Research on Piston Dynamics and Engine Performances of a Free-Piston Engine Linear Generator Coupling with Various Rebound Devices

: Free-piston engine linear generators (FPELGs) are an innovative linear power device that exhibits the distinctive dynamics of pistons and performance of free-piston engines. Furthermore, the single-cylinder/single-piston FPELG structure type has more advantages than other FPELG structure types, including a straightforward structure and ease of control. However, when coupled with various rebound devices, the operational characteristics and piston and engine performance of single-cylinder/single-piston FPELGs are quite different. Therefore, this paper aims to quantitatively compare the dynamics of the piston and engine performance of single-cylinder/single-piston FPELGs coupled with various types of rebound device. The results indicate that when the equivalent stiffness of the gas spring is greater than that of the mechanical spring, the operating frequency of the piston of the FPELG coupled with a gas spring will be higher than that when coupled with a mechanical spring. During the compression stroke, the piston velocity of a FPELG coupled with a mechanical spring changes linearly, while the piston velocity of a FPELG coupled with a gas spring changes nonlinearly. FPELGs coupled with gas springs have shorter compression and expansion durations compared to those coupled with mechanical springs. In addition, the indicated powers of FPELGs coupled with ideal gas springs and mechanical springs are 1.5 kW and 1.3 kW, respectively. However, due to leakage, the thermal efﬁciency of a FPELG coupled with an actual gas spring is reduced by approximately 2.5% compared with the FPELG coupled with the ideal gas spring. Furthermore, the operation frequency of the piston is positively correlated with the stiffness of the mechanical spring. In addition, as the stiffness of the mechanical spring increases, the combustion process of the engine becomes close to an isovolumetric process. The changes in piston dynamics and engine performance when increasing the initial gas pressure of the gas spring are similar to those when increasing the stiffness of the mechanical spring.


Introduction
In order to improve the conversion efficiency of energy devices and optimize their structure design, many academics are actively engaged in exploring innovative energy conversion apparatuses [1][2][3].As a typical representative, in recent years, free-piston engine linear generators (FPELGs) have garnered considerable attention from many institutions around the world.They directly combine a free-piston engine, a linear generator, and a rebound device [4].In addition, their operating principle is as follows: the hightemperature/high-pressure gas is produced in the cylinder of the free-piston engine; then, the gas drives the pistons and, reciprocally, the assembly connected with the magnetic rod Energies 2023, 16, 6570 2 of 20 of the linear generator; finally, the linear generator converts parts of the linear kinetic energy of the magnetic rod into electrical energy [5].Compared with conventional reciprocating engines, FPELGs have several advantages, such as the shorter energy transmission chain and higher power density [6].
Based on the structural features of the FPELG, there are three typical structure types of FPELGs, namely the single-cylinder/single-piston structure type, the dual-cylinder/dualpiston structure type, and the opposed-piston structure type [7].Due to its inherent advantages of simple structure and ease of control, compared with the other FPELG structure types, single-cylinder/single-piston FPELGs have emerged as the focal point of research attention.As is the case for all FPELG structure types, high-temperature/highpressure gas is produced during the power process, which then drives the movement of the piston and the magnetic rod of the linear generator.However, with respect to the nonpower process of the single-cylinder/single-piston FPELGs, a rebound device is required to overcome the compression resistance force [4].In view of this, the parameters of the rebound device can be modified to satisfy the various power requirements of the FPELG, arising from its structural features and operating principle.As a result, the dynamics of the piston and engine performances of single-cylinder/single-piston FPELGs have been proven to be highly sensitive to the characteristics of the rebound device.To date, mechanical springs and gas springs have emerged as typical choices for rebound devices in the single-cylinder/single-piston FPELGs around the world.Schematic diagrams of the single-cylinder/single-piston FPELGs coupled with the mechanical and gas springs are presented in Figure 1.
Energies 2023, 16, x FOR PEER REVIEW 2 of 21 the gas drives the pistons and, reciprocally, the assembly connected with the magnetic rod of the linear generator; finally, the linear generator converts parts of the linear kinetic energy of the magnetic rod into electrical energy [5].Compared with conventional reciprocating engines, FPELGs have several advantages, such as the shorter energy transmission chain and higher power density [6].
Based on the structural features of the FPELG, there are three typical structure types of FPELGs, namely the single-cylinder/single-piston structure type, the dual-cylinder/dual-piston structure type, and the opposed-piston structure type [7].Due to its inherent advantages of simple structure and ease of control, compared with the other FPELG structure types, single-cylinder/single-piston FPELGs have emerged as the focal point of research attention.As is the case for all FPELG structure types, high-temperature/high-pressure gas is produced during the power process, which then drives the movement of the piston and the magnetic rod of the linear generator.However, with respect to the non-power process of the single-cylinder/single-piston FPELGs, a rebound device is required to overcome the compression resistance force [4].In view of this, the parameters of the rebound device can be modified to satisfy the various power requirements of the FPELG, arising from its structural features and operating principle.As a result, the dynamics of the piston and engine performances of single-cylinder/single-piston FPELGs have been proven to be highly sensitive to the characteristics of the rebound device.To date, mechanical springs and gas springs have emerged as typical choices for rebound devices in the single-cylinder/single-piston FPELGs around the world.Schematic diagrams of the single-cylinder/single-piston FPELGs coupled with the mechanical and gas springs are presented in Figure 1.

