Parameter influence law analysis and optimal design of a dual mass flywheel

The influence of the dynamic parameters of a dual mass flywheel (DMF) on its vibration reduction performance is analyzed, and several optimization algorithms are used to carry out multiobjective DMF optimization design. First, the vehicle powertrain system is modeled according to the parameter configuration of the test vehicle. The accuracy of the model is verified by comparing the simulation data with the test results. Then, the model is used to analyze the influence of the moment of inertia ratio, torsional stiffness, and damping in reducing DMF vibration. The speed fluctuation amplitude at the transmission input shaft and the natural frequency of the vehicle are taken as the optimization objectives. The passive selection method, multiobjective particle swarm optimization, and the nondominated sorting genetic algorithm based on an elite strategy are used to carry out DMF multiobjective optimization design. The advantages and disadvantages of these algorithms are evaluated, and the best optimization algorithm is selected.

natural frequency corresponding to the frequency of fundamental mode of the powertrain system can be reduced, and the resonance of the vehicle at idle speed can be eliminated. 4 Hartmut et al. 5 developed the dynamic model of the DMF drivetrain system, analyzed the rotational speed fluctuation and angular acceleration of the engine, transmission input, and output shaft, and evaluated the vibration reduction effectiveness of the DMF using the angular acceleration amplitude of the secondary flywheel. Kang et al. 6 used several test methods and theories to establish a performance test platform for DMF, which provided experimental conditions for verifying the theoretical research, performance improvement, and parameter optimization of DMF. Chen et al. 7 analyzed how to distribute the moment of inertia on both sides of DMF and design the multistage torsional stiffness, and designed a DMF based on test vehicle information matching, which was verified using ADAMS software to have good working performance, but no design optimization was carried out. Cheng 8  The structure of this paper is as follows: In Section 2, a certain MPV vehicle is taken as the study model. First, the vehicle powertrain model is developed according to its structural parameters, and the torsional vibration of the vehicle is simulated. The vehicle test is carried out, and the simulation data are compared with the experimental data to verify the accuracy of the simulation model. In Section 3, the vehicle powertrain-system model is used to study the influence of the DMF's primary and secondary flywheel moment of inertia ratio, torsional stiffness, and damping on the torsional vibration of the drivetrain. In  reflect the fluctuation of engine torque in a working cycle. 9 Since it is difficult for the engine steady-state torque model to simulate the torsional vibration component, the Fourier series is usually superimposed on this basis to simulate the torsional vibration of the engine, 10 and different orders are selected according to different accuracy requirements. 11 However, in general, this method cannot simulate the torque fluctuation of the engine accurately.
According to the working principle of the engine, the key to obtain the transient output torque of the engine is to obtain the gas pressure data in the cylinder at different speeds and different throttle valve openings. 12 The gas pressure in the cylinder is obtained through a series of processes such as fuel injection, gas distribution, timing, combustion, and so forth. It is difficult to establish a mathematical model for the

| Force analysis of the vehicle
The branch model of the drivetrain system needs to consider not only the torsional degrees of freedom of the drivetrain system but also the translational degrees of freedom of the vehicle. Through the force analysis of the driving wheel and the driven wheel, the interaction between the translational mass of the vehicle and the drivetrain system is studied. The force analysis of the whole vehicle is shown in Figure 4.
By simplifying the two driven wheels of the rear axle of the vehicle into one, the rolling resistance of the driven wheels of the rear axle can be obtained from where f is the rolling resistance coefficient.
The vertical load distributions of the front and rear axles of the vehicle are given, respectively, as The air drag of the vehicle is where C D is the air drag coefficient, A is the frontal area of the vehicle, and ρ a is the air density.
According to the Lugre tire model, 13 where σ′ 0 is the equivalent stiffness of the bristles, σ′ 1 is the equivalent damping of the bristles, σ′ 2 is the relative damping of the friction surface, z i is the average deformation of the bristles, and v i r is the relative velocity of the friction surface.
According to Newton's Second Law, Equation (9) yields

| Vehicle powertrain model
This paper considers a certain MPV vehicle running on a level road and working in the first gear creeping condition as the study example, and develops the vehicle powertrain model as shown in Figure 5.
According to Newton's Second Law, the equation of motion of the powertrain is obtained as where J is the moment of inertia matrix, C is the torsional damping matrix, K is the torsional stiffness matrix, θ t ( ) is the angular displacement vector, and T t ( ) is the external moment vector.
The meanings and values of the parameter and symbols in the vehicle powertrain model are listed in Table 1. Some of these parameters are directly provided by the cooperative enterprises, and some are obtained by CAD software according to the materials and dimensions of the parts provided.

| Results of the test
The signals that need to be collected in the vehicle test are the speed signals at the DMF primary flywheel and the transmission input shaft.

