Research on Coupling Transfer Characteristics of Vibration Energy of Free Piston Linear Generator

Jingyi Tian， Huihua Feng， Yifan Chen and Shuochun Wang

（School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China）

Abstract: In order to clarify the mechanism and main influencing factors of the vibration energy coupling transmission with a dual?piston structure, a thermodynamic and dynamic coupling model of the free piston linear generator (FPLG) was established. The system energy conversion, vibra?tion energy coupling transmission, and influencing factors were studied in detail. The coupling transmission paths and the secondary influence mechanism from in?cylinder combustion on vibra?tion energy transmission were obtained. In addition, the influence of the movement characteristics of the dual?piston on the vibration energy transmission was studied, and the typical parameter vari?ation law was obtained, which provides theoretical guidance for the subsequent vibration reduction design of the FPLG.

Key words: free piston linear generator (FPLG)；coupled motion of dual?piston；vibration en?ergy transfer mechanism；analysis of influencing factors

Due to energy shortages and environmental degradation, while improving the fuel economy and emission performance of traditional engines,researchers are also researching new power sources that are more energy?efficient and envir?onmentally friendly[1?2]. The free piston linear generator (FPLG) is a new type of integrated power device aimed at the application of hybrid vehicles[3].

The test prototype of FPLG is shown in Fig. 1,which mainly includes two two?stroke ignition engines, a scavenging tank, an intake system, and a linear motor. The engine piston and the motor mover are fixedly connected by a connecting rod.The engine on both sides alternately generates explosive pressure to push the piston mover as?sembly to reciprocate and cut the magnetic in?duction line to generate electricity, converting fuel chemical energy into electrical energy out?put. FPLG cancels the crank connecting rod mechanism of traditional internal combustion en?gines. It not only has high comprehensive effi?ciency, but also has the advantages of large unit volume power density, strong fuel adaptability,simple mechanical structure and more[4], so its application prospects are optimistic. At present,research on the performance improvement and stable operation control of the new power source are being carried out worldwide.

The Toyota Technology Center (Toyota Central R&D Labs Inc.) has published a series of patents related to FPLG and built a single?cylin?der recovery FPLG prototype test platform to solve the problem of thermal load isolation[5?6].The German Aerospace Center (DLR) has been working on free?piston internal combustion en?gines since 2009[7]. At present, the commission?ing of a prototype capable of stable power gener?ation has been completed. Roskilly and Mikalsen of Sir Joseph Swan Centre, University of New?castle, UK, have been engaged in research work on FPLG engines and motor systems matching design and electronic control strategies since 2003[8]. Nanjing University of Science and Tech?nology, Shanghai Jiaotong University, Tongji University, and Beijing University of Technology have also conducted a series of research on the structural layout, combustion characteristics, and stable operation control of new power forms[9?12].

Fig. 1 FPLG test prototype

Researching energy transfer and conversion of new power systems, Sun Peng et al.[13]pro?posed a decoupling design method for system parameters by analyzing the power distribution,energy flow, and conversion process of FPLG sys?tems. Tian Chunlai et al.[14]established an energy model of a free?piston internal combustion gener?ator based on the law of conservation of energy.By analyzing the energy conversion characterist?ics, the energy changes under different compres?sion ratios and their effects on performance were obtained. The typical dual?piston FPLG has a large unbalanced inertial force during the move?ment process, so it will generate a large vibra?tion energy during the energy transfer process. In addition, the coupling transmission characterist?ics and influencing factors of FPLG vibration en?ergy are very different from traditional engines.At present, there are few reports about the re?search on the coupling transmission of FPLG vi?bration energy.

In this paper, based on the study of energy conversion mechanisms of the new power system,an FPLG multiphysics coupling simulation mod?el was established, and the model was verified by the test data. Through simulation analysis, the characteristics of the motion and vibration en?ergy excitation of the free piston were obtained.Then the transmission mechanism of vibration energy was analyzed to obtain the two main in?fluencing factors of energy transfer: the cylinder gas pressure and the compression energy. Based on the study of the cylinder gas pressure on the vibration energy transfer in different frequency bands, a secondary influence mechanism was ob?tained from the characteristics of in?cylinder combustion to the vibration excitation force and the vibration energy transfer. In addition, the in?fluence of typical structure, performance, and control parameters on vibration energy coupling transfer characteristics was obtained, which provides an important reference for FPLG vibra?tion reduction design.

