One of the most novel applications of the Stirling cycle is in the free piston configuration that was initially designed by W. Beale. In free piston Stirling engines (FPSEs), there are no mechanical linkages coupling the pistons or displacers, the motions of the reciprocating components follow the working gas pressure variations. Fillipo de Monte and G. Benvenuto have recently proposed a linearization technique of the dynamic balance equations. The aim of this paper is to predict the thermodynamic conditions for stable operation of FPSEs and their modeling. The equations of the angular velocity are solved analytically in terms of the working gas mass and the displacer-piston phase angle of the machine. Using the criterion of stable engine cyclic steady operation, a mathematically rigorous form is obtained for the main parameters of the engine. Furthermore, for simplicity reasons, thermodynamic magnitudes are obtained using the Schmidt analysis (isothermal model). (C) 2003 Elsevier Ltd. All rights reserved.