8 Continuous or distributed parameter systems 8.1 Introduction The representation of physical systems by means of discrete models in which properties like inertia, stiffness and damping are localized and identified with different elements like masses, springs and dampers is often very convenient and leads to satisfactory results in many circumstances In reality, however, one has to deal with, say, aircraft structures, pipelines, car bodies, various types of buildings, etc.; in other words, with structures which generally comprise cables, rods, beams, plates and shells, all of which are neither rigid nor massless Every material portion of the system may possess mass, stiffness and damping properties at the same time, and these properties may vary from point to point In these cases, whenever possible, one can resort to continuous models (we already encountered some examples of such models in Chapters and 5), where the displacement is a continuous function of both space and time and we are in presence of an infinite number of degrees of freedom Distributed parameter models are based on another idealization: the continuous elastic medium which, for its part, leads to a fundamental insight into the nature of mechanical vibrations: the so-called wave-mode duality In everyday engineering problems we often tend to think of vibrations in terms of modes and of, say, acoustical phenomena in terms of waves As a matter of fact, this distinction is somehow fictitious because we are just considering the same physical phenomena: that is, the propagation of a localized disturbance (mechanical in our case) which ‘spreads’ from one part of a medium into other parts of the same medium or into a different medium We are referring here to the propagation of mechanical waves which, as a matter of fact, represents a deeper level of explanation for mechanical vibrations Normal modes of vibration are in fact particular motions of a system (for which the system is, let’s say, particularly well suited) which ensue because of its finite physical dimensions in space and hence because of the presence of boundaries In other words, the superposition of travelling waves reflected back and forth from the physical boundaries of the medium ultimately result in the appearance of standing waves which, in turn, represent the normal modes of vibration of our system Copyright © 2003 Taylor & Francis Group LLC This book is primarily concerned with aspects of mechanical vibrations that not need a detailed discussion of the ‘wave approach’ and for this reason the subject of wave propagation and motion in elastic solids will only be touched on briefly whenever needed in the course of the discussion For our purposes, the importance of the considerations above is mostly a matter of principle, but it must be pointed out that the wave-mode duality has also significant implications in all fields of engineering where the interest lies in the study of interactions between sound waves and solid structures In essence, distributed parameter systems present many conceptual analogies with MDOF systems However, some important differences of mathematical nature will be clear from the outset First of all, the motion of these systems is governed by partial differential equations and, second, these equations must be supplemented by an appropriate number of boundary conditions Moreover, boundary conditions are just as important as the differential equations themselves and constitute a fundamental part of the problem; for short, one often says that the motion of a continuous system is governed by boundary value problems These problems are in general much more difficult to solve than their discrete counterpart (where boundary conditions enter only indirectly because they are implicitly included into the system’s matrices) and hence, for continuous systems, exact solutions are available only for a limited number of cases We will consider some of these cases and provide the exact solutions Nonetheless, for more complex systems we have to resort to approximate solutions which, in turn, are often obtained through spatial discretization and in the end—despite the fact that the techniques of analysis may be highly sophisticated (e.g the finite-element method)—bring us back to finite-degreeof-freedom systems On a more theoretical basis, we pass from the finite-dimensional vector spaces of the discrete case to infinite-dimensional vector spaces More specifically, we have to deal with Hilbert spaces, i.e infinite-dimensional vector spaces where an inner product has been defined and are complete with respect to the norm defined by means of the above inner product The reader can find in Chapter some introductory considerations on these theoretical aspects because it is important to be aware of the fact that the conceptual analogies with the discrete case rest on the fact that, broadly speaking, Hilbert spaces are the ‘natural generalization’ of the usual finitedimensional vector spaces At the beginning of this chapter we will consider a simple system—the flexible string—in some detail in order to gain some insight on the fundamental aspects of wave propagation, natural frequencies and modes of a continuous system which, with the appropriate modifications, can be taken as representative for other types of distributed parameter systems The beam in bending vibration will be considered next, before turning our attention to more general aspects of the differential eigenvalue problem and to the analysis of some two-dimensional systems Copyright © 2003 Taylor & Francis Group LLC 8.2 The flexible string in