F I G U R E 4 Force analysis of the vehicle
The test equipment used in the test mainly includes Hall-type speed sensors, a ROTEC torsional vibration tester, and a laptop equipped with a RASnbk control and analysis system, as shown in Figure 6.

| Results of simulation
The developed vehicle powertrain model is used to simulate and process the obtained simulation data to obtain the results presented in this section. Figure 9 shows   DMF not only has the function of a torsional vibration damper but also has the function of energy storage as an ordinary flywheel. Therefore, it is necessary to make sure that the total inertia of both flywheels remains unchanged. Figure 11    relationship, indicating that when the torsional stiffness is too large, the rotational speed fluctuation increases sharply. As can be seen from the figure, a reasonable selection interval for torsional stiffness should be between 1 and 5 Nm/(º).
where X i is the variable input of the ith design parameter.
The parameter configuration and performance of the DMF equipped with the test vehicle are shown in Table 2.

| Optimal design based on the PSM
The PSM refers to the simulation calculation of the DMF finite set of parameters using the vehicle dynamics model, and a set of parameters with relatively good performance is selected from the calculation results as the optimal solution of the optimal design. The method should first determine the design parameter variables to be  Table 3.
There are only four optimal solutions in this optimization design, and the selection is relatively simple. However, the number of Pareto optimal solutions is uncertain and has a large randomness, and a large number of Pareto optimal solutions may appear, so a quantitative index for the selection of the final optimization result is introduced. In this paper, the expectation value method (EVM) is selected to determine the final optimization scheme. Different weights are set for each optimization goal to obtain the final expected value. The formula is as follows: where i is the number of optimization objectives, j is the number of optimization scheme, and d ij is the normalized value.
where w i is the weight of the ith optimization objective, Y ij is the jth optimization result of the ith optimization objective, and high i and T A B L E 3 Value of the Pareto optimal solution for PSM To summarize, it can be seen that the advantage of the PSM is that the optimization process is relatively simple, and the entire multiobjective optimization design process can be completed only by performing simulation calculations on a limited number of samples, but the disadvantages are also obvious. On the one hand, there is no guarantee that the optimal solution in the sample is the DMF optimal design point. On the other hand, there are relatively few nondominated solutions to choose from, which may not fully satisfy the corresponding needs.

| Optimal design based on MOPSO
PSO was proposed by Kennedy and Eberhart in 1995 based on the foraging behavior of birds. 15 Bird flock foraging is essentially an optimal decision-making process, and PSM treats each bird as a particle with specific position and velocity information. In addition to recording the best position and current position that each particle has passed through, it also knows the best position the group has passed through at present. By adjusting the flight direction of the particles using the above information, the group will gradually be directed to the location of the food.
MOPSO is the application of PSO in multiobjective optimization, and its process steps are shown in Figure 16.
MOPSO is used for DMF multiobjective optimization. After 50, 100, 500, and 1000 iterations, the Pareto solution set is shown in Figure 17.

| Optimal design based on the NSGA
The NSGA was proposed by Srinivas and Deb 16  accuracy is improved, and the exploration performance is enhanced. The evolutionary calculation process is shown in Figure 18.
NSGA-II is used for DMF multiobjective optimization. After 50, 100, 500, and 1000 iterations of calculation, the Pareto solution set is shown in Figure 19. It can be seen from the figure that after 50  Table 6. Among them, NSGA-II with the high iteration number performs the best, and PSM is the least ideal.

| CONCLUSIONS
In this paper, the influence of the DMF dynamic parameters on its vibration reduction performance is analyzed, and the DMF multiobjective optimization design is studied. The following conclusions can be obtained: 1. The DMF dynamic parameters, including the moment of inertia ratio, torsional stiffness, and damping, have a significant influence on the vibration reduction effect. When the moment of inertia ratio increases, the DMF damping effect first increases and then decreases.
When the moment of inertia ratio is 0.9, the vibration absorption capacity is the best. The DMF vibration absorption capacity decreases with the increase of torsional stiffness and damping.
F I G U R E 18 Optimization process of NSGA-II. NSGA-II, nondominated sorting genetic algorithm based on an elite strategy.

F I G U R E 19
Optimization results under different iterations of NSGA-II. NSGA-II, nondominated sorting genetic algorithm based on an elite strategy.
T A B L E 5 Optimization scheme of NSGA-II 2. In this paper, PSM, MOPSO, and NSGA-II are used to optimize DMF.
The PSM optimization process is the simplest, but there is no guarantee that the best optimization solution will be obtained. The advantage of MOPSO is that it can converge at lower iterations, but the number of Pareto front points is small and the distribution is not uniform. The convergence of NSGA-II is greatly affected by the number of iterations. At a higher number of iterations, the number of Pareto frontier points is not only sufficient but also the convergence and distribution uniformity are improved. In this case, NSGA-II performs significantly better than other algorithms.
Through the research of this paper, one can obtain the influence law of DMF's dynamic parameters on its damping effect, and can choose a suitable algorithm to complete the DMF parameter optimization design.

ACKNOWLEDGMENTS
This project was supported by the National Natural Science Foundation of China (Grant No. 52075388).