Vibrations of a Beam Moving Over Supports with Clearance

H.P Lee

doi:10.1155/1994/405058

Abstract

The transverse vibration of a beam moving over two supports with clearance is analyzed using Euler beam theory. The equations of motion are formulated based on a Lagrangian approach and the assumed mode method. The supports with clearance are modeled as frictionless supports with piecewise-linear stiffness. A feature of the present formulation is that its complexity does not increase with increased number of supports. Results of numerical simulations are presented for various prescribed motions of the beam. The effect of support clearance on the stability of the beam is investigated.

Highlights

The mechanics of a stationary beam acted upon by moving loads have been studied in connection with machining processes and applications in the behavior of railway tracks and bridges under moving loads
The dynamics of a beam moving over supports with clearance has not been reported in the literature
The present analysis is an extension of an earlier work (Lee, 1992) for analyzing the dynamics of a beam moving over multiple supports without clearance

Summary

INTRODUCTION

The mechanics of a stationary beam acted upon by moving loads have been studied in connection with machining processes and applications in the behavior of railway tracks and bridges under moving loads. The earliest work on the vibration of a beam subjected to a constant moving load was presented by Timoshenko (1922). The dynamics of a beam moving over supports with clearance has not been reported in the literature. The present analysis is an extension of an earlier work (Lee, 1992) for analyzing the dynamics of a beam moving over multiple supports without clearance. The supports in that article were modeled as linear springs with very large stiffness. Each support with clearance was modeled as a spring with piecewise-linear stiffness. This assumption removes the necessity to compute the normal forces at the supports. It will be shown in the formulation that the complexity of the formulation does not increase with an increased number of supports

THEORY AND FORMULATIONS

The velocity at the point is

The Lagrangian of the beam involving wean be expressed as

Sinusoidal Longitudinal Motions

CONCLUSION