Influences of nonlinear parameters on the performance of a free‐piston Stirling engine

This study presents a thermodynamic–dynamic model of a free‐piston Stirling engine (FPSE) with nonlinear coefficients of spring, load damping, and pressure for working spaces. The effects of the nonlinear coefficients of hardening and softening springs on the movement and operating frequency of a displacer and piston are investigated. Subsequently, the influences of the nonlinear pressure terms of the working space and buffer spaces, as well as the nonlinear coefficients of the load on the amplitudes of the two pistons, frequency, and output powers are discussed. The results indicate that the low nonlinear coefficients of the spring have a minor effect on the amplitudes and frequency. As the high nonlinear coefficients of the hardening spring increase, the amplitudes and output power decrease, while the frequency increases. As the nonlinear terms of the softening spring increase, the amplitudes and output power reach maximum values, but the frequency is reduced by 3.55%. The effects of the nonlinear pressure terms on the amplitudes and output power are not evident. However, an increase in the nonlinear load leads to a significant decrease in the FPSE's performance.

engineering. 10,11Since 2003, NASA has conducted research on a 110-W FPSE for a radioisotope power system program, and a pair of 55-W FPSEs have been tested in a vacuum environment. 12,13Sunpower developed a 160-W FPSE as a portable power charger for military applications. 14][17] FPSEs eliminate driving mechanisms, and the movements of the displacer and power piston are determined by the periodic variations in the gas pressure in working spaces (compression and expansion spaces), compared with kinematic Stirling engines. 18,19In addition, the pressure of working spaces is based on the thermodynamic and dynamic characteristics of the FPSE system, and gap seals are used between the pistons and cylinders, resulting in a self-starting, compact structure, lower noise pollution, and no vibration. 20The FPSE system is a multidegree vibration system, and a strong coupling relationship exists between the thermodynamic and dynamic parameters.Simultaneously, spring stiffness, load damping, and gas pressure exhibit remarkable nonlinear behavior, causing instability of FPSE as well as inaccurate prediction of the pistons' amplitudes, operating frequencies, and output powers.Therefore, it is necessary to study the influences of nonlinear parameters on the performance of FPSE.
In recent decades, researchers have investigated the nonlinear characteristics of FPSEs using various methods.Considering the nonlinearity of spring stiffness, Tavakolpour-Saleh et al. proposed a multiple-scale perturbation technique to design an FPSE and presented analytical relationships to estimate frequency, piston stroke, and output power, among other factors.In addition, the effects of the mass and stiffness of the displacer and piston on the output power were determined.The results revealed a frequency, phase angle, and piston stroke, output power, and efficiency of 71.1 rad s −1 , 60°, 11.4 mm, 3.46 W, and 7.1%, respectively. 21Moreover, a genetic algorithm method was proposed to obtain the optimal values of the mass spring stiffness of two pistons, and the cross-sectional area of a displacer rod.The stroke of the piston and the output power were 6 mm and 3.40 W, respectively. 22In 2020, Zare et al. used an averaging-based Lyapunov method and described a function technique using a genetic algorithm to evaluate the performance of an FPSE.The nonlinear terms of the springs of the power piston were analyzed.Under stable conditions, the frequency and the displacement of the two pistons were 72.2 Hz, 5.8 mm, and 12 mm, respectively. 23,24Yang established a modified nonideal adiabatic model of a γ-type FPSE to obtain the dynamic behaviors of the displacer and power piston.
The thermodynamic and dynamic performances under different loading and charged pressures were obtained.The maximum output power was 595 W, while the frequency, charged pressure, and loading values were 66.26 Hz, 5 MPa, and 218 N s m −1 , respectively. 25Some researchers have considered the nonlinear terms of pressure and load.Majidniya et al. developed a nonlinear thermodynamic model of a β-type FPSE with a permanent magnet linear synchronous motor, where nonlinear pressure drops crossing the heater, regenerator, and cooler were taken into account.The results indicated that the nonlinear model was more accurate than the linear model, and the maximum strokes of the pistons were calculated. 26,27For the nonlinear terms of damping, Sim et al. used a linear and nonlinear dynamic model with an external load for a β-type FPSE to predict the performance of the system.They applied the root locus method to predict the operating parameters.The results showed that the linear damping coefficient was 162.4 N s m −1 as the nonlinear damping increased from 0.2 to 5 N s 3 m −3 . 28,29Ye et al. built a thermodynamic and dynamic model for a γ-type FPSE with nonlinear load damping.The effects of load damping and nonlinear terms on the two pistons' amplitudes and output power were determined.The initial positions of the displacer and piston had a minor effect on the pistons' amplitudes.As the charged pressure varied from 3.0 to 3.7 MPa, the frequency and output power increased linearly.Additionally, the output parameters were predicted using response surface methodology and an artificial neural network, and the results indicated that the amplitudes of the displacer and power piston, operating frequency, and output power increased with the hot-end temperature and spring stiffness of the displacer. 30,31Chen et al. developed and verified the numerical model of an FPSE with a nonlinear load.A dimensionless parameter was proposed to distinguish the behavior of critical oscillation.They found that the nonlinearity of the piston's amplitude was more evident than that of the displacer's amplitude.In addition, the operating frequency first increased rapidly and then increased slowly as the charged pressure, hot-end temperature, and piston spring stiffness increased. 32s shown in the literature above, many methods have been developed and applied by several researchers to study thermodynamic-dynamic analyses and validate models through experiments, including nonlinear dynamic modeling, key parameter prediction, and performance improvement.In addition, the relationships between key parameters and the operating frequency, pistons amplitudes, and output power of the FPSEs have been determined.However, only a single nonlinear behavior, such as load damping or spring stiffness, is considered during FPSE modeling, while the pressures of working and backspaces are equivalent to the linear expression related to pistons strokes in the literature above.However, an FPSE is a comprehensive system affected by many nonlinear factors.Therefore, it is necessary to analyze the influences of several nonlinear effects on the movement of the displacer and piston, operating frequency, and output power by building an accurate mathematical model of the FPSE system to determine the relationship between these nonlinear systems and performance, which is the key to improving the performance of FPSEs.
Accordingly, a coupled thermodynamic-dynamic model with several nonlinear coefficients is developed in this study.The effects of nonlinear terms on the performance of an FPSE are determined.First, a theoretical model considering the nonlinear coefficients of spring stiffness, load damping, pressure of working spaces, and buffer spaces is established.The movement and the frequency spectra of the displacer and power piston are studied, and the influences of nonlinear terms, including the hardening and softening spring, compression space, buffer space, and load damping, on the frequency, amplitudes of the two pistons, and output power are determined.The results obtained provide design guidelines and analytical tools for improving the performance of FPSEs.

