From: Hopf Instabilities in Free Piston Stirling Engines Date of download: 11/2/2017 Copyright © ASME. All rights reserved. From: Hopf Instabilities in Free Piston Stirling Engines J. Comput. Nonlinear Dynam. 2013;9(2):021003-021003-11. doi:10.1115/1.4025123 Figure Legend: Schematics of two different FPSE configurations are shown with the key parameters listed in Table 1. Heat addition takes place across the heater, heat removal occurs across the cooler, and the regenerator is in between these two elements. Additionally, the work extraction is represented by a damping element in the bounce space. (a) The FPSE in a β configuration. (b) The FPSE in a double acting α configuration; it was found that a minimum of three cylinders is required for a successful engine [4]. The expansion process and the compression process for each of the individual thermodynamic cycles is located in adjacent cylinders, which dynamically couples the associated pistons.

From: Hopf Instabilities in Free Piston Stirling Engines Date of download: 11/2/2017 Copyright © ASME. All rights reserved. From: Hopf Instabilities in Free Piston Stirling Engines J. Comput. Nonlinear Dynam. 2013;9(2):021003-021003-11. doi:10.1115/1.4025123 Figure Legend: Parametric plot of the nondimensional steady-state piston and displacer displacements. The parameter values are based on the Sunpower RE-1000, as reported in Ref. [2] and adapted to satisfy the condition of proportional damping. As predicted, in the limit, the system migrates to a limit cycle. The analytical approximation (dashed line) is shown for comparison with the numerical solution of Eq. (9) (solid line); the difference between the two is attributable to the neglected higher order terms.

From: Hopf Instabilities in Free Piston Stirling Engines Date of download: 11/2/2017 Copyright © ASME. All rights reserved. From: Hopf Instabilities in Free Piston Stirling Engines J. Comput. Nonlinear Dynam. 2013;9(2):021003-021003-11. doi:10.1115/1.4025123 Figure Legend: (a) Variations of the nondimensional piston amplitude (solid line) and amplitude ratio (dashed line). (b) Variation of the phase angle. (c) Variations of the corrected frequency (solid line) and uncorrected frequency (dashed line). (d) Variation of the nondimensional power. The variation in the parameter γ corresponds to a hot side temperature range of 585 ≤ Th < ∞ (K).

From: Hopf Instabilities in Free Piston Stirling Engines Date of download: 11/2/2017 Copyright © ASME. All rights reserved. From: Hopf Instabilities in Free Piston Stirling Engines J. Comput. Nonlinear Dynam. 2013;9(2):021003-021003-11. doi:10.1115/1.4025123 Figure Legend: (a) Variations of the nondimensional amplitude (solid line) and amplitude ratio (dashed line). (b) Variation of the phase angle. (c) Variations of the corrected frequency (solid line) and uncorrected frequency (dashed line). (d) Variation of the power output. The parameter αp is varied over a range corresponding to a mean pressure range of 57 ≤ Pm < 142 (bar).

From: Hopf Instabilities in Free Piston Stirling Engines Date of download: 11/2/2017 Copyright © ASME. All rights reserved. From: Hopf Instabilities in Free Piston Stirling Engines J. Comput. Nonlinear Dynam. 2013;9(2):021003-021003-11. doi:10.1115/1.4025123 Figure Legend: (a) Plot of the steady-state displacements of the three pistons with respect to time; all three plots are identical, except for a phase shift of 120 deg. The cylinder 1 response is depicted by a solid line, the cylinder 2 response is depicted by a bold dashed line, and the cylinder 3 response is depicted by a light dashed line. (b) The three-dimensional parametric plot of the steady-state displacements with the numerical solution is shown by using a solid line and the analytical solution is shown by using a dashed line.

From: Hopf Instabilities in Free Piston Stirling Engines Date of download: 11/2/2017 Copyright © ASME. All rights reserved. From: Hopf Instabilities in Free Piston Stirling Engines J. Comput. Nonlinear Dynam. 2013;9(2):021003-021003-11. doi:10.1115/1.4025123 Figure Legend: (a) Variation of the amplitude. (b) Variation of the frequency. (c) Variation of the power output. The variation of the parameter γ corresponds to a heater temperature range of 407 ≤ Th < ∞ (K).

From: Hopf Instabilities in Free Piston Stirling Engines Date of download: 11/2/2017 Copyright © ASME. All rights reserved. From: Hopf Instabilities in Free Piston Stirling Engines J. Comput. Nonlinear Dynam. 2013;9(2):021003-021003-11. doi:10.1115/1.4025123 Figure Legend: (a) Variation of the amplitude. (b) Variation of the frequency. (c) Variation of the power output. The variation in the parameter α corresponds to a charge pressure range of 0 ≤ Pm < 3.2 (bar).