^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

Since the driving motor is embedded in the wheel, the unsprung mass and the wheel rotational inertia of the in-wheel motor drive electric vehicle both increase, which not only affect the vehicle smoothness but also worsen the motor’s working condition due to its own vertical vibration. The evaluation index of in-wheel motor’s vertical vibration is introduced on the basis of vehicle smoothness analysis. The parameters’ optimization of vibration absorber and suspension are carried out, respectively, and the optimization results show the contradictory relationship between smoothness objective and the motor’s vertical vibration acceleration objective. Regarding the contradictory indices of the smoothness and the motor’s vertical acceleration as the objective function, a multiobjective optimization scheme is designed. Then, the orthogonal experimentation and fuzzy clustering method are applied in the multiobjective optimized design based on the game decision analysis, and the Nash equilibrium and cooperative competition game theory are used to optimize the parameters of suspension and vibration absorber. The optimized results verify the game relation between the optimization variables, the game optimization obtains better optimized results than traditional linear weight sum method, and the in-wheel motor functional stability and the vehicle smoothness can be both achieved. Compared with the traditional complex iterative process and the manmade preassurance weight allocation, the game optimization has the advantages of less iterations, faster convergence, and less influence by human experience.

New energy automobiles have become the forward-looking field in the car industry. The driven technology of the new energy automobiles and the traditional internal combustion engine cars shares a great deal of differences, among which the in-wheel motor drive electric vehicles embrace a promising prospect as a new car driven method. Compared with motor integrated power driven, the in-wheel motor drive has a big advantage, which is more flexible and has less complicated mechanic driven system. Nevertheless, it also has some disadvantages. The electric wheels in the in-wheel drive vehicles combine the electric machines, wheel hubs, retarding mechanisms, and brakes together, greatly increasing the unsprung mass and rotational inertia, worsening the vibration caused by the in-wheel motor drive, and causing the vibration between the neighbour parts and the rotors, which brings bad influence on the vehicles’ smoothness and the safety of the tires touching the ground. How to reduce and even get rid of this kind of bad effect has become one of the key problems needed to be solved in the development of in-wheel drive electric vehicles.

Focusing on the above problems, currently, the scholars have already conducted initial research on the related questions to reduce the influence of the in-wheel drive motor on vehicle vibration. References [

However, with the development of the in-wheel motor technology, the orientation of the in-wheel motor drive cars is not limited to the travel tool in the city. Its optimized analysis is not limited to the traditional smoothness optimization. As the core part of the in-wheel motor electric cars, the operation, and condition of the in-wheel motor directly exert influences on the safety and cruising ability of the car. The research on in-wheel motor has gained more and more attention. Wei Tong et al. [

Besides, the related optimized design analysis and research on in-wheel motors have progressed greatly [

Many related solutions have been put forward to overcome the negative impact brought by the torque ripple of the in-wheel motors on electric automobiles. While the suspension vibration and the in-wheel motor vibration are both affected by each other. Only observing the vibration of the suspension to get good ride comfort may lead the in-wheel motor over vibration. Due to the limited space in the wheel for the motor, the over vibration of the vehicle motor will shorten its using lifetime; meanwhile, this also affects the other parts working in the wheel or the vehicle. In this paper, we will discuss different suspension types with the in-wheel motor. During different suspension types, suspension parameters selection will get different vibration performance. Thus, the parameter optimization method and strategy are very important to improve the suspension performance. In this paper, it is focused on the two optimized schemes on the in-wheel motor drive vehicle, which are suspension parameters optimization and vibration absorber optimization, respectively. Based on vehicle’s smoothness analysis, the evaluation index of vertical vibration of the in-wheel motor is introduced. The parameters of suspension and vibration absorber are optimized, respectively, and the two optimization schemes are compared and analysed. Then, regarding the contradictory smoothness index and the in-wheel vertical acceleration as the two objective functions, the suspension parameters’ multiobjective optimization scheme is carried out, which could better consider the smoothness of the whole vehicle and the stability of the motor vibration. Finally, the orthogonal experimentation and fuzzy clustering method are applied in the multiobjective optimization design based on the game decision analysis, realizing the transfer of the multiobjective optimization to game-related problems and the division of the game player side. The Nash equilibrium and the cooperation and competition equilibrium are adopted, respectively, to solve the best solution of the related parameters with the profit function as optimization target. Compared with the traditional optimized result, the optimization of the game theory is further analysed, and the effect of the multiobjective optimization with the help of the game theory is validated.

