Generalized integral transform solution for free vibration of orthotropic rectangular plates with free edges

Abstract

Free vibration of orthotropic rectangular thin plates of constant thickness with two opposite edges clamped and one or two edges free is analyzed by generalized integral transform technique. Numerically stable eigenfunctions in exponential function forms of Euler–Bernoulli beams with appropriate boundary conditions are adopted for each direction of the plate. The governing fourth-order partial differential equation for the mode function of free vibration is transformed into a system of linear equations, by integral transform in both directions of the rectangular plate. The boundary conditions at free edges are satisfied exactly by considering the terms generated in the transformed equations by integration by parts, which are absent in the equations by traditional Rayleigh–Ritz method. The natural frequencies of free vibration of orthotropic rectangular thin plates obtained by the proposed integral transform solution are compared with available results in the literature and numerical solutions by finite element analysis, showing excellent agreement and high convergence rate.

Appendix

(a) CC (clamped edges):

The eigenfunctions $X_i(x)$ for a pair of clamped edges in the ‘x’ direction are given by solving problem (11) analytically [35,36,37]:

$$\begin{aligned} X_i(x)&= (-1)^{i}\cos (\mu _ix)-\sin (\mu _ix) \cot \left( \frac{\mu _i}{2}\right) ^{(-1)^i} \\&\quad -\frac{(-1)^i e^{-\mu _ix}}{1-(-1)^i e^{-\mu _i}}+\frac{e^{-\mu _i(1-x)}}{1-(-1)^i e^{-\mu _i}}. \end{aligned}$$

(32)

The transcendental equations for the eigenvalues $\mu _i$ are given by

$$\begin{aligned} (-1)^i \tan \left( \frac{\mu _i}{2}\right) =\frac{1-e^{-\mu _i}}{1+e^{-\mu _i}} \quad \text{ and } \quad i=1,2,3,\ldots \end{aligned}$$

(33)

(b) SF (simply supported and free edges):

The eigenfunctions $Y_{1j}(y)$ for a pair of simply supported and free edges in the ‘y’ direction are given by solving problem (12 and 13) analytically [35,36,37]:

$$\begin{aligned} Y_{11}(y)&= \root \of {3} y \quad \text{ and } \\ Y_{1j}(y)&= \root \of {2}\left( \sin (\phi _{1j} y)-\frac{e^{-\phi _{1j}}\sin (\phi _{1j})}{1-e^{-2\phi _{1j}}}e^{-\phi _{1j} y} \right. \\&\quad \left. +\frac{e^{-\phi _{1j}(1-y)}\sin (\phi _{1j})}{1-e^{-2\phi _{1j}}}\right) . \end{aligned}$$

(34)

The transcendental equations for the eigenvalues $\phi _{1j}$ are given by

$$\begin{aligned}&\phi _{11}=0, \\&\tan (\phi _{1j})=\frac{1-e^{-2\phi _{1j}}}{1+e^{-2\phi _{1j}}} \quad \text{ and } \quad j=2,3,4,\ldots \end{aligned}$$

(35)

(c) CF (clamped and free edges):

The eigenfunctions $Y_{2j}(y)$ for a pair of clamped and free edges in the ‘y’ direction are given by solving problem (12 and 14) analytically [35,36,37]:

$$\begin{aligned} Y_{2j}(y)&= \cos (\phi _{2j} y)-\frac{1+(-1)^{j} e^{-\phi _{2j}}}{1-(-1)^{j} e^{-\phi _{2j}}}\sin (\phi _{2j} y) \\&\quad - \frac{1}{1-(-1)^{j} e^{-\phi _{2j}}}e^{-\phi _{2j} y}+ \frac{(-1)^{j}}{1-(-1)^{j} e^{-\phi _{2j}}}e^{-\phi _{2j}(1-y)}. \end{aligned}$$

(36)

The transcendental equations for the eigenvalues $\phi _{2j}$ are given by

$$\begin{aligned} \cos (\phi _{2j})= \frac{-2 e^{-\phi _{2j}}}{1+e^{-2\phi _{2j}}} \quad \text{ and } \quad j=1,2,3,\ldots \end{aligned}$$

(37)

(d) FF (free and free edges):

The eigenfunctions $Y_{3j}(y)$ for a pair of two free edges in the ‘y’ direction are given by solving problems (12 and 15) given by [35,36,37]

$$\begin{aligned} Y_{31}(y)&= 1, \quad Y_{32}(y)=\root \of {3}(1-2y) \quad \text{ and } \\ Y_{3j}(y)&= \cos (\phi _{3j} y)-\frac{1+(-1)^{j} e^{-\phi _{3j}}}{1-(-1)^{j} e^{-\phi _{3j}}}\sin (\phi _{3j} y) \\&\quad + \frac{1}{1-(-1)^{j} e^{-\phi _{3j}}}e^{-\phi _{3j} y}- \frac{(-1)^{j}}{1-(-1)^{j} e^{-\phi _{3j}}}e^{-\phi _{3j}(1-y)}. \end{aligned}$$

(38)

The transcendental equations for the eigenvalues $\phi _{3j}$ are given by

$$\begin{aligned}&\phi _{31}=\phi _{32}=0, \\&(-1)^j \tan \left( \frac{\phi _{3j}}{2}\right) =\frac{1-e^{-\phi _{3j}}}{1+e^{-\phi _{3j}}} \quad \text{ and } \quad j=3,4,5,\ldots \end{aligned}$$

(39)