transverse motion The flexible string under tension—with some basic assumptions that will be considered soon—is the simplest model of continuous system where mass and elasticity are distributed over its whole extent From the discussion developed in preceding chapters we can argue that, in principle, we could (and indeed we do) arrive at a satisfactory description of its motion by considering it as a linear array of oscillators where masses are lumped at discrete points and elasticity is introduced by means of massless springs connecting the masses The greater the number of degrees of freedom, the better the approximation Furthermore, we could also work out an asymptotic solution by increasing the number of masses indefinitely and by letting their mutual distance tend to zero However, we adopt a different method of attack which contains more physical insight than the mathematical expedient of the limiting procedure; that is, we not consider the motion of each one of the individual infinite number of points of the string but only concern ourselves with the shape of the string as a whole So, let us consider a string of indefinite length (we want to avoid for the moment a discussion of the boundary conditions) which is stretched by a tension of T0 newtons and whose undisturbed position coincides with the xaxis Let us further assume that the displacements of each point of the string are wholly transverse in a direction parallel to the y-axis It follows that the string motion is specified by a ‘shape’ function y(x, t) where x and t play the role of independent variables: for a fixed time t1 the graph of the function y(x, t1) depicts the shape of the string at that instant while the graph of the function Fig 8.1 Displaced differential element of taut string Copyright © 2003 Taylor & Francis Group LLC fixed—represents the motion of the point located at x1 as time passes, i.e the time history of the particle at x1 Qualitatively, if we assume that any variation of the tension due to the transverse displacement of the string is negligible, we can apply Newton’s second law to the differential element of the string shown in Fig 8.1 to get the equation of motion in the vertical direction as (8.1) where µ (kg/m) is the mass per unit length of the string, which for the present we assume uniform throughout the length of the string If we further assume that the slope of the string is everywhere small—i.e or, in other words, the inclination angle θ is always small compared with one radian, we can write so that eq (8.1) becomes (8.2a) which we can choose to write in the form (8.2b) where has the dimensions of a velocity and, in the approximations above, is independent of both x and t Note that the small slopes approximation (or small-amplitude approximation) expressed by allows us to neglect all quantities of second and higher order in ∂y/∂x Only in this circumstance the net horizontal force on the differential element of string is zero and we can assume so that the displacement of each point is perpendicular to the x-axis and the tension T0 remains unchanged in passing from point x to point x+dx When the small slopes assumption ceases to be valid, the resulting differential equation is nonlinear Copyright © 2003 Taylor & Francis Group LLC Equation (8.2b) is the well-known one-dimensional differential wave equation which, by assigning the appropriate meanings to the quantities involved, represents a broad range of wave phenomena in many branches of physics and engineering (acoustics, electromagnetism, etc.) Obviously, in our case, the motion ensues because the string has been disturbed from its equilibrium position (‘plucked’, for example) at some time The tension then provides the restoring force but inertia delays the immediate return to the equilibrium position by overshooting the rest position Note also that, for the time being, no consideration whatsoever is given to dissipative damping forces and to the effect of stiffness that, although generally negligible, occurs in real strings All books on basic physics show that the general solution of the onedimensional wave equation is of the form (8.3) which is called the d’Alembert solution of the wave equation The functions f and g can be any two arbitrary and independent twice-differentiable functions whose forms depend on how the string has been started into motion, i.e on the initial conditions It is not difficult to see that f(x–ct) represents a shape—or a profile—which moves without distortion in the positive xdirection with velocity c, while g(x+ct) represents a similar wave (with a different shape if ) which moves in the negative x-direction with velocity c; the linearity of the wave equation implies that if both waveforms have a finite spatial duration, they can ‘pass through’ one another and reappear without distortion Most of us will be familiar with these ‘travelling wave’ phenomena from childhood games with ropes 8.2.1 The initial value problem In the case of a string of indefinite length, we can gain some further insight by considering its motion due to some initial disturbance This disturbance is specified by means of the initial conditions, i.e the functions that determine the shape and velocity of the string at t=0 Let these functions be (8.4) From the general solution, at t=0 we have (8.5) Copyright © 2003 Taylor & Francis Group LLC where the primes represent here the derivatives of the functions f and g with respect to their arguments The second of eqs (8.5) can be integrated to give (8.6) where the constant of integration C can be set to zero without loss of generality From