Prediction on onset conditions of alpha, beta, and gamma type free piston Stirling generators

The study of the onset characteristics of the free piston Stirling generator (FPSG) is essential for its development. In this study, the onset characteristics of the FPSG were discussed based on a thermodynamic model using the Routh stability criterion. First, the onset conditions of three types of FPSG, alpha, beta, and gamma, were analyzed. Then an experimental and numerical study of a 100 W beta‐type FPSG prototype was conducted. The impacts on the onset temperatures of spring stiffness, charging pressure, and external load were further explored. The results demonstrate that no matter how the parameters are altered, alpha‐type FPSG cannot start up. Whether beta and gamma type FPSG can start up depends on the design and operating parameters. For the prototype, as the displacer spring stiffness increases, the onset temperature rises first and then drops. When the displacer spring stiffness is 19.2 N/mm, the corresponding onset temperature is the lowest of 489 K. The FPSG is easier to start up at a higher external load and higher charging pressure. The experimental results of the impacts of the spring stiffness, charging pressure, and external load on the onset temperature agree well with the calculated results, with errors ranging from 1.7% to 9.12%. The proposed approach can play a guiding significance in studying FPSG onset conditions.

crankshaft mechanisms that connect the displacer and piston in the kinematic Stirling generator. 7 As a result, the FPSG has a simpler structure than a kinematic Stirling generator, which leads to lower noise, greater reliability, and a longer lifespan. Based on the arrangement of the displacer and piston, FPSG can be categorized into three types: alpha, beta, and gamma, as shown in Figure 1. 8 Two opposing pistons are situated in two cylinders in the alpha-type FPSG. 9 The displacer and piston of the beta-type FPSG are coaxial and housed in the same cylinder. 10 For gamma-type FPSG, the displacer and piston are positioned in two cylinders in parallel. 11 There are many theoretical studies of FPSG have been conducted using thermodynamic or dynamic approaches. 12 The articles which investigate the dynamic characteristics of FPSG are primarily focused on the output performance prediction and performance optimization of the generator. 13 Chmielewski et al. 14 developed a multidomain simulation model based on dynamics. The influence of thermodynamic and dynamic parameters such as the mass of the piston and hot end temperature on the theoretical output power were analyzed which had practical information. Formosa 15 developed a semianalytical dynamic model with the consideration of regenerator effectiveness and the pressure drop of the heat exchangers. The model was validated by the experimental data of the NASA RE-1000 Stirling engine prototype, indicating that the approach can be effective for the preliminary design of FPSG. Ye et al. 16 built an artificial neural network model to predict the dynamic behavior of the FPSG. The influence of dynamic parameters on the operating frequency, amplitude ratio, and phase angle was analyzed. The errors of the above-mentioned operating parameters in the simulation were 0.85%, 2.78%, and 3.19% compared with experimental data. Zare and Tavakolpour-Saleh 17 proposed a frequency-based design approach for FPSG with the usage of a genetic algorithm. The optimum values of design parameters such as the spring stiffness of the displacer and piston were obtained, as well as the displacer rod diameter. And the experiment results support the applicability of this approach. Majidniya et al. 18 investigated a nonlinear thermodynamic model for the free-piston Stirling engine with a linear synchronous machine and experimentally validated the model. However, sufficient conditions were not investigated.
Although dynamic characteristics of the FPSG in stable operation have been deeply analyzed, the in-depth analysis of the onset characteristics of FPSG is lacking. However, the study of the onset characteristics is vitally significant in the development of FPSG. Due to the strong combination of thermodynamics and dynamics of FPSG, 19 whether or not FPSGs can start up becomes a serious concern. 20 By studying the onset characteristics of FPSG, the key factors affecting the onset process can be understood, and the FPSG's ability to work can be predicted, avoiding time and money waste. Riofrio et al. 21 proposed a method to design the FPSG. The root locus analysis, gain, phase margin, and Nyquist stability criterion were used to ensure the generator had an unstable oscillatory response. Sim et al. 22 developed a dynamic model of FPSG and analyzed its instability. However, they did not deeply study the onset characteristics of FPSG. Zare and Tavakolpour-Saleh 23 discussed the onset conditions of the FPSG by using the averagingbased Lyapunov technique. They used energy variations as the basis for instability judgments and studied the impact of different parameters, such as the crosssectional area of the piston and displacer rod, the spring stiffness, and the mass of the displacer and piston, on the starting conditions. However, the interaction of these parameters was not taken into account, and the onset model is complicated. Mou and Hong 20 proposed three necessary conditions for the startup of FPSG and researched the startup mechanism and power distribution of the FPSG. The conclusion that alpha FPSG cannot start was first presented. But they did not further investigate the effect of key parameters on the onset characteristics. Additionally, there are few articles on the onset mechanism of alpha, beta, and gamma types of FPSGs.
In the present study, a simpler approach is proposed to analyze the onset conditions of FPSG. By adopting the Routh stability criterion, the onset characteristics will be acquired based on developed thermodynamic models of the FPSGs. First, the onset conditions of alpha, beta, and gamma-type FPSGs are analyzed. Second, the onset range of each parameter is investigated separately using a 100 W FPSG prototype. In addition, experiments are carried out and analyses of the impacts of displacer spring stiffness, piston spring stiffness, charging pressure, and external load on the onset temperature are made. The purpose of this study is to better understand the onset mechanisms of the FPSG.

