Abstract: In this paper a new linearization technique of the dynamic balance equations of a free-piston Stirling machine is developed. It takes into account the non-linear thermo-fluid-dynamic terms inherent in the machine, while keeping the linearity of the differential dynamic equations. This allows the equations of motion to be solved analytically and, therefore, it allows the algebraic relations linking the various machine parameters, established in a companion paper, to be suitably used. The casing motion is also considered. The following advantages are related to the proposed linearization methodology: 1) it gives the correct interpretation of the machine response to variations in the operating conditions because the considered non-linear terms have a stabilizing effect that cannot be ignored; 2) it can be used to predict the machine performance not only with more accuracy than conventional linear dynamic analyses that fully neglect the non-linearities, but in a more exhaustive way, allowing the piston stroke and, therefore, the delivered power to be calculated; 3) it enables the machine to be designed in the initial stages in such a way as to enhance its inherent stability. To illustrate these features, we have considered, as an example of a free-piston Stirling engine, the well known Space Power research Engine.