499 JOURNAL OF PROPULSION AND POWER Vol. 14, No. 4, July – August 1998 Re ections on Free-Piston Stirling Engines, Part 1: Cyclic Steady Operation F. de Monte* University of L’Aquila, L’Aquila 67040, Italy and G. Benvenuto† University of Genoa, Genoa 16145, Italy In spite of the conceptual simplicity, the design of the free-piston Stirling engines (FPSEs) is made dif cult by the necessity to accurately foresee the effect of the various geometric, dynamic, and ther- modynamic variables on their behavior. This paper describes a fully developed mathematical model able to characterize these machines taking into account all of the relevant physical aspects involved, including the casing motion. A closed-form solution of the governing equations is used, together with a theorem developed by the authors to derive a basic criterion for the FPSE cyclic steady operation. Because this criterion is expressed in an analytical form as a function of the different variables involved, it is possible to choose the engine design parameters to ensure a steady periodic state. The developed model may be used not only for design purposes but also to simulate theoretically the dynamic behavior of a built engine. Nomenclature A = cross-sectional area for moving element D = damping coef cient $ = damping matrix F = force I = unit matrix j = operator de ned as 21 Ï L = Laplace transform operator L 21 = inverse Laplace transform operator M = moving element mass p = pressure R = residue Re[s] = real part of s r = displacer-piston stroke ratio, X d /X p r c = casing-piston stroke ratio, X c /X p S = stiffness coef cient 6 = stiffness matrix S c = stiffness of the mounting springs s = complex number s k = kth root (eigenvalue) of W(s ) t = time V = volume V Ç = volumetric ow rate W(s ) = Wronskian X = moving element stroke x = displacement x = displacement vector x* = displacement de ned in the text a, b, g, d = coef cients de ned in the text Dp = pressure drop D x = constant de ned in the text m dc = displacer-casing mass ratio, M d /M c m pc = piston-casing mass ratio, M p /M c Received July 31, 1997; revision received Feb. 7, 1998; accepted for publication Feb. 16, 1998. Copyright Q 1998 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. *Ph.D., Assistant Professor, Dipartimento di Energetica, Localita ´ Monteluco. Member AIAA. †Full Professor, Dipartimento di Ingegneria Navale e Tecnologie Marine, Via Montallegro 1. f = piston-displacer phase angle fc = casing-displacer phase angle C = pressure drop per unit of volumetric ow rate v = angular frequency Subscripts b = bounce space c = compression space, casing d = displacer e = expansion space gs = gas spring H = gas spring hysteresis losses ld – l = load device– load subsystem lm = laminar ow in the regenerator p = piston r = displacer rod s = stop (no-running machine) tr = turbulent ow in the heater and cooler w = working gas, working gas circuit 0 = initial Superscripts Ç = rst-order derivative with respect to time ¨ = second-order derivative with respect to time ¯ = average over a cycle 9 = per unit of moving element mass ˜ = Laplace transformed Introduction C OMPARED with kinematic Stirling engines, the free-pis- ton Stirling engines (FPSEs) of recent design present some inherent features such as 1) a simpler mechanical design because of the absence of the drive mechanism; 2) the possi- bility to avoid any side force on the piston and, also, by using exure bearings, the possibility to eliminate all rubbing parts, reducing engine wear 1 ; 3) a high-energy conversion (thermal to mechanical) ef ciency; and 4) an easy starting and ef cient, reliable, and fail-safe power control 2 that make them particu- larly suitable for those applications for which the highest re- liability is basic. These applications include, for instance, remote area power generators, space power systems, and dish solar terrestrial sys-