spectra The Plancherel-Parseval theorem [8.20] leads directly to the definition of the mutual or cross-spectra of two distinct signals, according to the following formula:
which is written in agreement with the definition [A1.15] of the scalar product given in appendix 1. Starting from a pair of given signals X(t) and Y(t), it is thus possible to define two distinct mutual spectra, namely SXY(f) and 5re (/), which are connected to each other by complex conjugation. However, provided that we deal with real signals, as it is most often the case, it is easily checked that SXY (-f) = S*XY (/). In most applications, it is found convenient to use the following reduced cross-spectrum, which is restricted to the range of non negative frequencies:
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Discrete systems
The modulus CXY (/)
is called the coherence function. This quantity is of little use
when X and Y are deterministic functions, since it reduces simply to CXY (/) = 7 . However, we shall see in Volume 4 that when X and Y stand for random functions, instead of deterministic functions as it is the case here, the coherence function conveys useful information because it can vary from zero, in the case of totally incoherent signals, to one, in the case of fully coherent signals. On the other hand, the phase YXY (/) is called the phase function. This quantity conveys useful information independently of the deterministic or random nature of the signals. EXAMPLE. — Original and delayed signals Let X(f) stand for the original signal which is amplified and delayed to produce the new signal Y = aX (t-r}. The shift theorem [A7.5] gives:
This result clearly points out that both signals are fully coherent and that the phase function is proportional to the delay and to the frequency. As an application, we will see in Volume 4, that it can be used to measure the transport velocity of fluctuating quantities in a permanent flow. 8.2.5.4. Spectra and correlation functions Shifting once more back from the frequency to the time domain, the spectral quantities defined just above are made to correspond to temporal quantities called correlation functions, which may be defined as the inverse Fourier transforms of the spectra. Accordingly, the autocorrelation function of a signal is defined as:
This result is broadly known as the Wiener-Khintchine theorem. As an exercise, it is thought instructive to establish it by performing the following simple calculation. First, we have by definition:
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Substituting the two last expressions of [8.33] into the first, we get:
an expression which is first integrated with respect to frequency, to produce:
Integrating then [8.35] with respect to 0, we obtain the desired result:
In the same way, the mutual, or cross-correlation function between two signals is defined as the inverse Fourier transform of the cross-spectrum:
As a consequence of the definitions made just above, the spectra and crossspectra are also termed autocorrelation and cross-correlation spectra, respectively. As already mentioned, we shall come back to these concepts in Volume 4, when dealing with the description of stationary random processes. 8.2.5.5. Coefficients of correlation As could be expected from the correlation integrals [8.32] and [8.36], the correlation function provides us with a suitable tool for measuring the degree of similarity, in the sense of energy, or power, of a pair of signals, including the cases when they are shifted from each other by an arbitrary delay T. Indeed, the autocorrelation function is endowed with the following property:
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Discrete systems
In [8.37], the equality arises directly from the definition of the autocorrelation function. The inequality, however, is more subtle and proceeds from a more general theorem, known as the Schwartz theorem on the triangular inequality. In the present case, it may be formulated by starting from a signal Z(t), which is a linear superposition of a signal X(t) of finite energy and the anticipated signal:
where a and b are considered to be two free parameters. Calculating the energy of Zft), we get:
Now, as the energy is essentially positive, so is the quadratic form in a, or b. The discriminant of the equation in a, or b, is thus necessarily negative and so:
which is precisely an equivalent form of the inequality [8.37]. In accordance with [8.38], it is suitable to reduce the autocorrelation function in a dimensionless form by using the signal energy as a norm. The result is known as the coefficient of autocorrelation (or of auto-covariance if the signal is centred) noted pxx , which can vary from -7 and +7:
A similar calculation, carried out with a - b, leads to the coefficient of crosscorrelation (or of cross-covariance in the case of centred signals):
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Finally, when carried out with a = -b, the same calculation shows that pXY is suited to measure the similarity between two signals in terms of energy. Indeed, let us define the gap signal:
Its energy is written as:
Hence, it appears clearly that the larger pn is, the less is the energy of the gap signal. In particular, starting from two signals of same energy, it is found that £22 vanishes when pXY = 1. EXAMPLE 1. — Cross-correlation between two rectangular pulses Let us consider the signal X(t)= A{M(t)-K(t-0)}.
By definition the
function of autocorrelation of X(t) is:
and so the function and the coefficient of autocorrelation are found to be:
Let us consider now two rectangular pulses of distinct lengths 0l and 02. By a similar calculation to that above we obtain:
where Om = mm (0j, 02) EXAMPLE 2. - Decreasing exponential signals Let us consider the signal X (f ) = Ae~"eM(t). The function and the coefficient of autocorrelation are found to be:
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Discrete systems
Let Y and X be two exponentials with characteristic times of decay 0, and 02. The function and the coefficient of intercorrelation between Y and X are found to
be:
This time, the result depends upon the integration order as RXY(*) * ^rx(^) 8.2.5.6. Correlation of periodic signals As already emphasized, periodic signals have an infinite energy whereas those which correspond to physical quantities have a finite power. As a consequence, the concept of functions and coefficients of correlation can be extended to such periodic signals, provided power is used instead of energy. In this manner we define the autoand cross-correlation functions:
8.2.5.7. Functions approximated by truncated Fourier series Let X(t) be a function of period T, or even non periodic but considered over a finite time interval T only. Furthermore, X(t) is assumed to be bounded and to have, at most, a finite number of discontinuities and extrema on T. Such a function can always be approximated to by a trigonometric series of the type:
The aim is now to determine the coefficients of the series which provide us with the best approximation. Of course, what we mean by "best approximation" has to be specified in accordance with a suitable criterion concerning the behaviour of an error function. In this respect, it may be argued that a criterion based on the energy of the error signal is well suited for applications in mechanics. Therefore, the problem is to
Spectral analysis of deterministic time signals
minimize, with respect to the unknown coefficients [«] N ,[^L' ^e defined as the power of the gap signal:
error s
327
ignal
We already know that in such a problem one has first to make E stationary (cf. Chapter 3, subsection 3.2.1). The condition for optimizing the approximation [8.45] is thus:
Performing the calculation prescribed by [8.47], we come precisely across the coefficients of the Fourier series:
Now, it can be understood without further calculation that the stationary solution corresponds in fact to a minimum. Indeed, the error function [8.46] may be interpreted as the equation of a paraboloid in a N-dimensional space which has a unique stationary point (equations [8.47] are linear) and this point cannot be a maximum since the error can be made arbitrarily large. Therefore, it may be concluded that the Fourier series is the trigonometric series which provides us with the best approximation over the interval T of the original function, in terms of power (or energy). To conclude on this subject, it is noted that the approximation method based on the minimization of the energy of the gap signal is also broadly known as the least squares method. On the other hand, the procedure of minimisation [8.47] would hold if the function were expanded on any other basis of orthogonal functions.
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8.3. Digital signal processing 8.3.1. Sampling of a time signal Generally, the time-signal originating from a measurement arises as a function which is continuous. Numerical recording and processing of such signals are necessarily preceded by an operation of discretization, which consists of sampling the original signal to retain only a finite sequence of values, taken at successive discrete times: X(t)=^lx(tn]\,n = l,2,...,N. Ideally, the sampled signal is related to the original signal by the following transformation:
where the hat symbol (A) marks the distinction between the sampled and the original signal. Most often, the discrete values are evenly spaced in time, for instance t = (n - I)TS where TS is the sampling period. Therefore, the ideal transformation [8.48] may be viewed as the multiplication of the original signal by a periodic sequence of Dirac's impulses, broadly known as a Dirac's comb, which is truncated to N consecutive teeth. On the other hand, time signals resulting from numerical simulation on a computer are directly available as sampled signals, where the sampling period identifies with the time-step of computation, provided the latter is kept constant. In fact, digital processing presents three peculiarities of practical importance, which will be described here, starting from the case of a measured signal to be digitized. Generally, the signal is first obtained in analogue form as the output X(t) of a sensor, typically a voltage proportional to the physical quantity we are interested in. X(t) is then processed through a device called an analogue-to-digital converter (in short A-D). The A-D comprises a locked sampling circuit, which converts the analogue information into a sequence of discrete numerical values denoted X(tn), coded as binary numbers of fixed size. This operation calls for the three following remarks: 1. As the storage capacity of any electronic device is necessarily finite, the signal must be processed sequentially at successive time intervals, whose duration is denoted 26 . This first necessary operation thus consists of transforming the original signal X(t) into the truncated signal X(t;20), defined as the product of X(t) by the time-centred rectangular window:
X(t;2d) may be viewed as a sample of the original signal X(t).
i 2. Every individual A-D conversion takes a finite time T( , during which the input of the A-D converter is locked. Of course, rt cannot exceed the sampling period TS . In a similar way, a computed signal arising from a numerical simulation may be viewed as a signal sampled according to T(=rs=h where h is the time-step of the computation. Hence, to perform the sampling transformation [8.49], the ideal Dirac comb is replaced in practice by a comb made of rectangular pulses of duration T( , which are repeated at the sampling period TS . 3. The binary coding in words of fixed size leads to quantification of the numerical values of the sampled signal which can be made discernible. For instance, a word comprising 16 digits allows one to quantify the value of X according to 214 - 1 = 16 383 distinct levels. Indeed, one digit is used to specify the sign of the number and another digit, called the digit of parity, is used to detect possible coding errors. The coding dynamic is defined as the ratio of the maximum value which can be coded over the elementary step of coding. Usually, the coding dynamics is expressed in decibels, that is by using the logarithmic scale:
where it is noted that the relative scaling of the signal is defined in terms of energy, or power. According to [8.50], the dynamics provided by a word of 16 digits is equal to 84 dB. Of course, this last remark holds also in the case of a signal arising from numerical simulations. 8.3.2. The Shannon sampling theorem In order to make a proper use of the storage and data processing capabilities of the digital instrumentation, or computer, it is not advisable to scan a signal at a sampling frequency fs =l/Ts much higher than the largest frequency of interest with respect to the physical content. Therefore, one is naturally faced with the problem of determining what minimum value of fs is to be adopted when the aim is to analyse the sampled signal up to the highest frequency /„,. In order to answer such a question, it is appropriate to analyse how the Fourier transforms of the continuous original signal X(t) and that of the sampled signal X(t) are connected to each other. Thus, let us consider a signal X(i) which is first truncated on 20 to produce the sample X(t;20), which in turn is sampled to produce a sequence of2N+l discrete values Xn(nTs) where -N
and 26-2Nrs . Using the ideal sampling
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We thus obtain:
It is worth emphasizing that in relation [8.52] frequency can still be viewed as a continuous variable. Now, the point is to establish the mathematical connection between X(f) and X(f). With this object in mind, it is first noted that as the original and the sampled signals are restricted to the finite interval 20 = 2NTS, nothing prevents us from assuming that the truncated functions X(t;29) and X(t;20) are 26 periodic. Therefore, X(t;20) can be written in the non truncated form [8.48]:
The unlimited Dirac comb can be viewed as a periodic distribution, which can be expanded as a Fourier series:
If k denotes an integer index, it is easily found that:
The relation [8.54] enables us to rewrite the ideal sampling transformation [8.53] in the following equivalent form:
Now, starting from the form [8.56], it is possible to calculate once more the Fourier transform of the sampled and 26 periodic signal, based on the convolution theorem (see formula [A7.9] in Appendix 7), according to which it is found that:
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Furthermore, relation [8.21] provides us with the following results:
and:
whence the final result connecting the Fourier transforms of the continuous original signal and that of the sampled signal is deduced, as truncated in a time interval 20 :
The result [8.60] is of major importance in practice, as it shows that the discrete sampling operation modifies the Fourier transform of the original signal. Fortunately, starting from X (/;20), it is however possible to recover the desired information contained in X(/;20), provided suitable precautions are taken in the data processing. A priori, we can be faced with two distinct cases, which control the rules specified in the Shannon sampling theorem. First case: the spectral range of the original signal is finite The Fourier transform of the original, yet truncated, signal extends over a finite frequency range A/ = [-/<.,+/c] where fc stands for the cut-off frequency of the truncated signal. When this is the case, it becomes possible to recover X(/;2#) without any ambiguity from X(/;2#), provided the so called Shannon sampling rule is fulfilled:
This rule arises as a direct consequence of the formula [8.60]. Indeed, if fs < 2fc it is found that the sum of two consecutive components in the series [8.60] leads necessarily to a spectral overlapping, hence to an irreversible loss of information. In contrast, if the criterion [8.61] is fulfilled, no overlap takes place and the information contained in the original signal is preserved. Figure 8.17 shows schematically how overlap is avoided, or not, depending whether the sampling frequency is appropriate, or not.