Single-Cylinder/Single-Piston FPELG Structure Type Coupled with a Mechanical Spring
The Clark group of West Virginia University investigated the operation characteristics and the piston and engine performance of a single-cylinder-type FPELG and a dualcylinder-type FPELG coupled with a mechanical spring on the basis of simulation results and experimental results [8][9][10].It was found that the operation frequency of the piston and the power of the free piston engine are positively correlated with the spring stiffness, while the efficiency decreased with the increase in spring stiffness [11].Furthermore, a larger spring stiffness was beneficial for reducing cyclic fluctuations during the steady operating process [12].And compared with a FPELG without springs, the piston dynamic of the FPELG coupling with the mechanical spring was nearly closed to the sinusoidal motion [13,14].Moreover, the combustion characteristics of the FPELG coupling with the mechanical spring were also simulated, and the results demonstrated that the combustion starting timing advanced, the adverse work and heat transfer both increased, while the overall efficiency declined [15].Yingcong Zhou from Stony Brook University researched The Clark group of West Virginia University investigated the operation characteristics and the piston and engine performance of a single-cylinder-type FPELG and a dual-cylindertype FPELG coupled with a mechanical spring on the basis of simulation results and experimental results [8-10].It was found that the operation frequency of the piston and the power of the free piston engine are positively correlated with the spring stiffness, while the efficiency decreased with the increase in spring stiffness [11].Furthermore, a larger spring stiffness was beneficial for reducing cyclic fluctuations during the steady operating process [12].And compared with a FPELG without springs, the piston dynamic of the FPELG coupling with the mechanical spring was nearly closed to the sinusoidal motion [13,14].Moreover, the combustion characteristics of the FPELG coupling with the mechanical spring were also simulated, and the results demonstrated that the combustion starting timing advanced, the adverse work and heat transfer both increased, while the overall efficiency declined [15].Yingcong Zhou from Stony Brook University researched the thermodynamic performances of the FPELG with a high-stiffness helix spring.The thermodynamic performances and piston dynamics of the engine under various parameters Energies 2023, 16, 6570 3 of 20 were analyzed.It was found that the lower spring energy ratio resulted in decreased heat transfer and improved efficiency.Furthermore, the peak thermal efficiency could reach 46.7% [16].In addition, the gas exchange characteristics of the FPELG under HCCI combustion mode was simulated by using the computational fluid dynamics (CFD) model, and the results demonstrated that the optimal scavenging and trapping efficiencies can be obtained under the conditions of moderate input pressure and high scavenge ports inclination angles [17,18].
Zhaoping Xu from Nanjing University of Science and Technology focused on the performances of the FPELG under a four-stroke operation mode.The experimental results showed that the output power of the FPELG was 2.68 kW, and the generating efficiency could reach 33.4%.To further improve the output power and generating efficiency, the performances of the FPELG under an irreversible Miller thermal cycle were analyzed by using finite-time thermodynamics [19,20].In addition, the hierarchical hybrid control system was proposed based on the principle of energy balance to effectively control the piston motion, and it had been successfully applied to the prototypes [21].Chunlai Tian from Beijing Institute of Technology conducted research on the oscillation characteristic of the FPELG coupling with the mechanical spring.A nonlinear oscillation model of the FPELG was developed, and the results revealed that the FPELG could achieve a limit loop, based on the principle of energy conservation [22].Subsequently, the layered hybrid stability control strategy of the FPELG was proposed, and it was found that the FPELG could stabilize itself after a brief cycle of fluctuations [23].Chenheng Yuan from Chongqing Jiaotong University analyzed the combustion and emission characteristics of the FPELG coupling with the mechanical spring.It was found that the operation frequency of the piston and the compression ratio of the engine were positively correlated with the spring stiffness, respectively.In addition, the simulation results revealed that by increasing the spring stiffness within a suitable range, the excellent thermal efficiency of the free piston engine might be produced.However, excessive spring stiffness exacerbated pollutants, such as NOx and soot [24].

Single-Cylinder-Single/Piston Structure Type of the FPELG Coupling with the Gas Spring
Researchers at the German Aerospace Center focused on the engine performances of the FPELG coupling with the gas spring based on the simulation results and experimental results [25].The gas exchange process of the FPELG was thoroughly analyzed by using the CFD model and verified through the measured results [26].The results demonstrated that the three-valve arrangement, consisting of two intake valves and one exhaust valve, is an ideal choice for the FPELG [27].The FPELG was described as a highly efficient range extender for vehicles, and the results suggested that the FPELG, which played the role of the vehicle power unit, might show promising potential [28,29].Furthermore, the FPELG under a HCCI combustion mode could reduce emissions and increase energy conversion efficiency [30].
Researchers from Toyota Central R&D Labs Inc. conducted in-depth research on the output power and thermal efficiency of the FPELG under various combustion modes.Compared with spark ignition combustion mode, the thermal efficiency and the output power was 42% and 10 kW, respectively, when the premixed charged compression ignition combustion mode was applied on the FPELG [31].Furthermore, the experimental results highlighted that the ignition position was a crucial control parameter for the FPELG during the steady operation process.To improve the stabilization of the FPELG during the operating process, it was essential to closely align the piston trajectory with the reference piston curve [32].In addition, in order to improve the energy conversion efficiency of the FPELG, the resonant pendulum control method was proposed.The results showed that during the stable operation process, when the FPELG had high efficiency, the gas pressure of the gas spring depended on the input energy [33].
Chi Zhang group from the Chinese Academy of Sciences had proposed a decoupling design approach for the FPELG with the output power range from 10 kW to 25 kW and an interval of 5 kW.Furthermore, the experimental results demonstrated that the errors of the key operation parameters were less than 5% [34].Subsequently, the full cycle operation strategies of the FPELG were developed, including the starting process, the stable operating process, the fault recovery and stopping process [35].In order to improve the output power of the FPELG, the ladder-like electromagnetic force control strategy was proposed [36].And the simulation results demonstrated that the output power of the FPELG could increase by 7~10% when the input energy was constant.

Summary
According to the aforementioned discussion, it becomes evident that researchers around the world have mostly focused on the piston operation characteristics of the singlecylinder/single-piston structure type of the FPELG coupling with the mechanical spring or the gas spring.However, the specific advantages of the single-cylinder/single-piston structure type of the FPELG coupling with different rebound devices have not been reported.Consequently, in this paper, the dynamics of the piston and engine performances of the single-cylinder/single-piston structure type of the FPELG coupling with various rebound devices are quantitatively analyzed.Furthermore, the performance sensitivity of the FPELG coupling with the mechanical spring and the gas spring under various parameters is also investigated, respectively.The results from this paper provide valuable insights into the specific advantages of the single-cylinder/single-piston structure type of the FPELG coupling with different rebound devices, which can help in making informed decisions about selecting the most suitable rebound devices for specific application processes.