| Method description
In FPSG, both the displacer and piston are affected by spring, gas, and damping, as shown in the following. The forces on the displacer and the piston follow the expressions regardless of the type of alpha, beta, or gamma FPSG, the difference being that the expression for the gas force changes.
Among them, the gas force is closely related to the working pressure. The working pressure fluctuates due to the movements of the displacer and piston. The Schmidt isothermal model is incorporated into the ideal gas equation, and the equation of working pressure is obtained as follows: Linearizing Equation (3) and letting C 1 and C 2 represent the linearized coefficients of pressure variation with respect to the volume variation of the expansion and the compression spaces.
Based on the structure of the FPSG, the volumes of expansion and compression spaces are expressed as functions of the motion of the displacer and piston. Substituting the above functions and Equation (5), Equation (6) into Equation (4), the working pressure as a function of the displacer and piston motions are obtained, and thus the expressions for the gas force are obtained. Therefore, Equation (1) and Equation (2) can be transformed into the following form of state space, and the initial condition is listed.
The general form of matrix A s is described in the following. The unknowns in the matrix are represented by the structure parameters or operating conditions of the FPSG.
pp pd dp dd dp dd (9) Finally, the characteristic equation can be obtained in the following.
According to the Routh stability criterion, the necessary and sufficient condition for system stability is: (1) The coefficients of the characteristic equation are positive; (2) The values in the first column of the Routh matrix composed of characteristic equation coefficients are positive. In other words, if the generator meets conditions in the following, the generator cannot start up. According to the above-mentioned method, the onset characteristic of alpha, beta, and gamma-type FPSGs can be investigated. And the onset range of each parameter can be investigated separately. Furthermore, the impacts on the onset temperature of different parameters can be explored.

| Alpha type FPSG
A schematic representation of alpha-type FPSG is given in Figure 2, showing (A) the schematic layout of the FPSG and (B) the force schematic diagrams of the displacer and piston. The alpha-type FPSG consists of bounce space, displacer, piston, alternator, expansion space, compression space, heater, cooler, and regenerator. The two cylinders in the alpha-type FPSG are independent, and the displacer motion is independent of the piston motion.
According to Figure 2B, the equations which describe the dynamic behavior of the piston and displacer are given: The volumes of expansion space and compression space are related to the motion of the displacer and piston.

| Beta type FPSG
A conceptual diagram of beta-type FPSG is given in Figure 3. Figure 3A shows the schematic layout of the FPSG and Figure 3B demonstrates the force schematic diagrams of the displacer and piston. Expansion space, compression space, bounce space, heater, cooler, regenerator, displacer, piston, and alternator are all included in the beta-type FPSG. The piston and displacer are arranged in the same cylinder, and the volume of compression space is influenced by the motion of both the displacer and piston. Figure 3B shows the force schematic diagram of the displacer and piston and the force equilibrium equations are given: According to Figure 3, the volume of expansion space and compression space can be calculated.
The linearized working pressure is obtained with the motion of the displacer and piston as variables.

