Nonlinear combined resonances of FGM stepped sandwich plates under transverse and axial excitations

This paper addresses the nonlinear vibration problem of variable-thickness functionally graded structures and presents the first systematic investigation on the combined resonance characteristics and chaotic dynamic behavior of functionally graded material (FGM) stepped sandwich plates under combined transverse and axial excitations. A nonlinear dynamic model of the FGM stepped sandwich plate is established based on the third-order shear deformation theory (TSDT) and the von Kármán geometric nonlinearity assumption. The governing equations are derived via Hamilton’s principle, discretized into a single-degree-of-freedom Duffing equation using the Galerkin method, and solved by the introduced method of varying amplitudes (MVA). Numerical analyses reveal that increasing damping can effectively suppress the hardening spring nonlinearity. Longitudinal and axial excitations exhibit opposite regulatory effects on the multi-solution region, with the former compressing it while the latter expanding it. The initial phase angle significantly suppresses hysteresis while barely affecting the primary resonance. For the first time in stepped plates, the complete evolution path from single periodic motion to chaos via period-doubling bifurcations with increasing excitation amplitude is elucidated. These findings provide important theoretical guidance for the vibration safety design of lightweight variable-thickness structures in advanced engineering applications.