Bifurcation and chaos in parametrically excited contact vibrations

The rolling contact problem between two contacting cylinders is analysed. The contact force between the bodies is assumed to be given by the Hertzian contact law. The equation of motion is derived in nondimensional form which is nonlinear and contains harmonic parametric excitation terms induced by the surface corrugations and the relative rotation between the cylinders. The periodic solutions are obtained by a numeric-analytical algorithm based on FFT and Galerkin error minimization. An arc length based path following technique in conjunction with a stability analysis is used to obtain the response curves and the bifurcation points. Flip bifurcation boundaries are obtained for different values of surface waviness amplitude in a two dimensional parametric plane consisting of damping and rotational speed parameter. For low values of damping and high values of amplitude of surface waviness, the system exhibits chaotic behaviour attained through a period doubling route.