1 The transverse cross‐sections of the plate still remain plane in the presence of strains, and
2 Neither longitudinal nor transverse strains occur on the middle (neutral) plane.
These conditions are satisfied approximately when the flexural wavelength is at least equal to a six times the thickness h of the plate. Under this assumption, the wave equation for transverse vibration of the plate is [14]
where W(x,y,t) is the instantaneous transverse displacement, ρ is the density of the plate, B = Eh3/12(1 − ν2) is the stiffness of the plate for bending, ν is the Poisson's ratio, E is the modulus of elasticity (Young's modulus) and f(x,y,t) are the external forces referring to unit surface area of the plate. The sum of the first three terms of Eq. (2.65) is a double Laplacian of W, ∇4 W, expressed in rectangular coordinates. It is useful to define the solution for steady‐state conditions as
(2.66)
where W0(x,y) is the amplitude of the transverse displacement.
A vibrating structure, such as a plate, at any instant contains some kinetic energy and some strain (or potential) energy. The kinetic energy is associated with the mass and the strain energy is associated with the stiffness. In addition, any structure also dissipates some energy as it deforms. This conversion of ordered mechanical energy into thermal energy is called damping. A simple way to describe the energy loss is given by the analysis of the one‐dimensional system of Eq. (2.17). Thus, for sinusoidal vibration, use of a spring with an appropriately defined stiffness is completely equivalent to the use of an elastic spring and a dashpot. The internal losses in a plate arise not because of the motion of the plate as a whole body, but depend on the mutual displacements of the neighboring elements of the plate, and these are proportional to the changes in time of ∇4 W. Therefore, in order to account for energy dissipation in a plate, one may simply introduce a complex modulus of elasticity
(2.67)
where η is the internal loss factor of the plate. As an example, Figure 2.17 shows the computation of the velocity level response in decibels of the first resonance in a clamped‐clamped rectangular panel and the effect of varying its internal loss factor η.
Figure 2.17 Velocity level of a fully‐clamped rectangular plate as a function of frequency for selected values of η. The resonance of its first mode is shown. The plate is a plywood board with properties E = 6 × 109 N/m2, ρs = 6.5 kg/m2, S = a × b = 0.7 × 0.6 m2, and ν = 0.3. (
Source: from Ref. [16].)
The free vibration of plates has two important characteristics:
1 The velocity of propagation of flexural waves in the plate depends on frequency, and
2 The second component of the deflection, arising as a result of the stiffness of the plate, brings about additional changes, compared to a beam, in the distribution of the vibration.
As in the case of beams, plates can be fixed at their edges in various manners. Solutions for vibrating plates subjected to different boundary conditions have been discussed in several books for the 27 possible combinations [14, 17]. In this section we will discuss a uniform simply supported plate of length a and width b, mainly because this problem is illustrative and it has a closed‐form solution. In this case, the plate at its supports cannot perform motion perpendicular to its surface, but can rotate around the edge, and hence the displacement and bending moment are zero at the corresponding boundaries (see Eq. (2.55) for a beam)
(2.68)
The solution for the amplitude of flexural vibrations of a simply‐supported plate vibrating in mode (m,n), where m−1 and n−1 are the number of nodal lines in the x and y directions, respectively, can be obtained by separation of variables and is given by [14]
where Amn is the modal amplitude, km = mπ/a, kn = nπ/b and m and n are integers. Equation (2.69) gives the mode shapes of the simply‐supported plate. Figure 2.18 shows the first six mode shapes of a rectangular plate. Natural frequencies of each plate's mode (m,n) are given by
where ρs is the mass of the plate per unit surface area. Note that Eq. (2.70a) can be approximated by
(2.70b)
where fm,n is the characteristic modal frequency (Hz) and cL is the longitudinal wave speed in the plate material (m/s).
Figure 2.18 First six modes of a rectangular plate.
Thus, the total solution for the free vibration of a simply‐supported plate is
(2.71)
Note that for a square plate (a = b) ωmn = ωnm. Therefore modes (m,n) and (n,m) have the same frequency and they are called degenerate. This fact also happens when the length or width are integer multiple of each other.
Example 2.12
Determine the natural