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Figure 8.17. Problem of spectral overlapping of an under-sampled signal
Figure 8.18. Phenomenon of aliasing (for explanation see text)
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The spectral overlap phenomenon is further illustrated in Figure 8.18, which illustrates what it is meant by "aliasing". Figure 8.18 shows that a sinusoid at 10 Hz (full line), sampled at 9 Hz over 1s (dots) cannot be distinguished from a sinusoid at 1 Hz (dashed line). In agreement with [8.60], the aliased frequency fa is related to the actual frequency f by fa = f - fs , where fs is the sampling frequency. Second case: original signal of "infinite " bandwidth A priori, if the bandwidth of the signal is unlimited, the spectral overlapping is unavoidable, whatever the sampling frequency may be. Of course, no physically feasible signal is provided with an infinite bandwidth; furthermore, one is often led to diminish the spectral domain of a signal on purpose, by using a low-pass filter. Thus, the same sampling rule holds as in the former case, the cut-off frequency being that of the filter. To conclude on this point, it is worth emphasising that the Shannon criterion must be satisfied also when numerical simulations are performed, the cutoff frequency being identified with the reciprocal of the time-step of computation. Therefore, it is necessary to select a suitable time-step for describing satisfactorily the spectral content of the simulated signal, up to the highest frequency which is desired. 8.3.3. Fourier transforms of the original and of the truncated signals
Figure 8.19. Fourier transform of a rectangular pulse Shannon's theorem provides us with a criterion for sampling a signal, in such a A.
way that the Fourier transform X(f,20) of the sampled signal can reproduce exactly
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the same information as that contained in the Fourier transform X(f,20) of the original and truncated signal (such a Fourier transform is termed truncated or finite Fourier transform). However, the problem now is to find out the relation between the finite Fourier transform and the Fourier transform of the original signal, without any truncation. In order to investigate this let us return to the definition [8.49] of a truncated signal. The convolution theorem implies that:
Figure 8.19 shows the shape of the Fourier transform of the rectangular pulse used for windowing the original signal, which is marked by a central peak plus a series of progressively decaying undulations, or ripples. Due to such ripples, the convolution induces some distortion of X(f,20) with respect to X(f). The shorter 6 is, the larger is the distortion. The truncation effect is illustrated in Figure 8.20, which refers to the windowing of a cosine function at f1 = 1 H z . The convolution product is:
Figure 8.20. Fourier transform of the truncated cosine signal
This is precisely the same windowing effect which is conspicuous in the plots of the truncated Fourier series shown in Figure 8.9 and 8.10. As a final remark, it is possible to alleviate such defects by using other window shapes, smoother than a
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rectangular pulse. However, such windows have their own drawbacks as described in the same references as those already quoted above. 8.3.4. Discretization of the Fourier transform 8.3.4.1. Discrete finite Fourier transform and Fourier series Returning to the original function, truncated over 26 . If periodicity 29 is assumed, it can be expanded as a Fourier series:
The coefficients are given by [8.8] and performing now a Fourier transform of the series [8.63], we obtain, in terms of distributions:
This expression is a discrete version of the Fourier transform of the truncated signal. It indicates, in particular, that the smallest identifiable frequency separation contained in the truncated signal is precisely 6 f = 1/20, a result which is not surprising. 8.3.4.2. Definition and properties of the discrete Fourier transform When time data are processed at the sampling frequency fs , the only transform which is available is a finite sampled function X(f;20), and not X(f;20) as given by the infinite series [8.64]. Since the smallest frequency separation which can be accounted for by the sampled time signal X (t;20) is fs = 1/20, this value may be used as a suitable frequency-step to sample X (f;20). Of course, only the terms at frequencies less or equal to the cut-off value fs 12 , are retained in agreement with the Shannon criterion. This produces 2N+1 values of the sampled discrete Fourier transform, which is finally written as:
Relation [8.65] is obtained by substituting into formula [8.52] the current frequency f by the sampled value kfs and by adopting the normalization factor 1/2N.
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On the other hand, since the time samples {xn } are real quantities, the 2N+1 values of the discrete Fourier transform are interrelated by the condition of complex conjugation:
Such a result shows that starting from 2N+1 sampled values of a time signal, formula [8.65] provides us with N+1 sampled values of the Fourier transform, defined in the domain f > 0. It is thus possible to restrict the computation to the range f > 0, without any loss of information. This is to be expected, because the computation procedure [8.65] transforms 2N+1 real numbers into N complex numbers, plus the real number at zero frequency (k = 0). It may also be noted that if the case of an unbounded sequence of successive integer values k is considered, it can be concluded that the corresponding discrete Fourier transform becomes a sequence with periodicity 2N:
This also is in full agreement with the considerations made in the last subsection about the assumed periodicity of the truncated time signal. On the other hand, the inverse Fourier transformation is found to be given by:
A priori, examining the formulas [8.65] and [8.68], it could be concluded that the transformation requires a number of calculating operations which is proportional to (2N + 1)2 . Fortunately, in 1965, Cooley and Tukey made available a fast Fourier transformation (FFT) algorithm, which reduces this number significantly (number of calculating operations proportional to N Iog2 (N), provided N is an integral multiple fj
of 2 (N = 2m; m: integer). The availability of this algorithm [COO 65] has greatly promoted the use of spectral analysis since the late sixties, and it is now a basic technique of signal processing, in common use in mechanical engineering in particular. Modified FFT algorithms to which the limitation of N = 2m does not apply have also been developed, see for instance [BRI 74]. 8.3.4.3. Illustrative example It is worthwhile to conclude this chapter by applying the foregoing general notions to a specific example, at least in order to illustrate their practical relevance. Let us consider the following signal:
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X (0 = 2 {U (t- 8.042)- U(t- 8.342)}+ 0.06 sin(10^t)(l + 3 sin(16;t)) where time t is given in second. Hence, the highest frequency component of the signal is f3 = 13 Hz. It arises from the addition of the frequencies of the two sine functions present in the above analytical expression. Two other spectral lines are also expected, at 3 and 5 Hz respectively. Finally, the spectral continuum arising from the rectangular pulse is marked by a cut-off frequency of about 3.33 Hz.
Figure 8.21. Time-history of the signal to be processed
Figure 8.22. p Power spectrum of the signal suitably sampled
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Figure 8.23. Time-history of the under-sampled signal
Figure 8.24. Power spectrum of the under-sampled signal
The sampled function X(t) is displayed in Figure 8.21, over 20= 40 s. The sampling frequency fs is 57.2 Hz providing thus 2048 sampled values. This sampling rate is more than compatible with the Shannon criterion. It has to be emphasized that the continuous line conspicuous in the plot is merely due to an
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artefact of the plotting subroutine which has been activated on purpose, in order to help visualisation of the analytical function. Figure 8.22 is a plot of the PSD of the sampled signal. The frequency range of the discrete spectrum extends up to fs 12 . The frequency separation between two successive components is Sf = 1/20 = 12.5mHz . Figure 8.23 displays the sampled time signal obtained by adopting fs = l2.8 Hz, providing thus 572 sampled values which are "under-sampled" with respect to the Shannon criterion. Here the plot has been performed by deactivating the optional continuous line display of the MATLAB subroutine. Doing so, it can be seen that the original signal is hardly recognisable based on the sampled signal, as soon as N becomes small enough, as it could be verified even when using a sampling frequency in agreement with the Shannon criterion. Figure 8.24 is a plot of the corresponding PSD with the continuous line display activated. The sampling frequency is still sufficient to describe the spectral content of the pulse and the two lines at 3 and 5 Hz. However as expected, it is found insufficient to describe the line at 13 Hz. Indeed, on the plot of Figure 8.24 the line at 13 Hz is folded back to the much lower frequency f3'= f 3 - f s =0.2Hz, in full agreement with the principle schematically displayed in Figure 8.17. The above example is found suitable in order to illustrate the practical rules it is necessary to comply with when processing digital data. If the numerical signal originates from an analogue record, a necessary preliminary is to filter the analogue signal up to the cut-off frequency fc = fm before digitalisation, where fm stands for the maximum frequency of interest. Then the signal is sampled at the frequency fs = 2fm • For identifying spectral components separated in frequency by at least Sf , the length of the sampled signal must be is selected as:
Starting now from the results provided by a numerical simulation performed on the computer, the problem can be restated as follows. The practical object is to compute a signal X(t), whose spectral content is bounded by the maximum frequency fm. Moreover, spectral components separated by at least Sf must be identified. According to the foregoing considerations, it is necessary to perform the numerical simulation with the time-step of the dynamic integration algorithm set to:
The duration of the computed signal is still given by [8.69].
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Chapter 9
Spectral analysis of forced vibrations
The methods of spectral analysis are of special interest in the treatment of dynamical systems, in particular, but not exclusively, when they are linear. We shall first show that by considering the spectral content of excitation and natural modes of vibration (in short the "modal spectrum") it becomes possible to identify those modes which have to be retained in the dynamical model and those which can be neglected. This aspect of modelling is of crucial importance since the modal sequence of real material systems is a priori infinite, as detailed in Volume 2. On the other hand, for many applications in mechanical engineering the pertinent information is contained in the spectral properties of the response signals and little would be gained by embarking on solving the problem in the time-domain. Finally, even if the dynamical system is far from linear, spectral analysis still provides quite valuable information concerning the existence and the nature of nonlinearities present in the system on one hand, and the periodic or the chaotic properties of the response to periodic excitation, on the other.
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9.1. Introduction The mathematical tools described in the preceding chapter are applied here to analyse the spectral properties of the response of mechanical systems vibrating according to steady-state regimes. Section 9.2 is devoted to the analysis of the damped harmonic oscillator. It is shown that the spectral properties of the response can be fully understood by starting from those of the transfer function and of the excitation signal. In section 9.3, the methods of spectral analysis are applied to the case of MDOF linear systems. The most prominent feature of the spectral properties of the transfer functions is the presence of three distinct frequency domains, which require distinct levels of accuracy in their description, namely the domain of quasistatic responses, resonant responses and finally the domain of quasi-inertial responses. Clearly, the most demanding domain for modelling is that of resonant responses, in which the description of the vibration modes, including damping, is of crucial importance. General results are further discussed based on the analysis of two simple devices of practical interest. The first system is a vibration absorber which makes use of the principle of antiresonance. The second system is the shock absorber of a car suspension which was already described in Chapter 7, subsection 7.5.3. Finally, in section 9.4 another incursion into the nonlinear domain is considered, based on the classical example of the steady state responses of the Duffing oscillator to harmonic excitations. The study presented here demonstrates the profound differences which exist between the dynamical behaviour of linear and nonlinear systems. The most prominent feature, also encountered in many other nonlinear dynamical systems, is the possible occurrence of chaotic responses. For certain values of the internal coefficients of the oscillator and for certain values of the external harmonic forcing function, though the system is basically deterministic in nature, the response becomes so irregular that it is impossible to ascertain which dynamical state the system will have at a fixed time, as though the response were random. On the other hand, chaotic responses are also marked by a continuous spectrum, in contrast with the periodic, or pseudo periodic responses. Chaotic dynamics stems from the pioneering work of Poincare. However, it was not before the second half of the XXth century and the advent of numerical computers that the practical and philosophical implications of deterministic chaos were fully investigated. The reader interested in an historical survey and a qualitative introduction to the subject, is referred in particular to [BER 84] and [GLE 88]. 9.2. Linear (harmonic) oscillator 9.2.1. Spectra of excitation and response By transposing to the frequency domain the basic equation [7.23], which relates the response of a SDOF linear system to its excitation, we obtain in terms of Fourier transforms:
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Starting from [9.1] the following spectral relationships follow immediately:
As further investigated in the next subsection, these formulas show the filtering properties of the oscillator which acts as a narrow band filter, centred at the natural frequency, relative bandwidth being controlled by damping.
9.2.2. Spectral properties of transfer functions 9.2.2.1. General features of the displacement/force transfer function
Figure 9.1. Squared modulus of the displacement/force transfer function of the damped harmonic oscillator in loglog scales Starting from the formula [7.24], the transfer function of the damped oscillator, relative to the output/input ratio "displacement/force" is now written in terms of the reduced frequency z, as follows:
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f1 is the natural frequency and £, is the damping ratio of the linear oscillator. In accordance with the spectral relation [8.20], it is convenient to describe the complex quantity [9.4] by using its squared modulus and its phase:
The shape of |H (z)|2 is depicted in Figure 9.1 for several values of the damping ratio, by using logarithmic scales. If gl is sufficiently less than one, the curves are marked by a prominent and sharp peak centred at the resonant frequency z = 1. The peak value is given by:
Provided again that g1 is sufficiently small, the magnitude of the peak is reduced by a half when frequencies are shifted by Az = ±g1 from the resonant frequency:
Moreover, the area under the curve can be expressed analytically in terms of the oscillator coefficients:
When z is sufficiently smaller than one, it is found that the modulus of the transfer function is essentially constant and equal to \IK. But if z is sufficiently larger than one, |H| decays according to a z-2 law. The detailed shape of the resonance peak is better shown in Figure 9.2, where |H (z)|2 is plotted in the vicinity of resonance by using semi logarithmic scales.
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Figure 9.2. Squared modulus of the displacement/force transfer function of the damped harmonic oscillator in the vicinity of resonance
According to the phase curve of H, if z is much less than one, the phase between the response and the excitation is essentially zero. Thus, the response is "in-phase" with the excitation. At the opposite, when z is sufficiently larger than one, the phase is practically equal to -n. Of course, as far as phasing is concerned, -n is equivalent to +r, both values corresponding to an opposition of phase between the response and the excitation. As shown in Figure 9.3, most of the variation of the phase angle from zero to ±n takes place in a narrow range of frequencies centred at resonance, where the slope of ^(z) is proportional to g1. At resonance the phase lag is -n /2, a value which corresponds to a phase quadrature. Finally, it may be noted that the phase is varied from -n /4 to 3n /4 over the narrow frequency range Az = 1 + £1 .Therefore, the most suitable way to display graphically the resonant portion of the transfer function in detail is to make an Argand (or Nyquist) plot, in which the imaginary part is plotted versus the real part, as shown in Figure 9.4. It is left to the reader as an exercise to show that the Argand plot of the transfer function [9.4] is a circle, from which the parameters of the oscillator can be conveniently extracted.
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Figure 9.3. Phase function *t(z) in the vicinity of resonance
Figure 9.4. Argandplot of the transfer function
9.2.2.2. Spectral ranges of the oscillator response It is of interest to investigate the consequences of the response properties of the harmonic oscillator on the forced response to an excitation of which the spectrum is
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known. Starting from the first result [9.2], we can draw a few conclusions of major interest, based on the order of magnitude of the ratio f1/fc of the resonant frequency f1 over the cut-off frequency fc of the excitation. Let us examine first the case of a monochromatic excitation of pulsation We and magnitudeQ(E)0. As already shown in Chapter 7, the forced response is also harmonic, at pulsation we. Thus the three frequency ranges identified in Figure 9.5 are worth discussing.
