Nonlinear free vibration analysis of square plates with various boundary conditions

Abstract

Non-linearity in plate vibration problems may arise out of material and geometric non-linearity. The present study focuses only on geometric non-linearity. A new methodology is proposed that can be employed for plate structure problems having any combination of boundary conditions to determine the non-linear frequencies and mode shapes. Large amplitude vibration problem is analysed in two parts. The static problem corresponding to a uniform transverse loading is solved first and the dynamic problem is subsequently taken up with the known deflection field. Both these problems are formulated through energy method, the underlying principle being the extremisation of total energy of the system in its equilibrium state. The solution methodology employs an iterative numerical scheme using the technique of successive relaxation. The results of non-linear analyses are validated with the published results and excellent agreement is observed. The solution methodology can be applied to any kind of boundary condition as pointed out and the back-bone curves documented may be used by the practicing engineers as design curves.