The torque ripple of the in-wheel motor exerts influences on various functional parameters of the suspension system, especially on the unsprung mass wheel, deteriorating the ground-catching ability of the wheels and harm to the smooth travel of the vehicles. At the same time, the vertical vibration of the in-wheel motor will also deteriorate the motor’s working condition. In [

The simplified 1/4 vertical vibration model of the vehicle with the in-wheel motor is established as shown in Figure

Vertical vibration model of 1/4 in-wheel motor drive electric vehicle.

Because the tire damping is relatively low, the tire damping is neglected. The motion equation of the 1/4 vehicle vertical vibration model with in-wheel motor is built as follows:

The root-mean-square value of the car body’s acceleration is

In the above formula, _{q} (_{0}) is the average value of the road spectrum, _{0} is the reference space frequency, and _{0} is the input of the uneven degree of the road, _{j} (_{1} is the unsprung mass (mass of the wheel), _{2} is the sprung mass (mass of the 1/4 car body), _{3} is the in-wheel motor mass, _{1} is the wheel damping coefficient, _{2} is the vehicle suspension damping coefficient, _{1} is the tire stiffness, and _{2} is the vehicle suspension stiffness. The system parameters of the target vehicle are shown in Table

System parameters of the target vehicle.

Parameter | _{1} (kg) | _{2} (kg) | _{3} (kg) | _{1} (N/m) | _{2} (N/m) | _{1} (N/(m/s)) | _{2} (N/(m/s)) |
---|---|---|---|---|---|---|---|

Value | 45.4 | 317.5 | 50 | 192000 | 15000 | 50 | 1500 |

The introduction of the in-wheel motor brings about the severe vertical negative effect on the vehicle. On the premise of not changing the suspension structure and parameters of the initial design in Section

Vertical vibration model of the 1/4 in-wheel motor vehicle contained with vibration absorber.

The wheel, vehicle body, and in-wheel motor’s vertical displacement are _{1}, _{2}, and _{3}, respectively. With the origin of coordinates staying in the respective equilibrium points, the road displacement input is _{0} and its dynamic differential equation is

In the above formula, _{3} is the damping coefficient of vibration absorbing system between the motor and the wheel and _{3} is the spring stiffness between the motor and the wheel.

The in-wheel motor’s vertical displacement and the in-wheel motor’s vertical displacement relative to the wheel, that is, the root-mean-square value of the in-wheel motor dynamic deflection, are

Suspension could be used to buffer the impact on the in-wheel motor drive electric vehicle caused by the rough road and reduce the vibration to ensure the smooth travel of the vehicle. The suspension parameter optimization aims at fulfilling the realistic demand of the vehicle wheels to reach an optimal suspension parameter value. The in-wheel motor is the drive core of in-wheel motor drive electric vehicle. Its vertical vibration analysis is one of the crucial subjects in the research. The covibration frequency belt of the car-carried in-wheel motor has the characteristics of large changing range, big covibration peak, slow decreasing speed, and so on [

According to the standard GB/T 4970–1996, the car’s smoothness relies on the random input of driving trial method. The suspension parameter optimization scheme is set as follows: selecting the road average spectrum _{q} (_{0}) as 0.0000256 m^{3} (level C) and the random road speed 80 km/h as working encouragement. The evaluation index of vertical vibration of the in-wheel motor

_{3} and _{2} are solid connections, so the vibration of _{3} is basically the same as _{2}. Therefore, the vibration of _{3} is about the relative dynamic load of the wheel.

The variation range of the variables in the optimization process is

The introduction of the in-wheel motors brings about the severe vertical negative effect on the vehicle. Adoption of the method in 3.1 could have some improvement, but the challenges are great in terms of the parameter adjustment and matching involved in the vehicle stability in the method [_{3} and _{3} are optimized by using the same working condition, evaluation index, and objective function as the suspension parameter optimization scheme.

The vertical vibration acceleration can be calculated by equation (_{vao}, and the optimization comparison results are shown in Table

Comparison of the main evaluation indexes of three optimization schemes.

