A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic parameters

Abstract

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic distribution parameters is proposed. The partial differential equation with spatial varying parameters for nonlinear vibration of beams with periodic parameters under harmonic excitations is derived. The procedure of the multimode perturbation method includes three main steps: first, the nonlinear partial differential equation is transformed into linear partial differential equations with varying parameters by applying perturbation method; second, the linear partial differential equations are transformed into ordinary differential equations with multimode coupling by applying Galerkin method, where multiple vibration modes of the beams are used and the equations are suitable to nonlinear vibration of periodic structures with high parameter-varying wave in wide frequency band; third, the ordinary differential equations are solved by applying harmonic balance method to obtain vibration response of the nonlinear periodic beam, which is used for characteristics analysis of frequency response and spatial mode. Furthermore, the stability problem of nonlinear harmonic vibration as multidegree-of-freedom system with periodic time-varying parameters is solved by applying direct eigenvalue analysis approach. The proposed method can incorporate multiple vibration modes into response analysis of nonlinear periodic structures and consider mode-coupling effects due to structural nonlinearity and parametric periodicity. Finally, a nonlinear beam with periodic supports under harmonic excitations is studied. Numerical results on frequency response of the beam are given to illustrate an application of the proposed method, new frequency response characteristics, and influences of periodic parameters on structural response. The results have potential application to nonlinear structural vibration control and support damage detection of nonlinear structures with periodic supports.