| Model description
The FPSE is an external combustion engine in which the working gas inside the cylinders can be heated using external heat sources.Electrical heating is applied to the FPSE system.A schematic diagram of the FPSE is shown in Figure 1A.The main components are a power piston, a displacer piston, two stacks of springs for two pistons, four chambers (expansion and compression spaces, and two buffer spaces in each piston), and three heat exchangers (regenerator, heater, and cooler).
As displayed in Figure 1A, the piston and the displacer were arranged in two cylinders, and supported by the spring stacks, a linear generator that is attached to the piston is considered as an external load and is equivalent to the damping force.The periodic variations in the pressures within the inner chamber of the displacer and the outer chamber of the piston are considered.The basic design parameters for the FPSE are described in Refer 30.

| Assumptions
To derive the theoretical model of the studied FPSE, the following assumptions are made: (1) The working gas in the chambers is ideal gas.
(2) There is no leakage and the mass of the working gas remains constant.(3) The fluid temperatures in the compression and expansion spaces are equal to T h and T c , respectively.(4) The effective temperature of the regenerator can be defined as a linear distribution.

| Dynamic model
The free-body diagrams of the displacer and power piston are shown in Figure 1B.The two pistons are connected to the spring stacks and dampers.According to the assumptions mentioned above and by applying Newton's second law of motion, the displacements of the displacer and power piston are obtained by solving the following equations: where subscripts d and p represent the displacer and power piston, respectively; x x x , , and ̈are the displacement, velocity, and acceleration, respectively; F xd and F xp are the spring forces attached to the displacer and power piston, respectively; R is the damping coefficient; the pressures of the working gas in the compression and expansion spaces are p c and p e , while those in the buffer spaces of the displacer and piston are expressed as p d and p b , respectively; A d , A p , and A r are the cross-sectional areas of the displacer, power piston, and displacer rod, respectively; F load is the damping force applied to the power piston.
Normally, the displacement of two pistons and the deformation of the spring are relatively small, and the spring force is assumed to be proportional to x d and x p .However, there are nonlinear effects of spring forces during the testing of spring stiffness, even causing instability of the FPSE system or pistons to collide with the FPSE casing owing to the exceeding of the piston amplitudes.Therefore, the nonlinear terms of the spring stiffness should be considered.The series expansion of a nonlinear spring function can be expressed as a combination of the linear and nonlinear spring components.In this study, to capture the effects of variations in the nonlinear coefficients of the spring, a term that makes analyses intuitive was chosen.In addition, a nonlinear spring with a linear spring and a cubic nonlinear spring element were selected to perform a nonlinear analysis of the equations of motion of the two pistons.Therefore, the spring forces of the two pistons were expressed as shown in Radomirovic and Kovacic 33 and Dibesh 34 in Equations ( 1) and ( 2), as follows: (3) where σ d and σ p are the nonlinear coefficients of the spring supporting the displacer and power piston.A positive σ value indicates a hardening spring, while a negative value represents a softening spring composed of a material that is relatively soft and has low-stress intensity, characterized by light material and low strength.However, the characteristics of a hardening spring are opposite to those of a softening spring.
The gravitational forces acting on the power piston and displacer are ignored in Equations ( 1) and ( 2), compared with the forces resulting from the pressure variations in the working gas.
In Equation ( 2), F load can be described as a nonlinear expression 29 : ( ) where k alt and f c are the linear and nonlinear terms, respectively.
In Equation (1), p c − p e is the pressure drop caused by the three heat exchangers, which is equal to Δp and can be defined as follows 35 : where R pp and R dd represent the equivalent damping that causes a pressure drop through the heater, regenerator, and cooler.Therefore, the time-averaged output power is obtained and expressed as shown in Equation ( 7): where W net is the output power and ω is the angular frequency.
On the basis of isothermal assumptions and the ideal gas equation, the nonlinear expression for the instantaneous pressure of the working gas in the FPSE system can be expressed as follows: where M g is the total mass of the working gas; V r , V k , and V h are the dead volumes of the regenerator, cooler, and heater, respectively; V e and V c are the instantaneous volumes of the expansion and compression spaces, respectively; and R g is the gas constant of the working gas.The volumes of the expansion and compression spaces are functions of x d and x p .According to Figure 1B, the volume can be expressed as where V eo and V co denote the equilibrium volumes of the expansion and compression spaces, respectively.According to assumption (4), by substituting Equations ( 9) and (10) into Equation ( 8), working pressure p is obtained as follows: Let us define Substituting Equation ( 14) into Equation (11), the working pressure is represented based on a Taylor series of order three To examine the linear and nonlinear terms, coefficient τ 1 is substituted into Equation (15).To simplify the FPSE model, it is assumed that the pressure of the compression space is equal to p.
In Equations ( 1) and ( 2), p b and p d are calculated using Equations ( 17) and ( 18), as follows: where V do and V bo are the volumes of the displacer and piston buffer spaces in the median positions of the two pistons, respectively; γ is the ratio of the specific heat.Because the variation in p d is less than p b , the buffer space of the displacer p d is described as an expression of x d , and the nonlinear term of p b is considered in Equation (17).Equation ( 17) is expanded according to the Taylor series expansion of order three, as follows: Thus, the nonlinear term of p b is represented as where the coefficient τ 2 is substituted into Equation ( 19), representing the nonlinear term of p b .

| Equations of the nonlinear system
On the basis of the equations presented above, a group of nonlinear equations is derived, which is expressed in the form of a state space: where the first term on the right side is a linear term and f(X) is a nonlinear term. where p .Stiffness coefficient K and damping coefficient D are listed in Table 1.f x ( ) 3 and f x x ( , ) 1 3 are the nonlinear terms of p b and p c , respectively.
To solve nonlinear Equation ( 22), a coupled thermodynamic-dynamic model is built by applying the MATLAB software; thus, the movement of the displacer and piston, pressures of the four spaces, and operating frequency can be determined.

