Vibration of Continuous Systems, 2nd Edition

9Longitudinal Vibration of Bars

9.1 INTRODUCTION

A straight elastic bar can undergo longitudinal, torsional, and lateral vibration. Of these, the longitudinal vibration is the simplest to analyze. If x denotes the longitudinal (centroidal) axis and y and z represent the principal directions of the cross‐section, the longitudinal vibrations take place in the x direction, the torsional vibrations occur about the x axis, and the lateral vibrations involve motion in either the xy plane or the xz plane. These vibrations may be coupled in some cases. For example, if the cross‐section is not symmetric about the y or z axis, the torsional and lateral vibrations are coupled. If the bar is pre‐twisted along the x direction, the lateral vibrations in the xy and xz planes are coupled. We consider first the longitudinal vibration of a bar using a simple theory.

9.2 EQUATION OF MOTION USING SIMPLE THEORY

We consider a simple theory for the longitudinal vibration of bars based on the following assumptions:

1. The cross‐sections of the bar that were originally plane remain plane during deformation.
2. The displacement components in the bar (except for the component parallel to the bar's longitudinal axis) are negligible.

These assumptions permit the specification of the displacement as a function of the single space coordinate denoting location along the length of the bar. Although lateral displacement components exist in any cross‐section, the second assumption can be shown to be valid for ...