the first of eqs (8.5) and (8.6) it follows which establish the initial values of the functions f and g For the variable x with the appropriate arguments to get we replace or, alternatively (8.7a) which, by integration of the second term on the r.h.s., i.e can be written as (8.7b) Equation (8.7b)—which has also been obtained by using transform techniques in Section 2.3—physically represents identical leftward and rightward propagating disturbances containing separate contributions from the displacement and velocity initial conditions Copyright © 2003 Taylor & Francis Group LLC 8.2.2 Sinusoidal waves, energy considerations and the presence of boundaries Sinusoidal waves Out of the infinite variety of functions permitted as solutions by the wave equation (as a matter of fact, any reasonable function of x+ct or x–ct), it should be expected that sinusoidal waveforms deserve particular attention This is because, besides their mathematical simplicity and the fact that many real-world sources of waves are nearly sinusoidal, we can represent as closely as desired any reasonable periodic and non-periodic function by the linear superposition of many sinusoidal functions (Fourier analysis—see Chapter for more details) Mathematically, we can express an ideal sinusoidal wave of unit amplitude travelling along the string as (8.8) where k is the so-called wavenumber: when the quantity kx increases by 2π the corresponding increment in x is the wavelength so that or, equivalently, Note that, in the light of preceding chapters, the symbols may be a bit misleading: here k is not a spring constant and is not an eigenvalue However, these symbols for the wavenumber and the wavelength are so widely used that we adhere to the common usage: the meaning is generally clear from the context but precise statements will be made whenever some ambiguities may arise in the course if the discussion As far as time dependency is concerned, we already know that the period T is related to the frequency v by T=1/v and that the angular frequency ω is given by the fact that the wave moves to the right can be deduced by noting that, as time passes, increasing values of x are required to maintain the phase constant Two ‘snapshots’ of the waveform at times t0 and t0+T look exactly the same; this implies that the wave has travelled a distance in the time interval T so that (8.9) Moreover, it is evident that also in this case the exponential representation (Chapter 1) (8.10) is widely adopted and is often very convenient The wave of eq (8.8) is obtained by taking the imaginary part of eq (8.10) but, as stated in previous Copyright © 2003 Taylor & Francis Group LLC chapters, the real-part convention may be adopted as well, and the difference is irrelevant as long as consistency is maintained The general restriction of small amplitudes translates for harmonic waves into or, in other words, into (8.11) which states that the maximum amplitude must be much smaller than the wavelength One final word here to point out that the velocity of the propagation of the disturbance c must not be confused with the velocity of the individual particles of the string, i.e with ∂y/∂t; as a matter of fact, for a general waveform y(x, t)=f(x–ct), since it follows that the small-amplitude approximation requires (8.12) where it must be understood that the string particles move in a transverse direction while the waveform propagates along the string The word ‘propagation’ itself, as we shall see shortly, implies a transport of energy and momentum Energy considerations From the previous discussion, it is apparent that the kinetic energy in a differential element of string is given by and the kinetic energy in a segment between x1 and x2 is then Copyright © 2003 Taylor & Francis Group LLC which, for small deflections, can be approximated as (8.13) The calculation of the potential energy is a bit more involved because second order terms come into play The string must possess potential energy because some external work would have to be done to give it the deflected shape which, in turn, must locally stretch the string where the wave is present This local stretching, however, must excite longitudinal waves that propagate along the string as well as the transverse waves The coupling between longitudinal and transverse waves is expressed by nonlinear terms in the equation of motion, and precisely these terms is what we want to neglect This difficulty can be circumvented by assuming a negligible Young’s modulus (ideally E=0, i.e a string which is perfectly flexible) In this hypothesis we can consider the change in length of a portion of string of initial length dx: this is so that the potential energy between x1 and x2 is (8.14) because the stretching takes place against a force of tension T0 In the light of eqs (8.13) and (8.14), it is often convenient to speak of kinetic and potential energy densities (8.15) although these definition have a certain degree of arbitrariness because it is often difficult—and sometimes meaningless—to keep track of the location in space and time of a given amount of energy Copyright © 2003 Taylor & Francis Group LLC Two points are worthy of notice at this point: If we consider a general waveform f(x–ct), the kinetic and potential energy densities are given by respectively, and since we see that the two expressions are equal Moreover, we can consider a harmonic wave in the exponential form —where A is just the amplitude which we assume now to be different from unity—and calculate the average kinetic and potential energy densities Let us consider for example the potential