| Model validation
A 30-W FPSE was designed and operated steadily to verify the accuracy of the nonlinear thermodynamic-dynamic model. 30Figure 2 presents a schematic diagram of the experimental test system and the FPSE prototype.The experimental test system consisted of an FPSE, heating and temperature-control equipment, cooling equipment, loadcontrol equipment, and a parameters-testing system.The models and specific parameters of the primary instruments are listed in Table 2.
The steps of the experiment were as follows: (1) After baking the FPSE for 24 h to remove the gas, the air inside the FPSE was pumped out using a vacuum pump (Figure 2A); (2) the FPSE system was filled with helium gas for 2-3 times of gas replacement to ensure the purity of the gas and prevent oxidation of the internal components of the FPSE.(3) Thereafter, the FPSE was filled with high-pressure helium gas of 3.0-3.5MPa, after which the helium cylinders were closed.(4) The water chilling unit was turned on to ensure that the outlet temperature did not exceed 313 K. (5) A heat source with several electric heating rods was used.However, it is noteworthy that the heating power should be controlled to gradually increase the heater temperature.(6) An AC power supply for the excitation source was applied as the temperature of the heat source exceeded 473 K.The operating parameters were displayed on the parametertesting device.
Figure 3 shows the displacer and power piston which are made of titanium alloy.The yellow part of the pistons is a wear-resistant layer added to the surface of the piston to reduce the mechanical friction between the piston and cylinder.The masses of the displacer and piston are 0.074 and 0.26 kg, respectively, in this experiment.Linear and nonlinear terms k alt and f c are 18.4 N s m −1 and 147,566 N s m −3 , respectively.
Figure 4 presents the trend of the output power with an increase in the heating temperature.The output power increases linearly with the hot-end temperature, and the predicted values are higher than those of the experimental results; however, the relative errors are less than 5%.This is because the loss terms, such as the clearance seal and viscous damping of the working gas that consumes the output power, are ignored in the model building.In addition, the errors are caused by T A B L E 1 Stiffness-and damping-related coefficients.)

Stiffness-related coefficients
deviations between the design and experimental parameters.The mechanical damping of the pistons caused by their cylindricity and coaxiality during their movement at high frequency is set to be constant in the theoretical model.However, the mechanical damping varies with the operating conditions during the experiment.
However, within the heating temperature range, the simulation results are consistent with the experimental results.

| RESULTS AND DISCUSSION
As shown in Equations ( 3) and ( 4), the nonlinear coefficients of the springs supporting the displacer and power piston are considered in the FPSE model building.
In our previous study, the force of the spring is assumed to be proportional to the displacement of the power piston and displacer, which is expressed as the stiffness of the spring multiplied by the pistons' displacement.However, the nonlinear coefficients of the spring cause instability in the FPSE system, which becomes evident when the displacement of the pistons is large during FPSE testing; thus, the nonlinear coefficients of the spring should be considered.In addition, the effects of the hardening and softening springs on the performance of the FPSE are quite different.Thus, this section focuses on analyzing the effects of the hardening and softening springs on the system under low and high nonlinearities.