Index | Smoothness evaluation index, _{vao} | Root-mean-square value of the in-wheel motor acceleration |
---|---|---|

Suspension parameter optimization | 1.7577 | 0.3978 |

Vibration absorber optimization | 1.1717 | 0.4765 |

Multiple-objective optimization | 1.2009 | 0.4102 |

The comparison results of two optimization schemes show that the two schemes both improve the smoothness of electric vehicle. The vibration absorber optimization scheme is superior to the suspension parameter optimization scheme in terms of vehicle smoothness. In the suspension optimization, the in-wheel motor acceleration increased by 1.2% with a little deterioration, while in the vibration absorbing optimization, the in-wheel motor acceleration increased by 19.7% with bigger deterioration.

From the above analysis, it could be seen that the vibration absorbing scheme enjoys an obvious improvement in the vehicle’s smoothness compared with the initial design. However, the condition of the in-wheel motor deteriorates greatly. In suspension parameter optimization, the smoothness improvement is not that obviously compared with the initial design and the corresponding working condition of the in-wheel motor in this scheme deteriorates only a little.

For this circumstance, reference [_{3} is in the free vibration end and the system mass is reduced greatly compared with the former solid connection. Though part of the vibration is absorbed by damping and spring, generally speaking, the ability to resist the road input vibration is reduced. If the road input energy is large, the independent takeover of the vibration should be taken by _{3} but this would lead to worse working conditions.

Therefore, it is required to design a new multiobjective optimization of suspension parameter, by adjusting the parameters _{2}, _{2}, _{3}, and _{3} to enable both the suspension system’s smoothness and in-wheel motor vertical vibration acceleration in a better working condition.

Based on the analysis in Section _{vao} as the first optimal objective and the root-mean-square value of the in-wheel motor acceleration _{2}, _{2}, _{3}, and _{3}. To better compare the optimization results, the initial design parameter {_{3}, _{3}} = {45000, 4500} is set.

Firstly, the classic linear weight sum method is adopted to solve this multiple-objective optimization problem. It is shown as follows:

In the above formula,

The smoothness evaluation index _{vao} and the root-mean-square values of in-wheel motor acceleration

The changing ranges of the optimization variables in the multiple-objective optimization process are as follows:

Because there is not enough former theoretical support for solutions, it takes nine groups of different weight parameters for optimization, regarding the average value [

Multiobjective optimization weighting parameters.

_{1} | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.4 | 0.7 | 0.8 | 0.9 |

_{2} | 0.9 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.6 | 0.3 | 0.2 | 0.1 |

Compared with the suspension parameter optimization, the multiple-objective optimization smoothness evaluation index _{vao} is reduced by 30%, and compared with the vibration absorbing optimization, it increased by 2%. Compared with the suspension parameter optimization, the in-wheel motor acceleration increased by 3%, and compared with the vibration absorbing optimization, it is reduced by 20%. Compared with the initial design, multiple-objective optimization smoothness evaluation index _{vao} is reduced by 31%, and the in-wheel motor acceleration only increased by 4%. Compared with the suspension parameter optimization, the vibration absorbing optimization, and multiple-objective optimization, the solution of the multiobjective optimization scheme is more balanced, which is more comprehensive and reasonable than the original design.

Game means an action competitive or with protesting nature. From the perspective of methodology, the game theory is a vector optimization consisting of objective function vector and strategic vector. From the perspective of countermeasure, the game mainly studies the decision when the decision-makers act with each other and the equilibrium formed by the decision under some restrictions [

From the above sections, it could be seen that, in the suspension parameter optimization, the improvement of the smoothness is smaller compared with the initial design. The in-wheel motor working condition is only deteriorated a little bit, while in the vibration absorbing optimization, the smoothness is obviously improved compared with the initial design and the corresponding in-wheel motor’s working condition deteriorates greatly.

It could be seen that the smoothness evaluation index is in a relative resistance relation with the index, that is, the acceleration of in-wheel motors, meeting the requirement of the game equilibrium optimization. The optimization is actually to form an individual advantage and a decision-making process of taking profits separately from each side. By game decisions’ mutual compromise, the final optimization results eliminate the gaining contradictions between the two, thus forming an equilibrium scheme taking both the vehicle’s smoothness and the motor’s vibration smoothness into account.

