An averaging-based Lyapunov technique to design thermal oscillators: A case study on free piston Stirling engine

This article introduces a novel averaging-based Lyapunov approach to design complex oscillators such as free-piston Stirling engines (FPSEs). First, a class of thermal oscillators namely the FPSE is introduced. Then, the existence of limit cycle in dynamic response of the oscillator, which is a required criterion for steady operation, is investigated via the proposed technique. Accordingly, a systematic way containing two conditions is presented to justify the performance of the oscillator system. As the preliminary condition, instability of the equilibrium state of the oscillator is studied through the proposed averaging-based Lyapunov theorem (necessary condition). Subsequently, as the second condition, the existence of stable limit cycle in the oscillator response is investigated through the mentioned method (sufficient condition). Next, some useful equations for effective prediction of the oscillator performance are presented using the proposed averaging-based Lyapunov method. Finally, the simulation outcomes, as well as the experimental data, clearly reveal the effectiveness of the proposed methodology in designing the free piston Stirling oscillator, which can be further extended to other oscillators with periodic response such as engines, energy harvesters, vibration/acoustic isolators, and resonant absorbers as well.