| Effects of low nonlinear coefficients of the spring stiffness
First, the effects of the low nonlinear coefficients of spring stiffness σ d on the movement and frequency spectra of the displacer and power piston of the FPSE are considered.As shown in Figures 5 and 6, under a cubic spring term σ p of 0, a hardening spring is selected.Values of σ d = 0 and 10 2 m −2 are the alternative parameters used to evaluate the effects of low nonlinear coefficients of the spring added to the displacer.It should be noted that the amplitudes of the displacer and piston remain nearly unchanged following the addition of the low nonlinear coefficients of spring stiffness, meaning that the amplitudes of the two pistons (Figures 5A and 6A) are close to each other.In addition, the oscillation frequency for the two cases is almost the same.Thus, the amplitudes of the displacer and piston without nonlinear terms (σ d is set to 0) were 1.89 and 6.44 mm, respectively, the frequency was 63.3714 Hz, and the amplitudes and frequency with the added nonlinear terms (σ d of 100 m −2 ) were 1.88 mm, 6.44 mm, and 63.3713 Hz, respectively, which are largely unchanged, indicating that the low nonlinear coefficients of the spring stiffness of the displacer do not affect the movement of the two pistons and frequency.Next, nonlinear coefficients σ d are fixed at 0, and the effects of the low nonlinear coefficients of spring stiffness σ p on the movement and oscillation frequency of the two pistons are shown in Figures 7 and 8.As shown in Figure 7, there is no considerable difference following the addition of nonlinear coefficient σ p , as the amplitudes of the displacer and oscillation frequency are close to each other in the two cases.However, the movement of the power piston changes slightly, whereas the oscillation frequency remains constant, as shown in Figure 8, meaning that the low nonlinear coefficients of spring stiffness σ p for the piston did not affect the movement of the pistons.
The results illustrated above show that the amplitudes of the displacer and power piston are almost unchanged because the nonlinear coefficients of the spring stiffness are relatively small.Furthermore, the oscillation frequencies for the two cases are similar because the motions of the two cases almost overlap.It is concluded that the output parameters of the FPSE, including the output power, operating frequency, and amplitudes of the two pistons, are unaffected by low nonlinear coefficients of σ p and σ d .However, how will the movement of the two pistons, oscillation frequencies, and output power change with high nonlinear coefficients of spring stiffness for the displacer and power piston?What impact will the hardening and softening springs have on the performance of the FPSE?9A and 10A, the amplitudes of the displacer and power piston decrease slowly at a σ d of 10 4 m −2 and then decrease rapidly at a σ d of 10 5 m −2 , the amplitudes of the displacer at nonlinear coefficients σ d of 10 4 and 10 5 m −2 are 1.84 and 1.49 mm, respectively, while the piston's amplitudes are 6.37 and 5.61 mm, respectively, and the amplitudes of the two pistons without nonlinear spring stiffness coefficients are the largest.In Figures 9B  and 10B, the frequency spectra for the displacer and power piston shift to the right with an increase in high nonlinear coefficient σ d , indicating that the oscillation frequency has increased.Therefore, an increase in the high nonlinear coefficient of σ d for the hardening spring increases the oscillation frequency and decreases the amplitudes of the two pistons.In addition, the larger the nonlinear coefficients, the greater the amplitude reduction.
When designing FPSEs, the frequencies, amplitudes of the two pistons, and output powers are of practical importance.However, the effects of high nonlinear coefficients of the spring stiffness on these parameters have not yet been established.Therefore, it is important to investigate the influence of high nonlinear coefficients σ d and σ p on the performance of FPSEs. Figure 11 shows the sensitivity of the amplitudes of the two pistons, frequency, and output power to variations in σ d .
Increasing σ d may result in a decrease in the amplitudes of the displacer and piston and cause the output power to decrease as the frequency increases from 63.371 to 64.767 Hz.It can be concluded that increases in the σ d of the hardening spring have a negative impact on the pistons' amplitudes and output power while increasing the oscillation frequency.
