ON NON-AXISYMMETRICAL TRANSVERSE VIBRATIONS OF CIRCULAR PLATES OF CONVEX PARABOLIC THICKNESS VARIATION

摘要

This paper presents an approach for finding the solution of the partial differential equation of motion of the non-axisymmetrical transverse vibrations of axisymmetrical circular plates of convex parabolical thickness. This approach employed both the method of multiple scales and the factorization method for solving the governing partial differential equation. The solution has been assumed to be harmonic angular-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. Solving them, the solution resulted as first-order approximation of the exact solution. Using the factorization method, the first differential equation, homogeneous and consisting of fourth-order spatial-dependent and second-order time-dependent operators, led to a general solution in terms of hypergeometric functions. Along with given boundary conditions, the first diferential equation and the second differential equation, which was nonhomogeneous, gave respectively so-called zero-order and first-order approximations of the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson's ratio on the natural frequencies and mode shapes could be further studied using the first-order approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.