Impact Oscillations: Linear Theory Of Stability And Bifurcations

Abstract A unified approach to analysis of impact oscillations is proposed. Its mathematical essence is the continuous representation of the impulsive motion in some auxiliary variables. As a result an explicit formula for the fundamental matrix is obtained, which allows one to construct the characteristic equation. Then the stability and bifurcations can be studied by the usual techniques. The exceptional case of bifurcations, related to grazing impacts, is investigated by taking into account a non-zero impact duration. Due to this approach linearization is also possible, and the complex bifurcation becomes a sequence of ordinary ones. Some algebraic conditions are obtained which allow one to determine the type of the resulting motion. A mechanical example is considered: a disc with an offset centre of gravity bouncing on an oscillating surface.