Dynamic stability of an axially loaded beam on an elastic foundation with damping

The subject of this research is the dynamic stability of a simply supported beam which is subjected to a periodic axial load and supported laterally by an elastic foundation with damping. The Winkler, Hetényi and Pasternak foundation models are modified to include the viscous damping of the supporting medium. Stability studies are performed for the 1 2 subharmonic response by determining the conditions on the dynamic load which produce a periodic solution to the resulting differential equation of motion. Approximations to the critical dynamic loads and the regions of instability are employed to discuss the boundedness of the lateral displacements of the beam. Viscous damping is emphasized in the studies, which illustrate the effect of the foundation properties on the stability behavior. The results indicate that an increase in the foundation stiffness or damping increases the critical dynamic load and shifts the region of instability to a higher applied frequency. The mode of vibration associated with the smallest critical dynamic load is shown to vary with both the foundation stiffness and the flexural rigidity. A comparison of the Winkler model results with those obtained from the Pasternak or Hetényi models shows that the inclusion of the foundation's shear or flexural stiffness properties increases the effect of damping.