The eigenfuncions are written (8.154) where the constant A—which, a priori, can depend on both n and m—can be fixed by means of normalization. Different boundary conditions lead to more complicated calculations: for example, if our plate is simply supported at r=R the boundary conditions to be imposed on the solution (8.150) are, from eq (8.147a) at r=R, and in polar coordinates the bending moment M r is written explicitly Things are even worse for a completely free plate; in fact, in this case the boundary conditions read (eq (8.147c)) where M r is as above and the transverse shearing force Q r and the twisting moment M r θ are given by 8.8.2 Rectangular plates Due to its importance in many fields of applied engineering, let us now consider a uniform rectangular plate extending in the domain and The equation of motion of free vibrations is again (8.145) which, assuming a harmonic time dependence becomes eq (8.148a) for the function of the space variables As in the preceding case, this equation can be written as Copyright © 2003 Taylor & Francis Group LLC and we can express its solution as Obviously, it is now convenient to adopt a system of rectangular coordinates so that the Laplacian and biharmonic operators are written explicitly as The function u 1 satisfies the equation by separating the space variables and looking for a solution in the form we arrive at the two equations (8.155) where Equations (8.155) have the solutions so that (8.156) The equation satisfied by the function u 2 is implying that its solution can be obtained from eq (8.156) by replacing the trigonometric functions by hyperbolic functions. This means that we can write the complete solution u(x, y) as (8.157) Copyright © 2003 Taylor & Francis Group LLC where the values of the constants A j and parameters α and ß depend on the boundary conditions. The simplest case is when all edges are simply supported and we must enforce the boundary conditions (8.158a) (8.158b) where the conditions on the second derivative are obtained (eq (8.147a)) by noting that, in rectangular coordinates, the bending moments M x and M y are given by (8.159) By inserting the conditions of eqs (8.158a and b) into the solution (8.157) we obtain that only A 1 is different from zero so that (8.160) In addition we get the two characteristic equations (8.161) which imply and with n, m=1, 2, 3,…and (8.162) The corresponding eigenfunctions are (8.163a) where it is evident that and it is easy to see that the Copyright © 2003 Taylor & Francis Group LLC requirement yields the following mass-orthonormal eigenfunctions: (8.163b) The first few modes of a plate simply supported on all edges are shown in Fig. 8.8. It is interesting to note at this point that trying to enforce different boundary conditions—say free or clamped—on the solution (8.157) is not at all an easy task. This has to do with the fact that in order to apply a separation of variables to the eigenvalue problem we must limit ourselves to the six combinations of boundary conditions where two opposite edges are simply supported. Let us investigate this point a bit further. If we take a step back and write the solution in the form substitution into the eigenvalue problem leads to (8.164) and we can separate it into two independent equations if (8.165a) Fig. 8.8 A few lower-order modes for a rectangular plate simply supported on all edges. Copyright © 2003 Taylor & Francis Group LLC or (8.165b) or both. Let us suppose that eq (8.165a) holds, this implies and (8.166) If now we consider the boundary conditions of simply supported (SS), clamped (C) and free (F), along x=0 we have which come from the expression of eqs (8.147a–c) in rectangular coordinates by noting that M x is given in eq (8.159) and that the Kirchhoff condition reads (8.167) A set of similar conditions apply at x=a. Now, it is not difficult to show that only the SS conditions can be satisfied by a function of the form (8.166) and, more specifically, we need a sine function which satisfies i.e. If also eq (8.165b) holds, all sides are simply supported and an analogous line of reasoning yields Moreover, substitution of eqs (8.165a and b) and of into eq (8.164) yields (8.168) which can be solved for the frequency to give eq (8.162). When the edges at x=0 and x=a are simply supported and we exclude the case of the other two edges simply supported, we are left with five possibilities for which we must solve the equation (8.169) whose solution depends on whether or However, even if separation of variables is possible in these latter cases, the information on natural frequencies and mode shapes is not easily obtained and the interested reader is urged to refer to the wide body of specific literature on the subject. Copyright © 2003 Taylor & Francis Group LLC A final comment of general nature can be made on the orthogonality of the eigenfunctions. From our preceding discussion, we know that mass and stiffness orthogonality are guaranteed by the symmetry