Figure 9.5. Schematic identification of the spectral ranges of the oscillator response
1.
Quasistatic range of response: we«wl
In this low-frequency range, the transfer function is practically constant and equal to 1/K. This means that the oscillator responds to the excitation as a simple spring of stiffness coefficient K. Accordingly, the displacement and the force signals are simply proportional to each other, in the time as well as in the frequency domain, and both signals are "in-phase" with each other. As a consequence, the magnitude of the vibration is qs =Q(e)0/K, which corresponds precisely to the displacement induced by a static load Q(eQ . In short, in so far as we « w1, the dynamic internal forces of the oscillator can be safely neglected. 2.
Resonant range of response: we=w1
In this domain, which corresponds to that of the resonance peak, the closer to w1 is we the larger is the magnitude of the forced response. At resonance, the amplitude
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of the vibration is magnified by the factor 1/(2s1), when compared with the quasistatic case. Now, in terms of signal energy, at resonance it is found that: ; where In the resonant domain, the stiffness and the inertia force of the oscillator are practically cancelling out each other. As a consequence, the dynamical equilibrium of the forced oscillator is controlled essentially by the damping forces. 3.
Quasi-inertial range of response: we » wl
In this high-frequency range, the stiffness force of the oscillator becomes negligible in comparison to the inertia force. Accordingly, the transfer function can be approximated by - 1 / ( K z )
. However, by reason of the physical origin of this
approximation, it is preferable to rewrite this result in terms of mass and pulsation as -1/(Mw2e). Accordingly, the displacement lags by half a period the force signal and its magnitude decreases as w-2e. Let us consider now the case of a forcing spectrum which is broadbanded. Provided the energy (or the power) of the force signal contained in the resonant range is not negligible, the response is largely dominated by the resonant component, at least if damping is sufficiently small. Moreover, in so far as the variation of the spectral level of excitation can be discarded over a frequency range of a few w 1 g 1 , centred at the resonant frequency w1, the energy of the response is given with good accuracy by the following formula:
Comparison of formulas [9.10] and [9.11] brings out the important following result: when the bandwidth of the excitation spectrum is broad, the energy of the response is smaller by a factor gl than in the case of a monochromatic excitation spectrum, provided the energy of excitation in the resonant domain is the same in both cases. Such an attenuation is thus particularly large in the case of lightly damped systems. This result is easy to understand since, in the monochromatic case, the excitation energy is entirely concentrated at the resonance, whereas in the broad banded case, it is uniformly distributed over the whole resonant range, through which the transfer function undergoes large variations in magnitude. EXAMPLE. - The harmonic oscillator as a narrow band filter
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Figure 9.6. Sample function of the force signal
Figures 9.6 to 9.9 illustrate the narrowband filtering effect that an input signal (excitation) undergoes when processed through a linear oscillator. Here we deal with a discrete signal denoted Q(t;20), which is scanned at frequency/, during a time interval 29. It is formed by adding a scanned sine function of 12 Hz and amplitude 1N to the random signal produced by sampling at random values within a given real and finite interval (±30N in the present exercise). Actually, the sampling uses an algorithm which produces pseudo-random values. Because of the random or pseudorandom nature of the signal, the exciting force varies according to very different time-scales, and the presence of the sine component is completely blurred out, as seen in Figure 9.6. Anticipating here the description of random processes which will be presented in Volume 4, the signal just defined may be understood as a sample function, denoted Fk (t;20), of a stationary random process. Let us assume then that the randomness of the process can be suitably taken into account by performing the statistical average of a set of N independent sample functions. In practice they are obtained by applying the calculation described above over time N(20), where N is a sufficiently large integer (typically a few hundred). A statistical estimation of the spectrum of the process is produced by using the following ensemble averaging:
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Such a statistically averaged spectrum of the excitation signal is shown in Figure 9.7, in which the line at 12 Hz is clearly conspicuous, emerging from a broadband component, which is highly irregular, though its mean level is essentially constant within the frequency range explored.
Figure 9.7. Statistical estimate of the PSD of the force signal
Figure 9.8. Sample function of the oscillator response
Figure 9.8 shows a sample of the response of the oscillator, which is assumed to be tuned atf1 = lOHz, g1=4%. This output signal is much smoother than the input signal. It consists essentially of a vibration at 12Hz, highly modulated in amplitude. Modulation produces an envelope which involves time-scales much larger than those
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of the resonance and the exciting sine function. One reason for this is that the -2 displacement response of the oscillator decays quickly with frequency, as w-L in the quasi-inertial range, as can be seen in the power spectrum depicted in Figure 9.9.
Figure 9.9. Statistical estimate of the PSD of the oscillator response
NOTE. - Energy and r.m.s. value of the response It is known from the Plancherel-Parseval theorem (cf. Chapter 8 subsection 8.2.2.3), that the integral from zero to infinity of the response spectrum produces the response energy. The square root of energy crq = J£~ is termed root mean square value, or in short, r.m.s. value. In the case of a centred signal, the mean value of response is zero and the r.m.s. value is used to measure the mean value of the vibration magnitude, termed standard deviation and o>2q is termed variance of the signal. Making use of such a quantity is very convenient when performing design calculations, in terms of orders of magnitude. However, care has to be taken concerning the fact that the r.m.s. value is a global quantity which discards automatically the eventual significance of the extreme values occurring in the actual time-history of the response. This aspect of the problem will be further discussed in Volume 4, in relation to flow-induced random vibration.
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9.3. MDOF linear systems 9.3.1. Excitation and response spectra Transposing into the spectral domain the equation [7.66], which relates the response of the i-th degree of freedom to an excitation applied to the j-th degree of freedom, yields the following result expressed in terms of Fourier transforms:
From relation [9.12] the following spectral formulas arise immediately:
9.3.2. Interesting features of the transfer functions Transfer functions, as defined either through relation [9.12] or [9.14], present some new features in comparison with the case of SDOF systems, and are worth discussing, starting with an example. Let us consider for instance a chain of four identical mass-spring systems of the kind already described in Chapter 6. A transfer function of the type displacement/force is shown in Figures 9.10 and 9.11 as squared modulus and phase plots. In Figure 9.10, four resonance peaks are present, which may be identified with the resonant responses of the four natural modes of vibration of the system. However, it has to be stressed that such a one-to-one correspondence does not hold as a general result. Indeed, a resonance peak may disappear from a transfer function, either because damping is too high, or because the response is merged together with another neighbouring resonance peak, or finally because displacement of at least one degree of freedom specified by the indicial arguments i, j, vanishes for the mode considered: pn (i) or qn (j) = 0 (see for instance the Figures 9.12 and 9.14). As a consequence, the experimental identification of the natural modes of vibration of structures (termed modal testing) necessitates the measurement of several transfer functions, the location of point excitation and/or vibration sensors being varied, see subsection 9.3.3 for a short presentation of the basic principles underlying this type of measurement, for a thorough presentation of the technical aspects see for instance [EWI 00]. On the other hand, as also seen in Figure 9.10, HH* can display very pronounced minima lying between two successive resonant peaks. The reason for such a feature, known as an antiresonance is the cancellation effect, in a limited frequency range, of the contributions to the response of two neighbouring modes which vibrate nearly out-of-phase from each other with a similar magnitude. In subsection 9.3.4 an application of this effect will be described, which turns out to be of great interest in practice for vibration isolation.
Spectral analysis of forced vibrations
Figure 9.10. Squared modulus of a transfer function of the displacement/force
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type
Finally, Figure 9.10 indicates that |H (i, j,w)| becomes essentially constant in the low frequency range w «w1, response domain:
which is immediately identified with the quasistatic
As a physical quantity, H(i,j ;0) can be considered as a flexibility, which is the reciprocal of the stiffness. Its disappearance means that when the jth-DOF is excited no displacement of the ith-DOF takes place (the respective effect of i and j being obviously reversible). This occurs if all the modes present a node at the DOF concerned. Alternatively, in the quasi-inertial domain w » wN , the transfer function may be written as:
The phase of the transfer function is plotted versus frequency in Figure 9.11. As expected, displacement in the quasistatic domain is found to be in-phase with the exciting force. Then, sudden variations are observed which occur at the crossing of resonances. They are clearly controlled by the modal damping ratios. Of course, the smaller the damping, the more abrupt is the phase variation. Finally, in the quasiinertial domain, displacement is found here to be out-of-phase with the exciting
aa 4
Discrete systems
force. However, in other instances it can be in-phase, as the sign of the transfer function H (i,j;w. —> °°) depends in a rather intricate way upon the particularities of the mode shapes and modal masses.
Figure 9.11. Phase function of the transfer function This may be checked by considering again a chain of coupled oscillators. We consider here two transfer functions of a chain of 13 identical mass-spring systems. H(3,9;f) refers to a configuration in which the pair of the excited and responding particles are disposed symmetrically about the middle particle. All the modes of the chain are excited as can be seen in Figure 9.12, where 13 resonance peaks are detectable. The phase function plotted in Figure 9.13 shows that in the quasi-inertial range the displacement is in-phase with the excitation, as in the quasistatic range. H(7,7;f) refers to the case in which both excitation and response are disposed at the middle particle. At first, due to the peculiar location of excitation, only modes of odd rank are excited, as clearly shown in Figure 9.14. Then, displacement and force in the quasi-inertial domain are out-of-phase (\i/ = ±n), see Figure 9.15. This result is merely a direct consequence of formula [9.16], when particularized to the case of an excitation and a response relative to the same degree of freedom.
Spectral analysis of forced vibrations
Figure 9.12. Chain of 13 identical oscillators: |H (3,9; f)|2
Figure 9.13. Phase function of H (3,9; f)
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Discrete systems
Figure 9.14. Chain of 13 identical oscillators: \H (7,7;f)|2
Figure 9.15. Phase function of H (7,7;f)
As already mentioned, it is also useful to define other types of transfer functions than the displacement to force type, in particular because many kinds of vibration sensors are used for measuring quantities other than displacements. For instance, in
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many applications the vibration sensors which are found to be the most appropriate are accelerometers. When acceleration is substituted for displacement, the transfer functions take the form:
Figure 9.16. Squared modulus of a transfer function of the type acceleration/force
As clearly seen from Figure 9.16, such transfer functions tend to zero at low frequencies like w2 and become constant when w tends to infinity:
From the asymptotic result [9.18], an equivalent inertia may be defined as follows:
Meq (i,j) is found to be infinite, provided that at least one modal displacement pk(i) or pk (j) is zero. This is simply because, if such is the case, the system does not move according to the mode considered. Thus, in order to study the physical motion of the system it is necessary and sufficient to remove from the model all the modes which have a node either at particle i, or j.
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9.3.3. Basic principles of the measurement of transfer functions Let us consider a linear system, whatever its physical nature may be. Such a system is assumed to be excited by the input signal denoted F(j;t) applied solely to the degree of freedom j, and the response of the degree of freedom i is recorded, producing the output signal X(i;t). Discarding here the specific problem of actually measuring and recording the data, which have been already discussed in Chapter 8, section 8.3, the transfer function of the output to the input is formally defined as the following ratio of Fourier transforms:
For determining H X F ( i , j ; f ) experimentally, the basic idea is to rely on formula [9.20]. At first, suitable sensors are selected to measure F(j;t) and X(i ;t). In mechanical applications, piezoelectric transducers are widely used to measure forces and accelerations. Then, the analogue signals issued from the signal conditioning amplifiers associated with the sensors are processed through a spectrum analyser, which turns out to be a very sophisticated device, including a computing unit with various subroutines implemented in it. The signal processing comprises first a lowpass filtering to comply with the Shannon criterion. Then the signals are digitized at the desired sampling frequency and the discrete Fourier transforms are computed using an FFT algorithm. Finally, various subroutines may be selected optionally to compute and visualize the desired transfer functions and spectra, including statistical averages of all of these quantities. Indeed, though a transfer function can be produced directly by using the formula [9.20] on a single pair of measured output/input signals, there are several advantages to processing the measured signals as random quantities and to use the spectral relation:
The basic reason for this is that actual measurements are unavoidably spoiled to some extent by random noise arising from various sources; whence the interest of minimizing the noisy part of the signals through statistical averaging. Moreover, when using the relation [9.21] instead of [9.20] the statistical estimate of the transfer function becomes free from spurious signals with zero coherence between the input and the output. Finally, according to relation [9.21], a very large variety of signals may be used for exciting the system, including of course random or pseudo-random signals. The sole condition is to build a signal the spectrum of which includes the frequency range of interest. The signals in most widespread use for excitation are: 1.
The shock signals, which are produced by the impact of a hammer provided with a sufficiently stiff force transducer (using typically a piezoelectric crystal). The frequency range of the shock signal can be conveniently varied by an order of magnitude by changing the material of the impacting tip, modifying thus the shock stiffness, as already described in Chapter 5, subsection 5.3.3.
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2.
Electromagnetic shakers, in which an electrical input signal is supplied by a power amplifier, and then converted into a fluctuating magnetic field actuating a coil attached to the drive part of the shaker. The shape of the electrical signal can be varied at will. For performing accurate measurements, it is often preferred to use a sine function of which frequency is progressively swept through the frequency range of interest. Of course, the sweeping rate must be slow enough in such a way that only the response in the steady regime is actually recorded. Furthermore, the amplitude of the excitation is controlled independently, to be able to change the level of excitation as resonances are passed through. Such procedures for vibration measurements are very demanding in time, especially when the resonances of the system are at low frequencies and are poorly damped. To reduce the time for measurement, the shaker can be driven by using a pseudo-random signal with a flat spectrum up to the highest frequency of interest (cf. signal of Figure 9.5).
