Longitudinal Vibration Analysis of a Stepped Nonlocal Rod Embedded in Several Elastic Media

Tamer Elsayed* (Corresponding Author), Moustafa S. Taima, Said H. Farghaly

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)
2 Downloads (Pure)


Mechanical properties of 1D nanostructures are of great importance in nanoelectromechanical systems (NEMS) applications. The free vibration analysis is a non-destructive technique for evaluating Young's modulus of nanorods and for detecting defects in nanorods. Therefore, this paper aims to study the longitudinal free vibration of a stepped nanorod embedded in several elastic media.

The analysis is based on Eringen’s nonlocal theory of elasticity. The governing equation is obtained using Hamilton’s principle and then transformed into the nonlocal analysis. The dynamic stiffness matrix (DSM) method is used to assemble the rod segments equations. The case of a two-segment nanorod embedded in two elastic media is then deeply investigated.

The effect of changing the elastic media stiffness, the segments stiffness ratio, boundary conditions and the nonlocal parameter are examined. The nano-rod spectrum and dispersion relations are also investigated.

The results show that increasing the elastic media stiffness and the segment stiffness ratio increases the natural frequencies. Furthermore, increasing the nonlocal parameter reduces natural frequencies slightly at lower modes and significantly at higher modes.
Original languageEnglish
Pages (from-to)1399–1412
Number of pages14
JournalJournal of Vibration Engineering & Technologies
Issue number4
Early online date28 Mar 2022
Publication statusPublished - 1 Jun 2022


  • Nanorod
  • Stepped rod
  • Exact solution
  • Elastic media
  • Continuum mechanics
  • Dynamic stiffness matrix


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