1 FPLG System Energy Model

1.1 Mathematical model of system energy

1.1.1 Combustion heat

From the ideal gas state equation, the en?ergy released into the system by the combustion of the mixed gas in the cylinder of each side of the single?side engine is

Here Huis the low calorific value; ηcis the indicated efficiency, which represents the related energy loss in the process of incomplete combus?tion, heat transfer, and intake and exhaust; ηeis the mechanical efficiency, which represents the energy loss related to friction and vibration. α is the excess air coefficient, ηsis the scavenging effi?ciency, and ε is the compression ratio during steady?state operation.

1.1.2 Compression potential

During the compression stroke, the piston mover assembly relies on its own inertial force to overcome the resilience of the gas in the cylinder.The gas resilience force is Fcl. According to the first law of thermodynamics, when the piston reaches the dead center of the right cylinder, the compression energy of the system is Ec

The change of gas resilience force in the cyl?inder is non?linear. The closer to the dead ends on both sides, the more obviously the resilience changes with displacement, and the more drastic the change in corresponding compression poten?tial energy.

1.1.3 Kinetic energy

In a cycle, the kinetic energy of the piston mover assembly acts as a carrier for energy transfer. On the one hand, it will be affected by changes in the input and output energy of the system; on the other hand, it determines the movement of the piston mover assembly, which determines the energy transfer process between different forms. The kinetic energy at any mo?ment in the cycle is

where m is the mass of the moving component of the piston mover and v is the instantaneous velo?city.

1.1.4 Output energy

Under steady?state operating conditions, the electromagnetic resistance is approximately ex?pressed as

1.2 Thermodynamics and dynamic coupling simulation model

The power system model of the FPLG estab?lished by Simulink is shown in Fig. 2, in which the gas pressure, system friction force, and linear motor electromagnetic force in the engine cylin?ders on the left and right sides are given by each subsystem. The sum of the output forces of the four subsystems determines the instantaneous ac?celeration, velocity and displacement of the mov?ing components of the piston. The engine subsys?tem model includes two stages: start?up and steady?state operation. The steady?state opera?tion stage includes four working processes: com?pression, combustion, work, and exhaust scaven?ging, which are triggered by Stateflow discrete signals. The linear motor subsystem model in?clude two modes of motor and generator. The Runge?Kutta solver is used to solve the differen?tial equations. Iterative calculation uses a fixed step size with a step size of 10–6s.

The simulation process is shown in Fig. 3.Firstly, the parameters of the model variables are initialized. Then, the cylinder pressure and elec?tromagnetic resistance are calculated. Secondly,the acceleration, velocity and displacement of the piston are calculated. Finally, the program judges whether the simulation time has reached the pre?determined time. If the result is no, the program continues. If the result is yes, the simulation ends.

Fig. 2 FPLG operation process numerical simulation model

Fig. 3 Numerical simulation solution process

1.3 Simulation model verification

The main parameters are shown in Tab. 1.In order to verify the accuracy of the simulation model, the FPLG prototype was tested. The cyl?inder pressure signal obtained by the simulation calculation was compared with the experimental test results, as shown in Fig. 4.

Tab. 1 Main parameters of test prototype

Fig. 4 Comparison results of FPLG cylinder gas pressure

It can be seen that the cylinder pressure curve obtained by numerical simulation is basic?ally consistent with the experimental test, and the error of the peak pressure is within 6%, indic?ating that the simulation model can accurately reflect the FPLG piston movement and the ther?modynamic process in the cylinder.

2 Vibration Energy Transfer Mechanism

2.1 Coupled motion of dual-piston and vibration excitation of FPLG

The time domain and frequency domain characteristics of piston motion are analyzed, as shown in Fig. 5. Compared with the traditional engine (TE), the motion characteristics of FPLG pistons are quite different. The piston displace?ment curve of a TE is axisymmetric with respect to the top dead center position. However, the dis?placement of the FPLG shows obvious asym?metry, and has a relatively slow compression stroke and a rapid expansion stroke. The maxim?um difference in frequency domain is 10 dB. The peak value of FPLG velocity is 14% lower than that of TE, but the time to reach its peak value is 0.002 s earlier than that for TE. The maxim?um difference of velocity in the frequency do?main is 5 dB. It can be seen from the time?domain curve that the peak acceleration of the FPLG is close to 3 times larger than the TE. The difference of acceleration in frequency domain is nearly 20 dB.

