Loosely connected sructures may rattle: free play due to design tolerances, wear, or manufacturing error causes drastic force discontinuities (impacts) in periodically driven mechanisms. Previous investigators have examined impact oscillators described by linear equations in the non-impact regions, and have shown the existence of periodic limit cycles and chaotic motion. Experiments have confirmed the nature of their results. However, many useful systems, such as robot arms or linkages, include strongly nonlinear effects away from the impact condition. In order to study the effect of nonlinearity between impacts the linear restoring force of a rattling model is replaced with a sinusoidal, or pendulum-type, restoring force. A modified time grid is used to locate and model impacts accurately in numerical simulations, and a number of alternative Poincare maps are used to describe the complex motion exhibited by such innocent-looking systems.