Nonlinear Vibration Of Functionally Graded Materials Rectangular Plates

Functionally Graded Material is a new type of composite materials, which can be widely applied to material areas of various requirements through the ingenious combination of component materials. The nonlinear vibration problems of functionally graded plates and cylindrical shells is particularly common in engineering practice, the solution in mathematics of the nonlinear governing partial equation is very difficult, therefore, the study on the nonlinear vibration of functionally graded plates and cylindrical shells has important theoretical and practical significance.In this paper, nonlinear forced vibration of rectangular functionally graded plates subjected to external excitation is studied. Principal resonance and parameter vibration of FGM plates and other issues are focused discussed. Firstly, according to the classical plate theory and the physical properties parameters of functionally graded materials, the nonlinear forced vibration equations of rectangular functionally graded plates respectively in the harmonic excitation and parametric excitation were deduced by using Galerkin method.Secondly, for principal resonance of FGM rectangular plate with four clamped edges and principal parametric resonance of simply supported FGM rectangular plate, the amplitude-frequency response equations were respectively derived using multi-scale method. Based on the Lyapunov stable theory, the critical conditions of steady-state solutions'stability were gotten. For the strongly nonlinear principal resonance of simply supported FGM rectangular plate, based on parameter transformation, the Duffing equation of system was studied by using modified multi-scale method. The amplitude-frequency response equation was gotten.Finally, several typical functionally graded materials were analyzed. In principal resonance, the amplitude-frequency curves, amplitude-excitation curves of FGM rectangular plate with four clamped edges, the amplitude- frequency curves and time history curves and phase trajectory of simply supported FGM rectangular plate were derived under different situations. In principal parametric resonance, the amplitude-frequency curves and amplitude- damping response curves and the solution's existence regional diagram of the simply supported FGM rectangular plate were derived under different situations. The influence of different parameters on nonlinear resonance properties of system was discussed.