After determining the effects of the σ d of the hardening spring on the movement and frequency spectrum of the displacer and piston, different values of high nonlinear coefficients σ p for the hardening spring are obtained, as shown in Figures 12 and 13.The variation trends of the amplitudes for the two pistons and frequencies are consistent with the results shown in Figures 9 and 10.As the value of the σ p for the hardening spring increases to 10 5 m −2 , the amplitudes of the displacer and power piston decrease significantly, while the curves of the frequency spectra shift to the right, and these phenomena are particularly evident at a σ p of 10 5 m −2 .In addition, the effects of increasing σ d on the pistons' amplitudes and frequency are more significant than those of increasing σ p (see Figures 11 and 14).
The influences of the δ p of the hardening spring on the amplitudes of the displacer and piston, frequency, and output power are displayed in Figure 14.As σ p increases from 0 to 1 × 10 5 m −2 , the amplitudes of the displacer and piston decrease, and the output power decreases.This is because the output power is related to the pistons' amplitude and frequency, according to Equation (7).Because the decrease in the pistons' amplitudes is greater than the increase in the frequency, the output power exhibits a downward trend (see Figure 14).Increasing the σ p of the hardening spring as well as σ d resulted in decreases in the amplitudes and output power, while the frequency increased.In addition, the influence of the σ p of the hardening spring on the four output parameters is greater than that of σ d .Therefore, it is necessary to control the values of σ p when designing the spring supporting pistons.Generally, the nonlinear coefficients of a spring are adjusted by changing its internal profile, slot width, thickness, and material.Subsequently, the effects of the σ p and σ d of the softening spring on the FPSE's performance are illustrated in Figures 15-20.The absolute values of the σ p and σ d of the softening spring are analyzed.The movement and frequency spectra of the two pistons with respect to a high nonlinear coefficient σ d are shown in Figures 15 and 16.The amplitudes for the displacer and power piston decrease slowly at a σ d of 10 4 m −2 and then decrease rapidly at a σ d of 10 5 m −2 , and the amplitudes of the displacer and power piston are, respectively, 1.93 and 6.49 mm, respectively, at a σ d of 10 4 m −2, while they are 1.80 and 5.71 mm, respectively, at a σ d of 10 5 m −2 .However, the curves of the two pistons' frequency spectra shift to the left, indicating that the oscillation frequency decreases as the σ d of the softening spring increases.
To present the effects of the σ d of the softening spring on the amplitudes of the two pistons, frequency, and output power, five values for σ d ranging from 10 4 to   The maximum values of the amplitudes of the displacer and piston are determined, and the output power generated as the σ d of the softening spring is 2.5 × 10 4 m −2 .As shown in Equation ( 7), the output power produced by the nonlinear load is proportional to ω 2 , x p 2 , and x p 4 . Because the frequency decreases with the σ d of the softening spring, and the amplitude of the piston reaches a maximum, the output power also shows the same trend.However, an increase in the σ d of the softening spring increases the instability of the system.This phenomenon should be avoided during spring design.
According to these results, the effects of σ p are more significant than those of σ d .Therefore, the values of the σ p for the softening spring are selected in the range of 0-10 4 m −2 to avoid crashing of the FPSE system.The movement and frequency spectra of the two pistons at different σ p values for the softening spring are displayed in Figures 18 and 19.Increases in the σ p of the softening spring up to 10 4 m −2 resulted in increases in the amplitudes of the displacer and piston.However, further increases in the σ p value of the softening spring render the FPSE system unstable.In addition, the oscillation frequency at a σ p of 0 m −2 is 63.37 Hz, while the frequency at a σ p of 10 4 m −2 is reduced to 60.46 Hz.
Figure 20 shows how the σ p of the softening spring affects the amplitudes of the two pistons, oscillation frequency, and output power.For the softening spring, as σ p varies from 0 to 10 4 m −2 , the amplitudes of the displacer and the piston and the output power increase, but the frequency decreases linearly.Consequently, the increase in the σ p of the softening spring causes the amplitudes of the two pistons and output power to increase, while the oscillation frequency decreases.However, the FPSE system may be unstable under extreme conditions or even cause a crash as the σ p of the softening spring increases to a certain value.Through detailed analysis, it is concluded that high values of nonlinear coefficients σ d and σ p for both hardening and softening springs significantly affect the performance of FPSEs.Therefore, finite element analysis and spring stiffness testing are generally conducted before FPSE experiments to avoid the performance degradation caused by high nonlinearity.

