In many ways, a fundamental reference on the subject is Leissa [15]. This work—besides being a complete summary of all known results up to 1966—
contains a comprehensive set of results for the frequencies and mode shapes of free vibration of plates both according to the so-called ‘classical theory’
and with further complications such as anisotropy, variable thickness, in plane forces etc. For our part, since it is beyond the scope of this book to go into the
1 Bessel’s functions have been extensively studied and tabulated (e.g. Abramowitz and Stegun [10], Jahnke et al. [13] and the tables published by Harvard University Press [14]).
details of this rich and mathematically delicate subject, we will limit ourselves to a discussion of some of its fundamental aspects within the framework of the classical theory.
In essence, plates are the two-dimensional counterpart of beams, much in the same way as membranes are the two-dimensional counterpart of strings.
In other words, plates do have bending stiffness and the additional complications arise not only from the increased complexity of two- dimensional wave motion but also from the complex stresses that are set up when a plate is bent. In fact, when a plate element is bent, the material inside the bend becomes compressed and tends to expand laterally while the material outside the bend is stretched and tends to contract laterally, so that bending in one direction necessarily involves bending in a direction at right angles to it. It is well known that the ratio of the lateral extension (contraction) to compression (tension) is Poisson’s ratio v, which is approximately equal to 0.2–0.3 for most materials.
This ‘sideways effect’ was ignored in the case of beams because a beam element, in comparison with its length, is generally assumed to be thin enough as to make lateral bending negligible for most practical purposes.
Now, if we assume that our undisturbed plate lies in the x–y plane, the basic assumptions of the classical theory of plates vibrations can be summarized as follows:
1. Only the transverse displacement is considered.
2 The plate is thin, i.e., its thickness h is small compared to its lateral dimensions (say, where a is the smallest in plane dimension of the plate).
3. The stress in the transverse direction σz is zero. More specifically, since σz must vanish on the external layers at and h is small, then σz
is assumed to be zero for all values of z.
4. During bending, plane cross-sections remain plane and perpendicular to the midplane (just as in the Euler-Bernoulli beam development).
5. Only small deflections and slopes are considered (a maximum deflection of one-fifth of the thickness is generally considered the limit for small- deflection theory).
Given the above assumptions, it can be shown [see for example (e.g. Graff [8] or, for a variational approach, Meirovitch [9] and Gerardin and Rixen [16]) that the equation of motion for the free vibrations of a plate is given by (8.145) where the Laplacian of the Laplacian, is called the biharmonic operator, ρ is the mass density of the material (hence is the plate mass
density per unit area),
(8.146) is the plate bending stiffness and E is the modulus of elasticity of the material.
Now, our interest is mainly in eigenvalues and eigenfunctions of finite plates, and we do not pursue the subject o waves propagation in infinite plates. However, in this regard it is worth noting that the plate—like the beam—is a dispersive medium, meaning that waves of different wavelength trvel with different velocities. By contrast, in the case of finite plates, we note that eq (8.145) must be supplemented by appropriate boundary conditions in order to define a complete eigenvalue problem. The simplest types of boundary conditions are: simply supported (pinned), clamped and free. Without reference to any particular set of coordinates (rectangular, polar, etc.), we denote by s and n the coordinates in the tangential and normal directions to the contour and write the boundary conditions as:
• Simply supported edge,
(8.147a) where Mn is the normal bending moment per unit length.
• Clamped edge,
(8.147b)
• Free edge,
(8.147c) where Qn is the shearing force per unit length and Mns is the twisting moment per unit length about the direction n.
Note that the second boundary condition of eq (8.147c), i.e.
is by no means self-evident. In fact, at first glance, it would seem that the three stresses Qn, Mn and Mns should be independently equal to zero at a free edge; however, for a fourth-order equation, only two boundary conditions are required along each edge. It was left to Kirchhoff to show that Qn and Mns combine into a single edge condition as given above (e.g. Timoshenko and Woinowsky-Krieger [17] or Mansfield [18]).
For the above boundary conditions it can be shown with the aid of vector analysis that the biharmonic operator is symmetrical, so that the free vibration of plates also fits into the framework of symmetrical eigenvalue problems.
8.8.1 Circular plates
To be more specific, let consider the case of a uniform circular plate clamped at its outer edge r=R. As for the circular membrane of Section 8.6, the nature of the problem suggests the use of polar coordinates; assuming a harmonic time dependence we write the solution in the form and obtain for the space part of the solution
(8.148a) where Equation (8.148a) can be rewritten in the form
(8.148b) whose solution can be written as being the solution of and u2 being the solution of The equation for u1 is formally equal to eq (8.112) for the circular membrane (note, however, that now the constant γ does not have the dimension of a wavenumber), so that we can separate the variables, write
and arrive at the solution (Section 8.6)
(8.149a)
were n is an integer because—as for the circular membrane—continuity considerations require that
The function u2 can be explicitly obtained just by noting that its equation can be rewritten as we separate the space variables and arrive at the so-called modified Bessel equation for the function f2(r) so that we obtain
(8.149b)
where In and Kn are the modified Bessel functions of the first and second kind. They are related to Jn and Yn by and
Putting eqs (8.149a) and (8.149b) together we have
(8.150)
where the Bessel functions of the second kind Yn and Kn have been eliminated, since they have singularities at r=0. Note that for circular plates with a circular concentric hole of radius r=a<R these functions must be retained in the solution because their singular behaviour at the origin no longer plays a role; in this case, however, two additional boundary conditions must be specified at r=a.
The boundary conditions for our case (plate clamped at its outer edge) read
(8.151) so that we obtain the frequency equation
(8.152a) which, owing to the recursion relationships obeyed by Bessel’s functions, can be equivalently written as
(8.152b) Equation (8.152a or b) must be solved numerically: for each value of n there are an infinite number of roots which we can identify with an index m, where m=1, 2, 3,…. If we now define the frequency parameter
the natural frequencies can be written as
(8.153) where, for a given pair n, m there are two eigenmodes (except for n=0) so that all modes with are twofold degenerate. Furthermore, as in the case of the clamped circular membrane, we have n nodal diameters and m–1 nodal circles. The first few values of are as follows:
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 Mr 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 Mr is as above and the transverse shearing force Qr and the twisting moment Mrθ 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
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 u1 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 u2 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)
where the values of the constants Aj and parameters α and ò 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 Mx and My are given by
(8.159)
By inserting the conditions of eqs (8.158a and b) into the solution (8.157) we obtain that only A1 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
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.
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 Mx 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.
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 unm and ulk: 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 ulk when the load is acting. In other words, the loads q1 and q2 represent two systems of generalized forces while unm and ulk 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 q1 of forces under a distortion caused by a system q2 of forces equals the work done by the system q2 under a distortion caused by the system q1. 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
eigenvalue problems in which boundary conditions contain the eigenvalue can be found in Humar [19] and Meirovitch [6, 9].