Multi-objective optimization of active suspension system in electric vehicle with In-Wheel-Motor against the negative electromechanical coupling effects

Mechanical Systems and Signal Processing 116 (2019) 545–565

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Multi-objective optimization of active suspension system in electric vehicle with In-Wheel-Motor against the negative electromechanical coupling effects Zhe Li a,b, Ling Zheng a,b,⇑, Yue Ren a,b, Yinong Li b, Zhoubing Xiong c a

College of Automotive Engineering, Chongqing University, Chongqing 400044, China State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China c Changan Automotive Engineering Institute, Chongqing 400023, China b

a r t i c l e

i n f o

Article history: Received 3 March 2018 Received in revised form 30 May 2018 Accepted 1 July 2018

Keywords: Four wheel drive In wheel motor Electromechanical coupling Active suspension Multi-objective optimization PSO

a b s t r a c t This paper presents a multi-objective optimization control method of active suspension system for solving the negative vibration issues emerged from In-Wheel-Motor (IWM) in electric vehicles. An integrated model which considering electromechanical coupling between electromagnetic excitation in motor and transient dynamics in vehicle is established and developed. The characteristics of electromagnetic excitation are discussed and its influences on vehicle dynamics are analyzed. The key factors are formulated and selected as the objective criteria for multi-objective optimization approach. The Pareto solution set of optimal parameters in active suspension system is generated by Particle Swarm Optimization (PSO) method, a comparison in vehicle dynamic performances is made to verify the targeted optimization method. The simulation results indicate that the optimized active suspension system can effectively reduce the vertical component of unbalanced electromagnetic excitation by maintaining the relative eccentricity of driving motor in a reasonable interval meanwhile attenuate the sensitivity of the vehicle system to electromagnetic excitation. Furthermore, active suspension system also preserve dynamical advantages in vehicle by means of a balance between the ride comfort and the road holding. The proposed multi-objective optimization method of active suspension system demonstrates a potential application in engineering in order to solve vibration issues in electric vehicle with in wheel motor. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction With the development of electrification and intellectualization in vehicle, In-Wheel-Motor Electric-Vehicles (IWM-EVs), which eliminates mechanical transmission system and integrates fast and accurate multidimensional dynamic control, has brought significant attention in automotive industry. However, vibration and noise issues in IWM are generated due to a special arrangement of propulsion system. The driving motor is connected to the wheel directly without any torsion damper or decelerator and transmission chain in the driveline, which will result in emergence of new electromechanical dynamic issues.

⇑ Corresponding author at: College of Automotive Engineering, Chongqing University, Chongqing 400044, China. E-mail address: [email protected] (L. Zheng). https://doi.org/10.1016/j.ymssp.2018.07.001 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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In the past decades, research interests regarding IWM systems are mainly concentrated on the emerging vertical dynamic issues for IWM driven electric vehicle. One of the negative effects bought by power integration is the increased unsprung mass in electric vehicles. It will cause deterioration in tire road holding performance and ride quality evaluation. To solve this problem, the advanced dynamic-damper mechanism [1,2] and active control method [3] as well as design technology of driving motor [4] were proposed. The influence of intensified unsprung mass was abated from both aspects of vehicle structure re-design and motor structure re-design. Moreover, the negative electromechanical coupling in IWM system on vehicle dynamics has received great attention. The impact of unbalanced electromagnetic force on vehicle system dynamics was investigated in Refs. [5–7]. The results show that the vibration problems in motors not only causes deterioration of stability in vehicle but also has a considerable effect on the ride comfort in vehicle, and an appropriate control method will be desired. Some corresponding schemes to improve the vehicle vertical dynamics of IWM system had been proposed [8,9]. A novel rubber bushings IWM topology scheme was brought forward by Tan and Luo [10], aiming to absorb the vibration energy from the road surface and abate the MMG (motor magnet gap) deformation. The damping parameters in the proposed suspension system, the bushing stiffness, the bushing damping and the bushing deformation were re-matched by an optimal design subsequently [11]. The electromechanical coupling circle (Unbalanced Electromagnetic Excitation-Magnet Gap Deformation) has been demonstrated to be essential to the deterioration of vertical dynamic performance in this special structural layout in inwheel motor driven electric vehicle. However, to date, little research has taken account of this negative effect from driven motor or the dynamic electromechanical coupling process in vehicle dynamics system. In previous researches, either the vertical unbalanced excitation in motor was neglected, or a constant magnet gap deformation was assumed [5,7,10,11]. On the other hand, the control method proposed for the electromechanical coupling effect is indirect control and the effectiveness is limited [6], meanwhile the re-design of IWM configuration is complex [10] and, difficult to develop in practice. Thus, the dynamic electromechanical coupling process and its specific control methodology need further research. As well known, active suspension equipped with electromagnetic actuator can solve the contradiction between the ride comfort and the handing performance effectively. It demonstrates obvious dynamical advantages in its application process. In this paper, a multi-objective optimization method for the active suspension system is developed to solve the negative effect issue caused by the electromechanical coupling between IWM system and vehicle dynamic performance. By proposing an integrated working in a practical scenario, the electro-mechanical coupling process in IWM-EV is described in real-time. A comparison for dynamic performances in IWM-EVs is completed to verify the effectiveness of the proposed optimization approach. This paper is organized as follows: a literature is introduced in Section 1. After that, a Fourier series analytical Switched Reluctance (SR) in-wheel driven motor model is brought forward, followed by a preliminary discussion on the characteristics of unbalanced electromagnetic excitation. And then, the motor model is applied to subsequent developed real-time joint model of IWM-EV in Section 2. Then a Liner Quadratic Gaussian (LQG) controller for active suspension system is approached and a torque output time-varying case under random road roughness excitation is applied in the third section. Followed by, a multi-objective integrated parameter optimal process via PSO method after the targeted objective functions are formulated. Subsequently, the vehicle dynamic responses of the optimized results are compared and analyzed in both time and frequency domain. At last, some key conclusions are given in Section 5.

