The dynamic behavior of free piston Stirling engines is investigated in order to determine whether nonlinear analysis can provide a means of improving engine design. In this study, we consider the nonlinearity contributed by the working fluid pressure. By modeling the thermodynamics of the working fluid with the Schmidt model, the system can be mathematically described by a set of coupled nonlinear second order ordinary differential equations. Each oscillating element will contribute one such equation. These equations are derived for a number of altered engine configurations; root loci, which treat the hot side temperature as a parameter, are constructed in order to determine the effect on the eigenvalues of the various configurations. These root loci reveal which of the altered configurations show promise as a working engine; the analysis revealed that the existing beta configuration and a double acting three cylinder configuration will work. Having observed that the governing equations for these engines are weakly nonlinear, we apply perturbation analysis to the promising engine configurations. In order to expedite the analysis, we make the simplifying assumption of proportional damping so that we can decouple the linear part of the system using modal analysis. The analysis yields analytical approximations describing the system dynamics while incorporating the nonlinear behavior of the engine. Since perturbation analysis only gives accurate results when the system is weakly nonlinear, these analytical approximations are compared to numerical integrations in order to evaluate for what amplitudes and what parameter values they are accurate.