Non-linear vibrations of a beam with cantilever-Hertzian contact boundary conditions

The non-linear vibrations of a linear beam with cantilever-Hertizian contact boundary conditions are investigated. The method of multiple scales is used to analyze this problem in which it is assumed that the beam remains in contact with the moving surface at all times. One primary result from this analysis is the amplitude-frequency relation for the various flexural modes. The amplitude-frequency curves exhibit softening behavior as expected. The amount of softening is shown to depend on the linear contact stiffness as well as the specific mode. In addition, the associated non-linear normal modes of this system are derived. The modes include a non-linear modification to the linear, harmonic component as well as a static offset term and second and third harmonic components.