## 2 Dynamic Model of FPSE  
Figure 1 presents a conceptual diagram of a beta-type FPSE with an external load: (a) schematic layout of the engine and (b) free-body diagram of the pistons. The displacer piston (DP) and power piston (PP) are located on the same axis in one cylinder, and assembled, respectively, with a spring element--normally in the form of metal flexure spring. The heater and cooler supply and remove heat through their respective heat exchangers. The regenerator is positioned in between to improve thermal efficiency by preheating and precooling the working gas. The DP divides the working space into expansion and compression spaces. The DP motion circulates the working gas through heat exchangers, yielding the thermodynamic heating and cooling processes. The PP divides the total sealed space into the working space and bounce space. The PP motion produces power output of an external load device via the thermodynamic expansion and compression processes. The FPSE has two vibration systems of PP and DP. The two piston motions are coupled by the internal pressure, forming a two-DOF vibration system. Equations of motion (EOMs) are established from the free body diagram of the two pistons considering the internal pressure, spring force, and damping force, as shown in Figure 1b, and are given as follows:  
\[ \begin{array}{c}m_{p}\ddot{x}_{p}+c_{p}\dot{x}_{p}+k_{p}x_{p}=-\big{(}A_{p}- A_{r}\big{)}(P-P_{B})\\ m_{d}\ddot{x}_{d}+c_{d}\dot{x}_{d}+k_{d}\dot{x}_{d}=-A_{r}(P-P_{B})\end{array}  \tag{1}\]  
where \(m\), \(c\), and \(k\) denote the piston mass, damping coefficient, and stiffness coefficient, respectively, and the subscripts \(p\), \(d\), and \(r\) denote the PP, DP, and DP rod, respectively. The mechanical friction damping coefficients are expressed as \(c_{p}\) and \(c_{d}\), respectively. \(P\) and \(P_{B}\) are the instantaneous pressure values in the working space and the bounce space, respectively. The working space is assumed to have a uniform pressure [19]. The heat exchangers and regenerator have a flow friction force acting on the DP motion directly, while the external loads have a resistance force acting on the PP motion. In this study, only major direct damping is considered, such as gas flow friction damping of the heat exchangers acting on the DP and the external load damping acting on the PP. The minor friction damping caused by flow resistance at a narrow gap between the piston and cylinder is ignored [20]. The EOMs show that the FPSE is a forced response system in which the internal pressure acts as an excitation force. To solve the EOMs, the pressure terms are defined as a function of piston displacements using the state equation of ideal gas. The pressure and  
Figure 1: Conceptual diagram of a beta-type free-piston Stirling engine (FPSE) with an external load: (**a**) schematic layout of the engine and (**b**) a free-body diagram of the pistons. temperature in each working space are determined based on the ideal isothermal model [21], where the heat exchangers and regenerator are assumed to be perfectly effective and the temperatures are constant in each space. In addition, the engine working space is separated from the bounce space without leakage of the working gas [21]. The following formulation of equations for the pressure definition and linearization is based on Reference [11]. The ideal gas equation associates the working pressure with the volume changes in each working space, as follows:  
\[P=m_{gas}R_{gas}\bigg{(}\frac{V_{H}}{T_{H}}+\frac{V_{E}}{T_{E}}+\frac{V_{R}}{ T_{R}}+\frac{V_{C}}{T_{C}}+\frac{V_{K}}{T_{K}}\bigg{)}^{-1} \tag{2}\]  
where \(T\) and \(V\) are the temperature and the volume of the working spaces, respectively. The subscripts \(H\), \(K\), \(E\), \(C\), and \(R\) denote the spaces of the hot heat exchanger, cold heat exchanger, expansion, compression, and regenerator, respectively. Further, \(m_{gas}\) and \(R_{gas}\) are the mass and gas constant of the working gas, respectively.