| Gamma type FPSG
A schematic representation of gamma-type FPSG is given in Figure 4, showing (a) the schematic layout of the FPSG and (b) the force schematic diagram of the displacer and piston. There are several components in the gamma type FPSG, expansion space, compression space, bounce space, heater, cooler, regenerator, displacer, piston, and alternator. The configuration of gamma type FPSG is similar to that of beta type FPSG, and the main difference between the two types is whether the piston and displacer are arranged in one cylinder. Figure 4B shows the force schematic diagram of the displacer and piston of gamma type FPSG and the equations are given: According to Figure 4, the volumes of expansion space and compression space are given as follows: The linearized working pressure can be expressed in the following.
Substituting Equation (25) into Equations (21) and (22), and the unknowns in Equation (9) can be obtained, as follows: That is to say, the alpha-type FPSG always meets the Routh stability criterion, failing to start up. An alpha-type FPSG has been built in our laboratory (shown in Figure 5), the detailed design parameters are shown in Table 1. Regardless of the design and operation parameters changes, the alpha-type FPSG did not start up. This result supports the feasibility of the theoretical method.

| Beta type FPSG
Using the aforesaid method, whether or not the beta type FPSG met the Routh stability criterion was analyzed. The results show that only when parameters of beta type FPSG meet some conditions, can the generator start up (details are shown in Appendix B).
A 100 W beta-type FPSG prototype developed in the key laboratory of technology on space energy conversion is used as the research object. The detailed parameters of this prototype are shown in Table 2 and the photograph is shown in Figure 6. It was found that only when the prototype meets certain conditions, could the prototype start up. The results effectively support the above analysis. The detailed onset conditions are investigated in the next section.

| Gamma type FPSG
The configuration of gamma-type FPSG is similar to that of beta-type FPSG, as well as the force schematic diagram, which leads to the similarity in the onset F I G U R E 5 Alpha type free piston Stirling generator prototype.

