## Exact, Series-Type, and Approximate Methods for Transverse Vibration of Plates

We have obtained the equation of motion for the transverse vibration of a plate in Chapter 3 as D ∇ 4 w + ρ h ∂ 2 w ∂ t 2 = 0 where

ω is the transverse displacement of the plate

ρ is the density of the material of the plate

h is the thickness of the plate

D = E h 3 12(1 − ν 2 ) is the flexural rigidity

v is the Poisson’s ratio

E is the Young’s modulus of elasticity

∇^{4} is the biharmonic operator

^{4}w = ∇

^{2}(∇

^{2}w) where ∇

^{2}is the Laplacian operator, and this has been defined in Chapter 3 in the cases of rectangular, polar, and elliptical coordinates. For free vibration with circular (natural) frequency ω, we can write the motion of the plate in polar coordinates as w ( r , θ , t ) = W ( r , θ ) e i ω t and in Cartesian coordinates w ( x , y , t ) = W ( x , y ) e i ω t In general, this will be written in the form w = W e i ω t 106By substituting Equation 4.5 in Equation 4.1, we get ( ∇ 4 − β 4 ) W = 0 β 4 = ρ h ω 2 D Equation 4.6 is then written as ( ∇ 2 + β 2 ) ( ∇ 2 − β 2 ) W = 0 whose solution may be obtained in the form of two linear differential equations: ( ∇ 2 + β 2 ) W 1 = 0 ( ∇ 2 − β 2 ) W 2 = 0 One can write the solution of Equation 4.8 as the superposition of the solutions of Equations 4.9 and 4.10. Let W

_{1}and W

_{2}be the corresponding solutions. Then, one may have the solution W of the original differential Equation 4.8 as W = W 1 + W 2