energy density: we have from eq (8.15) where the bracket indicates the average over one period If now we resort to the phasor convention of Section 1.3, we get from which it follows that (8.16) By the same token, the reader is invited to calculate the average kinetic energy density, verify that and arrive at the same result by considering a harmonic wave in the form of eq (8.8) The equation of motion (8.2) can be obtained by substituting the kinetic and potential energy densities in eq (3.109) where the Lagrangian density is given by In addition, we may be interested in the flux of energy past a given point x; this rate of energy transfer is just the instantaneous power flow from any piece of the string to its neighbour Mathematically, it is obtained as the product of the vertical component of tension by the transverse velocity of the string at x (Fig 8.1), i.e (8.17) so that a positive value of P (watts) implies power flowing toward the positive Copyright © 2003 Taylor & Francis Group LLC Example 8.2 Let us now consider a uniform clamped-free rod of length L and mass per unit length µ excited by a tip load at the free end, i.e If the rod is at rest before the excitation occurs the first two terms on the r.h.s of eq (8.176) are zero and (8.197) so that eq (8.176) reduces to (8.198) because If we further assume that the explicit form of p(t) is a unit step (Heaviside) function θ(t), i.e we can substitute θ(t) into eq (8.198), perform the integration by noting that and obtain (8.199) Finally the displacement in physical coordinates is obtained as the superposition (8.200) where in the second expression we take into account the explicit form of Also, it is worth pointing out that eq (8.200) is dimensionally correct because, since we have assumed a unit force, the dimensions of w(x, t) are displacement per unit force (i.e m/N) Copyright © 2003 Taylor & Francis Group LLC At this point, it is interesting to note that the system above can be analysed either as: an excitation-free system with a time-dependent boundary condition at x=L, or a forced vibration problem with homogeneous boundary conditions As stated at the beginning of the preceding section, free-vibration problems with nonhomogeneous boundary conditions are often tackled by an integral transform (Laplace or Fourier) approach; however, the modal approach can also be adopted in consideration of the fact that a boundary value problem of type (1) can usually be transformed into a boundary value problem of type (2) (e.g Courant and Hilbert [20] or Mathews and Walker [21]) In general, it appears that in these cases a disadvantage of the modal approach—which is essentially a ‘standing waves solution’—is that the resultant series converges quite slowly and many terms must be included in order to achieve a reasonable accuracy By contrast, depending on how the inverse transformation is carried out, the Laplace transform method allows us the possibility to obtain a solution either in terms of standing waves or in terms of travelling waves (waves being reflected back and forth within the rod) This latter possibility—the travelling wave approach—leads to a solution in the form of a rapidly converging series, thus making this strategy more attractive However, on physical grounds, we may argue that the time scale in which we are interested suggests the type of solution to adopt; in fact, the travelling wave solution converges rapidly when we consider the short-term response of our system whereas, if the long-term response is desired, more and more terms are needed The situation is reversed for the modal solution: as time progresses, the terms corresponding to higher modes die out because of damping and we are left with a series in which, say, only the first two or three terms have a significant contribution We will not consider an integral transform strategy of solution here (the interested reader is referred to Meirovitch [22]) but, using the rod example above, we will show how a problem of type (1) can be transformed in a problem of type (2) Our rod problem can be formulated as a type (1) problem in the following form: (8.201a) (8.201b) Copyright © 2003 Taylor & Francis Group LLC where eq (8.201a) is the homogeneous equation of motion and eq (8.201b) are the boundary conditions Since one boundary condition (the second) is nonhomogeneous we assume the solution of our problem in the form (8.202) where the term —which we can define in compact notation as —is a so-called ‘pseudostatic’ displacement brought about by the boundary motion and v(x, t) is the displacement relative to the support displacement Mathematically, the function ust is chosen in such a way as to make the boundary conditions for v(x, t) homogeneous On physical grounds, the usual assumption made for the choice of ust is that no inertia forces (i.e., no accelerations) are produced by the application of the support motion; hence the name ‘pseudostatic’ For our case, this assumption implies that ust obeys the equation (8.203) from which follows (provided that (8.204) Moreover, given the expression (8.202), the boundary conditions (8.201b) become from which—if we want homogeneous boundary conditions for v(x, t)—it follows (8.205) Enforcing the boundary conditions (8.205) on the solution (8.204) leads to (8.206) Copyright © 2003 Taylor & Francis Group LLC The transformation of the problem (8.201) into a type (2) problem is complete when we determine the nonhomogeneous equation of motion for the relative displacement v(x, t): this is simply accomplished by substituting eq (8.202) into eq (8.201a) and results in (8.207a) where