A unified Fourier spectral method for nonlinear free vibration analysis of the laminated composite and sandwich beams with arbitrary restrained ends

Nonlinear free vibration analysis is necessary for understanding the vibration nature of the structures with large vibration amplitudes. The purpose of this paper is to provide a unified method to deal with the nonlinear free vibration analysis of the sandwich-like beams with arbitrary restrained ends and study nonlinear vibration characteristic of the beams with free vibration amplitudes. Von Kármán-type nonlinear vibration of the laminated composite and sandwich (LC-S) beam is studied using a modified Fourier spectral method. Nonlinear free vibration analysis under arbitrary boundary conditions is realized for the first time by a numerical boundary technique, which has the ability to study the exact influence of the boundary condition on nonlinear vibration behavior of the sandwich beams. The modified Fourier spectral method on basis of the Timoshenko beam theory is used to derive the characteristic governing equation with quadratic and cubic nonlinear terms. The assumed displacements are in a general form, which consists of a Fourier spectrum and supplementary polynomials. The assumed displacements are in a general form, which consists of a Fourier spectrum and supplementary polynomials. Such general form is capable of dealing with arbitrary boundary condition without need to satisfy them in advance. Numerical investigations are conducted to highlight the effect of the large vibration amplitude on natural behaviors of the LC-S beams. The effects of the boundary condition on natural nonlinearity of the LC-S beams are studied for the first time.