Analytical determination of critical velocity and frequencies of beam with moving mass under different supporting conditions

Abstract

For a moving mass-beam system, the critical velocity of the moving mass is a key parameter that relates to the vibration stability of the system. In fact, the critical velocity obtained by the commonly used assumed mode method (AMM) differs from the actual situation. In this study, an analytical procedure is introduced to determine the critical velocity and frequency of the moving mass-beam system. The influence of moving mass is considered in the modal functions of the beam, and the frequency equations of the system were obtained through the modal analysis method and Laplace transform. And beams with four types of boundary condition were analyzed, which are hinged-hinged (HH) beam, clamped-hinged (CH) beam, clamped-clamped (CC) beam, and cantilever (CF) beam. By solving the frequency equations, the vibration frequencies of the system can be obtained, and the critical velocity can be determined. The results of the proposed method were validated by the finite element method (FEM). Through some examples, it was found that the natural frequency and critical velocity obtained by AMM is relatively high. And the critical velocities of the same moving mass-beam system under different supporting conditions ranked in ascending order are as follows: vcrHH<vcrCH<vcrCF<vcrCC. It is also found that when the moving mass undergoes variable motion on a beam, the vibration frequency obtained with acceleration considered is higher than that obtained with acceleration neglected. The results of this article will be helpful for structural design and its dynamic analysis.