Numerical Models
From the aforementioned published literature, it is evident that the mechanical springs and the gas springs serve as typical types of the rebound devices that are employed in the single-cylinder/single-piston structure type of the FPELG.These two different rebound devices of the single-cylinder/single-piston structure type of the FPELG possess distinctive characteristics and have their own unique operating principles.In this paper, numerical models are employed to facilitate a more comprehensive quantitative analysis of the dynamics of the piston and engine performances of the FPELG coupling with the mechanical springs and the gas springs.
The moving components of the FPELG are collectively referred to as the mover, which includes the piston from the free piston engine, the magnet rod for the linear generator, and the connecting rod for the rebound device.According to the operation principle of the FPELG, the kinematics of the piston and rods are completely determined by the forces acting on it.The mover of the single-cylinder/single-piston structure type of the FPELG, regardless of whether the rebound device is the mechanical spring or the gas spring, is susceptible to the following forces: the gas pressure acting on the piston from the free piston engine, the friction force, the electromagnetic resistance force from the linear generator, and the rebound force from the rebound device.And then Figure 2 depicts all the forces acting on the piston and rods of the single-cylinder/single-piston structure type of the FPELG during the stable generating process.The reference point used as the origin (0 point) is denoted as the TDC (Top Dead Center) of the piston.And the positive value of the piston displacement indicates that the piston is positioned to the right of the reference point.Likewise, the positive value of the piston velocity indicates the rightward movement of the piston from the reference point, while the negative value of the piston velocity indicates the movement of the piston in the opposite direction.Based on Newton's second law, the dynamics equation of the piston is expressed where, x is the displacement of the piston and the rods of the linear generator and rebou device, m is the mass of the piston and the rods of the linear generator and rebound dev and A is the top area of the piston of the free piston engine.
As for the electromagnetic resistance force Fe, it can be described as follows [23]: where, kf is the thrust force constant coefficient, kε is the constant coefficient of the b EMF of the linear generator, Rs is the resistance of the coil, RL is the resistance of the ex nal load, and L is the inductance of the linear generator.
As for the friction force Ff, which is approximately proportional to the velocity of piston during the operation process, it can be expressed as where, Cf is the coefficient of friction.
As for the gas pressure in-cylinder from the free piston engine in this paper, the ze dimensional model is adopted to describe the free piston internal combustion engine d ing the operating process, which includes the compression process, combustion and h transfer process, expansion process, and exhaust process [37].
Referring to the previous paper [38], the gas pressure in-cylinder from the free pis engine can be calculated as 1) where, V is the instantaneous cylinder volume of the free piston engine, Qin is the h released during the combustion process, and Qht is the heat released during the comb tion process.
Referring to the Wiebe function, the heat released during the combustion process be derived as follows: Forces acting on the piston and rods of the single-cylinder/single-piston structure type FPELG.Note: p is the gas pressure in the engine cylinder, F f is the friction force, F e is the electromagnetic resistance force of the linear generator, and F s is the rebound force from the spring of the rebound device.
Based on Newton's second law, the dynamics equation of the piston is expressed as where, x is the displacement of the piston and the rods of the linear generator and rebound device, m is the mass of the piston and the rods of the linear generator and rebound device, and A is the top area of the piston of the free piston engine.
As for the electromagnetic resistance force F e , it can be described as follows [23]: where, k f is the thrust force constant coefficient, k ε is the constant coefficient of the back EMF of the linear generator, R s is the resistance of the coil, R L is the resistance of the external load, and L is the inductance of the linear generator.
As for the friction force F f , which is approximately proportional to the velocity of the piston during the operation process, it can be expressed as where, C f is the coefficient of friction.
As for the gas pressure in-cylinder from the free piston engine in this paper, the zero-dimensional model is adopted to describe the free piston internal combustion engine during the operating process, which includes the compression process, combustion and heat transfer process, expansion process, and exhaust process [37].
Referring to the previous paper [38], the gas pressure in-cylinder from the free piston engine can be calculated as where, V is the instantaneous cylinder volume of the free piston engine, Q in is the heat released during the combustion process, and Q ht is the heat released during the combustion process.
Referring to the Wiebe function, the heat released during the combustion process can be derived as follows: where, a and b are parameters in the Wiebe function, respectively.C d is the duration of the combustion process, t s is the combustion starting timing, Q is the input energy from the fuel in each cycle.The significant factors affecting heat transfer were found to be the gas temperature in-cylinder, heat transfer area, and flow pattern of the cycle [39].The heat transferred to the cylinder wall can be derived as follows: where, A c is the area of the in-cylinder surface in contact with the gas, T w is the average surface temperature of the cylinder wall, and h is the heat transfer coefficient [40].The Woschni heat transfer coefficient (h) can be expressed as As for the rebound force from the rebound device, the single-cylinder/single-piston structure type of the FPELG rebound force mainly consists of the rebound force from the mechanical spring or the gas spring.The rebound force from the mechanical spring is described according to Hooke's law, which does not take the mass or damping of the mechanical spring into account [41].Similarly, the rebound force from the gas spring is described according to the ideal gas equation of state [42].It is worth noting that the heat transfer loss and gas leakage of the ideal gas spring are ignored in this paper.To summarize, the rebound force from the rebound device can be described as follows: where, k s is the stiffness of the mechanical spring of the rebound device, p g is the gas pressure from the rebound-cylinder of the rebound device, and A g is the top area of the rebound piston of the rebound device.
In the simulation models, the gas spring of the rebound device is treated as an ideal air spring, and the heat transfer and gas leakage in the rebound-cylinder are neglected.The change in the gas pressure in the rebound-cylinder of the rebound device is caused by the fluctuation of the gas volume.The gas pressure of the rebound-cylinder can be described as follows: where, V g is the instantaneous rebound-cylinder volume of the gas spring.
Based on the aforementioned analysis, the zero-dimensional simulation models of the single-cylinder-single-piston type of the FPELG coupling with various rebound devices are established.These models mainly consist of several sub-models, including the piston dynamic sub-model, the electromagnetic resistance force sub-model, the friction sub-model, the gas pressure in-cylinder sub-model of the free piston engine, and the rebound force submodel of the rebound device.The numerical functions of these sub-models are summarized in Table 1.
piston engine sub-model of free Thermodynamic Compression Linear generator model Rebound device model

Main Parameters Specification
The parameters of the FPELG coupling with the mechanical spring and the gas spring are summarized in Table 2.It is worth noting that the stiffness of the mechanical spring is constant throughout the entire analysis process.On the contrary, as for the gas spring, the rebound force is dependent on the initial gas pressure and the bore of the rebound-cylinder, as described in the sub-model functions of the gas spring.In order to quantitatively compare the dynamics of the piston and engine performances of the FPELG coupling with various types rebound devices, the FPELG is operated under a certain operating condition.The intake air pressure of the free piston engine is set to be 1.2 bar, and the air/fuel ratio of the free piston engine is set to stoichiometric (the value is 1), the ignition timing is set to be 7 mm away from the cylinder head, and the fuel is gasoline.These standardized operating conditions can enable a fair and consistent comparison of the dynamics of the piston and the engine performances between the FPELG coupling with various types rebound devices.