Fig. 5 Comparison results of piston motion

Special motion characteristics of the FPLG directly influence excitation characteristics. The excitation force of the FPLG includes inertial force, cylinder gas pressure, electromagnetic force and friction force, as shown in Fig. 6. The amp?litudes of inertial force, cylinder gas pressure,electromagnetic force, and friction force of the FPLG are 9.28 kN, 10.55 kN, 2.38 kN and 0.13 kN,respectively, and the period is 0.04 s. In the fre?quency domain, the excitation force is concen?trated in the low frequency band below 500 Hz,and the fundamental frequency is 23 Hz. As the frequency increases, the amplitude gradually de?creases. Inertial force is the main force affecting the vibration of the whole machine, while the gas pressure, electromagnetic force, and friction force in the cylinder mainly affect structural vibration.

Fig. 6 FPLG vibration exciting forces

2.2 System energy conversion

The FPLG system transfers the heat energy generated by the fuel combustion in one cylinder to the opposite cylinder through the movement of the piston mover during the work process. Dur?ing the energy transfer process, the linear motor converts the mechanical energy generated by the piston mover into electrical energy. The process of system energy transfer and conversion is shown in Fig. 7.

Fig. 7 Schematic diagram of system energy transfer and conversion process

According to the principle of energy conser?vation, the conversion relationship of the energy of the FPLG in a cycle is

Here Eiis the input energy, Eclis the com?pression energy of the left cylinder, Eeis the elec?trical energy output, Ecris the compression en?ergy of the right cylinder, and Esis the energy lost, including the losses due to combustion and heat transfer, as well as mechanical losses due to friction and vibration. The comparison of the dis?placement?velocity loop is shown in Fig. 8, and the energy conversion and transmission process is divided into three stages.

Fig. 8 Comparison of displacement?velocity loop

In stage 1, the combustion heat energy and most of the compression energy are quickly con?verted into the kinetic energy of the piston mover. Therefore, the unbalanced inertial force makes the FPLG produce larger vibration. In stage 2, the cylinder pressure has exceeded the peak and the piston movement speed is at a rel?atively high level. The energy conversion form at this stage is mainly that the kinetic energy of the piston mover assembly is consumed by the load of the linear motor. In stage 3, the kinetic en?ergy of the piston mover is converted into the compression potential energy in the right cylin?der. Under the gas pressure of the right cylinder,the piston mover starts the reverse movement in the next cycle.

Combining the three stages above, the change of the kinetic energy of the mover will directly affect the transmission of vibration en?ergy, and the kinetic energy depends on two factors. One is the heat energy generated by the combustion of fuel, and the other is the compres?sion energy transferred by the opposed cylinder.

2.3 Vibration energy transmission paths

There are three main transmission paths of excitation energy in the traditional engine cylin?der, which are transmitted to the surface of the body along the piston crankshaft system, the side wall of the cylinder block, and the cylinder head.For FPLG, due to the coupled motion of the dual?piston, the excitation energy increases an?other transmission path, as shown in Fig.9.

Fig. 9 FPLG vibration energy coupling transmission paths

Taking the cylinder on one side of the FPLG as an example, there are four main transmission paths of the exciting energy of the gas burst pres?sure during the transmission to the base. The first is that the gas burst pressure is transmitted to the cylinder block along the cylinder head, and then to the scavenging box and base; the second is the gas burst pressure directly transmitted to the cylinder block, and then to the scavenging tank and base; the third is the excitation energy from the piston transferred to the cylinder block,and then to the scavenging box and the base; the fourth is the energy transmitted to the opposite piston through the mover, and to the opposite cylinder, the scavenging box, and the base.inder can be decomposed into static force and dy?namic force

Here the static force P1reflects the macro characteristics of the gas pressure in the cylinder,and the dynamic force P2reflects the character?istics of instantaneous pulsation and fluctuation.According to the law of conservation of mass,combined with the continuity equation, energy equation, and state equation, the static force P1can be obtained asn be obtained as

3 Influence of Cylinder Gas Pressure on the Vibration Energy Transmission

3.1 Mathematical description of the cylinder gas pressure

The influence of the gas combustion status in the cylinder on the vibration energy transmis?sion is mainly reflected in the change of the cyl?inder gas pressure. The time of pressure change in cylinder is similar to that of combustion in cylinder. In addition, the cylinder pressure fluctu?ates near the top dead center, so it has a transi?ent characteristic. The gas pressure P in the cyl?