| Effect of the nonlinear coefficients of load damping and pressure
Figures 21 and 22 show the movement and frequency spectra of the displacer and power piston with respect to nonlinear coefficients of load damping f c .As the nonlinear coefficients of load damping increase from 2 × 10 5 to 1 × 10 6 N s m −3 , the amplitudes of the displacer and piston first decrease rapidly and then slowly, while the oscillation frequency remains constant.However, nonlinear coefficient f c should be selected within an appropriate range based on the stroke limitations of the system.The horizontal position of the pistons' curves does not change, indicating that the variation in the nonlinear coefficient of load damping does not affect the phase angle and frequency.Figure 23 shows the effect of the nonlinear terms of the working pressure and buffer spaces on the movement and frequency spectra of the displacer and power piston.The amplitudes of the two pistons and the frequency are very close to each other in the two cases, indicating that the addition of the nonlinear coefficients of the p c and p b has little impact on the amplitudes and oscillation frequency of the two pistons.
The effects of nonlinear terms p c and p b on the pressures in the compression space, expansion space, and two buffer spaces are shown in Figure 24.Under the conditions of linear and nonlinear pressures, p c and p e are very similar, implying that the pressure drops caused by the three heat exchangers are relatively small.According to the plots of periodic pressures p e , p c , p b , and p d , the amplitude values for the four pressures are 0.312, 0.317, 0.138, and 0.0509 MPa with the linear pressure terms, respectively, while they are 0.310, 0.315, 0.134, and 0.0506 MPa with the nonlinear pressure terms, respectively, indicating that the amplitudes of the p e , p c , p b , and p d of the linear pressure models are greater than those of the nonlinear models.In addition, as shown in Figure 24B, the periodic variation in the pressure in the inner chamber of the displacer, p d , showed little change compared with those of p c , p e , and p b ; thus, p d is set to the charging pressure in the subsequent modeling.
Figure 25 shows a comparison of the calculated results for the two pistons' amplitudes and output powers between the linear and nonlinear pressure models under five nonlinear coefficients of load damping.With an increase in the nonlinear coefficient of load damping f c , the amplitudes of the displacer and piston decrease, and the output power also shows the same trend.As the f c (A) (B) F I G U R E 24 Pressure of working and buffer spaces with/without nonlinear pressure coefficients.(A) Pressures of compression and expansion spaces and (B) pressures of buffer spaces.