the r.h.s of eq (8.207) has clearly the dimensions of N/m and, for short, can be indicated with the symbol feff (effective force) Equations (8.207a), (8.206) plus the homogeneous boundary conditions (8.207b) constitute our type (2) boundary value problem which fits into the scheme of problems that can be more effectively tackled by the modal approach In this light, we expand v(x, t) in a series of eigenfunctions and calculate the normal coordinates as prescribed in eq (8.182) (note that, from eq (8.206), we get i.e (8.208) Upon substituting the explicit expressions of φj and in eq (8.208), the space integral within braces gives so that eq (8.208) becomes (8.209) Copyright © 2003 Taylor & Francis Group LLC Now, in the problem we are considering, we assumed that p(t) is the Heaviside function θ(t); since (eq (2.67a) or (2.84)) the time integral of eq (8.209) becomes where we take into account the properties of the Dirac’s delta function (eq (2.69)) and the explicit expression of hj The final steps consist of substituting this result in eq (8.209), writing explicitly the series expansion of v(x, t), i.e and putting it all back together into the solution (8.202) which becomes (8.210) This result must be compared with eq (8.200) and it is not difficult to show that they are equal This is due to the fact that the function (π2x/8L) can be expanded in a Fourier series as (the proof is left to the reader) so that—after performing the product in eq (8.200)—the first term is exactly (x/EA), i.e the function (x) of the pseudostatic displacement The advantage of including explicitly the pseudostatic displacement from the outset lies in the more rapid convergence of the series (8.210) as compared to the series (8.200), the pseudostatic displacement representing the average position about which the vibration takes place Example 8.3 In modal testing, we are often concerned with the response of a given system to an impulse loading So, consider the rod of Example 8.2 subjected to a unit impulse applied at x=L at t=0 The response in physical coordinates at x=L is given by eqs (8.185) and reads (8.211) Copyright © 2003 Taylor & Francis Group LLC This result should be hardly surprising because we know from Chapter (eq (5.42)) that the impulse response function is the time derivative of the Heaviside response function So, in this circumstance we could have ignored eq (8.185) by simply noting that the result (8.211) can be obtained by calculating the time derivative of eq (8.200) and by substituting x=L in it On the other hand, the receptance FRF can be obtained from eq (8.190): at x=L this is (8.212) In the light of preceding chapters, we expect that h(L, L, t) and H(L, L, ω) form a Fourier transform pair However, the Fourier transform of eq (8.211) does not exist, but we may note that the Laplace transform of eq (8.211) does exist and is given by where s is the (complex) Laplace operator and can be expressed as Hence, leaving aside mathematical rigour for a moment, we see that we can arrive at eq (8.212) by first taking the Laplace transform of eq (8.211) and then letting This mathematical trick is just for purposes of illustration and it would not be needed if the system had some amount of positive damping; as a matter of fact, this is always the case for real systems whose time response and FRFs (eq (8.211) and (8.212)) not go to infinity when Example 8.4 Consider now the case of a constant force P moving at a constant velocity V along an Euler-Bernoulli beam simply supported at both ends The engineering importance of this case is evident because this example can be used to model a number of common situations, the simplest one being a heavy vehicle travelling across a bridge deck We also make the reasonable assumption that the mass of the vehicle is small in comparison with the beam mass (the bridge deck) and it does not alter appreciably its eigenvalues and eigenfunctions Mathematically, the moving load can be represented as (8.213) and, with reference to eq (8.182), we obtain Copyright © 2003 Taylor & Francis Group LLC so that, assuming the beam at rest at t=0, we get (8.214) where now are the eigenfrequencies of our pinned-pinned Euler-Bernoulli beam A double integration by parts in eq (8.214) leads to (8.215) so that the displacement in physical coordinates is given by (8.216) The solution (8.216) needs further comments First, it is interesting to note that there are a series of values of velocity at which resonance may occur; they are (8.217a) and the time of passage tj at these values of speed is given by (8.217b) so that, calling the fundamental period of vibration of the beam, we have Furthermore, if we try to evaluate the response (8.216) at the critical speeds—i.e when — we run into an indeterminate 0/0 situation However, we can use L’Hospital’s rule and obtain (8.218) which, owing to the finiteness of the time of passage, is a bounded quantity and does not grow indefinitely with time Copyright © 2003 Taylor & Francis Group LLC We will not consider an example of a two-dimensional system (say, a membrane or a plate) but it is evident that, besides the added mathematical complexities, the extension of the modal approach to these systems follows the same line of reasoning Obviously, now the expansion (8.174) must include all modes and hence we must sum on all indexes, i.e the mode indexes and the degeneracy indexes For example, recalling the eigenfunctions of a circular membrane clamped at its outer edge (eqs (8.120) and (8.121)) the expansion in series of eigenmodes reads where the first series involves the Bessel functions of the