Nonlinear Vibration Analysis Of Nanoplate Structure With The Temperature Rise

In this paper,the method of multiple scales is mainly used to analyze the geometrical nonlinear vibration of the nanoplate structure with the temperature rise.The dynamic equation of the geometrical nonlinear vibration of the nanoplate is deduced.Taking four edges simply supported boundary conditions as an example,the relationship between the amplitude and the geometrical nonlinear vibration frequency of the nanoplate,as well as the influences of small-scale effects,temperature rise,and external excitations on the vibration characteristics are discussed.And then,the geometrical nonlinear vibration characteristics of nanoplates based on high-order strain gradients under external excitation with the temperature rise are studied.The main research contents of this paper are as follows:Firstly,based on the strain gradient theory,the von Kámán large deflection theory is employed to consider geometrical nonlinearity,and the stress-displacement relationship of the nanoplate geometrical nonlinearity of the second-order strain gradient theory and fourth-order strain gradient theory,and fourth-order strain gradient theory with the temperature rise are deduced respectively.The equilibrium equation is established by the force analysis of the nanoplate structure,and then the dynamic partial differential equation of the geometrical nonlinear vibration of the nanoplate is obtained.Secondly,applying the compatibility equation theory of Kirchhoff’s plate model,the compatibility equation of the geometrical nonlinear vibration of nanoplates with high-order strain gradient theory is deduced,and then the stress function is obtained.Finally,the obtained partial differential vibration equation is discretized to obtain the ordinary differential equation.The geometrical nonlinear vibration ordinary differential equations of nanoplates are solved by the method of multiple scales,and the effects of temperature,non-local parameters,amplitude,and external excitation on the geometrical nonlinear vibration frequency are discussed respectively.