(A) (B)
F I G U R E 25 Amplitudes of displacer and piston and output power with linear/nonlinear pressure coefficients versus nonlinear coefficients of load damping.(A) Amplitudes of two pistons and (B) output power.
varies from 2 × 10 5 to 1 × 10 6 N s m −3 , the amplitude of the displacer decreases from 3.26 to 1.46 mm, the piston amplitude decreases from 11.1 to 4.99 mm, and the output power decreases from 306 to 32 W. It should be noted that the output power is very sensitive to the f c .The values of nonlinear coefficient f c are related to the performance of the linear alternator.As shown in Figure 25, the amplitudes of the two pistons and output power increased by 1.23% and 2.19%, respectively, showing that the pressure nonlinear coefficient has no considerable influence on the three output parameters.

| CONCLUSIONS
In this study, a coupled thermodynamic-dynamic model with nonlinear coefficients of spring stiffness for displacer and piston, load damping, working spaces, and buffer spaces was introduced to predict the output parameters of a γ-type FPSE, which had major impacts on the performance.The influences of the σ d and σ p of hardening and softening springs, nonlinear pressure coefficients p c and p b , and the nonlinear coefficient of load damping f c on the movement of two pistons, frequency, and output power were discussed.The following conclusions were drawn: (1) As the low nonlinear coefficients of the spring stiffness of the displacer and piston increased from 0 to 10 2 m −2 , the movement of the displacer and piston as well as the frequency spectra of the two pistons were close to each other, and there was not much difference following the addition of nonlinear coefficients σ d and σ p , indicating that the low nonlinear coefficients of the two pistons' spring stiffness had little effect on the amplitudes of the displacer and piston and the oscillation frequency.(2) The amplitudes of the displacer and piston and the output power were reduced by 21.16%, 12.88%, and 44.83%, respectively, and the frequency was increased by 2.21% as the high nonlinear coefficients of the hardening spring supporting the displacer increased from 0 to 10 5 m −2 .In addition, an increase in the nonlinear coefficients σ p of the hardening spring resulted in a decrease in the amplitudes of the two pistons and output power.However, the nonlinear coefficient σ p of the hardening spring had a more significant influence on the performance of the FPSE than that of σ d .(3) On the basis of the analysis results of high nonlinear coefficients of σ d for the softening spring, the amplitudes of the two pistons and output power increased first and then decreased, and the maximum amplitudes of the displacer and piston and output power at a σ d of 2.5 × 10 4 m −2 for the softening spring were 1.97 mm, 6.52 mm, and 62.29 W, respectively, while the frequency decreases by 3.55%.In addition, the amplitudes of the displacer and piston and output power increased by 12.86%, 13.82%, and 29.5%, respectively, as the σ p of the softening spring increased, but the addition of the σ d and σ p of the softening spring rendered the FPSE system unstable.(4) As the nonlinear coefficients of load damping increased from 2 × 10 4 to 1 × 10 5 N s m −3 under the condition of the nonlinear pressure models, the amplitudes of the two pistons and the output power of the FPSE were reduced to 1.46 mm, 4.99 mm, and 32.0 W, respectively, but it had little effect on the frequency.However, the values of the nonlinear coefficients of load damping were also limited by the strokes of the two pistons and the performance of the linear alternator of the FPSE system.In addition, as the nonlinear pressure coefficients were added to the FPSE models, the variations in the amplitudes of the two pistons and four pressure, and the output power were not considerable.
On the basis of the results obtained above, the coupled thermodynamic-dynamic model used to consider the nonlinear coefficients of springs, load damping, and pressure of working spaces can provide theoretical guidance and a supporting model for improving the performance of FPSEs by identifying different nonlinear coefficients.In addition, the conclusions of this study can be applied to other types of FPSEs.
Owing to limited time and funding, more relevant experiments were not conducted.However, in our future study, a γ-type FPSE system will be used in domestic cogeneration by replacing the material structures of the heat exchangers in the FPSE and reducing the processing requirements, and further experiments will be conducted to validate the theoretical results.

1
Schematic and free body diagrams of FPSE.(A) Schematic diagram of γ-type FPSE and (B) free body diagrams of piston and displacer.FPSE, free-piston Stirling engine.

