Transverse Vibrations of Rectangular Plates of Linearly Varying Thickness

This paper presents an approach for finding the solution of partial differential equation describing the motion of transverse vibrations of rectangular plates of unidirectional linear varying thickness. The original partial differential equation consists of three operators: fourth-order spatial-dependent, second-order spatial-dependent, and second-order time-dependent. Using the method of multiple scales, the partial differential equation has been reduced to two simpler partial differential equations which can be analytically solved and which represent two levels of approximation. The first partial differential equation was a homogeneous equation and consisted of two operators, the fourth-order spatial-dependent and second-order time-dependent. The solution of this equation was found using the factorization method. This solution was zeroth-order approximation of the exact solution. The second partial differential equation was an inhomogeneous equation. The solution of this equation was also found and led to first-order approximation of the exact solution of the original problem. This way the first-order approximations of the natural frequencies and mode shapes are found. Various boundary conditions can be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.