Piston Dynamics
The piston displacement profiles of the FPELG coupling with the mechanical spring and the gas spring are displayed in Figure 3.It is worth noting that the piston of the FPELG coupling with the gas spring reaches the TDC earlier than that of the mechanical spring.The reasons for this are that in order to achieve the same length of the piston motion from TDC to BDC (Bottom Dead Center) in this paper, the equivalent stiffness of the gas spring needs to be stronger than that of the mechanical springs.As a result, during the single cycle process, the stored energy of the FPELG coupling with the gas spring is greater than that of the mechanical spring.It implies that the rebound force from the gas spring acting on the piston is stronger, compared to that of the mechanical spring.Consequently, the piston operation frequency of the FPELG coupling with the gas spring is faster than that of the mechanical spring.It can be seen that the piston dynamics of the FPELG are significantly influenced by the rebound parameters of the rebound device.
coupling with the gas spring reaches the TDC earlier than that of the mechanical The reasons for this are that in order to achieve the same length of the piston motio TDC to BDC (Bottom Dead Center) in this paper, the equivalent stiffness of the ga needs to be stronger than that of the mechanical springs.As a result, during th cycle process, the stored energy of the FPELG coupling with the gas spring is grea that of the mechanical spring.It implies that the rebound force from the gas sprin on the piston is stronger, compared to that of the mechanical spring.Consequen piston operation frequency of the FPELG coupling with the gas spring is faster th of the mechanical spring.It can be seen that the piston dynamics of the FPELG are icantly influenced by the rebound parameters of the rebound device.The piston velocity profiles of the FPELG coupling with the mechanical spr the gas spring are shown in Figure 4.The results reveal that regardless of whet FPELG is coupled with the gas spring or the mechanical spring, the piston peak of the FPELG during the expansion process is significantly faster than that dur compression process.When the FPELG is coupled with the mechanical spring, th peak velocities during the compression process and expansion process are 2.7 m 4.69 m/s, respectively.Similarly, when the FPELG is coupled with the gas spring, ton peak velocities during the compression process and expansion process are 2.9 m 4.9 m/s, respectively.Because the rebound energy of the FPELG during the comp process comes from the rebound device, it is significantly less than that of the com and released heat energy during the combustion and expansion process, which or directly from the fuel.It is worth noting that during the starting stage of the comp stroke, the piston acceleration of the FPELG coupling with the gas spring is grea that of the mechanical spring, because the stiffness of the mechanical spring is co while the equivalent stiffness of the gas spring is nonlinear.During the compressi cess of the FPELG, as the volume of the compression chamber of the gas spring in the pressure decreases, resulting in a decrease in the equivalent stiffness of the gas The piston velocity profiles of the FPELG coupling with the mechanical spring and the gas spring are shown in Figure 4.The results reveal that regardless of whether the FPELG is coupled with the gas spring or the mechanical spring, the piston peak velocity of the FPELG during the expansion process is significantly faster than that during the compression process.When the FPELG is coupled with the mechanical spring, the piston peak velocities during the compression process and expansion process are 2.7 m/s and 4.69 m/s, respectively.Similarly, when the FPELG is coupled with the gas spring, the piston peak velocities during the compression process and expansion process are 2.9 m/s and 4.9 m/s, respectively.Because the rebound energy of the FPELG during the compression process comes from the rebound device, it is significantly less than that of the combustion and released heat energy during the combustion and expansion process, which originates directly from the fuel.It is worth noting that during the starting stage of the compression stroke, the piston acceleration of the FPELG coupling with the gas spring is greater than that of the mechanical spring, because the stiffness of the mechanical spring is constant, while the equivalent stiffness of the gas spring is nonlinear.During the compression process of the FPELG, as the volume of the compression chamber of the gas spring increases, the pressure decreases, resulting in a decrease in the equivalent stiffness of the gas spring.On the contrary, the stiffness of the mechanical spring is constant, resulting in a linear change in the piston velocity of the FPELG coupling with the mechanical spring throughout the entire compression stroke.However, during the expansion stroke, the piston velocity profile of the FPELG coupling with the mechanical spring is identical to that of the gas spring, because the input energy is the same.On the contrary, the stiffness of the mechanical spring is constant, resulting change in the piston velocity of the FPELG coupling with the mechanical sprin out the entire compression stroke.However, during the expansion stroke, the locity profile of the FPELG coupling with the mechanical spring is identical to gas spring, because the input energy is the same.Similarly, the positive value of the piston velocity means the p FPELG operates during the expansion stroke, which moves from TDC to BDC observed that, regardless of whether the FPELG is coupled with the gas spring chanical spring, the piston peak velocity of the FPELG during the expansion significantly faster than that during the compression process.As discussed regardless of whether the FPELG is coupled with the gas spring or the mechan the piston acceleration of the FPELG during the expansion process is significan than that during the compression process.In addition, since the residence dur piston at the TDC is much shorter than that at the BDC, it means that the FPEL present for a shorter period near TDC, which has the potential advantage of re heat transfer losses and NOx emissions.Figure 5 reveals the piston displacement-velocity profiles of the FPELG coupling with the mechanical spring and the gas spring.The negative velocity of the piston indicates the piston of the FPELG operates during the compression stroke, which occurs from BDC to TDC.Similarly, the positive value of the piston velocity means the piston of the FPELG operates during the expansion stroke, which moves from TDC to BDC.It can be observed that, regardless of whether the FPELG is coupled with the gas spring or the mechanical spring, the piston peak velocity of the FPELG during the expansion process is significantly faster than that during the compression process.As discussed previously, regardless of whether the FPELG is coupled with the gas spring or the mechanical spring, the piston acceleration of the FPELG during the expansion process is significantly greater than that during the compression process.In addition, since the residence duration of the piston at the TDC is much shorter than that at the BDC, it means that the FPELG piston is present for a shorter period near TDC, which has the potential advantage of reducing the heat transfer losses and NOx emissions.

