A lumped parameter model is derived for studying hysteretic effects in resonant bar experiments on rock. The model uses equations of state obtained by approximating closed hysteresis loops in the stress-strain plane by parallelograms. The associated approximate nonlinear state relations have a sound speed (modulus) that takes two values. Assuming hysteresis and discrete memory to be the primary nonlinear mechanisms, periodic solutions corresponding to these equations of state are obtained analytically for single-frequency continuous wave drivers, and their frequency spectral densities are analyzed. In this simple approximation, if hysteretic contributions to the signal speed are a correction to the linear elastic signal speed (i.e., the parallelogram is narrow), the model predicts that the spectral density at even multiples of the source frequency is zero, and an approximate ''pairing'' of amplitudes is predicted for odd harmonic multiples. Comparison of the model spectrum with experimental data shows the model to be qualitatively correct. We conclude that hysteresis is an important mechanism in rocks. We consider the model to be a prototype. ¿ American Geophysical Union 1996