Game equilibrium optimization is represented by Nash noncompetitive game and cooperation-competitive game, the main differences of which lie in whether the game individuals abide by the unified agreement. The cooperative competition game focuses more on the whole profit and team cooperation, rather than the cooperative competition game, which only considers its self-interest to realize the maximization of its profits [

The Nash equilibrium game roughly consists of the following four steps [

By calculating the sensitivity of the variables in the game, it divides the optimization variables

In the strategic combination

Mark _{1}, _{2}, …, _{m} are the profit functions of the game _{1}, _{2}, …, _{m}, respectively; that is, ^{th} game side (

Let

Framework of Nash equilibrium game.

Choosing the same multiobjective optimization model, optimization target, optimization function, and restrictions with those in Section _{vao} for the game optimization calculation [

Firstly, conduct single objective optimization to the objective function _{vao},

Single objective optimization results.

Objective variables | _{vao} | |
---|---|---|

Suspension stiffness, _{2} (N/m) | 11000 | 13813 |

Suspension damping, _{2} (N/(m/s)) | 750 | 1493 |

Suspension stiffness, _{3} (N/m) | 55336 | 25000 |

Suspension damping, _{3} (N/(m/s)) | 1292 | 4500 |

Root-mean-square value of the body’s acceleration | 0.9038 | 1.2995 |

Root-mean-square value of suspension dynamic deflection | 0.0124 | 0.0081 |

Root-mean-square value of wheels’ relative loading | 0.2321 | 0.2633 |

Root-mean-square value of the in-wheel motor acceleration | 0.5087 | 0.383 |

Objective function, _{vao} | 1.1484 | 1.5709 |

The optimization target in this paper is the nonlinear optimization target, very sensitive to searching space. The calculation efficiency of using general strategy division is relatively low. Therefore, it uses the orthogonal test in the experimental division to calculate the sensitivity of the various optimization variables to the objective functions.

It is known that this optimization contains four variables and two objective functions. According to the orthogonal test rule [

The orthogonal experiment table of the various objective functions is set up, getting the sensitivities of various optimization variables to each objective function shown in Tables

Level table of _{vao} optimization factors.

Factors | _{1} | _{2} | _{3} | _{4} |
---|---|---|---|---|

Level 1 | 10950 | 700 | 55286.58 | 1242.84 |

Level 2 | 11050 | 800 | 55386.58 | 1342.84 |

Orthogonal experimental results of the _{vao} optimization.

