The nonlinear response of a one-dimensional oscillator with two-sided amplitude constraints or with a single-sided amplitude constraint which is preloaded against a stop and subjected to nonharmonic excitation is investigated. Positive clearance and preload systems are discussed. The amplitude and stability of the periodic responses are determined and a bifurcation analysis of these motions is carried out. Period-doubling bifurcations and degenerate impacts occur in our model. Some parametric regions are shown to possess chaotic motions. The stable linear motion can coexist with stable nonlinear motion or transient chaos. It is found that the degenerate impact can cause a sudden change in the response structure not only to a stable motion but also to chaos.