Two-layer model of the railway track: Analysis of the critical velocity and instability of two moving proximate masses

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In this paper, the widely used two-layer model of the railway track is analysed with focus on the critical velocity and instability of moving masses. The model is assumed to be infinite with no changes in properties in the longitudinal direction. As far as the new demonstrations and conclusions: it is shown that in the undamped case, the onset of instability matches the critical velocity, and this is valid for one or more moving masses. However, in a damped case, the situation is more complicated. Instability of one moving mass occurs always in supercritical velocity range, but when two proximate masses are moving, then under certain conditions, the dynamic interaction between them together with the damping can shift the onset of instability into the subcritical velocity range. This newly derived negative effect has already been identified for Pasternak viscoelastic foundation in previous author's work. All results in this paper are presented in closed-form semi-analytical way for dimensionless parameters. The results presented are validated by eigenvalue expansion method on long finite model. For this purpose, vibration modes and orthogonality condition are derived. Modal equations are coupled and therefore rearrangement of the terms involved is introduced to save computational time. An excellent agreement between the results obtained on the infinite and finite models is achieved, which validates the newly derived formulas and the conclusions drawn from them. In order to simplify the expressions derived, continuous supports are implemented, because differences in behaviour would only be noticeable under high frequencies. This was checked by finite element analysis in commercial software. Unlike the more common usage of the two-layer model, shearing springs are added providing bases for further generalisations.

Original languageEnglish
Article number107042
Pages (from-to)1-25
Number of pages25
JournalInternational Journal of Mechanical Sciences
Publication statusPublished - 1 Mar 2022


  • Contour integration
  • Critical velocity
  • Dynamic interaction
  • Instability
  • Mass-induced frequency
  • Moving masses
  • Semi-analytical solution
  • Two-layer model


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