On the use of variable-separation method for the analysis of vibration problems with time-dependent boundary conditions Articles uri icon

publication date

  • December 2012

start page

  • 2912

end page

  • 2924


  • C12


  • 226

International Standard Serial Number (ISSN)

  • 0954-4062

Electronic International Standard Serial Number (EISSN)

  • 2041-2983


  • In this article, the axial vibrations of a rod with a clamped end and the transversal vibrations of a cantilever beam, both with a time-dependent and non-harmonic force applied on their free ends, are analysed. These are problems in which the traction and the shear, for the rod and the beam, respectively, prescribed in the boundaries of the bodies vary with time. The problems can be solved by the method proposed by Mindlin and Goodman. However, it is usual to solve this problem by the classic variable-separation method (which does not properly fulfil the time-dependent boundary conditions). The displacements and the forces along the systems are derived from both cited methods, and the results are compared. These results highlight the importance of using the proper solution method for the vibration problems with time-dependent boundary conditions.


  • Industrial Engineering
  • Mechanical Engineering


  • axial vibrations; transversal vibrations; time-dependent boundary conditions; natural frequencies; variable-separation method