first kind of order zero (no degeneracy), while the second series involves the Bessel functions of order and the twofold degeneracy (expressed by eq (8.121)) which, in the expansion above, is taken into account by means of the summation index where we defined for convenience and 8.10 Final remarks: alternative forms of FRFs and the introduction of damping For the sake of completeness, two final remarks are needed before closing this chapter The first remark has to with an alternative approach for finding a closed-form solution of the frequency response function of a continuous system In essence, if we assume a harmonic excitation in the form (8.219) we know that there will always exist a steady-state response of our system in the form (8.220) so that, upon substituting eqs (8.219) and (8.220) into the appropriate equation of motion, the exponential terms cancel out and we obtain a linear differential equation in W(x, ω) together with the appropriate boundary conditions Then, if we consider that the FRF is, by definition, the multiplying coefficient of the harmonic solution when the response is measured at x=xm and the excitation is applied at the point x=xk—i.e then we have (8.221) Copyright © 2003 Taylor & Francis Group LLC By this method, the solution is not obtained in the form of a series of eigenfunctions (for example, like eq (8.200)) and one of the advantages is that it can be profitably used in the case of support motion where, in general, a set of orthonormal functions cannot be obtained As an example of this method we can consider the longitudinal vibrations of a vertical rod (see end of Example 8.1) subjected to a support harmonic motion of unit amplitude If, to be consistent with eq (8.220), we call w(x, t) the longitudinal rod displacement, the relevant equation of motion is (8.222a) with the boundary conditions (8.222b) Assuming a solution in the form of eq (8.220) leads to (8.223a) where Moreover, the boundary conditions for W are obtained from eq (8.222b) as (8.223b) It is then easy to show that the solution of the problem (8.223) is (8.224) which becomes unbounded (no damping has been considered) when i.e in correspondence of the eigenvalues of a clamped-free rod The same line of reasoning applies if we reconsider the system of example 8.2 and we assume a harmonic excitation of unit amplitude at the free end x=L Equations (8.222a) and (8.223a) still apply, but the boundary conditions Copyright © 2003 Taylor & Francis Group LLC for W are now (8.225) which must be enforced on the solution to give (8.226) Equation (8.226a) must be compared with the series solution obtained from eq (8.190), i.e explicitly (8.227) where the frequencies ωj are given by the first of eqs (8.49) The fact that eqs (8.226) and (8.227) are the same is not obvious at first sight, but it left to the reader to prove that it is indeed so (Hint: use the orthogonality property of the eigenfunctions φj of the clamped—free rod and calculate the inner product From eq (8.226) the FRF at x=L is obtained as (8.228) which must be compared with the series solution (8.212) The second remark has to with the fact that in none of the preceding sections have we taken into account the effect of energy dissipation However, the inclusion of damping—both in free and forced vibration conditions— leads to results that parallel closely the MDOF case As stated on a number of occasions, damping is difficult to define and the general assumption of viscous damping is mostly a matter of mathematical convenience rather than an effective explanation of the physical phenomenon In this light, if we call w(x, t) the function that represents the displacement of our continuous system, the general equation of motion (8.172) can be written as (8.229) Copyright © 2003 Taylor & Francis Group LLC where C is a linear homogeneous ‘damping’ operator which involves only space derivatives up to the order 2p (Section 8.7.1) Now, if we assume that we already solved the undamped free-vibration problem in terms of eigenvalues j and mass orthonormal eigenfunctions φj, we can follow the modal approach and expand w(x, t) as in eq (8.174) Then, taking the inner product we arrive at (8.230) which, in general, are a set of coupled linear differential equations unless the damping operator is of the ‘proportional’ form (8.231) At this point the parallel with the MDOF case is evident: if eq (8.231) applies, eqs (8.230) become uncoupled and we can define the modal damping ratios ζj by means of (8.232) which is the infinite dimensional counterpart of eq (6.142) Also, it is understood that now the modal FRFs are given by eq (7.31) As for the MDOF case, it may often be more convenient to introduce viscous damping at the modal level, without the need to specify a damping operator In other words, we obtain the uncoupled set of undamped equations first, and only at this stage we introduce the terms the values of ζj being chosen on the basis of experience and/or experimental measurements Alternatively, for a harmonic forcing excitation the model of structural damping can be introduced This is generally done by assuming a damping operator which is proportional to the stiffness operator so that the uncoupled equations read (8.233) where here γ denotes the structural damping factor In the case of general damping, i.e a damping operator which does not allow the uncoupling of the equation of motion, it may be interesting to note that, in principle, we can still adopt the approach described at the beginning of this section In other words, if we want to obtain the frequency Copyright © 2003 Taylor & Francis Group