2. The electromechanical coupling model As IWM-EV is composed of electric wheels with driven motor and vehicle dynamic system assembly, these critical parts are electromechanically coupled. This paper develops an integrated model for electric vehicle with IWMs in which two submodels are included: one is SR-motor model in IWM system, which presents electromagnetic excitation to wheels. The other is vehicle model which emphasizes the dynamic responses in vehicle and outputs the transient electromechanical parameters back to the SR-IWM model in real-time. The sub-models are integrated on the basis of installation relationship of IWM-EV.

2.1. SR-motor modeling SR-motor has wide application potentials in electric vehicle for its simple structure, high starting torque and wide operating range. These features make it easy to meet the needs of dynamic performance in electric vehicle. In this paper, SR-motor is adopted as the driving motor in IWM. Fig. 1(a) is a typical outer rotor SR-motor [12–14] which is mounted in the wheel hub to drive the vehicle directly. It is composed of an outer rotor, an inner stator, a winding coil and a supporting shaft. Magnetic circuit passes 6 outer rotor salient poles and 8 inner stator salient poles to form a loop by air gap. The power is 3 kW, the voltage is 220 V and the speed range is around 30–2000 rpm. SR-motor is a typical electromechanical multi-field system. Its properties can be analyzed by finite element method, empirical table or analytical method. In this paper, Fourier series based on the virtual work principle and the electromechanical energy conversion are adopted to establish the electromechanical multi-field characteristic equations which include energy conversion equation, electromagnetic coupling equation, mechanical driving equation as well as air-gap eccentric equation.

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(a): under normal circumstances

Frk_Z y

Rotor Stator Wingding

Frk

Rotor Stator Wingding

x

Frm

g1

Frt

Air gap Air gap

Shaft

Shaft

Frn

lg(m) =lg(n)

g2

Eq. (12) Frm=Frn

counterbalanced internal force

residual force=Frm - Frn

(b) in case of eccentricity Frm ' Frm

Frm

lg(m) Air gap

1/2Dr’

Frn

1/2Dr

lg(n)

Frn

Frn' Frm =Frm ' Frn=Frn'

lg(m) lg(n)

Eq. (12)

Air gap deformation Frm Frn

Fig. 1. Typical SR motor: (a) under normal circumstances; (b) in case of eccentricity.

2.1.1. Energy conversion equation Assume that the mutual inductance between the windings is negligible [15], the co-energy Wm of the phase winding can be expressed as the integral of the phase-flux linkage wðh; iÞ to the phase-current i, which is shown as follows:

Z

Wm ¼

i

wðh; iÞdi

ð1Þ

0

The generalized electromagnetic force can be expressed by a partial differential of the magnetic energy Wm to the generalized displacement x. The k-phase broadening force of the motor Fk is generated by electromagnetic attraction as:

Fk ¼

 Z i @W m  @wk ðh; iÞ di ¼ @x @x i¼const 0

ð2Þ

Generalized displacements include the relative angular displacement h, the air-gap lg and the stator core thickness lz respectively. The corresponding generalized forces in motor are the driving torque Te, the radial electromagnetic force Fr and the motor axial force Fh respectively:

