On normal form calculations in impact oscillators

Normal form calculations are useful for analysing the dynamics close to bifurcations. However, the application to non-smooth systems is a topic for current research. Here we consider a class of impact oscillators, where we allow systems with several degrees of freedom as well as nonlinear equations of motion. Impact is due to the motion of one body, constrained by a motion limiter. The velocities of the system are assumed to change instantaneously at impact. By defining a discontinuity mapping, we show how Poincare mappings can be obtained as an expansion in a local coordinate. This gives the mapping the desired form, thus making it possible to employ standard techniques. All calculations are algorithmic in spirit, hence computer algebra routines can easily be developed.