Dynamic behavior and stability of simple fractional systems

Numerous authors have demonstrated that problems arise over existence and uniqueness of solution in quasi-static contact problems involving large coefficients of Coulomb friction. This difficulty was greatly elucidated by a simple two-degree-of-freedom model introduced by Klarbring. In the present paper, the dynamic behavior of Klarbringʹs model is explored under a wide range of loading conditions. It is demonstrated that the dynamic solution is always unique and deviates from the quasi-static only in a bounded oscillation for sufficiently low friction coefficients. Above the critical coefficient, slip in one of the two directions is found to be unstable so that the system never exists in this state for more than a short period of time compared with the loading rate. In the limit of vanishing mass, these periods become infinitesimal but permit unidirectional state changes with discontinuous displacements. A revised quasi-static algorithm is developed from this limit and is shown to predict the dynamic behavior of the system within a bounded oscillation for large coefficients of friction.