 Z i @W m  @wðh; iÞ di ¼ @h @h i¼const 0  Z i @W m  @wðh; iÞ ¼ di Fr ¼  @lg @lg i¼const 0  Z i @W m  @wðh; iÞ ¼ di Fh ¼  @lz @lz i¼const 0

Te ¼

ð3Þ

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2.1.2. Electromagnetic coupling equation For the further Fourier series representation, it is necessary to quantify the polar position, then an assumption is given that when the poles are not aligned in the initial position, the corresponding angular displacement is 0°. The salient poles are aligned in p/Nr, in which Nr is the number of salient poles in rotor. The self-inductance of windings Lðh; iÞ which is associated with the rotor position and the winding current can be written as Fourier series [16,17]:

Lðh; iÞ ¼ L0 ðiÞ þ L1 ðiÞ cosðNr h þ u1 Þ þ

1 X

Ln ðiÞ cosðnNr h þ un Þ

n¼2;3;

¼

ð4Þ

1 X Ln ðiÞ cosðnNr h þ un Þ n¼0

Where un ¼ np; the coefficients Ln of the Fourier series can be derived from winding inductances such as alignment, misalignment, and other positions [18]. Here, the pole-to-pole alignment position inductance La, the pole-to-groove misaligned position inductance Lu, and the semi-alignment position inductance Lm [19] are taken as the Fourier series, they can be written as:

  1 1 ðLa þ Lu Þ þ Lm 2 2 1 L1 ðiÞ ¼ ðLa  Lu Þ 2 1 1 L2 ðiÞ ¼ ½ ðLa þ Lu Þ  Lm  2 2

L0 ðiÞ ¼

ð5Þ

Besides, the air-gap in pole-to-groove misaligned position is relatively large, the corresponding inductance Lu under these circumstances can be considered as a constant [20]. La, Lm can be expressed as:

La ðiÞjh¼ p ¼

N X

Nr

an i

n

n¼0

Lm ðiÞjh¼ p

2Nr

ð6Þ

N X n ¼ bn i n¼0

According to Eq. (4), the winding inductance of the k-phase can be written as:

Lk ðh; ik Þ ¼

1 N N X X X 1 1 n 2 n Ln ðik Þ cosðnNr h þ un Þ ¼ ½cos2 ðNr hÞ  cosðNr hÞ an ik þ sin ðNr hÞ bn ik þ Lu ½cos2 ðNr hÞ þ cosðNr hÞ 2 2 n¼0 n¼0 n¼0

ð7Þ Therefore, the winding flux of k-phase can be expressed as:

Z wðh; ik Þ ¼ 0

ik

Lk ðh; ik Þdik ¼

N N X X 1 1 n 2 n ½cos2 ðNr hÞ  cosðNr hÞ cn i þ sin ðN r hÞ dn i þ Lu ik ½cos2 ðNr hÞ þ cosðNr hÞ 2 2 n¼0 n¼0

ð8Þ

Where, cn ¼ an1 =n and dn ¼ bn1 =n are integral coefficients of an and bn respectively. 2.1.3. Electromagnetic force and eccentric equations The partial differential of the angular displacement to the flux is derived according to Eq. (8): N N X X @wðh; ik Þ n n ¼ sinðN r hÞ en ik þ sinð2Nr hÞ f n ik @h n¼0 n¼0

ð9Þ

Here,

1 1 Nr cn ; e0 ¼ 0; e1 ¼ Nr ðc1  Lu Þ; f n ¼ Nr dn  en ; f 0 ¼ 0; 2 2 1 f 1 ¼ Nr ð2d1  c1  Lu Þ: 2

en ¼

Furthermore, the output electromagnetic force in SR-motor can be divided into two parts: the radial force and the tangential force. The tangential force supplies driven power in a form of torque for motor. By substituting Eq. (3) into Eq. (9), the torque output of k-phase is obtained:

Z

ik

Tk ¼ 0

N N X X @wðh; ik Þ 1 1 n n dik ¼ sinðNr hÞ en1 ik þ sinð2Nr hÞ f n1 ik @h n n n¼1 n¼1

ð10Þ

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The total torque developed is the sum of the torques generated by all individual phases:

Te ¼

4 X Tk

ð11Þ

k¼1

The radial force of k-phase can be obtained by ignoring the end effect and mutual inductance [21]:

