Three dimensional vibrations of a beam on an elastic half-space: Resonance interaction of vertical longitudinal and lateral beam waves

Three-dimensional vibrations of a Euler-Bernoulli beam on an elastic half-space are investigated. In the model the beam has a finite width and the half-space and beam deflections are equal along the centre line of the beam. It is shown that the vertical and longitudinal beam vibrations are uncoupled from the lateral ones. The dispersion relations for the lateral and vertical-longitudinal waves in the beam are derived and the respective dispersion curves are plotted. These curves can cross each other due to the different equivalent stiffnesses of the half-space in vertical and lateral directions and different vertical and lateral bending stiffnesses of the beam. The existence of a crossing point implies that if the vertical-longitudinal and lateral beam vibrations are coupled for some reason (half-space inhomogeneity, beam asymmetry, etc.), the energy of the vertical vibrations of the beam can be resonantly transferred into the energy of lateral vibrations. This transfer will take place if the frequency of vibrations is close to the frequency determined by the crossing point. The dependency of the frequency of the crossing point on axial compressional stresses in the beam is studied. It is shown that this frequency decreases as the stresses increase.