Thermoelastic Harmonic Resonance Of Functional Gradient Rectangular Thin Plate

Functional gradient materials are the new materials that combine different organic and inorganic materials in an ingenious way,such as metals and ceramics,and the material properties are obviously different from traditional composite materials.The applications of functional gradient material components are becoming more and more widespread.Rectangular plates and their combined structures are common in practical engineering,therefore,the study of nonlinear dynamics of functional gradient rectangular plates has important theoretical significance.In this paper,superharmonic,subharmonic and joint harmonic-parametric resonance problems of functional gradient rectangular thin plates in a thermal environment are investigated based on the physical neutral surface.For the functional gradient rectangular thin plate subjected to transverse simple harmonic excitation force in thermal environment,based on the large deflection theory and classical plate theory,the kinetic energy,potential energy and external virtual work of the plate are derived by considering the geometric nonlinearity,the effect of temperature change and the main parameters of the material distributed by power law along the plate thickness direction.Hamilton variational principle is applied to establish the nonlinear vibration equations of the functional gradient rectangular thin plate.The superharmonic resonance and subharmonic resonance problems of the functional gradient rectangular thin plate subjected to transverse simple harmonic excitation forces in a thermal environment are investigated.For the ceramic-metal functional gradient rectangular thin plate with four-sided simple support or four-sided clamped support boundary conditions,the corresponding Duffing-type nonlinear vibration differential equations are obtained by Galerkin integration assuming different displacement solutions.The amplitude-frequency response equations of the steady-state motion are obtained by solving with the multiscale method.The stability analysis of the resonant solutions is carried out by using the Liapunov stability theory.The amplitude-frequency characteristics,amplitude-plate surface temperature,and amplitude-excitation force amplitude curves of the plate under warming and cooling conditions are obtained by arithmetic examples.The results of analytical and numerical solutions,and those based on the physical and geometric neutral surface are compared and analyzed,and the effects of the gradient index,excitation force amplitude and temperature on the harmonic resonance of the system are analyzed.The joint harmonic-parametric resonance problem of the functional gradient rectangular thin plate subjected to the combined effect of transverse simple harmonic excitation force and axial alternating excitation in a thermal environment is studied.The joint resonance mechanical model of the plate and the joint resonance nonlinear vibration equation are established.Application Galerkin integrate to obtain the differential equation of vibration at different boundaries and solve by multi-scale method and variable transformation method to get the amplitude-frequency response equation of the system.Through the arithmetic example,the amplitude-frequency graphs and amplitude-excitation force amplitude graphs for different conditions are obtained;the results are compared and the effects of gradient index,temperature and other factors on the joint resonance of the system are analyzed.