This equation describes the combustion gas pressure oscillation.

3.2 Spectrum characteristics of force in the cylinder

The cylinder pressure distribution of FPLG is shown in Fig.10.

It can be seen from Fig.10 that the spec?trum of the gas pressure level in the FPLG cylin?der can be divided into three parts.

Fig. 10 FPLG cylinder pressure time?frequency distribution

In the low frequency range of 0–100 Hz, the maximum in?cylinder pressure level is mainly af?fected by the integral of the internal pressure curve of the cylinder and the peak cylinder pres?sure. The main excitation source is static force.Due to the high structural rigidity of the FPLG cylinder internal components, the vibration en?ergy generated by the static force excitation in the low frequency has obvious attenuation char?acteristics.

In the medium frequency range of 100 –1 000 Hz, the pressure level in the cylinder de?creases linearly with a logarithmic law, and the decreasing slope is affected by the rate of pres?sure increase. The force in the cylinder presents an overall pulsating characteristic. The vibration energy in the medium frequency will be transmit?ted to the body.

In the high frequency range above 1 000 Hz,the rise rate of the cylinder pressure and the high frequency oscillation of the local gas are the main factors that affect the pressure amplitude in the cylinder. There will be a peak pressure level in the cylinder in this frequency domain.

3.3 Influence mechanism of the exciting forces on vibration energy

There is a significant difference between the combustion processes and gas flow states of the FPLG and traditional engines. These character?istics will indirectly affect the transmission of vi?bration energy by affecting the in?cylinder force characteristics. Therefore, the mechanism of sec?ondary effect of cylinder force on vibration en?ergy is described, as shown in Fig.11.

Fig. 11 The second?order influence mechanism of FPLG vibra?tion energy transmission

In the process of energy transfer, the coup?ling motion of the dual?piston has an effect on the compression energy and compression ratio,which will affect the combustion of the next cycle. The gas flow and combustion stage charac?teristics will directly affect the force factor in the cylinder. The peak cylinder pressure reflects the characteristics of the static force in the cylinder,which will affect the transmission of low?frequency vibration energy. The peak pressure rise rate reflects the pulsating characteristics of the dynamic force in the cylinder, which mainly affects the transmission of medium frequency vibration energy. The rate of the change of pres?sure rise mainly reflects the fluctuation charac?teristics of the dynamic force in the cylinder,which mainly affects the transmission of high?frequency vibration energy.

4 Influence of Dual-Piston Movement on Vibration Energy Transmission

4.1 Mass of piston mover

The influence of dual?piston motion on vi?bration energy transfer is mainly reflected in the impact on the compression energy. The accumu?lated compression energy at the end of a work?ing cycle has a strong relationship with the max?imum kinetic energy converted in the process of dual?piston motion, which is directly affected by the system parameters. In order to obtain the parameter design method to reduce the vibration energy transfer, three typical structure, perform?ance and control parameters which have signific?ant influence on the motion of the dual?piston are selected in this paper, which are the mass of the piston mover, the electromagnetic resistance coefficient, and the ignition position.

The movement and exciting force character?istics that change with the mass of the piston mover are shown in Fig.12. As the mass in?creases from 3 kg to 7 kg, the system compres?sion energy increases from 4.2 J to 5.34 J, an in?crease of 27%. The rise rate of peak pressure in?creases from 0.15 MPa/s to 0.23 MPa/s, an in?crease of 53%.

Fig. 12 Motion law of the system under different mass of piston movers

The compression ratio of the piston mover and the peak cylinder pressure increases as the mass increases, resulting in increased vibration of the whole machine. In addition, as the peak speed of the piston motor decreases, the output electric energy converted from the motor de?creases, so at the end of the operation cycle, the compressed energy converted from the system to the opposed cylinder increases. With the in?crease of compression ratio, the rise rate of peak pressure also shows an increasing trend, which directly responds to the excitation of system vi?bration. Therefore, in order to reduce the trans?mission energy to the cylinder vibration, the mass of the piston movers should be reduced ap?propriately. However, if the mass is too small,the accumulated compression energy will be too small to reach the ignition position, resulting in shutdown failure. Therefore, it is necessary to consider the dynamic and vibration performance of the system when the mass of the piston is matched.