1 2 Lk ðh; ik Þ F rk ¼  ik 2 lg

ð12Þ

However, the radial force does not contribute to rotational output power in motor. It is the substantial source of noise and vibration. In general, the radial force may be zero under perfect geometrical and manufacturing condition. In fact, the external incentive such as road roughness excitation and uneven load act directly on the elastic structure between the hub and motor bearings, which maintain air-gap of motor, intensifying air-gap deformation and generating motor eccentricity. Although the bearing stiffness in motor is relatively large, the absolute position offset of air-gap caused is small. However, it has the same magnitude as the air-gap of rotating motor, the relative deformation is still considerable [10]. On the other hand, it is very difficult to ensure manufacturing accuracy. Therefore, a non-uniform air-gap in rotating process is unavoidable. It is revealed that although the rate of relative eccentricity in 30%–60% does not reduce the function of motor [22,23]. But it brings unbalanced radial force which may further aggravate the magnet gap uniformity. Two types of deformation in magnet gap are depicted in Fig. 2, situation (a) is chosen as a research object to facilitate the study. The relative eccentricity is defined to describe the magnet gap deformation:

e¼

Dg lg

ð13Þ

Where lg is the air-gap without eccentricity; Dg is the absolute offset of the stator and rotor in the radial direction. In the actual operating condition, the electromagnetic force is distributed along the stator radius. Each pair of opposite phases has different eccentricity. This paper considers the simplest vertical offset case in which the eccentricity of the four phases are assumed in the same e (Dr = Dr0 ) to facilitate the study, as shown in Fig. 1(b). On the other hand, the unbalanced radial force in the k-phase can be expressed as follows:

F urk ¼ F rm  F rn ¼

1 2 Lk ðh; ik Þ 1 2 Lk ðh þ p; ik Þ i  i 2 lg ðnÞ 2 lg ðmÞ

ð14Þ

Where, lg(n) = (1  e)lg; lg(m) = (1 + e)lg. For an 8/6 SR-in wheel driven motor depicted in Fig. 1, |m-n| = 4. The component of the unbalanced radial force in the vehicle vertical direction is yielded:

F rk

Z

¼ F urk cos h

ð15Þ

Therefore, the total vertical unbalanced radial force is derived:

Fr

Z

¼

4 X F rk

ð16Þ

Z

k¼1

Rotor

Stator

eccentricity

(a)

(b) Shaft

Air gap

Fig. 2. Magnet gap deformation.

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2.2. The vehicle modeling Fig. 3(a) illustrates IWM system structure. The motor stator is fixed on the wheel shaft and mounted on the motor bearing, and its vertical displacement and mass are denoted by zus and mus respectively; The motor rotor is fixed on the wheel hub, mounted on the hub bearing, has a vertical displacement of zur and total mass of mur. They are connected by the hub and the motor bearings which is simplified and equalized as a spring in the vertical direction, denoted by km. The corresponding vehicle vertical dynamic model with IWM system is shown in Fig. 3(b) in which the bearing free gap and the lubricant membrane are ignored. Moreover, the actuator force from active suspension system is denoted by Fa in Fig. 3(b). 2.2.1. Driving equation The rotation motion equation in wheel is written as follows:

_ ¼ T e  F tx Re  M t It x

ð17Þ

_ is the wheel angular velocity; Mt is the rolling resistance In Eq. (17), It is the rotational inertia of the total wheel; x moment generated by the tire; Re is the wheel rolling radius; Ftx is the reaction force between the tire and the road which can be obtained by the Magic Formula (MF) [24]; Te is the driving torque. 2.2.2. Vertical vibration equations According to Fig. 3(b) and Newton’s second law, the vibration equations in a quarter vehicle model can be deduced as:

8 > <

ms€zs þ ks ðzs  zus Þ þ cs ðz_ s  z_ us Þ  F a ¼ 0 mur €zur þ kt ðzur  zg Þ þ ct ðz_ ur  z_ g Þ þ km ðzur  zus Þ þ F r Z ¼ 0 > : mus€zus þ ks ðzus  zs Þ þ ct ðz_ us  z_ s Þ þ km ðzus  zur Þ  F r Z þ F a ¼ 0

