Analytical approach for dynamic instability analysis of functionally graded skew plate under periodic axial compression

Abstract

Analytical studies on the dynamic instability analysis of a functionally graded (FG) skew plate subjected to uniform and linearly varying in-plane periodic loadings with four different types of boundary conditions are presented. The total energy functional of the FG skew plate is formulated based on Reddy's third order shear deformation theory (TSDT) and this functional is mapped from the physical domain to computational domain using transformation rule. The boundary characteristics orthonormal polynomials (BCOPs) are generated for different boundary conditions using Gram–Schmidt process, which satisfy the essential boundary conditions of skew plates in the computational domain. The energy functional is converted into a set of ordinary differential equations (Mathieu–Hill equations) using Rayleigh–Ritz method in conjunction with BCOPs. The solution of Mathieu–Hill equations describes the dynamic instability behavior of skew plate. The instability regions are traced using Bolotin method. The effect of skew angles, power-law distributions, span-to-thickness ratios, aspect ratios, boundary conditions and static load factors on the instability region of FG skew plates are presented. The result indicates that the width of instability region become narrow with the increase in skew angle. Moreover, the time history response and corresponding phase plot in the unstable and stable region is studied to identify the instability behavior such as existence of beats, bounded and unbounded response, and effect of forcing amplitude and its frequency on the response.