F I G U R E 2
Test system of FPSE.(A) Schematic diagram of the test system and (B) FPSE prototype.DC, direct current; FPSE, free-piston Stirling engine.T A B L E 2 Main equipment parameters of the test system.

F I G U R E 3
Photos of the displacer and piston.(A) Displacer and (B) power piston.F I G U R E 4 Heating temperature versus output power.coefficients of the spring stiffness

Figures 9
Figures 9 and 10 show the variations in high nonlinear coefficients of σ d with the movement of the two pistons and the oscillation frequency.In this study, a hardening spring is used, and the values of σ d and σ p are varied within the ranges of 10 −4 and 10 −5 m −2 .As shown in Figures9A and 10A, the amplitudes of the displacer and power piston decrease slowly at a σ d of 10 4 m −2 and then decrease rapidly at a σ d of 10 5 m −2 , the amplitudes of the displacer at nonlinear coefficients σ d of 10 4 and 10 5 m −2 are 1.84 and 1.49 mm, respectively, while the piston's amplitudes are 6.37 and 5.61 mm, respectively, and the amplitudes of the two pistons without nonlinear spring

6 7 8 9
Movement and frequency spectrum of displacer versus low nonlinear coefficient σ d .(A) Displacer movement and (B) displacer frequency spectrum.Movement and frequency spectrum of piston versus low nonlinear coefficient σ d .(A) Piston movement and (B) piston frequency spectrum.YE ET AL.Movement and frequency spectrum of displacer versus low nonlinear coefficient σ p .(A) Displacer movement and (B) displacer frequency spectrum.Movement and frequency spectrum of piston versus low nonlinear coefficient σ p .(A) Piston movement and (B) piston frequency spectrum.Movement and frequency spectrum of displacer versus high nonlinear coefficient σ d of hardening spring.(A) Displacer movement and (B) displacer frequency spectrum.

FFF
I G U R E 10 Movement and frequency spectrum of power piston versus high nonlinear coefficient σ d of hardening spring.(A) Piston movement and (B) piston frequency spectrum.(A) (B) F I G U R E 11 Four output parameters versus high nonlinear coefficient σ d of hardening spring.(A) Amplitudes of displacer and piston and (B) frequency and output power.YE ET AL.I G U R E 12 Movement and frequency spectrum of displacer versus high nonlinear coefficient σ p of hardening spring.(A) Displacer movement and (B) displacer frequency spectrum.(A) (B) F I G U R E 13 Movement and frequency spectrum of power piston versus high nonlinear coefficient σ p of hardening spring.(A) Piston movement and (B) piston frequency spectrum.I G U R E 14 Four output parameters versus high nonlinear coefficient σ p of hardening spring.(A) Amplitudes of displacer and piston and (B) frequency and output power.

15 F
Movement and frequency spectrum of displacer versus high nonlinear coefficient σ d of softening spring.(A) Displacer movement and (B) displacer frequency spectrum.I G U R E 16 Movement and frequency spectrum of power piston versus high nonlinear coefficient σ d of softening spring.(A) Piston movement and (B) piston frequency spectrum.

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I G U R E 17 Four output parameters versus high nonlinear coefficients σ d of softening spring.(A) Amplitudes of displacer and piston and (B) frequency and output power.(A)(B) F I G U R E 18 Movement and frequency spectrum of displacer versus high nonlinear coefficients σ p of softening spring.(A) Displacer movement and (B) displacer frequency spectrum.I G U R E 19 Movement and frequency spectrum of power piston versus high nonlinear coefficients σ p of softening spring.(A) Piston movement and (B) piston frequency spectrum.

10 5 m
−2 are selected, and the results are shown in Figure17.The two pistons' amplitudes and output power first increase and then decrease gradually as σ d varies from 0 to 10 −5 m −2 , resulting in a decrease in the oscillation frequency in the range of 63.371-61.124Hz.

F
I G U R E 20 Four output parameters versus high nonlinear coefficients σ p of softening spring.(A) Amplitudes of displacer and piston and (B) frequency and output power.
coefficient, N s m −1 kg −1 [D] matrix of damping coefficients f frequency, Hz F force f c load damping nonlinear coefficient, N s m −3 k spring stiffness, N m −1 K stiffness coefficient per unit mass, N m −1 kg −1 [K]matrix of stiffness coefficients k alt load damping coefficient, V s m −1