Analysis on Nonlinear Vibrations of a Stepped Beam Considering Effects of Axial Inertia Force

Abstract

Thin walled beams are utilized in the parts of automobiles and micro machines in order to erduce the weight and to keep high rigidity. However, under lateral periodic load, nonlinear vibrations may be generated in specific frequency domains. Therefore, it is important to analyze nonlinear vibrations of the beams with various distribution of cross section along its axis. In this study, analytical results is presented on nonlinear vibrations of a stepped beam considering effects of axial inertia force. The beam is divided into a few segments to express the geometry of the stepped beam. The deflection of the beam is expanded with the mode share function that is expressed with the product of truncated power series and trigonometric functions. Taking the axial displacements, the deflection, slopes, bending moments and shearing forces at the node of the segments as unknown variables, nonlinear coupled ordinary differential equations are dirived with Galerkin procedure. Different from previous analysis, the equation of motion of the beam is formulated as an expression including the axial inertia force of the end-mass. Nonlinear periodic response are calculated with the harmonic balance method, which are compared with the previous experimental results.