of the eigenvalue problem; however, it may be of interest to approach the problem from a different point of view. Let us consider two different eigenfunctions, say u nm and u lk : the equations (8.170) are identically satisfied. Now, since from static classical plate theory the differential equation of static deflection is written we can interpret the first of eqs (8.170) as the equation of the static deflection of the plate under the action of the load and, by the same token, we can say that our plate assumes the deflected shape u lk when the load is acting. In other words, the loads q 1 and q 2 represent two systems of generalized forces while u nm and u lk are the displacements caused by such forces. We now invoke Betti’s theorem which states that: For a linearly elastic structure the work done by a system q 1 of forces under a distortion caused by a system q 2 of forces equals the work done by the system q 2 under a distortion caused by the system q 1 . In our case, this translates mathematically into (8.171a) where we had to integrate over the plate domain ⍀ in order to obtain the work expressions required by the theorem. Equation (8.171 a) gives (8.171b) and hence, since we assumed (8.171c) For our purposes, we can finish here our treatment on the free vibration of continuous systems referring the interested reader to the specific literature on the subject. In particular, an interesting discussion on one-dimensional Copyright © 2003 Taylor & Francis Group LLC eigenvalue problems in which boundary conditions contain the eigenvalue can be found in Humar [19] and Meirovitch [6, 9]. 8.9 Forced vibrations and response analysis: the modal approach The action of external time-varying loads on a continuous system leads to a nonhomogeneous partial differential equation. In general, in the light of the preceding sections, it is not difficult to obtain the governing differential equation also because—once the mass and stiffness operators have been introduced—its formal structure is similar to the matrix equation governing the forced vibrations of MDOF systems. However, now the boundary conditions come into play and we must pay due attention to them. Two general methods of solutions can be identified: 1. integral transform methods, which are particularly well suited to systems with infinite or semi-infinite extension in space (note, in fact, that in these cases the concept of normal mode loses its meaning) and for problems with time-dependent boundary conditions; 2. the mode superposition method. Here, we concentrate our attention on method 2, which we can call ‘the modal approach’. The relevant equation of motion is written as (8.172) where and, for brevity of notation, we indicate with x the set of space variables. Note that now f is a forcing function representing an external action (it is the counterpart of vector f of eq (7.1)) and must not be confused with the f functions of the preceding sections in this chapter, where this symbol was often adopted to identify a general function to be used for the specific needs of that part only. Equation (8.172) must be supplemented by the set of p boundary conditions (8.173) Now, by virtue of the results given in Section 8.7.1, the general response w can be expressed in terms of a superposition of eigenfunctions φ i multiplied by a set of time-dependent generalized coordinates y i (t), i.e. (8.174) Copyright © 2003 Taylor & Francis Group LLC which is the infinite-dimensional counterpart of eq (7.2). If we substitute the solution (8.174) into eq (8.172) and then take the inner product of the resulting equation by we get which, owing to the orthogonality relationships (8.128), reduces to the infinite set of 1-DOF uncoupled equations (8.175) whose solution is given by (Chapter 5 and eq (7.7)) (8.176) where y j (0) and j (0) are the initial jth modal displacement and velocity, respectively. Moreover, it is not difficult to show that we can obtain these quantities from the inner products (8.177) where and are the initial conditions in physical coordinates. The analogy with eq (7.5b) is evident. A few remarks can be made at this point. 1. First, it is apparent that the first step of the whole approach is the solution of the differential (symmetrical) eigenvalue problem satisfying the appropriate boundary conditions. The resulting set of eigenvalues and orthonormal eigenfunctions make it possible to express the general solution to the free vibrations problem (i.e. eq (8.172) with f=0) as in eq (6.50), which reads (8.178) For example, suppose that we want to consider the longitudinal motion of a uniform clamped-free rod of length L and µ mass per unit length. Let the Copyright © 2003 Taylor & Francis Group LLC initial conditions be such that and Since the mass orthonormal functions are given by we can calculate the rod free motion by means of eq (8.178). To this end we must first evaluate the inner product (the calculation is not difficult and is left to the reader) and then obtain the free motion as where we know from eq (8.49) that 2. When expressed in normal coordinates, the kinetic and potential energy of free vibration assume the particularly simple forms (8.179) so that the Lagrangian has no coupling terms between the coordinates and is simply the Lagrangian function of an infinite number of independent harmonic oscillators. 