A general analytical approximation for nonlinear vibrations analysis of continuous systems using renormalization group method

We propose a general procedure based on renormalization group method (RGM) and Green’s function method to determine the analytical approximation of nonlinear continuous systems with general linear and nonlinear operators. The nonlinear operator has a general form (not necessarily quadratic and cubic form) and can be function of displacement, velocity and acceleration or combination of thereof. In addition, the procedure can handle all types of resonances and there is no restriction on the number of modes which are considered in the analysis. Furthermore, for a specified form of the linear operator (e.g. wave operator), the nonlinear and non-homogeneous boundary conditions can be considered in the analysis. Although, the analysis is general, but it is assumed that the boundary data is small and therefore the boundary layer is not observed. To show the effectiveness of the procedure, some examples are presented.