Integrated Vibration Control of In-Wheel Motor-Suspensions Coupling System via Dynamics Parameter Optimization
In order to integrally control the vibration of in-wheel motor- (IWM-) suspensions coupling system of an electric vehicle, a novel nonlinear dynamics model of the coupling system, which consists of the motor magnetic gap (MMG), is established. Synthesizing subtargets of the vertical vibration acceleration of bodywork, the vertical deformation of tire, the suspension travel, and the vertical fluctuation of MMG, a composite optimization mathematical model is set up. Based on artificial fish swarm algorithm (AFSA), a novel dynamics parameter optimization method is proposed to search the optimal parameter combination existing in the nonlinear dynamics model. Simulation analyses demonstrate that the proposed optimization method is superior to genetic algorithm (GA) under the same optimization conditions, and it can significantly decrease the fluctuation of MMG and improve ride comfort.
With the increase of energy crisis and environmental pollution, it has become an important research goal to develop zero emissions of electric vehicles (EVs). Among them, the IWM-driven EVs have received a general attention due to their advantages of simple vehicle layout, high integrated characteristic, flexible drive feature, and so on.
However, this kind of driven form also brings new vibration problems that need to be solved. In practice, the MMG of IWM generates fluctuation under road excitation. Electromagnetic force is a function of MMG [1–3], for which the changes of electromagnetic force caused by the fluctuation of MMG deteriorate the ride comfort of an electric wheel vehicle [4, 5]. Furthermore, because the outer rotor of IWM is fixed on a hub, the fluctuation of MMG affects the adhesion performance between tire and ground . In addition, motor, brake, and other components are highly integrated in a wheel which makes the unsprung mass increase and leads to poor dynamics behaviors . In view of the reasons mentioned above, some research studies have been done to reduce the vibration of electric wheel vehicle. An advanced dynamics damping motor system was proposed by Shao et al. , which isolates the IWM vibration by an in-wheel spring damping element that has internal damping effect. This structural form breaks through the concept of traditional suspension system and provides theoretical support for the improvement of suspension structure. Except for changing systematic structure, Pang et al.  argued that it is also one of the most effective ways to adjust dynamics parameters for improving the performance of suspension system. The experimental results from a quarter car test rig which were done by Mitra et al.  show that mass and damping coefficient are the most influential parameters for ride comfort. Therefore, many scholars focus their attention on optimizing the mass, damping coefficient, and stiffness coefficient of suspension system and achieve many significant results in enhancing ride comfort by using GA [10–14]. Essentially, the dynamics parameter optimization of suspension system belongs to a multiobjective optimization problem (MOOP) which can be solved by using multiobjective optimization algorithm (MOOA) [15, 16]. Gadhvi et al. , respectively, used nondominated sorting genetic algorithm (NSGA), strength Pareto evolutionary algorithm (SPEA), and Pareto envelope-based selection algorithm (PESA) as tools for suspension optimization. It is proved that the results of the three MOOAs are different from each other in optimization effect. Thus, an appropriate MOOA plays an important role in solving MOOPs. The behaviors and evolutions of natural creatures bring enlightenments to practical optimization problems [17, 18]. Recently, the AFSA has been quickly developed. The main behavior of fish used by the AFSA is fish stay near their swarm to protect themselves from predators and meanwhile maintain safe distances from nearby partners to search food without collisions . The basic idea of AFSA adopted by He et al.  is partly derived from the aforementioned fish behaviors. The AFSA has been successfully applied to many different fields, such as the scheduling of renewable energy sources in a microgrid , a multiobjective fuzzy disassembly balancing problem , and dynamic weapon target assignment problem  because of its advantages of parallelism, global feature, traceability, and so on. Since the AFSA has a good ability to solve the above MOOP problems, it is a viable means to optimize the key dynamics parameters of the IWM-suspensions coupling system based on the AFSA.
