Stability assessment and oscillation prediction in free-piston linear generator using an approximate analytical method

The application and deployment of high-efficiency, multi-fuel compatible Free-piston linear generators are hindered by the difficulty in rapidly analyzing the stability and performance of their inherently nonlinear dynamics. This study proposes an innovative approximate analytical method, which transforms the system dynamics from the time domain into the frequency domain using the describing function. Unlike conventional methods, the proposed method uniquely integrates describing function theory with a trust-region-reflective algorithm, which effectively solves the transcendental equations that typically limit analytical analysis. This fusion retains the physical insight of an analytical approach while ensuring a fast and robust solution. The developed model accurately predicts the stability boundaries, such as stall and collision, and reveals that design and operating parameters like moving component mass, fuel energy, and ignition position exhibit unique, coupled, and decoupled control authorities over the output response. The predictive capability of the method is rigorously validated against two structurally and parametrically distinct prototypes, including methanol and gasoline-fueled, with prediction errors consistently below 4% for both operating frequency and stroke. The proposed methodology provides a computationally efficient and reliable tool that bridges the gap between complex theory and practical engineering, facilitating the accelerated design, optimization, and control of FPLG systems.