Free Vibration Studies of Classical Beams/Rods with Lumped Masses at Boundaries Using an Approach Based on Wave Vibration

Abstract

Vibrations in beams/rods with lumped masses at boundaries are normally not covered in introductory vibration textbooks. An effort is made here to formulate a concise and systematic approach to the study of the relatively complex vibration problems pertaining to such structures. An exact analytical solution is obtained for bending, torsional, and longitudinal vibration analysis based on a wave vibration approach. The reflection matrices corresponding to bending, torsional, and longitudinal incident vibration waves at the end mass are derived from classical vibration theories; these are then assembled with propagation matrices and reflection matrices at classical boundaries to form a systematic approach to the analysis of free vibration. Numerical examples are presented, comparisons with results available in the literature are made and good agreements are reached.