Steady-state vibrations of an elastic beam on a visco-elastic layer under moving load

The steady-state response of an elastic beam on a visco-elastic layer to a uniformly moving constant load is investigated. As a method of investigation the concept of "equivalent stiffness" of the layer is used. According to this concept, the layer is replaced by a 1D continuous foundation with a complex stiffness, which depends on the frequency and the wave number of the bending waves in the beam. This stiffness is analyzed as a function of the phase velocity of the waves. It is shown that the real part of the stiffness decreases severely as the phase velocity tends to a critical value, a value determined by the lowest dispersion branch of the layer. As the phase velocity exceeds the critical value, the imaginary part of the equivalent stiffness grows substantially. The dispersion relation for bending waves in the beam is studied to analyze the effect of the layer depth on the critical (resonance) velocity of the load. It is shown that the critical velocity is in the order of the Rayleigh wave velocity. The smaller the layer depth, the higher the critical velocity. The effect of viscosity in the layer on the resonance vibrations is studied. It is shown that the deeper the layer, the smaller this effect.