4.2 Electromagnetic resistance coefficient

The electromagnetic resistance coefficient is another important performance parameter that affects the motion of the piston mover. The change of electromagnetic resistance coefficient,the piston mover movement, and excitation char?acteristics of FPLG are shown in Fig.13. When the electromagnetic resistance coefficient in?creases from 200 N/ms–1to 300 N/ms–1, the com?pression energy decreases from 5.35 J to 4.75 J, a decrease of 11%. The rise rate of peak pressure decreased from 0.22 MPa/s to 0.14 MPa/s, a de?crease of 36%.

The change of the electromagnetic resist?ance coefficient will not affect the input energy of the system. In the case of equal compression en?ergy and fuel combustion chemical energy of the left cylinder, the difference of the electromagnet?ic resistance coefficient will lead to the change of the output electrical energy of the system, which will change the compression energy converted to the opposed cylinder. It can be seen from the ve?locity?displacement curve that with the increase of the electromagnetic resistance coefficient, the operating speed drops obviously when it moves to the position of the opposite dead center. Fi?nally, the compression energy of the cylinder is reduced, which affects the operation state of the next working cycle. At the same time, the com?pression ratio, peak cylinder pressure, and rise rate of peak pressure all show a downward trend,which is conducive to reducing the coupling transmission of vibration energy to the opposed cylinder. However, when the electromagnetic res?istance coefficient is too large, the operating fre?quency of the system will decrease, and the out?put electric energy will not follow the continu?ously increasing trend.

Fig. 13 Motion law of the system under different electromagnetic resistance coefficients

4.3 Ignition position

Ignition position is one of the key control parameters affecting combustion and energy con?version in the cylinder. As the ignition position changes, the piston mover’s movement and the excitation characteristics of FPLG are shown in Fig.14. With the increase of ignition position from 25 mm to 29 mm, the compression energy of the system increases from 4.32 J to 5.30 J, an increase of 23%. The rise rate of peak pressure decreases by 26% from 0.19 MPa/s to 0.14 MPa/s.The rise rate of peak pressure increases first and then decreases.

The change of ignition position mainly af?fects the combustion process, thus affecting the input energy of the system. When the distance between the ignition position and midpoint in?creases, with the increase of the fuel?air equival?ent ratio, the fuel molecules in unit increase, the combustion speed of mixed gas accelerates, and the peak pressure, rise rate of peak pressure, and compression energy all show a rising trend.However, if the fuel?air equivalent ratio is too large, the mixture in the cylinder will be too rich,resulting in the decrease of the gas pressure and the pressure rise rate. At the same time, the op?erating frequency of the piston mover will also be reduced, and the output electrical energy through the motor will also be reduced. From the velo?city?displacement curve, it can be seen that with the increase of ignition position, the maximum velocity of the piston mover decreases faster and faster, so the compression energy accumulated in the opposed cylinder shows a continuously in?creasing trend at the end of an operation cycle.

Fig. 14 Motion law of the system under different ignition positions

5 Conclusions

In this paper, a new type of coupled power source is studied. The mechanism of vibration en?ergy transfer of a dual?piston structure is ob?tained. The main influencing factors of energy transfer and the law of parameter change are cla?rified. The specific work is as follows.

① The FPLG piston mover has a faster ex?pansion stroke and a slower compression stroke,which is quite different from the traditional en?gine. The displacement of FPLG shows obvious asymmetry, the peak value of FPLG velocity is 14% lower than TE, and the peak acceleration is three times that of traditional engines.

②The second?order influence mechanism of vibration energy coupling transfer is obtained:For low?frequency vibration energy, the main in?fluence factor is the static force in the cylinder,which is reflected by the peak cylinder pressure;for medium frequency vibration energy, the main influence factor is the fluctuating characteristic of dynamic force, which can be expressed by the rate of pressure rise; high?frequency vibration is mainly affected by the high?frequency oscillation of the gas pressure in the cylinder.

③Three typical parameters are selected to study the influence of dual?piston motion on the coupled transmission of vibration energy. When the three parameters increase from the minimum to the maximum, the compression energy of the system changes 27%, 11%, and 23% respectively.Compression energy and peak pressure rise rate increase with the increase of mass, and decrease with the increase of the electromagnetic resist?ance coefficient. When the ignition position in?creases, the compression energy increases con?tinuously, while the peak pressure increases first and then decreases. Therefore, it is necessary to balance the vibration energy transfer and the stable operation of the system in the vibration reduction design of the FPLG.