LLC response of our (nonproportionally) damped system, we can assume a harmonic excitation and a harmonic response in the forms of eqs (8.219) and (8.220), substitute them in the equation of motion and arrive at a linear ordinary differential equation for the function W(x, ω) However, this differential equation has constant but complex coefficients and an analytical solution is often impossible to obtain 8.11 Summary and comments This chapter has dealt with the free and forced vibrations of continuous parameter systems As a matter of fact, real-world vibrating systems are systems whose physical properties (mass, stiffness and damping) are continuously distributed although—very often—the modelling scheme one chooses to adopt is a discrete parameter model which lends itself more easily to the computer implementation of the necessary calculations The rationale behind this choice is that continuous systems—which, mathematically speaking, have an infinite number of degrees of freedom—can be considered as the limit of finite DOF systems as the number of degrees of freedom tends to infinity Then, provided that a sufficient number of DOFs is used in the appropriate modelling scheme, the finite DOF model can approximate the continuous system under investigation within an acceptable (and often good or very good) degree of accuracy In this light, it would seem that a specific analysis of continuous systems is not strictly necessary from a practical point of view However, this analysis is of fundamental nature in its own right because it provides physical insight on the nature of the ‘proper modes of vibration’ of a structure (i.e the eigenvectors in the finite DOF representation) which extends over a finite domain of space In fact, these are specific motions of the system which arise from the superposition of travelling waves that propagate back and forth within the domain of space occupied by the structure Ultimately, it is the presence of physical boundaries which is responsible for the onset on these ‘standing waves’ Our analysis starts with the study of one of the simplest continuous systems, the flexible string in transverse motion Under some basic assumptions which are at the basis of the ‘classical’ treatment of the subject, we consider first the topic of transverse waves travelling along a string of infinite length, and only at a later stage (Section 8.3) we turn our attention to strings of finite length By adopting the well-known method of separation of variables, we arrive at the identification of the natural frequencies and proper modes of vibration of the system and obtain a solution of the relevant equation of motion in terms of these quantities It is at this stage of the investigation that we point out the important role played by boundary conditions in the analysis of the vibrating characteristics of continuous systems Copyright © 2003 Taylor & Francis Group LLC Next, Section 8.4 deals with the free longitudinal (axial) and torsional vibrations of rods of finite length and the discussion develops along the same line of reasoning of the flexible string case This is due to the fact that, mathematically speaking, the three cases are formally similar—that is, given the appropriate meaning to the mathematical symbols in each case, the equation of motion is always in the form of a one-dimensional wave equation The same does not apply to the case of flexural vibrations of beams, a type of system whose equation of motion is in the form of a fourth-order differential equation (Section 8.5) Moreover, as far as travelling waves are concerned, the beam is a so-called ‘dispersive’ medium, where this term indicates the fact that waves of different frequency travel at different speed so that a flexural pulse (i.e a given waveshape which is a superposition of a number of different sinusoidal components) will suffer distortion as it propagates along the beam Restricting our attention to the analysis of a so-called ‘Euler-Bernoulli’ beam of finite length, we consider the natural frequencies (eigenvalues) and proper modes (eigenfunctions) of some of the most common configurations encountered in engineering practice, the various configurations differing in the nature of the boundary conditions These are: (1) the pinned-pinned beam, (2) the cantilever beam, (3) the clamped-clamped beam and (4) the free-free beam As expected from the developments of preceding chapters, we point out that the free-free (unrestrained) beam, besides the elastic modes of vibration, shows two rigid-body modes at zero frequency Then, with the above results in mind, Sections 8.5.1 and 8.5.2 deal with three types of complications: the first with the effect of an axial force (as, say, in the case of a prestressed beam) and the second effect of rotatory inertia and shear deformation The situation in which the latter two types are taken into account (one generally speaks of a Timoshenko beam in this case) is of practical importance either when the beam is not sufficiently slender or we want to consider high-order modes of vibration Moreover, the corrections introduced also eliminate the nonphysical result of infinite wave velocity at high frequencies encountered in the case of an Euler-Bernoulli beam Proceeding in order of increasing complexity, Section 8.6 deals with the two-dimensional counterpart of the flexible string—i.e the flexible membrane—and investigates in detail the case of a circular membrane of radius R The discussion then turns to more theoretical aspects of the analysis of continuous systems in general (Sections 8.7, 8.7.1 and 8.7.2), with the scope