Singularities in soft-impacting systems

In this paper, the character of the normal form map in the neighbourhood of a grazing orbit is investigated for four possible configurations of soft impacting systems. It is shown that, if the spring in the impacting surface is relaxed, the impacting side of the map has a power of 3/23/2, but if the spring is pre-stressed the map has a square root singularity. The singularity appears only in the trace of the Jacobian matrix and not in the determinant. Under all conditions, the determinant of the Jacobian matrix varies continuously across the grazing condition. However, if the impacting surface has a damper, the determinant decreases exponentially with increasing penetration.It is found that the system behaviour is greatly dependent upon a parameter mm, given by 2ω0/ωforcing, and that the singularity disappears for integer values of mm. Thus, if the parameters are chosen to obtain an integer value of mm, one can expect no abrupt change in behaviour as the system passes through the grazing condition from a non-impacting mode to an impacting mode with increasing excitation amplitude. The above result has been tested on an experimental rig, which showed a persistence of a period-1 orbit across the grazing condition for integer values of mm, but an abrupt transition to a chaotic orbit or a high-period orbit for non-integer values of mm. Finally, through simulation, it is shown that the condition for vanishing singularity is not a discrete point in the parameter space. This property is valid over a neighbourhood in the parameter space, which shrinks for larger values of the stiffness ratio k2/k1k2/k1.

► The character of the normal form map near grazing is investigated for four configurations of soft impacting systems. ► If the spring is pre-stressed, the map has a square root singularity. Else the map has a power of 3/23/2 on one side. ► The singularity appears only in the trace and not in the determinant of the Jacobian. ► The system behaviour is dependent upon parameter m=2ω0/ωforcing, and the singularity disappears for integer values of mm. ► This disappearance of the singularity has been validated experimentally.