3.
Servo-controlled hydraulic jacks. This kind of device is typically used to drive shaking tables according to prescribed time-histories of displacement, or acceleration. Actually, such tables are generally conceived as very sophisticated and expensive test facilities which are used to test large and heavy structures subjected to various kinds of excitations, including namely, shocks, seismic shakes and harmonic signals.
9.3.4. Response spectra resulting from an MDOF excitation So long as the response of a system remains in the linear domain, the principle of superposition holds. Accordingly, the response q i (t) of a given degree of freedom i, to the force vector |Q(e)(t)| is obtained as the sum of the partial responses to each component Q(e) (t), where j=1,2,.. N. Since the Fourier transformation is also linear, the response in the frequency domain is again the sum of the partial responses:
As a definition, the autocorrelation spectrum (ESD, or PSD) of the response is given by:
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As a consequence of the quadratic nature of the above operation, it is found that the response spectrum is related to the auto- and cross- spectra of the excitation signals, through the following formula:
The actual calculation of such a formula may be rather tedious, in particular because the product of the transfer functions generates coupling terms between all the natural modes of vibration of the system. It is thus worth mentioning that such crossed terms can be generally neglected, provided the resonant frequencies are sufficiently apart from each other, the relevant criterion being given by the following inequality:
Thus, provided the condition [9.25] is fulfilled, each individual resonance peak of response is practically independent from the others. Accordingly, it becomes possible to calculate the energy, or the power, of the physical response simply by summing up the partial contributions of each individual resonance:
Or, in terms of standard deviations, or r.m.s. values:
9.3.5. Vibration absorber using antiresonant coupling In practice, it often occurs that monochromatic and resonant excitation of poorly damped systems is inevitable. If no specific action were taken, excessive vibration would occur, leading generally to a failure of the component, typically by fatigue. An efficient and elegant way to reduce the response to a safe level is to perturb the system by coupling it to a "small resonant oscillator", tuned at the natural frequency of the resonance to be cured. By "small" it is meant that the mass and the stiffness coefficients of the oscillator, though tuned to the proper ratio, are much less than the corresponding generalized quantities of the resonant mode to be perturbed. Indeed, as shown below, the effect of the perturbing oscillator is to replace the undesirable resonance of the initial system by a pair of closely spaced resonances, of which the contributions to the response cancel out each other over a fairly small frequency range centred at the natural frequency of the unperturbed system. In other words, the
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perturbed system has an antiresonance centred at the resonance of the initial system. This kind of vibration absorber is found to be particularly efficient precisely in the case of small damping. However, its range of application is restricted to the case of monochromatic, or fairly narrow bandwidth excitations.
Figure 9.17. Schematic picture of the antiresonant vibration absorber
How it works can be understood by using a model restrained solely to the resonant mode of the initial system, plus the perturbing oscillator, as shown in Figure 9.17. As the excitation is assumed to be monochromatic and in resonance with a given mode of the initial system, the response of the other modes may be neglected. The perturbed model is governed by the two following equations:
There would be no difficulty to solve directly the above system by eliminating the displacement Y of the perturbing oscillator between the two equations. However, we prefer to use the modal method to illustrate the underlying physics of the problem. Qualitatively, the important point is that the two modes arising from the coupling of the two oscillators have quite similar properties. In particular, the two oscillators being individually tuned at the same angular frequency w0, the modal frequencies of the perturbed system are found to bracket w0 in a fairly narrow interval [w1,w2]. The response of M to a monochromatic excitation of M at w0 is given by the following modal series:
In this step, to simplify the algebra, it is assumed here that the damping ratios of the two perturbed modes are the same and equal to that of the non perturbed mode, noted £0. However, the physical mechanism of the antiresonance would be the same even if distinct damping ratios would be considered. Indeed, the major point is that
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the two modal contributions cancel each other in a frequency interval of the order of 28(O = w2-w1 because the sign of the real part of the denominator is changed from one term to the next. The quantitative analysis is left to the reader as an exercise. Writing down the vibration equations in terms of the dimensionless parameter of perturbation:
It will be shown that the natural frequencies of the perturbed system can be written as:
The modal shapes and the generalized masses are found to be:
Then the following expression provides the ratio of the response at wO of the perturbed system to that of the initial system, in terms of complex amplitudes:
Finally, the efficiency of the attenuator may be measured by the ratio:
Such results are illustrated by considering a numerical example. Let there be a system having an undesirable mode of vibration with the following properties:
This mode is excited by a resonant force of magnitude 1N. The response magnitude is thus 10cm. In order to reduce it significantly, the system is perturbed by
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adding a mass-spring system, connected at an antinode of the mode to be modified. Let us consider first the case m- 0.1 Kg, k - 10 N/m. Numerical computation provides the following modal properties, which are in fair agreement with the analytical results quoted above: 1.
In-phase mode
2.
Out-of-phase mode:
In Figure 9.18 |X(w)| is plotted versus frequency, referring both to the initial system (dashed line) and the perturbed system (full line). The antiresonance is particularly well pronounced, resulting in an attenuation of vibration amplitude by two orders of magnitude.
Figure 9.18. Magnitude of the forced response of the main oscillator K, M (dashed line: initial system, full line: perturbed system)
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Figure 9.19. Magnitude of the forced response of the main oscillator K, M
Another calculation performed with a "larger" perturbing oscillator k = 50 N/m, m - 0,5 kg, produces similar results. The major difference with the former case is that the interval w2-w1 is larger, providing thus the opportunity to widen somewhat the antiresonant domain, see Figure 9.19. The modal properties are found to be: 1.
In-phase mode:
2.
Out-of-phase mode:
The magnitude of the response at the antiresonance now is 0.2 mm only. Finally, it can clearly be seen in Figures 9.18 and 9.19 that the antiresonant attenuator cannot be very useful in the case of broadband excitation.
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9.3.6. Shock absorber of a car suspension
Figure 9.20. Spectrum of the spatial transverse unevenness of the road profile
The device of interest here is the same as that already described in Chapter 7, subsection 7.5.3.2, where its absorbing efficiency at the crossing of a single bump was discussed. Another problem concerning the comfort of the passengers is that of steady vibrations, which are induced permanently by the irregularities of the road, and transmitted from the wheels to the car body. The roughness of the road profile is characterized by a continuous spectrum, of which a typical example is plotted in Figure 9.20, according to standard ISO data referred in [GEN 95]. It is analytically formulated as the following power law where the variable is the "spatialfrequency" A, expressed in cycle/m:
Such a law implies in particular that at wavelengths larger than six meters, the PSD of the irregularities decays more slowly than at shorter wavelengths. Now, this spectrum is converted into a spectrum expressed in m2 / Hz by introducing the cruising velocity V of the car, expressed in m/s. Indeed, it is easy to show that:
This equivalent spectrum for a cruising velocity of about 108 km /h is shown in Figure 9.21.
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Figure 9.21. Equivalent spectrum versus frequency for a cruising velocity V= 30 m/s
The r.m.s. value of the displacement impressed to the wheel is about 1 cm. On the other hand, the transfer functions calculated in Chapter 7, subsection 7.5.3.2, are expressed in term of circular frequency as:
characterizes the transmission of the vertical displacement from the wheel to the body, and Hzz (Z0,Z1;w)characterizes the transmissibility of the vertical displacement from the contact point to the wheel. In terms of acceleration, the spectral response of the car body to the irregularities of the road profile is thus given by the following formulas:
where The squared modulus of the transfer function Haa (Z 2 ,Z 0 ;f)is plotted in Figure 9.22. On such a plot, the two resonances of the model are still clearly detectable, though the system is highly damped. Figure 9.23 shows the acceleration spectrum of
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the car body, as expressed in g2 I Hz. It is worth noting that the r.m.s. value of the acceleration is about ag = O.O8g .
Figure 9.22. Squared modulus of the transfer function Haa(Z2,Z0 ;f)
Figure 9.23. Spectrum of the acceleration of the car body
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Discrete systems
9.4. Forced vibrations of Duffing's oscillator As shown in Chapter 5 subsection 5.2.1, the frequency of the free vibrations of nonlinear oscillators depends on the vibration amplitude, in contrast to the linear case. Therefore, it is also interesting to investigate the properties of response of nonlinear oscillators to an external harmonic force. In particular, the existence and uniqueness of steady periodic regimes of response may be questioned. Further, it is also of major interest to investigate the properties of such periodic responses depending on the frequency and magnitude of the excitation. As general analytical methods to deal with such problems are lacking, the discussion will be restricted here to the specific example of Duffing's oscillator, which is classically used to illustrate the major differences between the forced responses of linear and nonlinear oscillators. The equation to be discussed is written in the following dimensionless form:
In the next subsections, we identify first the existence of periodic solutions by using an analytical approximation, which is based on a simple physical consideration of energy. In this manner, important properties concerning nonlinear resonances can be disclosed. Then, by having recourse to numerical simulations, not only the results of the analytical approximation will be confirmed, but new types of non-periodical solutions will also be seen. Such solutions are highly sensitive to the values of the coefficients of equation [9.28] and to the initial conditions, so that the dynamical state of the system can become practically unpredictable, after a certain time, though the system remains basically deterministic in nature. Such dynamical behaviour is said to be chaotic. Since the early sixties, a very large amount of literature has been devoted to chaotic dynamics in nonlinear systems of various kinds. As an introduction to this quite fascinating topic, the interested reader is referred in particular to [BER 84], [THO 86], [MOO 87], [RAS 90], [ARG 94]. 9.4.1. Periodic solutions and nonlinear resonances 9.4.1.1. Ritz Galerkin method Having in mind the object of identifying the possible occurrence of periodical solutions in equation [9.28], we use a variant of the method originally proposed by Ritz to obtain approximate solutions of variational problems. Returning to the problem introduced in Chapter 3 subsection 3.2.3, which consists of finding a function y0(x) which produces a stationary value of the functional [3.12]:
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The Ritz method consists of assuming that the solution can be approximated by a linear superposition of a finite set of suitably chosen base functions ysk(x):
Now, substituting y 0 ( x ) into [9.29], the problem is conveniently reduced by determining the N coefficients qk which make [9.29] stationary:
Since the qk are linearly independent variables, in accordance with [3.6], they are given by the following set of algebraic equations, which may be linear, or not:
The reader interested in the mathematical aspects of the method is referred to [SAG 61]. It presents marked analogies to the modal projection method already introduced in a different context in chapter 7, and also with the technique of least squares curve fitting, which was outlined in Chapter 8 subsection 8.2.3.7. Returning to the present problem, the approximate solution is assumed to be of the form:
The major difference with the Ritz method is that here Q is also considered as an unknown, since the assumed period of the actual solution is not known a priori. Substituting [9.32] into equation [9.28], we get:
In the linear case, it is possible to identify immediately the coefficients q1 and q2, in a unique way. Of course, the solution thus found coincides with the steady forced response of the harmonic oscillator. However, in a nonlinear case, we must
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Discrete systems
proceed otherwise. The basic idea is that for periodic solutions to occur, it is necessary that energy is balanced in such a way that the work of the internal forces is opposite to that of the external force, at least when calculated over one cycle, as detailed in the next subsection. 9.4.1.2. Relationship between pulsation and amplitude of the oscillatory response The key to working out the energy balance of the assumed periodic motion is to pre-multiply the equation [9.33] by the base functions sin(/2r) and cos(/2r), then to integrate this expression over the assumed period 2n/Q, which is still unknown. In carrying out such an integration we take advantage of the orthogonality of the base functions. Such an analytical technique is known as the Ritz Galerkin method. This may be interpreted as writing down an equilibrium balance of all the forces present in equation [9.28], for a virtual displacement expressed in terms of judiciously chosen base functions. The detailed calculation presented here is taken from [ARG 91]. Proceeding term by term, the following intermediate results are produced:
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Further, it can be shown that:
Whence the following algebraic system:
Now, the mathematical way to solve the nonlinear system [9.34], is to define the new variables p and F, as follows:
Indeed, the system [9.34] can be thus re-written as a linear system, which allows q1 and q2 to be expressed in terms of F. After some elementary algebra we obtain:
From equation [9.35] which defines p, it is now possible to relate the pulsation to the magnitude of the vibration: where The relation [9.37] can be re-written as the algebraic equation:
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Discrete systems
As F is proportional to p, equation [9.38] is cubic with respect to p and quadratic with respect to A. Therefore, it is convenient to express A in terms of p:
Since A = Q2 is necessarily a positive quantity, it is found that the magnitude of the vibration is necessarily bounded by the following condition:
This condition may be expressed as:
Incidentally, it can be noted that in the linear case, we already know that at resonance we have:
and the above inequality is automatically verified, since it reduces to g2 > 0. As p is necessarily positive, the condition [9.41] can be transformed into:
By calculating the roots of the expression [9.43] equated to zero, it is an easy task to show that the required inequality holds so far as:
The internal coefficients a,n of the oscillator being fixed, the nonlinear resonance is defined by the condition of maximum amplitude:
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which in turn determines the frequency of the nonlinear resonance by using [9.39]. After some elementary algebra the following result is obtained:
Figure 9.24. Limitation of the vibration amplitude due to the nonlinearity
Qualitatively, the major results arising from the formulas [9.45] and [9.46] may be summarized as follows: 1.
The magnitude of the nonlinear resonant response can be substantially less than in the linear case, as shown in Figure 9.24, where the ratio of p ( n ) to p(0) is plotted versus n. It can be noted that the nonlinear magnitude tends asymptotically to the linear one if n tends to zero, as expected.