To sum up, the chassis structure of IWM-driven EV is different from that of traditional vehicle, for which some researchers have designed new in-wheel spring-damping elements to reduce control vibration of the chassis and improve the ride comfort. In addition to this method, other researches mainly focus on the vibration reduction of suspension system by using advanced optimization methods and different control methods. Although the influences of dynamics parameters on suspension system are revealed and the developments of optimization methods provide new ways for improving ride comfort, the research on the dynamics parameter optimization of IWM-suspensions coupling system is insufficient, and the dynamics parameters of IWM are usually overlooked, which is an obstacle to the integrated vibration control of IWM-suspensions coupling system and brings about unreasonable dynamics parameter combinations. This is an important problem that needs to be urgently resolved.
In this research, synthetically considering the dynamics parameters of IWM-suspensions coupling system, a novel nonlinear dynamics model, which consists of the MMG, is established. Based on the AFSA, a novel dynamics parameter optimization method is proposed by constructing optimization mathematical model and constraints. Eight key dynamics parameters of the IWM-suspensions coupling system are optimized, and the optimal parameter combination is obtained. The vibration control effect of the proposed method is numerically validated by comparing to GA’s optimization results under different road excitations. The optimization method proposed by this paper has important application prospects in the field of vehicular multisystem integrated dynamics design and control, e.g., the integrated design and control of electric wheel-steering-suspension system.
2. Model and Problem Formulation
2.1. Dynamics Modeling
The motor bearing that supports the rim is subjected to circumferential, radial, and axial forces which results in the relative displacement between the stator and the rotor, and the fluctuation of MMG further deteriorates the vibration of vehicle. In this part, the fluctuation of MMG caused by road excitation is mainly considered, and the incentive role of torque fluctuation produced by motor structure, processing error, and principle factor is neglected because it is far less than the road excitation . According to the actual structure of IWM-suspensions coupling system described in Figure 1, a quarter-car nonlinear vibration model is established and shown in Figure 2. Different from the traditional one, this nonlinear IWM-suspensions coupling system uses a suspension element to elastically isolate the overall IWM mass from unsprung mass (see Figure 1).
Currently, nonlinear components are widely applied to vehicle systems because of their predominant damping performances. Most vehicles universally use nonlinear suspensions and tires to reduce their vertical vibrations. Therefore, the nonlinear stiffness characteristics of the suspension 1 and tire are taken into account. The elastic restoring forces models of suspension 1 and the tire are, respectively, expressed as where and , respectively, represent the vertical elastic restoring forces of suspension 1 and the tire. and are, respectively, the stiffness coefficients of suspension 1 and the tire. and are respective vertical deformations of suspension 1 and the tire. and are nonlinearity coefficients.
Based on formulae (1) and (2), referring to the quarter-car nonlinear vibration model shown in Figure 2, the motion equation of IWM/suspensions coupling system is written as follows:where denotes the total mass of the tire and rim. is the total mass of the knighthead and brake clamp. is the mass of the quarter bodywork. and , respectively, denote the mass of IWM’s stator and rotor. is harmonic road excitation. , , , , and , respectively, represent the vertical displacements of , , , , and . and , respectively, denote the vertical stiffness and damping coefficients of the tire. and , respectively, denote the vertical stiffness and damping coefficients of hub bearing. and , respectively, denote the equivalent vertical stiffness and damping coefficients of suspension 1 (see Figure 1). and , respectively, denote the stiffness and damping coefficients of suspension 2 between motor stator and bodywork. and , respectively, represent the stiffness and damping coefficients of the bearing between motor stator and rotor. and , respectively, are the stiffness and damping coefficients of suspension 3 between motor stator and knighthead. and , respectively, denote the vertical stiffness and damping coefficients of the bushing between the motor rotor and the rim.
2.2. Model and Method of Optimization
The AFSA has a good ability to overcome local extreme value and obtain global optimal solution in search space . Here, in accordance with the experimental results by Mitra et al. ; eight key dynamics parameters of IWM-suspensions coupling system are chosen as design variables. They are , respectively.
