Dynamic stability of frictional slip.

Abstract

Numerous authors have demonstrated that problems arise about the 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. The dynamic behavior of Klarbring's 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 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. By extending Klarbring's model to three-dimensional space, the elastically supported mass is allowed to have three translational degrees of freedom which can make contact with a rigid Coulomb friction support. Similar to the two-dimensional model, a critical coefficient of friction is identified above which the quasi-static solution can be non-unique. A numerical solution of the corresponding dynamic problem shows that the state realized then depends on the initial conditions. Even below the critical coefficient of friction, the dynamic solution can deviate from the quasi-static even for arbitrarily small loading rates. Typical dynamic responses include limit cycle oscillations in velocity, oscillations involving a brief period of zero velocity (stick), and 'stick-slip' motion in which the mass spends significant periods in a state of stick, interspersed with short rapid slip periods. All these non-steady motions involve non-rectilinear motion, even in cases when the time derivative of the applied load is constant in direction. A perturbation analysis is performed on the quasi-static frictional slip solution and predicts instability for certain slip directions, depending on functions of the off-diagonal stiffness components and the coefficient of friction. These results are presented in dimensionless form in stability diagrams. It is also shown that quasi-static slip that is stable when there is no damping can be destabilized if the damping matrix has sufficiently large off-diagonal components.