2.
The nonlinear resonance frequency varies with the amplitude of the excitation.
In the present example, Qres is found to increase with a, a result which is not surprising since the effect of the nonlinearity present in equation [9.28] is to increase the stiffness of the oscillator, provided n > 0, as already evidenced in Chapter 5, subsection 5.2.1.
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9.4.1.3. Nonlinear resonance peak
Figure 9.25. Peak of nonlinear resonance
It is also of interest to investigate how the amplitude of the nonlinear response behaves when the pulsation Q of the harmonic excitation is continuously varied, sweeping through the resonant range. A typical result is shown in Figure 9.25, which is shaped as a "curved resonance peak". Excitation being maintained at a fixed amplitude (a = 1 ), the response is a decreasing function of damping, as in the linear case. Then, in contrast to the linear case, the peak is bent, here to the right because n > 0. The practical consequences of such a feature will be further discussed in the next subsection. 9.4.1.4. Hysteresis effect The hysteresis effect related to a bent peak of resonance is best evidenced when the nonlinear oscillator is excited by a sinusoidal signal, of which the frequency is progressively varied through the resonance peak. The results obtained in terms of time-histories are illustrated in Figures 9.29 and 9.30. They can be conveniently explained starting from the plots of Figure 9.26, in which the arrows indicate the direction of frequency sweeping. 1.
In the upper plot, frequency is progressively increased, at a sufficiently slow rate for observing at each frequency a steady vibration regime. Starting from a frequency below the resonance frequency, the magnitude of the response is
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increasing monotonically up to the top of the peak, (point A in the figure). When frequency is further increased, even by a very tiny increment, the magnitude of the vibration jumps down abruptly from point A to point B.
Figure 9.26. Sweeping of the resonance peak
In the bottom plot, excitation frequency is progressively decreased starting from a value higher than the resonance frequency. Now, the magnitude of the vibration increases monotonically up to the point C, where the slope of the resonance curve goes to infinity. When the frequency is further decreased, the magnitude of the vibration jumps abruptly from point C to point D.
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Therefore, inside a certain frequency range bracketing the nonlinear resonance, the magnitude of the response to a given excitation is found to be dependent on the previous history of the excitation signal, eventually by a large amount. 9.4.2. Numerical simulations and chaotic vibrations The numerical integration of Duffing's equation using an explicit algorithm, such as the method of the central differences, is quite straightforward. The main problem is the choice of an appropriate value of the time-step which has to be sufficiently less than the critical value [5.68] - which was derived in Chapter 5 in the framework of linear oscillators - in order to cope properly with the stiffening effect of the nonlinearity investigated here. 9.4.2.1. Periodic motions
Figure 9.27. Phase portraits of periodic solutions
When using numerical simulations, the first step is to check the existence of periodic responses, such as those predicted by the approximated analytical solutions of subsection 9.4.1. Then the various responses of the oscillator to sinusoidal forces
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of fixed frequency and variable amplitude can be computed. The results referring to the eventual steady states of the response are suitably visualized as phase portraits. Jndeed, if motion is periodic, the phase portrait reduces to a closed curve, as soon as the length of the simulated time-history is sufficient to be in a steady regime of response. Periodicity of the motion can also be checked by performing a spectral analysis of the response, the spectral signature of a periodic signal being a sequence of lines in harmonic progression, as seen in Chapter 8.
Figure 9.28. A typical sample of time-history of periodic motion (here amplitude of the displacement -full line - was magnified by a factor 100 to adapt the scale of the figure to the magnitude of the excitation force (dashed line))
A few phase portraits of periodic motions are shown in Figure 9.27. As indicated in the figures, damping ratio is fairly large g = 0.1, to save computing time, the coefficient of the cubic term is also fairly large n = 1, to produce a sufficiently strong nonlinearity and finally the excitation frequency is the same as that of the linear resonance, for contrasting the behaviour of the linear and the nonlinear oscillators. From such plots, it is found that motions are indeed periodic, coinciding with the period of excitation, in agreement with the analytical calculation made above. Furthermore, it is found that the shape of the phase cycle is more complicated when excitation amplitude is increased. The secondary loops which become apparent in the phase portrait correspond to secondary oscillations within the global cycle at the driving frequency, as seen in the time-history of Figure 9.28.
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Discrete systems
Figure 9.29. Response to a sine excitation swept from the low to the high frequencies
Figure 9.30. Response to a sine excitation swept from the high to the low frequencies
Finally, the hysteresis effect is also evidenced in the time-histories shown in Figures 9.29 and 9.30.
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9.4.2.2. Chaotic motions
Figure 9.31. Ueda 's oscillator: phase portrait £0 = 3.0 ; £0 = 4-0
The periodic motions described above are not the only ones possible. Indeed, from a systematic study in which the parameters of the oscillator and of the harmonic excitation are varied, other solutions can be obtained, comprising periodic motions with periods distinct from that of the excitation, arid even motions which are not periodic. Moreover, such non periodic motions are found to be highly sensitive to the initial conditions. Such features have been studied in a systematic way by Ueda [UED 80]. Suffice to quote here a particular case of this study referring to the equation:
The equation is provided successively with slightly distinct initial conditions:
The phase portrait resulting from a numerical simulation with the first set of initial conditions is shown in Figure 9.31. On such a plot, the contrast with periodic motion is immediately apparent. Indeed, the phase path is no more a closed cycle and as time progresses it is found to fill progressively a whole domain of the phase plane.
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Discrete systems
Figure 9.32. Ueda 's oscillator: chaotic time-histories
Figure 9.33. Ueda's oscillator: spectrum of a chaotic time-history
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On the other hand, sensitivity of the chaotic motions to initial conditions is illustrated in Figure 9.32, which superposes two time-histories differing from each other only by the initial conditions: £0 = 3.0; £0 = 4.0 (full line) and £0 = 3.05; £0 = 4.05 (dashed line). It can be immediately recognized that separation between the two motions increases rapidly as time elapses. From a practical point of view, this implies that knowledge of the equation of motion of such a deterministic system is not sufficient to be able to forecast the dynamical state of the system at a given time. Indeed, the parameters of a real system cannot be controlled to the degree of accuracy which would be required to obtain a predictable dynamical behaviour. Such motions are thus termed chaotic, and the kind of chaos met in such nonlinear systems is called deterministic chaos, to draw a clear distinction from the dynamic systems which are governed by random processes. Finally, it is worthwhile to check the non-periodicity of the chaotic solutions by performing a spectral analysis of the time-history. Indeed, the spectral signature of chaotic signals is a continuous spectrum, as shown in Figure 9.33.
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Appendices
Appendix 1: Vector spaces Al.l. Definition of a vector space Quantities X,Y,.., called vectors, belong to a vector space (V) (also called linear space), if the following axioms are satisfied:
3. There exists a unique null vector O such that there exists a unique vector noted such that: Moreover, to every vector X and every number a, there corresponds a vector aX such that:
A1.2. Dependence and independence of vectors X is said to be to be a linear combination of N vectors {^j,,^,...^}, if there exist numbers (a,, «2, ...aN} such that:
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Discrete systems
be a set of N vectors and consider the linear combination:
H —i
If equation [A1.2] has no other solution than a, = a2 =,... = aN = 0, the vectors are independent from each other, otherwise they form a dependent set. A13. Dimension of a vector space 1. The vector space is N-dimensional if at least one set of N independent vectors may be defined, whereas any set of N+1 vectors is found to be a dependent set. 2. If for every positive integer N, we can find N independent vectors, the vector space is co -dimensional. 3. A set of vectors {0>,,0>2,...^} is said to be to be a basis for the vector space, if: [ is an independent set; b) every vector of the space can be written as a linear combination of the type [Al.l]; c) If the vector space is W-dimensional, any set of Af independent vectors forms a basis. A1.4. Metric spaces The objective here is to endow vector spaces with a metric structure enabling us to measure the length of a vector. Al.4.1. Definition Any set of elements {jc,^,...| (not necessarily vectors) is known as a metric space if to each pair of elements x,y, corresponds a real number d(x,y), called metric, or distance function of the space, which satisfies the following conditions:
Appendices
385
Al.4.2. Convergence Let {JCB },« = 1,2,... be a sequence defined in a metric space and x be an element of this space; it is said to be that xn converges to x if d(xn,x} approaches 0 as n approaches oo . This is formally stated as: lim*,, = jc, or xn -> x , if to each £ > 0, there exists a positive integer TV such that
n—»o>
Al.4.3. Cauchy sequence and complete space {xn},« = 1,2,... is known as a Cauchy sequence if to each 8 > 0 there exists an inteeer N such that: whenever; Therefore, if (xn} is a Cauchy sequence, we can write that:
Of course, a converging sequence is a Cauchy sequence, but the converse is also true only if the space is complete, that is if the limit x also belongs to the space.
A1.5. Normed vector spaces A vector space is normed if to each vector X there corresponds a real value function \X\, known as the norm of X, which fulfils the following conditions:
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Discrete systems
Of course, it can be easily checked that a normed vector space is automatically a metric space with the distance function:
Relation [A1.7] defines the natural metric generated by the norm. Incidentally, a normed vector space which is complete in its natural metric is called a Banach space.
A1.6. Scalar, or inner, product The objective here is to endow vector spaces not only with a metric structure, but also with the notion of angle between vectors (in particular to answer the question whether or not two vectors can be said to be parallel, or perpendicular to each other). Al.6.1. Definition of the scalar product A scalar product <X,Y> on a complex vector space is a complex-valued function of ordered pairs of vectors, which fulfils the conditions:
where the asterisk marks the complex conjugation. From the conditions [A1.8], it can be inferred that:
and the Schwarz triangular inequality:
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387
Vectors are said to be parallel to each other whenever:
Vectors are said to be orthogonal (or perpendicular) to each other if:
whereas
A.l.6.2. Orthogonal projection of a vector It may be shown without difficulty that an orthogonal set of vectors is independent. On the other hand, the projection of a vector X on the line generated by a vector Y is given by:
X and Y being a pair of known vectors, the following decomposition of X is unique:
where, of course, it is assumed that
Al.6.3. Hilbert spaces On the other hand, <X,X > norm of the scalar product space:
can
be interpreted as a norm, known as the natural
A vector space provided with a scalar product and which is complete in the metric of the natural norm is called a Hilbert space.
388
Discrete systems
A1.7. Hilbert space of square integrable functions The set of all complex-valued functions f(t) defined on the domain a
(a,b).
Appendices
Appendix 2: Vector and multiple products of vectors The vector (or external) product of Q by r is noted:
The Cartesian components of the vector product are written as:
They can be also written in tensor form, as expressed in indicial notation:
The permutation symbol eijk is given by: , whenever at least two indices are repeated. is an even permutation. t is an odd permutation. The notation [A2.4] is convenient for proving the following relations:
389
390
Discrete systems
Appendix 3: Euler's angles and kinetic energy of rotating bodies Euler's angles
Figure A3.1. The rotations defining the Eulerian angles
Starting from the frame Oxyz, the counter clockwise (direct) rotation by an angle
A3.1. Application: kinetic energy of a rigid body of revolution Let us consider a rigid body of revolution, rotating about it centre-of-mass G, which is assumed to be fixed. Let (fl) be an inertial Cartesian frame with axes Gx,Gy,Gz and (#') an accelerated frame, rotating with the body. Moreover, the axes Gx',Gy',Gz'of the Cartesian coordinates in (#') are assumed to be a set of principal axes of inertia of the body, Gz' being taken as the axis of symmetry. The coefficient of inertia about Gz' is denoted J and / = J/2, about the two other axes. The kinetic energy of the rotating body is thus (cf. Chapter 2):
Appendices
391
Figure A3.2. Inertia! and corotating frames linked to the centre-of-mass of the rigid body In (/?), the angular speed of the body is described by the components 0,
Collecting these partial results, the following components of £2 in (/?') are found:
392
Discrete systems
The kinetic energy of the rotating body is written as:
A3.2. Rotating body of revolution vibrating about the spin axis
Figure A3.3. Rocking modes of the spinning axis of the body of revolution
Let us consider a body of revolution spinning at constant angular speed Q about the axis of revolution Gz of the inertial frame. Gz vibrates with a small amplitude according to rocking modes about G, in the planes Gzy and Gxz. The corresponding angular displacements are denoted \fsx,\i/y, as shown in Figure A3.3. In that case, kinetic energy takes a particularly interesting form, which is derived here according the calculation made in [GIB 88]. Indeed, provided the Euler's angle 0is small, the third formula [A3.2] may be simplified as:
Furthermore, it is easily shown that:
Appendices
From the formulas [A3.5], it is also inferred that:
Substituting relations [A3.6] in [A2.3], the kinetic energy becomes:
393
394
Discrete systems
Appendix 4: Hermitian and symmetrical matrices A4.1. Eigenvalues Let us consider a N x N Hermitian matrix; there is no difficulty in proving that the eigenvalues of [H] are real:
Transposition of the above relation gives:
As the norm ||[^]| is necessarily positive, this implies that An = A*n; hence AB is real. A4.2. Orthogonal basis of the eigenvectors Whenever all the eigenvalues /ln are distinct, orthogonality of the eigenvectors is very easy to prove. Indeed, by using again the invariance of [//] when transposed, we can write that:
or, in matrix notation:
then necessarily Orthogonality implies automatically the linear independence of vectors. Indeed, let us assume, to the contrary, that the eigenvector Vk is a linear combination of the others. Vk = ^ a , V ( . Then, we would verify that: .-j. /.