3. The final remark has to do with the apparent discrepancy according to which our system may be equally well described by a continuous system of coordinates w(x, t) or by a discrete one, i.e. y j (t). The general feeling is that this second set of coordinates cannot describe the same number of degrees Copyright © 2003 Taylor & Francis Group LLC [...]... Mechanics, 33, 335–340, 196 6 8 Graff, K.F., Wave Motion in Elastic Solids, Dover, New York, 199 1 9 Meirovitch, L., Principles and Techniques of Vibrations, Prentice Hall, Englewood Cliffs, NJ, 199 7 10 Abramowitz, M and Stegun, I., Handbook of Mathematical Functions, Dover, New York, 196 5 11 Page, C.H., Physical Mathematics, D van Nostrand, Princeton, NJ, 195 5 12 Bazant, Z.P and Cedolin L., Stability... New York, 199 4 17 Timoshenko, S and Woinowsky-Krieger, S., Theory of Plates and Shells, McGrawHill, New York, 195 9 18 Mansfield, E.H., The Bending and Stretching of Plates, Pergamon Press, 196 4 19 Humar, J.L., Dynamics of Structures, Prentice Hall, Englewood Cliffs, NJ, 199 0 20 Courant, R and Hilbert, D., Methods of Mathematical Physics, Vol 1, Interscience, New York, 196 1 21 Mathews, J and Walker,... Oxford University Press, 199 1 13 Jahnke, E., Emde, F and Losch, F., Tables of Higher Functions, McGraw-Hill, New York, 196 0 14 Anon., Tables of the Bessel Functions of the First Kind (Orders 0 to 135), Harvard University Press, Cambridge, Mass., 194 7 15 Leissa, A.W., Vibration of Plates, NASA SP-160, 196 9 16 Gerardin, M and Rixen, D., Mechanical Vibrations: Theory and Applications to Structural Dynamics,... xm and xk, we get from eq (8.189b) (8. 190 ) which is the (undamped) continuous systems counterpart of eq (7.28) Note that from eqs (8.185b) and (8. 190 ) we get (8. 191 ) which show that the reciprocity theorem holds Moreover, the generalization Copyright © 2003 Taylor & Francis Group LLC of the second of eqs (8. 191 ) to a general FRF function H(ω) other than receptance is straightforward and reads (8. 192 )... from all eigenvectors other than pk Inserting eq (9. 9) into eq (9. 3) we get (9. 10) and noting that we can expand the ‘error’ x as (9. 11) so that, owing to the orthogonality properties of eigenvectors, eq (9. 10) reduces to Copyright © 2003 Taylor & Francis Group LLC where the denominator can be expanded according to the binomial approximation to give (9. 12) The symbol o(ε3) means terms of order ε3 or... Journal of Analytical and Experimental Modal Analysis, 3(2), 49 56, 198 8 4 Irvine, M., Cable Structures, Dover, New York, 199 2 5 Kolsky, H., Stress Waves in Solids, Dover, New York, 196 3 Copyright © 2003 Taylor & Francis Group LLC 6 Meirovitch, L., Analytical Methods in Vibrations, Macmillan, New York, 196 7 7 Cowper, G.R., The shear coefficient in Timoshenko’s beam theory, ASME Journal of Applied Mechanics,... 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 1 Morse, P.M and Ingard, K.U., Theoretical Acoustics, Princeton University Press, 198 6 2 Elmore, W.C and Heald, M.A., Physics of Waves, Dover, New York, 198 5 3 Nariboli, G.A and McConnell,... excitation and a harmonic response in the forms of eqs (8.2 19) 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. .. eigenvalues and the eigenvalue separation property When no orthogonality constraints are imposed on the choice of u (such as in the discussion that leads to eq (9. 5)) we may note that, as our trial vector ranges over the vector space, eqs (9. 6) and (9. 8) always hold This leads to the important conclusions that Rayleigh quotient has a minimum when u=p1 and a maximum when u=pn, so that we can write (9. 13) and. .. and are given by (9. 1) respectively Hence, implies (9. 2a) where the pjs (j=1, 2,…, n) are the mass orthonormal eigenvectors On the other hand, a symmetrical continuous system leads to the same result if we consider the parallel between MDOF and continuous systems outlined in Sections 8.7 and 8.7.1 The continuous systems counterpart of eq (9. 2a) reads (9. 2b) where the eigenfunctions φj (j=1, 2, 3,…) are . Humar [ 19] and Meirovitch [6, 9] . 8 .9 Forced vibrations and response analysis: the modal approach The action of external time-varying loads on a continuous system leads to a nonhomogeneous partial. points x m and x k , we get from eq (8.189b) (8. 190 ) which is the (undamped) continuous systems counterpart of eq (7.28). Note that from eqs (8.185b) and (8. 190 ) we get (8. 191 ) which. is not difficult and is left to the reader) and then obtain the free motion as where we know from eq (8. 49) that 2. When expressed in normal coordinates, the kinetic and potential energy of