Engine Performances
The pressure-volume diagrams of the FPELG coupling with the mechanical spring and the gas spring are illustrated in Figure 6.It can be observed that regardless of whether the FPELG is coupled with the gas spring or the mechanical spring, the combustion process is similar to the constant-volume process.However, the peak pressure in-cylinder of the FPELG coupling with the gas spring is higher than that of the mechanical spring.This is due to the fact that the kinetic energy of the piston of the FPELG coupling with the gas spring is greater than that of the mechanical spring, as discussed above.

Engine Performances
The pressure-volume diagrams of the FPELG coupling with the mechanica and the gas spring are illustrated in Figure 6.It can be observed that regardless of the FPELG is coupled with the gas spring or the mechanical spring, the combust cess is similar to the constant-volume process.However, the peak pressure in-cyl the FPELG coupling with the gas spring is higher than that of the mechanical spri is due to the fact that the kinetic energy of the piston of the FPELG coupling with spring is greater than that of the mechanical spring, as discussed above.

Engine Performances
The pressure-volume diagrams of the FPELG coupling with the mechanica and the gas spring are illustrated in Figure 6.It can be observed that regardless of w the FPELG is coupled with the gas spring or the mechanical spring, the combust cess is similar to the constant-volume process.However, the peak pressure in-cyl the FPELG coupling with the gas spring is higher than that of the mechanical spri is due to the fact that the kinetic energy of the piston of the FPELG coupling with spring is greater than that of the mechanical spring, as discussed above.In order to quantitatively compare the engine performances of the FPELG c with various types of rebound devices, the duration of each process is displayed in 7. Similar to the conventional reciprocating engine, the durations of each process, ing the compression process, the combustion process, the expansion process and In order to quantitatively compare the engine performances of the FPELG coupling with various types of rebound devices, the duration of each process is displayed in Figure 7. Similar to the conventional reciprocating engine, the durations of each process, including the compression process, the combustion process, the expansion process and the gas exchange process, are significantly different.From Figure 7, it can be seen that the duration of the compression process for the FPELG coupling with the gas spring is shorter than that of the mechanical spring.Due to the same physical structure of the free piston engine, the piston velocity of the FPELG coupling with the gas spring is faster than that of the mechanical spring when the stroke length is constant.On the other hand, regardless of whether the FPELG is coupled with the gas spring or the mechanical spring, the combustion duration remains the same.It is worth noting that the compression ratio (TDC is the same) and the initial conditions are the same for both the FPELG when coupled with the gas spring and the mechanical spring.According to the operating principle of the FPELG, for the duration of the expansion process, the piston velocity profile is asymmetric, which means that the duration of the expansion process is shorter than that during the compression process.Furthermore, according to the properties of the gas spring and the mechanical spring, as the resistance force from the gas spring is smaller than that from the mechanical spring, the expansion duration of the FPELG coupling with the gas spring is shorter than that of the mechanical spring.In other words, unlike the mechanical spring, the equivalent stiffness of the gas spring is nonlinear.As the volume of the compression chamber of the gas spring decreases, the pressure increases, and hence the equivalent stiffness of the gas spring changes throughout the entire expansion process, increasing from a small value to a large value.On the contrary, the stiffness of the mechanical spring remains constant throughout the whole operating process.With regard to the duration of the gas exchange process, it can be observed that as the piston operation frequency of the FPELG coupling with the gas spring is faster than that of the mechanical spring, the gas exchange duration of the FPELG coupling with the gas spring is shorter than that of the mechanical spring.And due to the same physical structure of the free piston engine, the piston operation distance during the gas exchange stage is constant.exchange process, are significantly different.From Figure 7, it can be seen that the duration of the compression process for the FPELG coupling with the gas spring is shorter than that of the mechanical spring.Due to the same physical structure of the free piston engine the piston velocity of the FPELG coupling with the gas spring is faster than that of the mechanical spring when the stroke length is constant.On the other hand, regardless of whether the FPELG is coupled with the gas spring or the mechanical spring, the combustion duration remains the same.It is worth noting that the compression ratio (TDC is the same) and the initial conditions are the same for both the FPELG when coupled with the gas spring and the mechanical spring.According to the operating principle of the FPELG for the duration of the expansion process, the piston velocity profile is asymmetric, which means that the duration of the expansion process is shorter than that during the compression process.Furthermore, according to the properties of the gas spring and the mechanical spring, as the resistance force from the gas spring is smaller than that from the mechanical spring, the expansion duration of the FPELG coupling with the gas spring is shorter than that of the mechanical spring.In other words, unlike the mechanical spring the equivalent stiffness of the gas spring is nonlinear.As the volume of the compression chamber of the gas spring decreases, the pressure increases, and hence the equivalent stiffness of the gas spring changes throughout the entire expansion process, increasing from a small value to a large value.On the contrary, the stiffness of the mechanical spring remains constant throughout the whole operating process.With regard to the duration of the gas exchange process, it can be observed that as the piston operation frequency of the FPELG coupling with the gas spring is faster than that of the mechanical spring, the gas exchange duration of the FPELG coupling with the gas spring is shorter than that of the mechanical spring.And due to the same physical structure of the free piston engine, the piston operation distance during the gas exchange stage is constant.Due to the diverse properties of the gas spring and the mechanical spring, the performances of the FPELG coupling with various springs are different.The piston operation frequency of the FPELG coupling with the gas spring is faster than that of the mechanical spring and is the same as the peak gas pressure in cylinder.However, the thermal efficiency of the FPELG coupling with the gas spring is 32.5%, and it is 32% for the mechanical spring.This is because, when the input energy is constant, the indicated work of the FPELG coupling with the gas spring is approximately the same as that of the mechanical spring.Furthermore, as the piston operation frequency of the FPELG coupling with the gas spring is faster than that of the mechanical spring, the indicated power of the FPELG coupling with the gas and the mechanical spring is 1.5 kW and 1.3 kW, respectively.As for the gas spring, the key drawback is the gas leakage that occurs during the actual operating process.Consequently, the performances of the FPELG coupling with the ideal gas spring (the gas leakage of the ideal gas spring in cylinder are neglected) and the actual gas spring (the gas leakage of the actual gas spring in cylinder are not neglected) are quantitatively analyzed, as shown in Table 3.It has been observed that the peak gas pressure of the FPELG coupling with the ideal gas spring is higher than that of the actual gas spring.On the contrary, the thermal efficiency of the FPELG coupling with the actual gas spring is reduced by approximately 2.5%, compared to that of the ideal gas spring.At the same time, the indicated power and the output electric power of the FPELG coupling with the actual gas spring are both reduced by 10% and 12.5%, respectively.Due to the gas leakage of the actual gas spring, the rebound energy of the ideal gas spring is greater than that of the actual gas spring.The energy distribution of the FPELG coupling with various types of rebound springs is depicted in Figure 8.It is found that the indicated thermal efficiency of the FPELG coupling with the mechanical spring, the ideal gas spring, and the actual gas spring is 32%, 32.5%, and 30%, respectively.It is worth noting that the indicated work of the FPELG coupling with the ideal gas spring is roughly the same as that of the mechanical spring, and the input energy is also the same.Nevertheless, the indicated thermal efficiency of the FPELG coupling with the actual gas spring is less than that of the ideal gas spring.This discrepancy can be attributed to the gas leakage inherent of the actual gas spring, which diminishes its rebound force.Furthermore, due to the uniform construction of the free piston engine, the heat transfer loss of the FPELG coupling with various types of rebound devices is equivalent.And the same holds true for the fuel combustion loss and the friction loss.Consequently, compared with the mechanical spring and the ideal gas spring, the other loss of the FPELG coupling with the actual spring is much higher.

