Steady state responses of an infinite beam resting on a tensionless visco-elastic foundation under a harmonic moving load

Abstract

In this paper we study the steady state responses of an infinite beam resting on a tensionless visco-elastic foundation under a harmonic point force moving with a constant speed. The equation of motion is formulated with respect to a coordinate system moving with the point load. The beam and the foundation are defined as in contact when both the deflection and the “contact force” are negative. We replace the infinite beam by a finite beam with sufficient length and expand the beam deflection into a harmonic series. Newmark method is used to integrate the discretized equation. For a tensionless foundation, the contact condition between the beam and the foundation changes with time. Therefore, the matrices involved in the discretized equation of motion must be updated at every time step. In the bifurcation diagram we change the excitation frequency and record the beam deflection at the loading point via Poincaré sampling. When the dimensionless amplitude of the harmonic force is high, say 15, one can observe periodic, quasi-periodic, and chaotic solutions. We have also found that reducing the amplitude of the harmonic component or adding a time-invariant component can simplify the bifurcation diagram to include only periodic solutions.