Nonlinear vibration of an initially stressed plate based on a modified plate theory

Nonlinear partial differential equations for the vibrating motion of a plate based on a modified higher order plate theory with seven kinematic variables are derived. The present seven-variable modified higher order plate theory satisfies the stress-free boundary conditions. Using these derived governing equations, the large amplitude vibrations of a simply supported thick plate subjected to initial stresses are studied. The Galerkin method is used to transform the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge–Kutta method is used to obtain the ratio of linear to nonlinear frequencies. Frequency ratios obtained by the present theory are compared with the Mindlin plate theory results and Lo's 11-variable higher order plate theory results. It can be concluded that present modified plate theory predicts frequency ratios very accurately. Also, the benefit of significant simplification can be observed as compared with the Lo's higher order plate theory. The effects of initial stress and other factors on frequency ratio are investigated.