Large Amplitude Free Vibration of a Rotating Nonhomogeneous Beam With Nonlinear Spring and Mass System

Abstract

This paper investigates the free out of plane vibration of a rotating nonhomogeneous beam with nonlinear spring and mass system. The effect of nonhomogeneity of the beam appears both in the governing equations and in the boundary conditions, but the nonlinear spring-mass effect appears in the boundary conditions only. The solution is obtained by applying the method of multiple time scales directly to the nonlinear partial differential equations and the boundary conditions. The results of the linear frequencies match well with those obtained in open literature. The effect of the nonhomogeneity of the stiffer beam (β=0.01) reduces the frequencies of vibration of the beam. A possible physical explanation of this reduced frequency of the nonhomogeneous beam is discussed. A subsequent nonlinear study of the nonhomogeneous beam indicates that the mass of the spring and its location also have a pronounced effect on the vibration of the beam. The effect of the nonhomogeneity of the beam on the relative stability of the nonlinear vibration of the beam with spring-mass system is also studied.