Appendices
395
a result which is simply impossible, since the norm of Vk is strictly positive. As a consequence, to the N distinct eigenvalues N eigenvectors correspond, forming an orthogonal set. If N is the dimension of the Hilbert space on which the hermitian matrix is operating, the eigenvectors form an orthogonal basis of the space. We analyse next the case of multiple eigenvalues. It suffices to consider the case of multiplicity 2. Let W1be an eigenvector solution connected to \ = /^ = A:
Furthermore, it verifies the orthogonality condition:
Let W2 be a vector perpendicular to Wt and to the whole set of Vn. It may be used to complete the orthogonal basis W, W2, V3, ..., V N } . The aim is to prove that W2 is another eigenvector related to A.. With this object in mind, let us define the following vector:
Orthogonality of the basis implies:
whence we deduce that:
As the same reasoning may be applied to ffj, the desired result is reached, accordine to which:
Finally, as W} and W2 are two distinct eigenvectors connected to the same eigenvalue A,, any linear combination of the two vectors is also an eigenvector connected to A . Thus, it may be concluded that to an eigenvalue of multiplicity 2, a whole plane of eigenvectors corresponds, defined by the linear form a1 W2 + a2 W2. Moreover, this plane is perpendicular to the set
396
Discrete systems
connected to all the other eigenvalues. W1,W2 form an orthogonal basis of such a plane and it is pointed out that there exists an infinity of other such bases; they can be obtained simply by transforming W1, W2 through a plane rotation of arbitrary angle. Clearly, the reasoning presented here in the particular case of a twofold eigenvalue may be extended to the case of any multiplicity m. The linear form a1 W2 + ... + am Wm generates an hyperplane of dimension m. To conclude this paragraph, it can thus be stated that it is always possible to build an orthogonal basis, and even an orthonormed basis, if desired, in the TV-dimensional Hilbert space of dimension N, on which [H] is operating, which is formed by an orthogonal set of N eigenvectors. A4.3. Transformation of coordinates It is first noted that:
The eigenvector matrix [0], columns of which are N independent and normed eigenvectors, enables us to transform [H] into a similar matrix which is diagonal through the following formula: and 0 otherwise Transformation [A4.13] is known as a similarity transformation. Two matrices related to each other by a similarity transformation are said to be to be similar to each other. They have the same eigenvalues. Similarity can also be interpreted as a change of generalized coordinates:
whence the following formulas of coordinates transformation are deduced:
A4.4. Sign of the symmetric matrices A symmetric matrix [S] being given, we define the quadratic form:
[S] is said to be positively definite whenever a is a definite and positive scalar, for all [?]*[()]. [S] is said to be negatively definite whenever a is a definite and negative scalar, for all [?]*[0]. In the frame of the eigenvectors of [S], [A4.16] becomes:
As the q\ can take any arbitrary value, it is immediately inferred that: •
The necessary and sufficient condition for [S] to be positive definite is that all the eigenvalues are positive.
•
The necessary and sufficient condition for [S] to be negative definite is that all the eigenvalues are negative.
Another possibility is that the eigenvalues are of the same sign, except at least one of them, which is zero. In such cases, the matrices are said to be non negative, or non positive, according to the sign of the non zero eigenvalues. Finally, when the eigenvalues are not all of the same sign, the sign of the matrix is not defined.
T89
Discrete systems
Appendix 5: Grout's and Choleski's decomposition of a matrix A5.1. LU-decomposition of a non symmetric matrix Any regular NxN
matrix [A] can be written as the product of a right-triangular,
or upper triangular, matrix [U] and a left-triangular, or low-triangular matrix [Z,], based on the Gaussian method of elimination:
For instance if TV = 4, [A5.1] is expressed as:
The decomposition is not unique. A5.1.1. Croat's decomposition The linear equations involved in the system [A5.2] are:
We have thus at our disposal N2 linear equations involving N(N +1) unknowns. Such a system is clearly underdetermined and the N superfluous unknowns can be specified in an arbitrary way. One possible and rather natural choice is to let Lit -1. The remaining unknowns are then easily computed according to the following sequence of equations, which is repeated for every j:
Appendices
399
When applying such a procedure, it is verified that the numerical values of the elements Lik,Ukj, were made available before being used to compute those of Ly^Ujj. On the other hand, it can be noted that the elements 4, of the matrix to be transformed are used once only, hence the coefficients L^U^ of the transformed matrix can be conveniently stored at the place of Aij . Obviously, it is also useless to store the coefficients Lii = 1. Finally, it is realized that whenever i = j, the equations [A5.5] and [A5.6] become identical to each other, except that [A5.6] involves a division by Us . As a consequence, the suitable choice of the pivot for the column can be made after having computed every possible candidate. Division of all coefficients is then performed at once. A5.1.2. Computation of the determinant and of the reciprocal matrix The LU-decomposition of \A\ enables us to compute det[/l] very easily, since:
Now, the determinant of a triangular matrix is obtained by multiplying all the diagonal elements with each other. Thus, det[Z,] = 1 and relation [A5.7] becomes:
However, to produce the proper sign of the determinant, it is necessary to take into account which parity arises from the row permutations involved in the optimised choice of the pivot. Actual computation of the reciprocal matrix [A]~l can be carried out in the most suitable way by solving successively N forced systems of the type:
where [^/l
stands for they-th column of [A]-1 and [Ij] denotes they-th column
of the identity matrix. More generally, any forced problem is solved according to two steps carried out successively, both of them involving the solution of a triangular system. The principle of the method is as follows:
Letting [£/][A"] = [Y], the two following systems are solved successively:
400
Discrete systems
A5.2. Choleski's decomposition of a symmetric matrix Here it is shown that a regular and symmetric matrix [S], can be written as:
where [U] is again an upper triangular matrix. Indeed, starting from the decomposition [A5.1], [5] is written as:
[U] is obtained by normalizing the rows of [U] by the corresponding diagonal element Uu. Of course, as [S] is assumed to be regular, none is zero. [D] is the diagonal matrix gathering the diagonal terms of [U]. Equations [A5.5] and [A.5.6] mav be now written as:
whenever whenever It is worth noting that the permutation of the indices / and j under the summation sign, when shifting from one formulation of Uti to the other, is indifferent since it is correctly reflected in the summation index. Thus, the following decomposition is reached:
Appendices
401
Finally, to find the result [A5.12] it suffices to redefine [U] (or equivalently Ul)as:
402
Discrete systems
Appendix 6: Some basic notions about distributions A6.1. Functions described as functionals Lety =f(x) be a real function of the real variable x. We are used to considering/ as a mapping of the line Ox of the real numbers on itself. Indeed, it specifies a correspondence between a real number x, called the independent variable, and another number y, known as the value of the function, of the type y{ = /(*,), which can be put in a more concrete form by defining a table of correspondence of the type:
Now, there exists another point of view for describing/ which is proving to be more general and often preferable for applications to physics than the classical one, recalled just above. Indeed, a physical device is required in any case for measuring the values taken by a function which describes a physical quantity. However, one has to realize that instrumentation is unable to establish any point-to-point correspondence between measured quantities, but only a functional correspondence which express the action on the measurement system. In mathematical terms, one is led to consider the action of fix) on a set of auxiliary functions (p(x], which are selected judiciously, as detailed in subsection A6.2. Assuming that / is defined for any real value of x, action is defined as:
/is then described according to the following table of correspondence:
A6.2. Vector subspace of test functions For (p to be a suitable test function defined in the real domain, it is required that: 1.
Appendices 2.
403
q> may be differentiated up to any arbitrary order. It may be easily checked that the set of all possible {(p} are forming a vector
subset (or linear manifold) (;£>) . Moreover, whenever cp € (j£>) , the derivatives to any order of (p belong also to (£>).
EXAMPLE. otherwise Furthermore, a restraining criterion for convergence in (;£>) is stated, according to which a sequence {
All the (pj vanish outside a common finite interval, independently ofy;
2.
q)j and their derivatives of any order k, converge uniformly to cp and to the corresponding derivatives, wheny approaches infinity.
NOTE. - Extension to several variables More generally, to accommodate the description of multi-variable functions as functional, the test functions can be defined over a space of real numbers with dimension n, denoted (R"} , where n is, of course, a positive integer. For the sake of brevity, they will be denoted (p (r), where the position vector r is defined in a ndimensional Euclidean space. A6.3. Distributions A distribution T is a continuous and linear functional, built on the vector space (D). This means that to any test function (p e (£>), Tlets correspond a scalar (real, or complex, number) denoted T,,, in accordance with the following properties:
The distributions also form a vector space, denoted (D'). In particular, the addition of two distributions, and the product of a distribution by a scalar are defined by:
404
Discrete systems
A6.4. Regular distributions Let f ( r ) be a function of n real variables, which is locally integrable. It gives rise to a distribution through the scalar product:
It can be proved that this linear functional is continuous and that two locally integrable functions, which are not equal, almost everywhere generate distinct distributions. A6.5. Singular distributions All the distributions which are not generated by locally integrable functions are said to be singular. This is the case in particular of the Dirac distribution 8, which is defined as follows:
It may be noted that if 8 is viewed as a symbolic function, it is zero almost everywhere; consequently, this is also true as the product 6(r)(p(r) is concerned. Nevertheless, whether or not a distribution is regular, it is convenient to write it by using the scalar product [A6.6]. In the case of a singular distribution, the integration involved in the scalar product [A6.6] is purely symbolic in nature. The truly operative scalar product is that specified by the relation of definition of the singular distribution. For instance, in the particular case of 8 (f)), relation [A6.7] can thus be
A6.6. Operations with distributions 1.
Translation
Appendices
405
EXAMPLE. 2.
Scale expansion (or similarity)
EXAMPLE. 3. Multiplication by a locally integrablefunction
For {/,g#>} to be a distribution, it is necessary that g(p be indefinitely derivable. EXAMPLE 1. - Multiplication by a constant
EXAMPLE 2. - Multiplication of S (;c) by a function Whenever/can be differentiated up to any order, we have:
4.
Product of two distributions
The important point here is to realize that it is not always possible to define the product of two distributions, as illustrated in the following example. EXAMPLE. — 8(r} as expressed in Cartesian coordinates One may write:
with specific the meaning:
As a matter of fact, the product involved in [A6.16] is to be understood as a tensor product, according to which each 1-dimensional 8 is acting on a specific direction of the Euclidean space, distinct from the others. In consequence, a more suitable notation for such a product would be:
406
Discrete systems
where n is the symbol of intersection. Indeed, a relation such as S2(x) = S(x)6(x)
would be totally meaningless, if
taken in the sense of an ordinary product. However, adopting the tensor product defined in [A6.17], the following meaningful result is inferred:
5.
Differentiation of distributions
Of course, relation [A6.19] is obtained as the result of an integration by parts. It provides the suitable means to define the derivative of any distribution, regular or not. Furthermore, relation [A6.19] can be iterated to any arbitrary number of differentiations:
where
EXAMPLE. - Heaviside 's step distribution and derivatives of it As the definition of the Heaviside's step distribution, we have:
Obviously K(x], as taken in the sense of a function, cannot be differentiated at x = 0. To define the derivative of U(x) in the sense of distributions, we consider the following scalar product:
which is integrated by parts, to provide the meaningful result:
from which it is immediately inferred that in terms of distributions, we have:
Appendices
407
Repeating once again the same calculation, concerning the second derivative, we obtain:
Then we may infer that:
Accordingly S(x)can also be differentiated to produce another singular distribution known as Dirac 's dipole, which is formally defined as:
6.
Change of variable: Dirac's distribution of a function Let us start from the definition:
Now, we consider the neighbourhood of a value x0 such as /(;c0) = 0 and /'(;c0)*0. Since f(x) varies monotonically in the vicinity of x0, it is possible to perform the following change of variable:
At this step, the pertinent point is that only the vicinity of u = 0 is of importance in the integral. Indeed, based on the definition [A6.7] of S (u), we have:
whenever /'(.x 0 )>0, integration is carried out in the direction of the increasing values of u. Of course, it is performed in the reverse direction if/'(* 0 )<0.
408
Discrete systems
Considering now a function provided with only n simple zeros xk, & = 1,2,...«, from the result [A6.24], the following more general relation is deduced:
EXAMPLES. -
To conclude on the subject, it is worth stressing that when at least some zeros of /(jc) are multiple, £[/(*)] becomes meaningless.
Appendices
409
Appendix 7: Laplace transformation A7.1. Laplace transform A7.1.1. Differentiation and integration theorems The differentiation theorem for the first derivative is expressed as:
which can be proved without any difficulty. Indeed, it suffices to integrate by parts the relation [7.19] which is used to define the Laplace transformation, see chapter 7. In the same way, concerning the second derivative, we obtain:
Finally, these results can be extended to a derivative of any order n as:
This relation is the keystone of the Laplace transformation method for solving linear differential equations with constant coefficients. Indeed, it enables us to replace a derivation by a simple algebraic operation. Integration theorem is an immediate corollary of [A7.1], which states that:
A7.2. Shift theorem: time delay
A7.3. Shift theorem: advanced time
410
Discrete systems
Proof of shift theorems follows easily from the relation of definition [7.19]. A7.4. Laplace transform of a periodic function on the interval
where X(t)ll(i)
stands for the periodic function X(t], restricted on the interval
[0,+oo). To prove relation [A7.7] use is made of the shift theorem [A7.6] together with the series expansion of the delay factor (l - e~sT \ .