2.2.1. Mathematical Model of Optimization
Ride comfort is an important consideration in the design of vehicular suspension system, particularly the performance evaluation indices of vehicular suspension system [25, 26]. Traditional evaluation indices, such as the vertical vibration acceleration of bodywork, the vertical deformation of tire, and suspension travel, can influence the ride comfort of a vehicle. However, the fluctuation of IWM’s MMG is neglected which can deteriorate the ride comfort. So, here the fluctuation of IWM’s MMG is added to the indices.
Weighted-sum method is used to indicate the importance of each index and connect subtargets into a composite optimization objective. Root-mean-square (RMS) value is used to reduce the impact of contingency factors. The mathematical model of optimization problem is written assubject towhere , , , and which, respectively, denote the vertical vibration acceleration of bodywork, the vertical deformation of tire, the suspension travel, and the vertical fluctuation of IWM’s MMG. is an objective function. t1 = 0 s, t2 = 10 s; this time period includes the transient and steady-state response of dynamics system. , , , and are, respectively, the weighted value of each performance index which are assigned to be 0.25, 0.2, 0.2, and 0.35. and are constraints. and , respectively, denote the suspension limit travel and the limit value of MMG.
2.2.2. AFSA-Based Optimization Method and Procedure
The number of artificial fishes (AFs) in fish swarm is defined as N and each AF that includes the 8 dynamics parameters (i.e., ) is denoted by (i = 1, 2, …, N). So, can be written as , where each element in the vector represents a dynamics parameter. In this way, the fish swarm can be expressed by an 8 × N matrix which is shown in Figure 3. Each column of the matrix represents an artificial fish and its position.
By substituting AFi into formula (4), the corresponding Yi = F(f) can be obtained which is regarded as food concentration in the current position (also means the fitness in GA). As shown in Figure 4, Step is the AF’s movement size, Visual is AF’s field of view. Try-number is defined as the maximum number of attempts of foraging behavior. To avoid overcrowding with the surrounding partners, a congestion factor delta is set up which is expressed as follows:where α represents the degree of approximation of extremes and it is a coefficient between 0 and 1. nlimit is the maximum upper limit of the AF’s number existing in the field of view. Here we choose α = 0.9 and nlimit = 2.
The initial data of each AF in artificial fish swarm must be defined in advance which can be randomly generated within their definition domains [10, 27]. Without losing of generality, the i-th AF is denoted as . After initializing an artificial fish swarm, the AFs in the fish swarm search the optimal solution by iterations. During each iteration, the AFs update themselves by the following steps.
(1) Random Moving Behavior and Foraging Behavior. Random moving behavior is a kind of default foraging behavior during which randomly selects a state within its visual (formula (7)) and then moves one step to a new position (formula (8)). In the foraging behavior, the AF always tries to move to a better place based on the food concentration, which lays the foundation of convergence.where is an uniformly distributed random number over the interval [−1, 1] and .where (Pseudocode 1).
(2) Swarm Behavior. This behavior enables to move as much as possible to the center of nearby partners and avoid egregious overcrowding. The center of nearby partners is given bywhere is the number of nearby partners that is calculated by using the formula . The criteria for judging congestion are given by formula (10), which decides whether to follow formula (11).where represents the food concentration of .where is an random number over the interval (0, 1) (Pseudocode 2).
(3) Following Behavior. searches for partners within its visual and selectively moves to the one with the highest food concentration. If formula (12) is satisfied, moves according to formula (13):where is the AF with the highest food concentration (Pseudocode 3).
The dynamics parameter optimization procedure of nonlinear IWM-suspensions coupling system is shown in Figure 5.
