Analytical investigation of free piston Stirling engines using practical stability method

The existence of a stable limit cycle in the dynamical system (sufficient condition) of free piston Stirling engines (FPSEs) is the main challenge for designers to achieve. Therefore, this paper provides a novel analytical and parametrical scheme based on the practical stability and dynamic error to evaluate this issue. In this regard, the dynamic error equations of the engine are first extracted based on the target design criteria. Afterward, the practical stability theorem is implemented on the dynamic error of the engine. The main finding of this approach has led to the provision of nine parametric conditions to satisfy the sufficient condition (i.e. the most important condition for a stable oscillation) of the FPSE. Indeed, these conditions help the designers to specify the allowable ranges for the engine parameters in which the sufficient condition is met for the first time. In other words, a novel parametrical design criterion is provided for FPSE. In addition to this important achievement, this novel technique has been able to present a design criterion (value of ultimate bound) to evaluate the design accuracy for the first time. According to this design criterion, minimizing the ultimate bound value results in a more accurate engine design. In this regard, increasing the power piston mass, hot gas temperature, and mass of the gas inside the engine helps the designers get closer to their design targets. Besides, increasing the value of stiffness and mass of the displacer piston creates an optimum condition in which the value of the ultimate bound of the engine is reached its minimum amount. Finally, the presented method is then evaluated through the specifications of two experimental case studies, including SUTech-SR-1 and B10-B. The outcomes showed that this technique could well predict the sufficient condition of the FPSEs.