A7.5. Convolution in the time domain
A7.6. Ordinary product in the time domain
A7.7. Theorem of the initial value
A7.8. Theorem of the final value
Appendices
411
A7.9. Calculation of the inverse Laplace transform The inverse Laplace transformation is defined by the integral:
where the straight line passing through the points a + ico and a - ico leaves on the left side all the singular points of X(s)est. Such a line is known as a Bromwich contour. When all the singularities of the Laplace transform are poles, the inverse Laplace transform can be easily calculated as a direct application of the residue theorem. A pole s0 of X(s) makes X(SQ) singular, but in the vicinity of s0 X(s) remains uniform and s0 is a regular point of the reciprocal function 1 / X(s). For instance, simple pole
has two simple poles
,
and
has a
and a two-fold pole
When all the singularities are poles, the Bromwich contour can be conveniently replaced by any closed contour without any multiple point, which encircles the poles. The value of the integral [A7.12] is then obtained by using Cauchy's integral formulas, which may be written as follows: If SQ is a simple pole and (£*) a contour enclosing s0, we have:
If .SQ is a «-tuple pole and (
In each case, the value of the integral is equal to the residue of the pole. Then, the inverse Laplace transform is obtained by summing all the residues of X(s)est.
EXAMPLE 1.F (s) has two poles s, = -A + ico, s2 = -A - ico. The residues of F(s)sst are:
412
Discrete systems
Summing the two residues, we obtain:
EXAMPLE 2. The residue related to the two-fold pole is:
whence
When singularities other than poles occur, calculation of [A7.13] is more difficult In practice, tables of transform or specialized software of formal calculus may be used. The following table provides a few results very often met in dynamical problems.
Appendices
Function or distribution
Laplace transform
413
414
Discrete systems
Appendix 8: Modal computation by an inverse iteration method A8.1. The eigenmode problem related to the natural modes of vibration The problem to be solved is initially formulated as:
[/'(/I)] is a symmetric N x N matrix. The natural pulsations are deduced from the roots of the characteristic polynomialdet[P(A)1 = 0, which is of degree N. For convenience, the presentation is restricted first to the special case of N distinct positive roots. Then, we discuss how to reduce the general problem to this special case. A8.2. N simple positive roots of the characteristic equation The method used to compute the modal frequencies and the modal shapes of the system [A8.1] involves the following successive steps. /.
Isolation of the individual roots Using Choleski's decomposition (cf. Appendix A5.2]), the matrix is written as:
Then we obtain:
Since all the diagonal elements of the triangular matrix [U] are equal to 1, the result [A8.3] reduces to:
Letting A be a value selected inside the interval [A n ,A n+1 ], equation [A8.4] shows that n elements of the diagonal matrix [D(A)] are necessarily negative. This property provides us with a convenient tool to define an interval Akn enclosing one and only one eigenvalue /l n .
Appendices
415
A8.2.1. Iterative computation of the modal shape It is first realized that the desired modal shape [pn ] cannot be computed by solving directly the homogeneous system [A8.1], since det[/)(An)] = 0. However, starting from a given value such as in * An, it is possible to compute the solution of the forced system:
With the purpose of computing [pn ], it will be shown that a suitable choice for
where
are the limits of the interval enclosing the isolated eigenvalue
It can be proved that solution of the system [A8.5], when iterated over the indexy converges necessarily to [
It is rather easy to understand such a statement without entering into the details of a formal proof. Indeed, In, as produced by formula [A8.6], is sufficiently close to An for the response to any forced problem to be dominated by the nearly resonant mode, provided the forcing vector is able to excite it, as stated by condition [A8.7]. Thus [£>]] is already closer to [
where £ is the accepted relative error. It is important to note that from one iteration to the next, the solution has to be normed, because it stands for a nearly resonant response of an undamped system.
416
Discrete systems
It is also clear that the special form selected concerning the right member of the system [A8.5] is quite relevant to increase the rate of convergence of the iterative process, since the forcing vector is thus made nearly orthogonal (with respect to the mass matrix) to all the modal shapes except [
A8.2.2. Computation of the modal frequency The last iterate \
A8.3. Extension to the non standard cases A8 J.I. Null modal frequencies Null eigenfrequencies occur as a consequence of the singularity of the stiffness matrix. The singularity of [K] can be conveniently removed by using the so called spectral shift method, according to which the initial system [A8.1] is replaced by the following shifted system:
H is the parameter of the spectral shift. It can be easily checked that the modal shapes of the systems [A8.1] and [A8.10] are the same and that the eigenvalues are transformed by the following spectral shift:
A8.3.1J. Repeated eigenvalues Occurrence of several modes at exactly the same natural frequency is a consequence of some kind of symmetry taking place in the physical system to be analysed. The most expeditious way for solving a problem of this type is to break the symmetry by perturbing the initial system; for instance some DOF can be blocked. However, to perform properly
Appendices
417
such a change, some physical judgement is required in order to avoid uncontrolled effects of the perturbation. To conclude on the subject of this appendix, it is worth to stress that many other algorithms than the inverse method presented here are available to compute the modal properties of the conservative mechanical systems. Let us mention in particular the Lanczos algorithm, which is broadly implemented in the finite element programs devoted to structural analyses; on this point the interested reader may be referred, in particular, to [LAN 56], [ARG 91] and [GER 97].
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[LAN 54] LANDAU, L. & LIFCHITZ, E., Physique Theorique, Tome I: Mecanique, Editions de la Paix, Moscow, Russia, 1954. [LAN 76] LANDAU, L. & LIFSHITZ, E., Mechanics, Pergamon Press, 1976. [LAN 86] LANDAU, L. & LIFSHITZ E., Theory of Elasticity, Pergamon Press, 1986. [MAG 66] MAGNUS, W. & WINKLER, S., Hill's Equation, John Wiley & Sons Ltd, 1966. [MAR 87] MARPLE, S.L., Digital Spectral Analysis with Applications, Prentice Hall, 1987. [MAX 96] MAX, J. & LACOUME, J.C., Methodes et techniques de traitement du signal et applications aux mesures physiques, 5th edition, Masson, France, 1996.
422
Discrete systems
[MEI 67] MEIROVITCH, L., Analytical Methods in Vibration, Macmillan, 1967. [MEI 70] MEIROVITCH, L., Methods of Analytical Dynamics, McGraw-Hill, 1970. [MOO 87] MOON, F.C., Chaotic Vibrations, John Wiley and Sons Inc., 1987. [NAY 73] NAYFEH, A., Perturbation Methods, John Wiley and Sons Inc., 1973. [NAY 79] NAYFEH, A. H. & MOOK, D.T., Nonlinear Oscillations, John Wiley and Sons Inc., 1979. [NIT 89] NITSCHE, J.C.C., Introduction University Press, UK, 1989.
to Minimal
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[OSS 86] OSSERMAN, R., A Survey of Minimal Surfaces, Dover, 1986. [PAP 62] PAPOULIS, A., The Fourier Integral and its Applications, McGraw-Hill, 1962. [PRE 89] PRESS, W.H., FLANNERY, B.P., VETTERLING, T, Numerical Recipes, Cambridge University Press, UK, 1989. [PRE 90] PREUMONT, A., Vibrations aleatoires et analyse spectrale, Presses polytechniques et universitaires romandes, 1990. [RAS 90] RASBAND, S.N., Chaotic Dynamics of Nonlinear Systems, John Wiley & Sons, 1990. [ROC 71] ROCARD, Y., Dynamique generate des vibrations, 4th edition, Masson, France, 1971. [SAG 61] SAGAN, H., Boundary and Eigenvalue Problems in Mathematical Physics, John Wiley & Sons, 1961. [SAG 89] SAGAN, H., Boundary and Eigenvalue Problems in Mathematical Physics, Dover, 1989. [SCH 50] SCHWARTZ, L., Theorie des distributions, Vol. I, Hermann, 1950. [STA 70] STAKGOLD, I., Boundary Value Problems of Mathematical Physics, 5th edition, Macmillan, 1970. [TIM 70] TIMOSHENKO, S.P. & GOODIER, J.N., Theory of Elasticity, 3rd edition, McGraw-Hill, 1970. [THO 86] THOMPSON, J.M.T. & STEWART, H.B., Non Linear Dynamics and Chaos, John Wiley and Sons Inc., 1986. [UED 80] UEDA, Y., Steady Motions Exhibited by Duffmg's Equation: A Picture Book of Regular and Chaotic Motions, in Nex Approaches to Nonlinear Problems in Dynamics, Holmes (ed), p. 311-322, SIAM Philadelphia, USA, 1980. [WEA 90] WEAVER, W., TIMOSHENKO, S., YOUNG, H., Vibration Problems in Engineering, 5th edition, John Wiley and Sons Inc., 1990.
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[WHI 44] WHITTAKER, E.T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Dover, 1944. [YOU 01] YOUNG, W.C., BUDYNAS, R.G., ROARK, R.J., Roark's Formulas for Stress and Strain, McGraw-Hill, 2001. [ZEM 65] ZEMANIAN, A., Distribution Theory and Transform Analysis, McGrawHill, 1965.
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Index
3-D rotation 28,49 accelerated frame 35, 46, 127, 131 acceleration 21, 35 action 80 of a force 38 of the Lagrangian 103-04 of a moment 27, 38 of a physical quantity 22 adjoint matrix 51 admissible 40 variation 95 aliasing 333 analogue-to-digital converter 328 analytical mechanics 39, 40 angular displacement 29, 38, 48 frequency 143 momentum 26-27, 34, 38, 45, 106, 129 velocity 29, 44 antinode 363 antiresonance 342 Argand plot 345 articulated bar 160, 220,223-24 articulation 88 attracting point 152,164 attractor 168 autocorrelation function 322, 324 autocorrelation spectrum 323, 359 autonomous oscillator 139
backward mode 230 balance of energy 58 of forces 24 of moments 30 basis transformation 197 beats 263 Bernoulli, J. 59,81 bifurcation 93,220 Brahe,T. 27 broadbanded spectrum 348 buckling 153 instability 220 modes 220 Campbell diagram 231 catenoid 99 Cauchy's sequence 51 central force 22 centre-of-mass 23-24, 27-28 centrifugal force 29, 31, 37, 121 centripetal acceleration 37, 50 force 121 reaction 20 chain of oscillators 209, 226, 276, 309, 354-56 chaotic motion 368,379 response 342 characteristic equation 142, 146, 199 Choleski's decomposition 195,201
426
Discrete systems
chopped signal 296 coding dynamics 329 coefficient of autocorrelation 324 of cross-correlation 324 coherence 358 function 322 complex amplitude 147, 199 conjugation 50 modal shapes 229 vector 147 conditional stability 188 configuration space 4, 84 conjugate component 42 force 55 conservative force 56, 58 oscillator 142, 144 system 58, 103-04, 193 constitutive law 16 constrained function 112, 115 Lagrangian 118 system 81, 111, 116, 117, 192 constraint 8 condition 9, 19,60, 112 reaction 19,31,38,59,60, 119 contact force 17, 19 continuous medium 4 system 2 convolution product 268, 296 theorem 313 coordinate system 4-5 transformation 5 Coriolis acceleration 37, 50 force 29,34,37, 112, 129, 135 corotating frame 105 correlation function 322 coupled pendulums 55, 195
system 202,272 coupling 192 term 55 critical damping 150 coefficient 142 load 93 for buckling 221-22 spin velocity 130 time-step 170 value 87,219 cross-correlation function 323 spectrum 323, 360 cross-spectrum 321 cut-off frequency 302, 317, 331 cyclic variable 103,106 d'Alembert's principle 24, 59 damped oscillator 151, 164, 262, 343 damping 58,266,296 force 348 matrix 281,286 ratio 148,353 decay time 258 definite negative matrix 52 positive matrix 52, 194 positive quantity 43 degree of freedom 4, 6, 9 delay theorem 255 deterministic chaos 381 transient 240 differential condition 11, 116 constraint 116 digital processing 328 Dirac 125 comb 328,330 distribution 183-84 impulse 314 pulse 244, 257, 305 direction cosine 49 discrete Fourier transform 298, 335, 358
Index
signal 349 spectrum 318 system 2, 10 discretization 328 disk 11,32,234 dispersion equation 211 dispersive wave 211 dissipation 58,280 dissipative force 58 divergence 153 double pendulum 40, 65-66 Duffmg's oscillator 164, 175, 342, 368 dynamic equation 20 law 16 dynamical equilibrium 20, 24 instability 151 Earth's gravity 23 eigenpulsation 143 eigenvalue 51, 52, 84 equation 84 eigenvector 45, 51, 197 Einstein's convention 6 elastic buckling 87 contact 18 foundation 67-68 matrix 219 stop 179 electromagnetic shaker 359 energy 38 functional 147, 198 spectrum 315-16 envelope 265, 320, 350 equation of motion 20 equivalent inertia 357 equivalent stiffness 17 error signal 326 Euclidean space 3,41,51 Euler acceleration 37, 50 angles 45, 236, 237 force 37, 129 -Lagrange equations 64, 103, 116
427
explicit algorithm 169,179,376 extended Lagrangian 79, 80, 109 external force 16,23,25,57,240 loading 240, 250 potential 58, 64 extremum value 84 fast Fourier transformation 336 filtering 302, 349 finite difference 169 finite rotation 44-45 first integral 106, 154 first variation 82 fixed frame 21 point 187 flexibility 353 fluid-elastic force 58 fluid-structure system 58 fly-wheel 234 force 16,20 of inertia 24 of viscous damping 142 forced dynamical problem 196 motion 106, 140,252 oscillator 196 vibration 239 et al forward mode 230 Fourier integral 298 series 298-300 transformation 298,313,342^3 free flight 18, 186 function 115 modes of rigid body 219 motion 103, 140 system 117 vibration 252,297 frequency 143 functional 81,94-95 of action 102 notation 41,51 vector 242,243
428
Discrete systems