of focusing the attention on some unifying characteristics of the problem Recalling some important results given in Chapter 2, we note that the ideas developed in Chapters and for finite DOF (discrete) systems can be extended, at the price of additional mathematical complexity, to the case of infinite DOF (continuous) systems The generalized eigenvalue problem becomes now a differential eigenvalue problem where the system’s matrices are replaced by appropriate linear symmetrical operators, the finitedimensional vector space of the discrete case becomes an infinite-dimensional complete linear space with an inner product (a so-called Hilbert space and, Copyright © 2003 Taylor & Francis Group LLC more specifically, the Hilbert space L2(⍀), where ⍀ is an appropriate finite domain of physical space) and the expansion theorem in terms of eigenvectors becomes a series expansion in terms of eigenfunctions whose convergence, in the general case, is understood in the L2-sense These considerations retain their validity even when, for more complex systems, we are not able to obtain a solution in closed form because of the increasing mathematical complexity of the problem As an example of these difficulties, we consider in Section 8.8 the free vibration of thin plates, a type of system which, qualitatively, represents the two-dimensional counterpart of the beam Although it is still possible to obtain a closed-form solution for some types of boundary conditions—and we so for a circular plate clamped at its outer edge and a rectangular plate simply supported on all sides—it is clear that the problem soon becomes impracticable and not amenable to an analytical solution When this is the case, we have to resort to a discrete model with a finite number of degrees of freedom, which makes it possible to obtain an approximate solution and takes us back to the subject of MDOF systems Finally, also providing a number of worked-out examples, Sections 8.9 and 8.9.1 deal with the forced vibration of continuous systems, with particular attention to the modal approach, i.e the strategy that allows the analyst to express the solution of the problem as a series expansion in terms of eigenmodes Clearly, this is not the only method of attack (and sometimes it may not even be the best) but this choice is often preferred in the field of engineering vibrations because: In the study of the response of a vibrating system to a given excitation there is the possibility of including only a limited number of modes and neglect all higher-order modes which not significantly contribute to the response The technique of experimental modal analysis (Chapter 10) allows the experimental measurement of the lowest-order eigenfrequencies and eigenmodes of a given vibrating system and makes it possible to compare experimental and theoretical results (these latter having been obtained, typically, from a finite-element model) References Morse, P.M and Ingard, K.U., Theoretical Acoustics, Princeton University Press, 1986 Elmore, W.C and Heald, M.A., Physics of Waves, Dover, New York, 1985 Nariboli, G.A and McConnell, K.G., Curvature coupling of catenary cable equations, International Journal of Analytical and Experimental Modal Analysis, 3(2), 49–56, 1988 Irvine, M., Cable Structures, Dover, New York, 1992 Kolsky, H., Stress Waves in Solids, Dover, New York, 1963 Copyright © 2003 Taylor & Francis Group LLC Meirovitch, L., Analytical Methods in Vibrations, Macmillan, New York, 1967 Cowper, G.R., The shear coefficient in Timoshenko’s beam theory, ASME Journal of Applied Mechanics, 33, 335–340, 1966 Graff, K.F., Wave Motion in Elastic Solids, Dover, New York, 1991 Meirovitch, L., Principles and Techniques of Vibrations, Prentice Hall, Englewood Cliffs, NJ, 1997 10 Abramowitz, M and Stegun, I., Handbook of Mathematical Functions, Dover, New York, 1965 11 Page, C.H., Physical Mathematics, D van Nostrand, Princeton, NJ, 1955 12 Bazant, Z.P and Cedolin L., Stability of Structures, Oxford University Press, 1991 13 Jahnke, E., Emde, F and Losch, F., Tables of Higher Functions, McGraw-Hill, New York, 1960 14 Anon., Tables of the Bessel Functions of the First Kind (Orders to 135), Harvard University Press, Cambridge, Mass., 1947 15 Leissa, A.W., Vibration of Plates, NASA SP-160, 1969 16 Gerardin, M and Rixen, D., Mechanical Vibrations: Theory and Applications to Structural Dynamics, John Wiley, New York, 1994 17 Timoshenko, S and Woinowsky-Krieger, S., Theory of Plates and Shells, McGrawHill, New York, 1959 18 Mansfield, E.H., The Bending and Stretching of Plates, Pergamon Press, 1964 19 Humar, J.L., Dynamics of Structures, Prentice Hall, Englewood Cliffs, NJ, 1990 20 Courant, R and Hilbert, D., Methods of Mathematical Physics, Vol 1, Interscience, New York, 1961 21 Mathews, J and Walker, R.L., Mathematical Methods of Physics, 2nd edn, Addison-Wesley, Reading, Mass., 1970 22 Meirovitch, L., Elements of Vibration Analysis, McGraw-Hill, New York, 2nd edn, 1986 Copyright © 2003 Taylor & Francis Group LLC ... the finite-element method)—bring us back to finite-degreeof-freedom systems On a more theoretical basis, we pass from the finite-dimensional vector spaces of the discrete case to infinite-dimensional... correction is more important that the rotatory inertia correction These considerations are of general nature and retain their validity for types of boundary conditions other than the pinned-pinned... the forces on these elements For instance, if the string has a non-negligible mass m attached at x=0, the boundary condition reads (8.25) Copyright © 2003 Taylor & Francis Group LLC or, say, for