ð18Þ

Where ms and zs are sprung mass and its vertical displacement respectively; ks, kt and cs, ct are suspension/tire stiffness and suspension/tire damping respectively; Fr_Z represents the vertical component of unbalanced electromagnetic excitation, namely vertical unbalanced radial force in SR-motor. The stochastic road irregularities zg can be expressed by the filtered white noise model [25]:

pffiffiffiffiffiffiffiffiffi z_ g ðtÞ ¼ 2pf o zg ðtÞ þ 2p G0 v wðtÞ

ð19Þ

Where f0 is a low cutoff frequency that reflects the actual road situation; G0 is the roughness coefficient; v is the vehicle velocity; w(t) is the white noise. In this paper, the B-class road displacement spectrum is adopted as actual road excitation. As shown in Fig. 4. 2.3. The electro-mechanical integrating The magnet gap deformation is defined as relative eccentricity. In Fig. 3(b), the coordination position between the stator and the rotor in motor model can be described as zus  zur. According to Eq. (13), the relative eccentricity of motor air-gap in the vehicle system is expressed as:

(a)

(b) zs

mS Fa

kS zus

stator

housing

km zur zg

cS mus

Fr_Z rotor

hub an d tire

mur

kt

ct

Fig. 3. In-wheel motor structure: (a) typical IWM structure; (b) vehicle dynamic model.

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Fig. 4. B-class road road irregularities power spectral density (PSD).

vref Vehicle speed

(a)

Driving torque Vehicle vibration and noise response

Driving model

Driving Torque Calculate

Vehicle Model

SR-IWM Model

Vehicle speed

Vibration model

Fr_Z

road surface

Unbalanced Force Calculate

Speed/rotor position/ current

roughness excitation Joint model

(b)

Vehicle dynamic responses

Vehicle model Road excitation

Electromagnetic excitation Eq. (16) Eq. (20)

Vehicle speed

Driving model Vehicle speed + Reference speed

error

Power converter

In-wheel motor

Driving torque Eq. (11)

Eq. (37)

Phase A Phase B Phase C Phase D

PWM duty cycle signal PID controller

PWM driving cycle signal PWM signal generator

Fig. 5. Flow chart of the electromechanical coupling model: (a) logical diagram; (b) control diagram.

e¼

Dg zus  zur ¼ lg lg

ð20Þ

As a result, the electromechanical coupling model is established based on the installation relationship between the driving motor and the vehicle suspension system. The total logical diagram of the proposed model is obtained in Fig. 5(a). The electromechanical coupling model is composed of Eq. (11) and Eqs. (16)–(20). The control diagram is depicted in Fig. 5(b). The operation mechanism of it can be explained as follows: the unbalanced radial excitation from driving motor is calculated

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by three feedback signals in SR-motor model in real-time, which is current, speed and rotor position and one feedback signal in vehicle model—relative eccentricity induced by road roughness excitation, then it is applied to the vehicle vibration model to observe its negative effects on vehicle dynamic responses. Besides, the motor speed in the SR-motor model is adjusted by Proportion Integration Differentiation (PID) controller, which calculates the Pulse Width Modulation (PWM) duty cycle based on the vehicle speed difference between the current and the reference. All parameters for the SR-motor model and vehicle model are listed in the Appendix below. 2.4. Characteristics of unbalanced electromagnetic excitation 2.4.1. Radial force and unbalanced radial force According to Eq. (12), radial magnetic attraction is quadratic with the winding current, even if a small current can produce a large radial force. Fig. 6 demonstrates the output characteristics of SRM. It can be seen that the peak in the radial force can reach 8 kN when the winding current is 20 A. Once the winding current goes up or down, the radial force will decreased rapidly. On the other hand, the unbalanced radial force is the difference between the radial forces generated by antipoles in the SR-motor and is inversely proportional to the air-gap according to Eq. (14). It is noted that, due to the eccentricity, an increase in the radial force from one direction may lead to a decrease in radial force from the opposite direction, which further increases the magnitude of the unbalanced radial force. The vertical component of unbalanced radial force calculated by Eqs. (16) and (17) are shown in Fig. 7. It can be seen that the relative eccentricity and the vehicle drive current requirement are two key influencing factors to the unbalanced radial force. Consequently, higher eccentricity or drive current demand will lead to greater unbalanced radial force amplitude. In Addition, the excitation frequency of the unbalanced radial force is related to the vehicle speed. The higher vehicle speed is, the higher frequency of the unbalanced radial force excitation in motor is applied to vehicle system. 2.4.2. Coupling effects between road surface excitation and unbalanced radial force In an actual operation condition of the elastic IWM structure, it is clear that the vertical unbalanced excitation in motor is caused by air-gap change. The transient coupling relationships among the road surface excitation, the relative eccentricity and the vertical unbalanced electromagnetic excitation of IWM system under 15 m/s on B-class road are shown in Fig. 8. It can be argued that in the constant speed condition, without regard of electromechanical coupling effects, road roughness excitation is the main influencing factor for vertical unbalanced electromagnetic excitation because of the constant torque demand for driven motor. The magnitude of the unbalanced radial force varies with the relative eccentricity as the road surface roughness varies. However, it should be note that the electromechanical coupling intensifies the non-uniform air-gap

Fig. 6. Output characteristics of SRM: (a) radial force; (b) unbalanced radial force.