FPELG Performances Sensitivity Discussion
According to the operating principle of the single-cylinder/single-piston structure type of the FPELG, the rebound force of the rebound device is required to overcome the resistance force during the compression process.And the piston dynamics and engine characteristics during the compression process are determined by the properties of the rebound device.As a result, the parameter changes of the rebound device have a direct effect on the performances of the FPELG.Therefore, it is essential to discuss the performance sensitivity of the FPELG with various stiffnesses of the mechanical spring and various initial gas pressures of the gas spring.

Performances Sensitivity of the FPELG under Various Stiffness of the Mechanical Spring
The piston displacements of the FPELG coupling with the mechanical spring under various stiffness are depicted in Figure 9, with a stiffness range of 15 kN/m to 23 kN/m and an interval of 2 kN/m.It is obviously observed that the TDC position is positively correlated with the stiffness of the mechanical spring.As a result, the compression ratio increases as the stiffness of the mechanical spring increases, which improved the combustion efficiency of the free piston engine.However, the operation stroke length of the piston is inversely proportional to the stiffness of the mechanical spring.In addition, the operation frequency of the piston increases with the increase in the stiffness of the mechanical spring.Due to the increase in the stiffness of the mechanical spring and the rebound force, the piston velocity increased during the compression process.

FPELG Performances Sensitivity Discussion
According to the operating principle of the single-cylinder/single-piston structure type of the FPELG, the rebound force of the rebound device is required to overcome the resistance force during the compression process.And the piston dynamics and engine characteristics during the compression process are determined by the properties of the rebound device.As a result, the parameter changes of the rebound device have a direct effect on the performances of the FPELG.Therefore, it is essential to discuss the performance sensitivity of the FPELG with various stiffnesses of the mechanical spring and various initial gas pressures of the gas spring.

