Analytical optimization of displacer trajectory for ideal beta-type stirling engine cycles with sinusoidal piston motion

This study develops a generalized displacer motion equation for a beta-type Stirling engine. The proposed equation approximates the ideal Stirling cycle while maintaining sinusoidal piston motion to ensure stable power extraction. The displacer trajectory is modeled with a Fourier series and optimized. The resulting trajectory is then generalized as a piecewise function to improve applicability across different geometries. This approach improves control of working fluid distribution, allowing the expansion and compression processes to more closely approach isothermal conditions. The results show that the optimized Fourier trajectory achieves 91.9 % of the ideal Stirling-cycle work output, outperforming conventional drive mechanisms, where it only achieves 59.8 % for crank mechanism, 66 % for Scotch yoke, and 68.5 % for rhombic drive. For practical implementation, the optimized Fourier trajectory is generalized using a piecewise formulation. The generalized trajectory maintains approximately 80–90 % of the ideal Stirling-cycle work over a range of compression ratios without requiring re-optimization. These results demonstrate that the proposed approach provides both high thermodynamic performance and improved adaptability compared with conventional Stirling engine drive mechanisms.