Nonlinear forced vibration of in-plane bi-directional functionally graded materials rectangular plate with global and localized geometrical imperfections

Abstract

This research aims to investigate the nonlinear vibration of the in-plane bi-directional functionally graded (BDFG) plate with global and localized geometrical imperfection subjected to a transverse harmonic excitation. The material properties of plate are varying along two in-plane directions. The global and localized geometrical imperfections are simulated in the form of products of trigonometric and hyperbolic functions. Based on the von Kármán’s nonlinear plate theory, coupled nonlinear partial differential equations governing in-plane and transverse displacements of BDFG plate are derived employing Hamilton’s principle. The continuous nonlinear model of imperfect BDFG plate is discretized using the Galerkin scheme, resulting in the reduced order model. The pseudo-arclength continuation technique is employed to trace the periodic motion of plate and construct frequency-/force-response curves. Dynamic responses are solved using numerical integration. Numerical results of ceramic-metal BDFG plates are presented to examine the effects of system parameters, e.g. functional gradient parameters, external excitation parameters and damping coefficients. Particularly, influences of global and localized geometrical imperfections are highlighted. Results show that the variation of gradient parameter and the existence of geometrical imperfection change the nonlinearity of resonant response. Cyclic-fold, period doubling and torus bifurcations of the periodic solution are detected as excitation parameters varying. Periodic, quasi-periodic and chaotic motions of plate are also explored.