Performances Sensitivity of the FPELG under Various Stiffness of the Mechanical Spring
The piston displacements of the FPELG coupling with the mechanical spring under various stiffness are depicted in Figure 9, with a stiffness range of 15 kN/m to 23 kN/m and an interval of 2 kN/m.It is obviously observed that the TDC position is positively correlated with the stiffness of the mechanical spring.As a result, the compression ratio increases as the stiffness of the mechanical spring increases, which improved the combustion efficiency of the free piston engine.However, the operation stroke length of the piston is inversely proportional to the stiffness of the mechanical spring.In addition, the operation frequency of the piston increases with the increase in the stiffness of the mechanical spring.Due to the increase in the stiffness of the mechanical spring and the rebound force, the piston velocity increased during the compression process.
In order to directly evaluate the engine performances of the FPELG coupling with various stiffnesses of the mechanical spring, the pressure-volume diagrams are depicted in Figure 10.It can be observed that as the stiffness of the mechanical spring increases, the combustion process of the engine is nearly closed to the isovolumetric process.In addition, the area covered by the pressure-volume diagram became large with the increase in the stiffness of the mechanical spring.It means that the indicated work of the engine, i.e., the pressure-volume-diagram-covered area, is positively relative to the stiffness of the mechanical spring, as shown in Figure 11.Furthermore, when the input energy is constant, the indicated thermal efficiency of the FPELG also increased.These results demonstrate that the stiffness of the mechanical spring is a critical factor influencing the performances of the PFELG coupling with the mechanical spring.In order to directly evaluate the engine performances of the FPELG coupling with various stiffnesses of the mechanical spring, the pressure-volume diagrams are depicted in Figure 10.It can be observed that as the stiffness of the mechanical spring increases, the combustion process of the engine is nearly closed to the isovolumetric process.In addition, the area covered by the pressure-volume diagram became large with the increase in the stiffness of the mechanical spring.It means that the indicated work of the engine, i.e., the pressure-volume-diagram-covered area, is positively relative to the stiffness of the mechanical spring, as shown in Figure 11.Furthermore, when the input energy is constant, the indicated thermal efficiency of the FPELG also increased.These results demonstrate that the stiffness of the mechanical spring is a critical factor influencing the performances of the PFELG coupling with the mechanical spring.In order to directly evaluate the engine performances of the FPELG coupling with various stiffnesses of the mechanical spring, the pressure-volume diagrams are depicted in Figure 10.It can be observed that as the stiffness of the mechanical spring increases, the combustion process of the engine is nearly closed to the isovolumetric process.In addition, the area covered by the pressure-volume diagram became large with the increase in the stiffness of the mechanical spring.It means that the indicated work of the engine, i.e., the pressure-volume-diagram-covered area, is positively relative to the stiffness of the mechanical spring, as shown in Figure 11.Furthermore, when the input energy is constant, the indicated thermal efficiency of the FPELG also increased.These results demonstrate that the stiffness of the mechanical spring is a critical factor influencing the performances of the PFELG coupling with the mechanical spring.The equivalent rotational speed of the FPELG coupling with the mechanical spring under various stiffnesses is listed in Table 4.As the stiffness of the mechanical spring increases, the equivalent rotational speed of the FPELG also increases.This indicates that the stiffness of the mechanical spring can affect the output power of the free piston engine.Concurrently, the peak gas pressure also increases with the increase in the spring stiffness, further demonstrating the impact of the stiffness of the mechanical spring on the output power of the free piston engine.In addition, as spring stiffness increases, the piston peak velocity increases slightly.According to the mechanics principle, as the gas pressure in the cylinder increases, the force acting on the piston also increases, resulting in the piston moving more quickly.These demonstrate that the spring stiffness can also affect the operational stability of the free piston engine.According to the operating principle of the engine, one of the crucial factors describing the engine performances of the free piston engine is the compression ratio.The compression ratio steadily rises with the increase in mechanical spring stiffness, and the indicated efficiency of the FPELG increases.The equivalent rotational speed of the FPELG coupling with the mechanica under various stiffnesses is listed in Table 4.As the stiffness of the mechanical sp creases, the equivalent rotational speed of the FPELG also increases.This indica the stiffness of the mechanical spring can affect the output power of the free piston Concurrently, the peak gas pressure also increases with the increase in the spring s further demonstrating the impact of the stiffness of the mechanical spring on the power of the free piston engine.In addition, as spring stiffness increases, the pist velocity increases slightly.According to the mechanics principle, as the gas pre the cylinder increases, the force acting on the piston also increases, resulting in th moving more quickly.These demonstrate that the spring stiffness can also affect erational stability of the free piston engine.According to the operating principl engine, one of the crucial factors describing the engine performances of the fre engine is the compression ratio.The compression ratio steadily rises with the inc mechanical spring stiffness, and the indicated efficiency of the FPELG increases.Similar to the impact of the mechanical spring as the rebound device, the rebound force from the gas spring has a direct effect on the dynamics of the piston and engine characteristics of the FPELG coupling with the gas spring.According to the operating principle of the gas spring, the rebound force from the gas spring depends on the initial gas pressure.As a result, the performance sensitivity of the FPELG under various initial gas pressures of the gas spring is analyzed.In addition, the dynamics of the piston and engine characteristics are compared with the initial gas pressure of the gas spring range from −20% to 0% (which represents 1 bar) with an interval of 5%, while other parameters are constant.Figure 12 reveals that the compression ratio of the free piston engine is positively correlated with the initial gas pressure of the gas spring.This is because the equivalent stiffness of the gas spring increases with the increase in the initial gas pressure.Therefore, the changing trend of the gas spring with various initial gas pressures is similar as those of the mechanical spring with various stiffnesses.Furthermore, as the initial gas pressure of the gas spring increases, the operating frequency of the piston also increases.from −20% to 0% (which represents 1 bar) with an interval of 5%, while other parameters are constant.Figure 12 reveals that the compression ratio of the free piston engine is positively correlated with the initial gas pressure of the gas spring.This is because the equivalent stiffness of the gas spring increases with the increase in the initial gas pressure.Therefore, the changing trend of the gas spring with various initial gas pressures is similar as those of the mechanical spring with various stiffnesses.Furthermore, as the initial gas pressure of the gas spring increases, the operating frequency of the piston also increases.The pressure-volume diagram of the FPELG coupling with the gas spring under various initial gas pressures is shown in Figure 13.It is worth noting that as the initial gas pressure of the gas spring increases, the combustion process of the free piston engine is nearly closed to the isovolumetric process.According to the above discussion, the increase in the initial gas pressure of the gas spring is similar to the increase in the stiffness of the mechanical spring.Furthermore, the indicated work of the free piston engine, i.e., the pressure-volume-diagram-covered area, became large with the increase in the initial gas pressure of the gas spring, as shown in Figure 14.And when the input energy is constant, the indicated thermal efficiency of the FPELG also increases.Hence, the initial gas pressure of the gas spring plays a pivotal role in determining the performances of the PFELG coupling with the gas spring.The pressure-volume diagram of the FPELG coupling with the gas spring under various initial gas pressures is shown in Figure 13.It is worth noting that as the initial gas pressure of the gas spring increases, the combustion process of the free piston engine is nearly closed to the isovolumetric process.According to the above discussion, the increase in the initial gas pressure of the gas spring is similar to the increase in the stiffness of the mechanical spring.Furthermore, the indicated work of the free piston engine, i.e., the pressure-volume-diagram-covered area, became large with the increase in the initial gas pressure of the gas spring, as shown in Figure 14.And when the input energy is constant, the indicated thermal efficiency of the FPELG also increases.Hence, the initial gas pressure of the gas spring plays a pivotal role in determining the performances of the PFELG coupling with the gas spring.