| Onset characteristics of the FPSG
To further comprehend the onset characteristics of the FPSG, experiments, and calculations are carried out by using the prototype shown in Figure 6. The impact of hot end temperature T h , displacer spring stiffness k d , piston spring stiffness k p , charging pressure P 0 , and external load R o on the onset characteristics are investigated. Figure 7 illustrates the relationship between the instability of FPSG and hot end temperature. According to the above-mentioned Routh stability criterion, the FPSG can start up as long as either value is less than zero. As shown in Figure 7, as the hot end temperature rises, the value of delta 3 falls, which indicates that the generator tends to become unstable and can withstand larger damping. When the hot end temperature reaches the onset temperature of 557 K, delta 3 is smaller than zero, indicating that the generator satisfies the onset conditions. The experimental onset temperature under the same conditions is 591 K with an error of 5.7%. The main reason for the error is the idealism of the model.
It is worth noting that the magnitudes of the seven variables d 1 , d 2 , d 3 , d 4 , delta 1 , delta 2, and delta 3 are so different that the data for the vertical coordinates are processed for ease of presentation in the figures. The processing is: If d 1 > 0, value = lg(d 1 ); If d 1 < 0, value = −lg(−d 1 ); The other six parameters are handled in the same way and this processing is used for the data in Figures 8a, 9a, 10A, and 11A. Figure 8 displays the impacts of the spring stiffness of the displacer on the onset characteristics. As shown in Figure 8A, there is an onset range of the displacer spring stiffness, above which the dynamic stability of the system results, causing the generator fails to start up. The onset range of the displacer spring stiffness is from 11.6 to 42.5 N/mm at the hot end temperature of 800 K. As seen in Figure 8B, the calculated onset temperature drops first and then starts to climb as the displacer spring stiffness rises. When the displacer spring stiffness is 19.2 N/mm, the calculated lowest onset temperature is 489 K. This is because the phase difference and the displacer spring stiffness are tightly connected. There is an optimal phase difference that minimizes the heat required for the generator to start up. Experiments are carried out to test the onset temperature of the displacer spring stiffness at 14.5, 21.6, and 28.8 N/mm, respectively. The onset temperature is 530 K at a displacer spring stiffness of 21.6 N/mm, which is the lowest value. The onset temperature variation trends in the experiments are consistent with the calculated one with an error of 1.7%-6.4%. Figure 9 displays the impacts of the spring stiffness of the piston on the onset characteristics. It is depicted in Figure 9A that the system enters an unstable state when the piston spring stiffness is within a certain range, allowing the FPSG to start up. When the piston spring stiffness is more than 42.5 N/mm, the FPSG cannot onset. This is due to the fact that when the spring stiffness of the piston increases, more heat is required for the gas force to push the piston. The hot end temperature in Figure 9A is 800 K, which cannot provide enough energy. This situation is well displayed in Figure 9B. As the piston spring stiffness increases, the onset temperature of the generator gradually rises during the simulation. The experimental onset temperature of three-piston spring stiffness shows a trend that consistent with the simulation. The relative errors between the experimental and calculated results are 3.23%, 9.12%, and 5.2%, respectively. Figure 10 illustrates the numerical impact of the charging pressure on the onset characteristics. As depicted in Figure 10a, when the hot end temperature is 800 K, the generator can start up when the charging pressure rises to 2.2 MPa. However, when the charging pressure rises to a certain value, the generator will enter a state where it can no longer start up. Since the higher the charging pressure, the more working gas is involved in the cycle, and more heat is absorbed from the hot end under the same temperature which enforces the system to become unstable. However, the hot end temperature of 800 K is unable to supply the energy required for the cycle as the charging pressure rises higher, preventing the generator from starting up. Figure 10B gives the influence of the charging pressure on the onset temperature in the experiment and calculation. The calculated onset temperature is discovered to decrease initially before rising as the charging pressure increases. When the charging pressure is 6.75 MPa, the lowest calculated onset temperature is 437 K. This is due to the fact that as the charging pressure rises, the thermal relaxation loss of the regenerator enlarges, raising the onset temperature of the FPSG when the charging pressure exceeds 6.75 MPa. The errors between the experiment and calculation range from 6.95% to 7.73%. For instance, the experimental and calculated onset temperatures under the charging pressure of 4.2 MPa are 561 and 522 K, respectively. It is worth noting that although increasing the charging pressure within a specific range is beneficial for the onset, the pressure-bearing capacity of the FPSG must be taken into account during the experiment. Figure 11 displays the impacts of the external load on the onset characteristics. As shown in Figure 11, with the increase of the external load, the system becomes more unstable and the onset temperature decreases in the simulation. This occurs because the external load is inversely proportional to the electromagnetic damping and piston damping is made up of electromagnetic damping and mechanical damping. As a result, a higher external load reduces piston damping, which makes the FPSG easier to start up. The onset temperature of the generator has been experimented with for external loads of 90, 100, 110, and 120 Ω, respectively. The trend of the experimental onset temperature, which falls from 678 to 605 K when the external load decreases, is consistent with the simulations. The relative errors of the experiment and simulation are 7.96%, 6.48%, 5.73%, and 3.97%, respectively.
The research on FPSGs' onset prediction in this paper is significantly valuable. As seen in this subsection, the onset condition is sensitive to variations of various parameters. If whether an FPSG with certain design parameters will start up can be effectively predicted, the waste of time and money can be avoided.

| CONCLUSIONS
In this study, a simple and precise approach to forecast the onset features was proposed. The onset conditions were discussed using the Routh stability criterion and the thermodynamic model of the FPSG. The onset conditions of the FPSGs of the alpha, beta, and gamma types were explored. Besides, based on the laboratory's 100 W beta-type FPSG prototype, further theoretical and experimental studies on the onset characteristics were carried out. The results demonstrate that alpha-type FPSG cannot start up regardless of changes made to the design and operating parameters. As for the beta and gamma type FPSG, the FPSG can only start up when the design and operating parameters fall within the onset range. From the perspective of the onset temperature, the FPSG is easier to start up at a larger external load and higher charging pressure. As the displacer spring stiffness increases, the onset temperature increases first and then decreases. When the displacer spring stiffness is 19.2 N/mm, the corresponding onset temperature is the lowest of 489 K. The impact of spring stiffness, charging pressure, and external load on the onset temperature was investigated experimentally. The results show great agreement with those predicted, and the errors are about 1.7%-9.12%. The outcomes illustrate that the proposed method can be applied to the onset conditions analysis of the FPSG. The method may play a reference role in the design of FPSG to avoid the waste of time and money.