Nonlinear vibrations of moving functionally graded plates containing porosities and contacting with liquid: internal resonance

Nonlinear vibrations of longitudinally moving functionally graded material (FGM) plates containing porosities and contacting with liquid are investigated. The FGM plates contain porosities because of technical issues during the FGM preparation. Two types of porosity distribution, namely even and uneven distribution, are taken into account. The liquid is assumed to be incompressible, inviscid and irrotational. The velocity potential along with Bernoulli’s equation is employed to describe the dynamic liquid pressure on the plates. Considering geometric nonlinearity described by the von Kármán nonlinear theory, the governing equation of the system is developed by utilizing D’Alembert’s principle. Then, the Galerkin method is employed to reduce the governing equation into a set of ordinary differential equations. An approximately analytical analysis on the nonlinear steady-state response is performed by means of the harmonic balance method; the stability of steady-state response is examined by a perturbation analysis. To have an intuitive understanding of parameter effects, this study retains all dimensional parameters in the analytical results. The possibility of appearance of internal resonance is confirmed by linear analysis of the system. For moving porous FGM plates contacting with liquid, a distinctive internal resonance phenomenon can occur between the first two modes, as we shall see. Different from the common internal resonance, the present internal resonance is very sensitive to external force and can be motivated by quite small force. Results are presented to examine the effects of different parameters on the frequency response of the system.