Conclusions
In this paper, the dynamics of the piston and engine performances of the single-cylinder/single-piston type of the FPELG coupling with the mechanical spring and the gas spring are compared quantitatively.Furthermore, the performance sensitivity of the FPELG is discussed regarding the mechanical spring under various stiffnesses and the gas spring under various initial gas pressures, respectively.Consequently, the conclusions are as follows: (1) When the equivalent stiffness of the gas spring is stronger than the stiffness of the mechanical spring, the piston operation frequency of the FPELG coupling with the gas spring is faster than that of the mechanical spring.The piston peak velocity of the FPELG during the expansion process is significantly faster than that during the compression process, regardless of whether the FPELG is coupled with the gas spring or

Conclusions
In this paper, the dynamics of the piston and engine performances of the singlecylinder/single-piston type of the FPELG coupling with the mechanical spring and the gas spring are compared quantitatively.Furthermore, the performance sensitivity of the FPELG is discussed regarding the mechanical spring under various stiffnesses and the gas spring under various initial gas pressures, respectively.Consequently, the conclusions are listed as follows: (1) When the equivalent stiffness of the gas spring is stronger than the stiffness of the mechanical spring, the piston operation frequency of the FPELG coupling with the gas spring is faster than that of the mechanical spring.The piston peak velocity of the FPELG during the expansion process is significantly faster than that during the compression process, regardless of whether the FPELG is coupled with the gas spring or the mechanical spring.Furthermore, the piston peak velocity of the FPELG coupling with the gas spring is faster than that of the mechanical spring.The piston velocity of the FPELG coupling with the mechanical spring changes linearly during the compression stroke, while the changing trend of the piston velocity of the FPELG coupling with the gas spring is nonlinear during the compression stroke.(2) The combustion process is similar to the constant-volume process, regardless of whether the FPELG coupling with the gas spring or the mechanical spring.The compression duration and expansion duration of the FPELG coupling with the gas spring are shorter than these of the mechanical springs.The thermal efficiency of the FPELG coupling with the ideal gas spring is 32.5%, compared to 32% for the mechanical spring.And the indicated power of the FPELG coupling with the ideal gas spring and the mechanical spring is 1.5 kW and 1.3 kW, respectively.However, compared to the ideal gas spring, the thermal efficiency of the FPELG coupling with the actual gas spring under leakage reduces by approximately 2.5%.And the indicated power and output power of the FPELG coupling with the gas spring under leakage both reduce by 10% and 12.5%, respectively.(3) As for the FPELG coupling with the mechanical spring, the TDC position is positively related to the stiffness of the mechanical spring, while the piston operation stroke length is inversely proportional to the stiffness of the mechanical spring.The operation frequency of the piston increases with the increase in the stiffness of the mechanical

Figure 2 .
Figure2.Forces acting on the piston and rods of the single-cylinder/single-piston structure t FPELG.Note: p is the gas pressure in the engine cylinder, Ff is the friction force, Fe is the electrom netic resistance force of the linear generator, and Fs is the rebound force from the spring of the bound device.

Figure 3 .
Figure 3. Piston displacement profiles of the FPELG coupling with various types rebound

Figure 3 .
Figure 3. Piston displacement profiles of the FPELG coupling with various types rebound devices.

Figure 4 .
Figure 4. Piston velocity profiles of the FPELG coupling with various types rebound de

Figure 4 .
Figure 4. Piston velocity profiles of the FPELG coupling with various types rebound devices.

Figure 5 .
Figure 5. Piston displacement-velocity profiles of the FPELG coupling with various typ bound devices.

Figure 5 .
Figure 5. Piston displacement-velocity profiles of the FPELG coupling with various types of rebound devices.

Figure 5 .
Figure 5. Piston displacement-velocity profiles of the FPELG coupling with various typ bound devices.

Figure 6 .
Figure 6.Pressure-volume diagram of the FPELG coupling with various types of rebound

Figure 6 .
Figure 6.Pressure-volume diagram of the FPELG coupling with various types of rebound devices.

Figure 7 .
Figure 7. Duration of each stroke of the FPELG coupling with various types of rebound devices.

Figure 7 .
Figure 7. Duration of each stroke of the FPELG coupling with various types of rebound devices.

Energies 2023 , 21 Figure 8 .
Figure 8. Energy distribution of the FPELG coupling with various types of rebound devices.

Figure 8 .
Figure 8. Energy distribution of the FPELG coupling with various types of rebound devices.

Figure 9 .
Figure 9. Piston displacement profiles of the FPELG under different mechanical spring stiffnesses.

Figure 10 .
Figure 10.P-V diagrams of the FPELG coupling with different stiffnesses of the mechanical spring.

Figure 9 .
Figure 9. Piston displacement profiles of the FPELG under different mechanical spring stiffnesses.

Figure 9 .
Figure 9. Piston displacement profiles of the FPELG under different mechanical spring stiffnesses.

Figure 10 .
Figure 10.P-V diagrams of the FPELG coupling with different stiffnesses of the mechanical spring.Figure 10.P-V diagrams of the FPELG coupling with different stiffnesses of the mechanical spring.

Figure 10 .
Figure 10.P-V diagrams of the FPELG coupling with different stiffnesses of the mechanical spring.Figure 10.P-V diagrams of the FPELG coupling with different stiffnesses of the mechanical spring.

Energies 2023 ,Figure 11 .
Figure 11.Indicated work of the FPELG coupling with different stiffnesses of the mechani spring.

Figure 11 .
Figure 11.Indicated work of the FPELG coupling with different stiffnesses of the mechanical spring.

Figure 12 .
Figure 12.Compression ratio and operation frequency of the FPELG coupling with the gas spring under various initial gas pressures.

Figure 12 .
Figure 12.Compression ratio and operation frequency of the FPELG coupling with the gas spring under various initial gas pressures.

Energies 2023 , 21 Figure 13 .
Figure 13.Pressure-volume diagrams of the FPELG coupling with the gas spring under various initial gas pressures.

Figure 13 .
Figure 13.Pressure-volume diagrams of the FPELG coupling with the gas spring under various initial gas pressures.

Figure 13 .
Figure 13.Pressure-volume diagrams of the FPELG coupling with the gas spring under various initial gas pressures.

Figure 14 .
Figure 14.Indicated work of the FPELG coupling with the gas spring under various initial gas pressures.

Figure 14 .
Figure 14.Indicated work of the FPELG coupling with the gas spring under various initial gas pressures.

Table 1 .
Sub-models functions of the FPELG coupling with various types rebound devices.

Table 2 .
Simulation parameters of the FPELG coupling with various types rebound devices.

Table 3 .
Performances of the FPELG coupling with various types of rebound devices.

Table 4 .
Performances of the FPELG coupling with different stiffnesses of the mechanical s 4.2.Performances Sensitivity of the FPELG under Various Initial Gas Pressure of the Ga

Table 4 .
Performances of the FPELG coupling with different stiffnesses of the mechanical spring.