space 41 fundamental frequency 300,319 Galilean frame 20 transformation 21 gap 18 gear 216-17 generalized 192 coordinates 4, 39 displacements 5, 55, 101 forces 29,38,41,42,48,60,89 inertial forces 48 mass 147 momentum 67, 106 reaction 119 stiffness 73, 147 velocity 6 Gibbs oscillations 301 gradient transformation matrix 6 gravity 23 Green's function 241, 267, 273, 296 gyroscopic approximation 34 coefficient 132 coupling 193,228,229 force 112, 132, 136 matrix 228 Hamilton 39 principle 79, 101 harmonic component 300 form 199 oscillation 265 oscillator 139,249,253,342 sequence 321 vibration 147 wave 213 Heaviside step 243,251,257 Hermitian matrix 50 Hilbert's space 51 Hill's equation 177 holonomic conditions 9, 224 constraints 8, 112, 115, 119
homogeneous system 199 hydraulic jack 359 hyperstatic equilibrium 122 system 126 hysteresis effect 374-75, 378 identity matrix 51 image 247 domain 248,250 impact 18, 19 constraint 186 force 181 signal 317 stiffness coefficient 19 impacting oscillator 179 implicit algorithm 169, 286 impulse 241, 244 impulsive load 246 loading 250 response 241,250,273 indifferent equilibrium 85, 195,218 inertia 45 coefficient 140 force 29,30-31,43, 141 matrix 46, 53 inertial frame 20,35,46, 127, 130 of reference 43, 44 principle of Galileo 20 initial condition 9, 141, 143,240,250, 267, 379 initial displacement 251 inner product 41 in-phase mode 197, 198, 205, 226 input signal 249, 350, 358 instability 85 instantaneous rotation 44, 49 centre of 14, 61 integrable function 241 integral of action 242 interaction forces 22, 57 interference 214 internal force 16, 19, 20, 24
Index
internal Lagrangian 64, 65, 67 invariant of motion 27 inverse Fourier transformation 313 inverse Laplace transform 240,241 transformation 240, 247, 248, 252 inverse matrix 52, 195 inversion of a matrix 195-96 inverted pendulums 86 iterative solution 210 Jacobian determinant 6 Jacobian matrix 6, 42 Kepler, J. 27 kinematical constraint 2, 8, 19,40 continuity 9 kinetic energy 43, 46, 48 Lagrange, J. L. 31, 39 Lagrange's equations 59,61,64-67, 101, 192 function 64 multiplier 8, 111, 113, 114,216 Lagrangian 64, 85, 103, 104 Lanczos, C. 81 Laplace transform 240,247,253 transformation 240, 247 variable 248 law of action and reaction 22 gravitation 23 inertia 20 least action 79 mechanical behaviour motion 21 least squares method 327 lever arm 71 limit cycle 168,263 linear displacement 38 elasticity 17 momentum 21, 24, 38 operator 51
429
superposition 10, 209 system 194-95,358 velocity 29,44,46 linearized system 194 load factor 90,219,228 local extremum 83 maximum 84 minimum 84, 85 longitudinal wave 211,278 low-pass filter 302, 333, 358 mass density 2 matrix 43,65, 192, 193 -normalisation 148,200 -point 2 -spring system 25, 108, 128, 133, 140,274 material body 2 law 16,20,59 Mathieu's equation 177 MATLAB 210,301 matrix of constraints 215 rotation 49 viscous damping 109 matrix inversion 196 Maupertuis 79 Maxwell's theorem 271, 273 mean power 297 value 300 of acceleration 189 measured signal 358 mechanical action 22 energy 18,41,57, 107, 142, 143 equilibrium 24, 59 interaction 22 oscillator 140 system 2,20, 140 method of residues 249, 265 mixed formulation 216 mixed modal shape 217-18
430
Discrete systems
modal 192 analysis 146, 152-53, 196 basis 202,203,271 damping 281 ratios 281 mass 147,354 matrix 202 projection 200,271,281 reaction 216 series 271 shape 147, 199,200,354 stiffness 147 testing 352 vibration 213 mode 192-93 of elastic buckling 193,221 moment 16,26 of a force 26 of inertia 29,32 monochromatic excitation 348 motion invariant 56, 67, 103, 104 mutual energy 47 independence 9,10 interaction 23 kinetic energy 131 natural coordinates 203, 272 frequency 147,296 mode of vibration 146-47,199 norm 51,243 pulsation 143, 199 nearly periodical function 321 negative damping 151, 167 stiffness 130, 144 Newmark's algorithm 188,285 Newton, I. 20 equations 67 laws 20 Newtonian mechanics 3, 20 nonconservative force 58, 167 oscillator 148 system 57, 109
nonholonomic conditions 14 constraints 8, 11, 112, 116 non inertial coordinates 35 nonlinear oscillator 154 resonance 368, 374, 376 stiffness 30 system 57, 342 non-symmetrical mass matrix 48 numerical damping 175,188 numerical instability 172 orthogonal basis 52 orthonormal basis 52,202,300 matrix 49, 52 transformation 52 out-of-phase mode 198,205,226 output signal 249, 350, 358 overcritical damping 150 oscillator 152 value 92 overdamped oscillator 150 parametric instability 177 pendulum 176-77 resonance 177 particle 2 pendulum 3, 8, 134, 163, 219 inverted 86 perfect constraints 20,59,60,61 periodic function 298,318 motion 57, 377 oscillation 145 signal 297 perturbing oscillator 361 phase angle 143 curve 345 function 322, 354, 355, 356 portrait 144, 151-52,376,377, 379
Index shift 210
speed 211,310 trajectory 144, 152 Plancherel 313 -Parseval theorem 351 plane of phase 144 plane rotation 28-29, 36, 48 point-to-point mapping 5 polar coefficient of inertia 53 polar coordinate 28-29 pole 250 position vector 4 potential of constraint 119 energy 53-54, 140 sink 162 power 61 spectral density 318 prescribed motion 108,127,130 prestressed matrix 219 oscillator 140
state 130,219 string 156 principal axis of inertia 45, 53 coefficients of inertia 53 frame of inertia 53 principle of least action 39, 81 of relativity 21 of virtual work 60, 61, 63, 80, 101 proportional damping 281,290 pseudo-pulsation 153 pulsation 143 quadratic form 43, 46, 52, 65, 83, 141, 194 quarter-car model 283, 286 quasi-inertial domain 353, 354 range 348,351,354 quasi-static approximation 25, 194 response 289
431
radius vector 10, 26 random process 349 random signal 349 Rayleigh quotient 147,200 reaction 32 of constraint 40 force 112 rectangular pulse 253, 269, 315-16, 318,325
reduced damping 148 reference frame 5, 20-21, 46, 127 transformation 5,1,29 regular distribution 241 regular matrix 195 relative acceleration 37, 50 displacement 127 energy 47 kinetic energy 131 mass matrix 47 motion 7 velocity 7, 36 relaxation oscillations 168-69 repelling force 85 repulsing point 152,162 residue 254 theorem 252 resonance peak 344, 347^18, 352, 354, 374 resonant frequency 348 range 347^18,374 response 266 restoring force 17, 85 rheonomic constraint 9, 112, 127 rigid bar 87 body 2,9, 10,20,200 connection 61 rod 206 string 19 wheel 12,31 rising time 258, 301 Ritz-Galerkin method 370 r.m.s. value 351, 360 road profile 365, 366
432
Discrete systems
rocking mode 124,234-35 moment 71 motion 70 rolling contact 11 without sliding 14, 33, 61 rotating frame 48-49, 227 mass-spring 129, 133 rigid body 44 system 227 rotation matrix 135 mode 208 saddle point 83, 85 sample function 349 sampling 349 period 328 scalar product 41, 51 quantity 41 Schwartz theorem 324 scleronomic condition 215 constraints 9, 216 second variation 83 secular term 302 seismic excitation 108, 128 shake 127 self-adjoint matrix 51 self-sustained oscillation 167 self-sustaining oscillator 167 sensitivity to initial conditions 164 separatrix 154, 162 Shannon criterion 358 sampling theorem 298, 331 shift theorem 322 shock 19 absorber 342, 365 duration 181 signal 358 signal energy 243,314,315,348
similar matrix 52, 172 similarity 52 simple pendulum 30, 55, 57, 140 singular distribution 125, 244 sink of potential 155 sinusoidal modulation 296 skew-symmetric matrix 132 slenderness ratio 69, 71, 73, 207 sliding with friction 61 small motion 66 soap film 100 space metrics 51 spectral analysis 298,314,377 density of energy 315 domain 352 line 319 overlap 331-33 spectrum 349 analyser 358 spherical pendulum 105 spring 16-17 square integrable function 243 stable equilibrium 141,192,194 standing wave 213, 279-80 static equilibrium 24-25, 66, 141 instability 87, 153 law 16 stability 85,221 statics 24-25, 194-95 stationary action 79 point 82, 84 value 95 statistical average 349, 358 steady regime 263, 266, 359 stiff impact 179 stiffened articulation 90 stiffness coefficient 17, 140-41 force 141 matrix 65,87, 193 stress-wave 312 string tension 30-31 strong law of action and reaction 23
Index
subcritical damping 150,250 value 92 superposition principle 269 support reaction 124 symmetrical matrix 45, 50, 53, 194, 201,202 tension 30,31,40 tensioned string 155 tensor 46 test function 242 threshold of instability 195 time -differential condition 12 -history 3, 142 -scale 149 -step 170-71 torque 26, 34, 34-35 transfer box 270 function 240, 249, 251, 268, 270, 273, 343, 348, 352, 358 matrix 241,269,281 transient excitation 241,252 transient regime 263 translation mode 208 transport energy 46 force 112 kinetic energy 131 motion 8 velocity 7,21,36,278 transposed form 6 transverse wave 227 travelling wave 211, 309 triangular matrix 195-96 decomposition of a 195 truncated Fourier series 301-02,326 transform 334 function 330 signal 328 sine 260 truncation 301
unconditionally stable 189 unconstrained system 8, 112, 119 uncoupling 192, 196 unilateral condition 19,61 constraint 19 contact 11 support 124 unit eigenvector 51 upside-down pendulums 220 Van der Pol oscillator 167,263 variable of Laplace 239 variation 86 operator 80,95 variational calculus 81 derivative 97 equation 98 formulation 59 principle 39, 80 vector space 3, 51 velocity 6 vibration 66, 192 absorber 360-61 isolation 352 mode 352, 360 sensor 356-57 virtual displacement 60, 81, 84 variation 95 work 59-61 viscous damper 19 damping 18, 164 coefficient 140 model 280 wave 210,298 wavelength 365 wheel 33-34, 135,234 whirling mode 193,227 white spectrum 317 Wiener-Khintchine theorem 322 windowing 334 work 20,38,41
433
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Series synopsis: Modelling of Mechanical Systems
This important new series of four volumes has been written by an eminent authority in the field, and is suitable for practitioners, academics and students alike. The series presents the general methods that provide a unified framework to model mathematically mechanical systems of interest to the engineer, analysing the response of these systems. Whilst focusing on linear problems, non-linear problems are also included. A significant feature of the books is the treatment of the mathematical techniques used to perform analytical studies and numerical simulations. Francois Axisa writes in a clear, concise and accessible style, often from a practical rather than a theoretical standpoint, providing a thorough insight into the physical and mathematical aspects of modelling, and emphasizing essential aspects of the subject of most use in mechanical engineering.
Volume I. Discrete Systems The study of discrete systems constitutes the cornerstone of all mechanical systems, linear or non-linear. This volume covers the formulation of the equations of motion and a systematic study of free and forced vibration of discrete systems. The book explores analysis techniques using generalized coordinates and kinematical conditions, Hamilton's principle and Lagrange equations, linear algebra in Ndimensional linear spaces, and the orthogonal basis of natural modes of vibration of conservative systems. Also included are the Laplace transform and forced responses of linear dynamical systems, the Fourier transform and spectral analysis of excitation and response for deterministic signals. Volume II. Continuous Systems The focus in volume II is on flexible solids and in particular on the elastodynamic models of the basic structures, which are commonly encountered in mechanical engineering, such as straight beams and plates, arches and thin shells. The author formulates equilibrium equations which are then solved by extending to
436
Discrete systems
the continuous case the methods already described in Volume I. The study of the beams provides a good opportunity to introduce, with minimum mathematical effort, the basic principles of two distinct methods for discretizing continuous systems, namely finite elements and modal models. Both models are of paramount importance in practice, and can be of great benefit when used to predict response. Volume HI. Fluid-structure Interaction The dynamical behaviour of fluid-structure coupled systems are covered in volume III. The study is restricted to structures and fluids that vibrate about a static and stable state of equilibrium. Mathematical aspects of modelling are the same as those described in progression in Volumes I and II. However, the presence of a fluid vibrating in contact with a structure does enlarge considerably the physics involved in fluid-structure coupled problems. The book describes in detail aspects such as the inertial effects of dense and poorly compressible fluids, free surface effects and coupling with gravity waves, acoustic waves and acousto-mechanic coupling, energy dissipation via radiation, and viscous friction in the fluid. Volume IV. Flow-induced Vibration Flow-induced vibration of structures coupled to a fluid in permanent flow gives rise to problems which differ in several important aspects from that of the systems the terms, m studied in Volumes mes. I-III. In practical terms, In practical the major difference is that, if even a tiny fraction of kinetic energy stored in the flow is transferred to the vibrating structure, strong vibration can be excited, often leading to catastrophic failures. Modelling these problems raises major difficulties because it requires a detailed description of the steady and the fluctuating part of highly turbulent flows. As is often the case in fluid mechanics, the state of the art in this field still relies to a large extent on semi-empirical knowledge. This book is a comprehensive introduction to the mechanisms leading to flow-induced vibration. Topics covered are fluid elastic instabilities in internal flows such as pipes and rotating machines, random vibration induced by turbulence and related acoustic noise, instabilities of profiled structures and bluff bodies in external flow.
