Transverse Free Vibration of Axially Moving Stepped Beam with Different Length and Tip Mass

Abstract

Axially moving stepped beam (AMSB) with different length and tip mass is represented by adopting Euler-Bernoulli beam theory, and its characteristics and displacements of transverse free vibration are calculated by using semianalytical method. Firstly, the governing equation of the transverse free vibration is established based on Hamilton’s principle. The equation is cast into eigenvalue equation through the complex modal analysis. Then, a scheme is proposed to derive the continuous condition accordingly as the displacement, rotation, bending moment, and shear force are all equal at the connections of any two segments. Another scheme is to derive frequency equation from the given boundary conditions which contain a tip mass in the last segment. Finally, the natural frequency and modal function are calculated by using numerical method according to the eigenvalue equation and frequency equation. Due to the introduction of modal truncation, displacement and, the free vibration solution can be obtained by adopting modal superposition after Hilbert transform. The numerical examples illustrate that length, velocity, mass, and geometry affect characteristics and displacements significantly; the series of methods are effective and accurate to investigate the vibration of the AMSB with different length and tip mass after comparing several results.