Exact vibration of Timoshenko beam combined with multiple mass spring sub-systems

Tamer A. El-Sayed*, Said H. Farghaly

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)


This paper deals with the analysis of the natural frequencies, mode shapes of an axially loaded beam system carrying ends consisting of non-concentrated tip masses and three spring-two mass subsystems. The influence of system design and sub-system parameters on the combined system characteristics is the major part of this investigation. The effect of material properties, rotary inertia and shear deformation of the beam system is included. The end masses are elastically supported against rotation and translation at an offset point from the point of attachment. Sub-systems are attached to center of gravity eccentric points out of the beam span. The boundary conditions of the ordinary differential equation governing the lateral deflections and slope due to bending of the beam system including developed shear force frequency dependent terms, due to the sub-system suspension, have been formulated. Exact formulae for the modal frequencies and the modal shapes have been derived. Based on these formulae, detailed parametric studies are carried out. The geometrical and mechanical parameters of the system under study have been presented in non-dimensional analysis. The applied mathematical model is presented to cover wide range of mechanical, naval and structural engineering applications.

Original languageEnglish
Pages (from-to)989-1014
Number of pages26
JournalStructural Engineering and Mechanics
Issue number6
Publication statusPublished - 25 Mar 2016


  • Combined system
  • Eccentric mass
  • Exact solution
  • Sub-system
  • Timoshenko beam
  • Vibration frequencies


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