Factors | _{1} | _{2} | _{3} | _{4} | _{gop} |
---|---|---|---|---|---|

Experiment 1 | 10950 | 700 | 55286.58 | 1242.836 | 1.1389 |

Experiment 2 | 10950 | 700 | 55286.58 | 1342.836 | 1.139 |

Experiment 3 | 10950 | 800 | 55386.58 | 1242.836 | 1.156 |

Experiment 4 | 10950 | 800 | 55386.58 | 1342.836 | 1.1559 |

Experiment 5 | 11050 | 700 | 55386.58 | 1242.836 | 1.1438 |

Experiment 6 | 11050 | 700 | 55386.58 | 1342.836 | 1.1438 |

Experiment 7 | 11050 | 800 | 55286.58 | 1242.836 | 1.1602 |

Experiment 8 | 11050 | 800 | 55286.58 | 1342.836 | 1.1601 |

_{j1} (_{gop}) | 1.14745 | 1.141375 | 1.14955 | 1.149725 | |

_{j2} (_{gop}) | 1.151975 | 1.15805 | 1.149875 | 1.1497 | — |

_{j} (_{gop}) | 0.004525 | 0.016675 | 0.000325 | −2.50E-05 | — |

_{j} (_{gop}) | 4.53E-05 | 0.00016675 | 3.25E-06 | −2.50E-07 | — |

Level table of

Factors | _{1} | _{2} | _{3} | _{4} |
---|---|---|---|---|

Level 1 | 13763.02 | 1443.72 | 24950 | 4450 |

Level 2 | 13863.02 | 1543.72 | 25050 | 4550 |

Orthogonal experimental results of the

Factors | _{1} | _{2} | _{3} | _{4} | _{gop} |
---|---|---|---|---|---|

Experiment 1 | 13763.02 | 1443.72 | 24950 | 4450 | 0.3834 |

Experiment 2 | 13763.02 | 1443.72 | 24950 | 4550 | 0.3832 |

Experiment 3 | 13763.02 | 1543.72 | 25050 | 4450 | 0.394 |

Experiment 4 | 13763.02 | 1543.72 | 25050 | 4550 | 0.3938 |

Experiment 5 | 13863.02 | 1443.72 | 25050 | 4450 | 0.399 |

Experiment 6 | 13863.02 | 1443.72 | 25050 | 4550 | 0.395 |

Experiment 7 | 13863.02 | 1543.72 | 24950 | 4450 | 0.3839 |

Experiment 8 | 13863.02 | 1543.72 | 24950 | 4550 | 0.3847 |

_{j1} ( | 0.3886 | 0.3886 | 0.3838 | 0.390075 | — |

_{j2} ( | 0.3891 | 0.39065 | 0.39545 | 0.389175 | — |

_{j} ( | −0.00105 | 0.00205 | 0.01165 | −0.0009 | — |

_{j} ( | −0.0000105 | 0.0000205 | 0.0001165 | −9E-06 | — |

We get the test variables _{1}, _{2}, _{3}, and _{4}, that is, optimization variables of _{2}, _{2}, _{3}, and _{3}, and the sensitivities to the two objective functions are separately shown as

The sensitivity of each variable corresponds to one by one and influence factors

Based on the fuzzy clustering method [29], the fuzzy similar equivalent matrix is obtained as follows:

Choose the confidence level 0.92 to divide the matrix. From the fuzzy clustering method, the strategic set of game side _{1} is _{1} = {_{1}, _{2}, _{4}}, that is, {_{2}, _{2}, _{3}}, and the strategic set of game side _{2} is _{1} = {_{3}}, that is, {_{3}}. The main parameters in the initial design are used as the initial and feasible strategy combination

According to step (3), construct the profit function

Optimization results of Nash equilibrium optimization. (a) Spectrometer of root-mean-square value of vehicle body acceleration. (b) Spectrometer of root-mean-square value of the suspension dynamic deflection. (c) Spectrometer of root-mean-square value of the wheel relative loading. (d) Spectrometer of root-mean-square value of motor acceleration.

Results comparison by various optimization schemes.

Evaluation index | Suspension parameter optimization | Vibration absorber optimization | Multiobjective optimization | Nash equilibrium | Cooperative competition |
---|---|---|---|---|---|

Suspension stiffness, _{2} (N/m) | 11437 | 15000 | 11002 | 11600 | 11061 |

Suspension damping, _{2} (N/(m/s)) | 1236 | 1500 | 750 | 1414.2 | 778 |

Vibration absorber stiffness, _{3} (N/m) | — | 56631 | 45773 | 53232 | 45403 |

Vibration absorber damping, _{3} (N/(m/s)) | — | 1325 | 2857 | 1398 | 2182 |

Root-mean-square value of the body’s acceleration | 1.3418 | 0.9264 | 0.9283 | 0.9274 | 0.9303 |

Root-mean-square value of suspension dynamic deflection | 0.0181 | 0.0125 | 0.0126 | 0.0129 | 0.0122 |

Root-mean-square value of wheels’ relative loading | 0.3978 | 0.2328 | 0.2599 | 0.233 | 0.2456 |

Root-mean-square value of the in-wheel motor acceleration | 0.3978 | 0.4765 | 0.4102 | 0.4221 | 0.4028 |

Smoothness _{vao} | 1.7577 | 1.1717 | 1.2009 | 1.1734 | 1.1881 |

Iterations | — | — | — | 7 | 11 |

The cooperative competition game optimization is conducted using formula (

Results of cooperative competition game optimization. (a) Spectrometer of root-mean-square value of vehicle body acceleration. (b) Spectrometer of root-mean-square value of the suspension dynamic deflection, (c) Spectrometer of root-mean-square value of the wheel relative loading, and (d) Spectrometer of root-mean-square value of motor acceleration.

According to the above results’ comparison, the evaluation index presents more reasonable equilibrium relative to the suspension parameter and vibration absorber optimization, which are more close to multiple-objective optimization, that is, the traditional weighting optimization scheme. But compared with the traditional and complex iterative process and the manmade preassurance weight allocation, the game optimization has the advantages of less iterations, faster convergence, and less influenced by human factor, which is more efficient. Meanwhile, vertically comparing the optimization parameters in the different optimizations, it is easy to find that although the game optimization results are similar, the optimization parameters _{2}, _{2}, _{3}, and _{3} are not close to each other.