3. Optimization Results
The overall dynamics parameter optimization procedure is executed by Matlab software. In order to easily choose Step and Visual and accelerate convergence, during the optimization process, all the design variables are reduced to two digits and then the optimization results are zoomed out to the same multiple. The GA-based multiparameter optimization scheme is used for comparison with the proposed optimization method, which includes coding, crossing, mutation, and selecting operations . Road excitation is simulated by the harmonic function described in formula (15).(1)The parameters involved in the proposed optimization method are set up as below:(i)Number of AF: 15(ii)Max iteration: 1000(iii)Step: 0.5(iv)Visual: 5(v)Try-number: 3(vi)Delta: 0.5556(2)The selected optimization conditions for GA:(i)Number of individuals: 15(ii)Max iteration: 1000(iii)Crossover rate: 0.9(iv)Mutation rate: 0.5(v)Selecting rate: 0.8
For the established motion equation of nonlinear IWM-suspensions coupling system, the nonlinear coefficients and are both assigned to be 0.5. The dynamics parameter optimization results are shown in Table 1, where the original dynamics parameters of nonlinear IWM-suspensions coupling system are listed .
The larger the fish swarm, the greater the ability to jump out from the local extreme value and the faster the convergence rate, but the amount of each iteration calculation is also greater. Numerical experiments show that when the number of AF is 15, the amount of calculation and the convergence rate reach a good balance. The changes of food concentration calculated by the proposed method are shown in Figure 6. It can be seen that the food concentration increases from 5.693 × 10−3 to 6.491 × 10−3 step by step within 1000 iterations. From the whole optimization procedure, the food concentration breaks though local extreme values for four times and eventually reaches a global extreme value. This phenomenon interprets the global convergence characteristic of the proposed optimization method. The iteration process of each parameter is shown in Figures 7(a)–7(h) from which it is easy to see that c3 achieves convergence faster than other parameters. c3 and c4 finally converge to the boundaries of their definition domains.
4. Simulation Analyses of Optimization Control Effects
In this Section, numerical simulations are executed to compare and verify the optimization control effects of the proposed method and GA. Rough and bump road excitations simulated by harmonic functions  are used to evaluate dynamic performances of the nonlinear IWM-suspensions coupling system by analyzing the vertical vibration acceleration of bodywork, bodywork travel, tire dynamic load, suspension travel (y3−y2), and the MMG of IWM. The parameters involved in simulations are listed in Table 1.
4.1. Response Analyses under Rough Road Excitation
The function of rough road profile in time domain can be expressed as follows:where a = 0.1 m and l = 15 m. They, respectively, denote depth and length of the rough road profile. The vehicle velocity is = 60 km/h.
From the data shown in Tables 2 and 3, it can be seen that although the both optimization methods can improve dynamic performances of the nonlinear coupling system, compared with the GA, the optimization control effect of the proposed method is more obvious. Under rough road excitation, the nonlinear coupling system with the dynamics parameters optimized by the proposed method has better dynamic indicators, especially the maximum IWM’s MMG and the maximum suspension travel which are, respectively, decreased by 50% and 75%. Response comparisons of the nonlinear coupling system optimized by different methods are shown in Figures 8(a)–8(e).
Under rough road excitation and using different optimization methods, the bodywork vertical vibration acceleration, the bodywork travel, the suspension travel, the tire dynamic load, and the IWM’s vertical MMG of nonlinear IWM-suspensions coupling systems are depicted in Figures 8(a)–8(e). It can be seen that the dynamic responses of the optimized nonlinear IWM-suspensions coupling system are improved and their amplitudes are reduced. The vibrations experienced by passengers in the vehicle are reduced with the decreases of bodywork vertical vibration acceleration and bodywork travel, and the ride comfort is meanwhile enhanced. The decreased vertical fluctuation of the MMG indicates that the relative displacement between the stator and the rotor changes smoothly, which reduces unbalanced electromagnetic force waves, decreases motor wear, and extends motor bearing life. Lower tire dynamic load can extend tire life and reduce the frequency of tire replacement. From Tables 2 and 3 and Figures 8(a)–8(e), it can also be clearly seen that although dynamic characteristics of the GA-optimized coupling system have been improved, the minimum value of all the response indicators is always obtained by the proposed method which indicates that the proposed AFSA-based method has better optimization control effects.
