Acta Mechanica Sinica

, Volume 34, Issue 4, pp 676–688 | Cite as

Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects

  • Hai-Sheng Zhao
  • Yao ZhangEmail author
  • Seng-Tjhen Lie
Research Paper


Considerations of nonlocal elasticity and surface effects in micro- and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions: hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short, explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.


Fredholm integral equation Natural frequency Nonlocal elasticity Surface effects Timoshenko beam 



The authors would like to thank the School of Civil and Environmental Engineering at Nanyang Technological University, Singapore for kindly supporting this research topic.


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Civil and Environmental EngineeringNanyang Technological UniversitySingaporeSingapore

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