The basic optimal parameters _{2}, _{2}, _{3}, and _{3} in Nash equilibrium optimization are basically those values between the suspension parameter optimization and the vibration absorber optimization or close to that of either group of parameters. In Nash equilibrium optimization, the specific comparison results _{2} (11437 < 11600 < 15000), _{2} (1236 < 1414 < 1500), _{3}, and _{3} are close to those in vibration absorber optimization, while the optimization parameters _{2}, _{2}, _{3}, and _{3} in the cooperative competition optimization are more close to those in multiobjective optimization, which is due to the different profits focused by the game sides. In Nash equilibrium strategy, the game sides do not have a favour towards the objectives and they only focus on their own profits and the competition relations among the various objectives.

Nash equilibrium optimization, suspension parameters, and vibration absorber optimization have the same optimal parameters _{2}, _{2}, _{3}, and _{3} and the same profits (optimization objective), but they adopt different optimization strategies (suspension optimization and vibration absorber optimization), leading to the profits of each side conflict to each other (smoothness evaluation index and in-wheel motor acceleration have formed a certain contrast). Therefore, the Nash equilibrium could be totally seen as the competition and game between two game strategies of suspension parameters optimization and vibration absorber optimization, and the parameters allocation stays between the two values of the two optimizations. In cooperative competition equilibrium game strategy, the game sides focus on the whole profit of all the game sides and the objectives remain as cooperative and competitive relationship. The parameters are coordinated better. Similar to the idea of the traditional weight allocation, the parameters’ allocation is close to the multiobjective optimization naturally.

To view in a comprehensive way, Nash equilibrium game strategy and cooperative competition game strategy both achieve the optimization design effect of the in-wheel motor, fulfilling the complex design demand of the in-wheel motors and revealing the effectiveness and efficiency of game optimization.

The two common in-wheel motor smoothness optimization models, the suspension parameter optimization model and vibration absorber optimization model, are set up. Based on vehicle’s smoothness analysis, the evaluation index of in-wheel motors’ vertical vibration is introduced. The suspension stiffness and damping parameters are optimized for the suspension parameter optimization model, and the stiffness and damping parameters of the vibration absorber are optimized for the vibration absorber optimization model. The comparison and analysis of two optimization results show that, compared with the suspension parameter optimization scheme, the vibration absorber optimization scheme can better improve vehicle smoothness, but the corresponding vertical vibration acceleration deterioration of in-wheel motor worsens more.

Comparison analysis between the suspension optimization scheme and the vibration absorber optimization scheme is conducted. On this basis, a multiple-objective optimization scheme is designed. The linear weighting method is used to weigh the smoothness index and the in-wheel motors’ vibration acceleration index, and the parameters of suspension stiffness

The game optimization is applied to deal with the conflict of the two optimization objectives. Taking advantage of the orthogonal experiment combined with fuzzy clustering method, the multiple-objective optimization is converted to game equilibrium optimization. Based on the sensitivity of the optimization variables to optimization objectives, the different game strategic combinations are divided to conduct game confrontation. Using the main suspension and vibration absorber stiffness and damping coefficients as the optimization variables and the minimum profit function as objective function, the Nash equilibrium and cooperative competition game optimizations are adopted to optimize the parameters of suspension and vibration absorber, and the results are more ideal than that by linear weighting multiobjective optimization.

Combined with the game theory, the internal game relation between different optimization results is analysed. The game optimization has the advantages of the greater convergence and high efficiency. The different optimal equilibriums of the suspension vibration and in-wheel motor vibration have been achieved by the two different game strategies. Compared with the traditional linear weighting method, this method does not require the weighting of different objectives, which has greater value applied in engineering. Although the optimization design can significantly promote the in-wheel motor drive vehicle vertical vibration performance, the parameters in the system cannot be dynamically regulated according to the vehicle driving conditions. To design the controllers to achieve the suspension and absorber parameters’ real-time regulation while driving can further reduce the vehicle vertical vibration. Meanwhile, the motor vibration has the coupling with the suspension system vertical vibration, so paying attention to the coordinated controller design of the motor and suspension considering their characteristics, so as to further reduce the in-wheel motor drive vehicle vertical vibration and improve the system comprehensive performance, is especially worth studying.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This research was supported by the Major Education and Teaching Reform Research Project of Anhui Provincial Department of Education (2019zdjg11); the Natural Science Research Projects in Anhui Universities (KJ2019A1070); the Transportation Vocational Education Research Project (2018YJ203) and Open Fund of State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures (KF2020-27).