4.2. Response Analyses under Bump Road Excitation
Bump road excitation is adopted to analyze dynamic response characteristics of the nonlinear IWM-suspensions coupling system optimized by different methods, and its profile is described by the function below:where h and , respectively, denote height and length of the bump road profile. Here we choose = 0.1 m, = 5 m, and the vehicle velocity = 25 km/h.
Tables 4 and 5, respectively, give the computation results of response indicators of the nonlinear coupling system optimized by different methods and the descending percentages of some important indicators from which it can be clearly confirmed that the nonlinear coupling system optimized by the proposed method achieves the maximum descending percentages of response indicators compared with the GA-optimized one. The maximum values of bodywork acceleration, bodywork travel, tire dynamic load, suspension travel, and IWM’s MMG are individually reduced by 28%, 18%, 39%, 74%, and 53%, which indicates that the nonlinear coupling system optimized by the proposed method has better dynamic performance under bump road excitation. The response comparisons of the nonlinear IWM-suspensions coupling system optimized by different methods are shown in Figures 9(a)–9(e).
It can be seen from Figures 9(a)–9(e) that the original nonlinear IWM-suspensions coupling system cannot attenuate vibration rapidly, while the GA-optimized and the proposed method-optimized coupling systems have vibration damping performance under bumpy road excitation. Compared with the GA-optimized nonlinear coupling system, the proposed method-optimized one can weaken vibration amplitudes quickly and has shorter time to reach steady state. The proposed AFSA-based optimization method can well suppress the vibration of nonlinear IWM-suspensions coupling system on bumpy road, simultaneously limit the suspension travel within its allowable range, and more effectively reduce the bodywork vertical vibration acceleration and amplitude. Therefore, the nonlinear IWM-suspensions coupling system optimized by the proposed method has better synthesized dynamic performance which indicates the proposed AFSA-based optimization method can well improve the ride comfort of electric wheel vehicle on bumpy road.
Based on the established novel dynamics model of nonlinear IWM-suspensions coupling system which includes the MMG of IWM, the AFSA-based multiobjective optimization mathematical model and method are proposed to optimize multiple dynamics parameters of the coupling system. Under rough and bump road excitations, numerical results show that the dynamic performances of the proposed method-optimized coupling system are well improved and the optimal dynamics parameter combinations are obtained. More importantly, by using the proposed optimization method, the MMG’s vertical fluctuation and the bodywork vertical vibration acceleration can be effectively reduced which can decrease electromagnetic force waves and motor wear, extend bearing life, and significantly improve the ride comfort of electric wheel vehicle. Compared with the GA-optimized system and the original system without optimization, the system optimized by the proposed AFSA-based method has the best dynamic characteristics, which verifies the methodological optimization control effect, global convergence characteristic, and superiority than the GA under the same optimization conditions. The methodology investigated by this paper gives a valuable means to the optimization control of IWM-suspensions coupling system. As future work, it is beneficial to discuss the differences of different optimization methods, e.g., DE, PSO, the proposed AFSA-based method, and so on and study the parameter selection method of the proposed optimization algorithm so that we can improve the dynamic performance of IWM-suspensions coupling system further.
The source codes are given as supplementary materials named MATLAB.zip.
Conflicts of Interest
The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
This study was funded by the National Key Research and Development Program of China (grant no. 2016YFD0700800) and the Shaanxi Province Key Research and Development Program of China (grant no. 2017NY-176). The authors would like to appreciate all the authors listed in the references and would also like to thank the funding organizations that provided financial support.
The description of supplementary materials is “Matlab Codes of AFSA, GA, and Simulation.” (Supplementary Materials)
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