Fig. 7. Characteristics of the vertical unbalanced excitation: (a) under different eccentricity; (b) under different vehicle speed.

Z. Li et al. / Mechanical Systems and Signal Processing 116 (2019) 545–565

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Fig. 8. Coupling effects between road surface excitation and unbalanced radial force.

which further in turn increase the magnitude of the unbalanced electromagnetic excitation and worsen motor vibration as a comparison is made between vehicle model with unbalanced radial force and without unbalanced radial force. 2.4.3. The influence of unbalanced electromagnetic excitation on vehicle dynamics The effect of the unbalanced electromagnetic excitation on vertical dynamics can be regarded as an inertial force for the whole vehicle system. The vibration transfer characteristics of vehicle sprung and unspung mass to the unbalanced radial force, the vibration modes and vibration energy distributions can be derived from the state space form of Eq. (18), as shown in Fig. 9 and Table 1. As discussed before, unbalanced radial force is a series of periodic harmonic excitation related to wheel speed. While in vehicle system, it is noted that the unbalanced radial force mainly excites three major vibration modals around the resonance frequencies of sprung and unsprung masses. Its influences are mainly concentrated in the second and third resonance frequencies about 10 Hz and 65 Hz respectively, which are approximately equal to the resonance frequencies of tire and motor stator. Moreover, in these frequencies, 93.93% and 95.122% of the total vibration energy are included in vehicle unsprung masses. This implies that the unbalanced electromagnetic excitation has a greater influence on the unsprung masses in the relatively high frequency range. 3. Multi-objective optimization for active suspension system 3.1. Active suspension controller Active suspension system is required to balance the contradiction between the ride comfort and the handing performance in vehicle. The dynamics in vehicle is directly influenced by the control method advised in active suspension system. In recent years, great achievements have been made on active suspension control theory, such as back stepping control [26,27], sliding-mode control [28,29], intelligent control [30,31] as well as active disturbance rejection control [32]. However, the nonlinear output characteristics of actuator restricts the application of these methods. By using the adaptive vibration control strategy [33], the effect of actuator dynamic was considered. A filter-based adaptive control strategy is subsequently proposed to overcome the ‘‘exploration of terms” problem existing in standard back stepping control. In Ref. [34], a novel bioinspired dynamics-based nonlinearity approach was developed to deal with the nonlinear suspension

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Fig. 9. Vibration transfer characteristics to the unbalanced radial force in SRM: (a) vertical acceleration; (b) suspension deflection; (c) tire bounce.

Table 1 Vibration energy distribution of the IWM-EV model. Energy distribution (%) Modal order

(1)

(2)

(3)

Natural frequency (Hz) D O F

1.268 73.632 12.955 13.413

64.958 6.070 50.412 43.518

8.121 4.878 53.173 41.949

Sprung mass Motor stator &housing Motor rotor, rim &tire

properties. By introducing an asymmetric suspension structure inspired by limb motion dynamics of a Grus japonensis as the reference model for active tracking control of nonlinear suspension systems, the proposed active controller was able to employ and modify the inherent nonlinearities existing in vehicle suspension systems to become beneficial to vibration isolation. The results indicated that the active suspension advised by the proposed control method acquired a lower energy consumption than the standard adaptive control approach, which demonstrated the effectiveness perspective on robust controller design with a green or sustainable feature. While in the case of electromechanical coupling analysis, the influence parameters on electromechanical coupling process are the major concern in the controller design of the active suspension. For helping facilitate the study and achieving low performance-cost rate of the controller, a classic Liner Quadratic Gaussian (LQG) controller based on system state estimator is designed to regulate a minimal input power for the active suspension. For the controller design purposes, the Eq. (18) is written as the standard form of state space:

X_ ¼ AX þ BU þ GW

ð21Þ

Y ¼ CX þ DU þ HW Select the following state variables to distinguish the outputs

X ¼ ½z_ s zus  zs zg  zur zur  zus z_ us z_ ur 61 T

ð22Þ

The outputs concerned include vehicle body movement responses, suspension deflection, tire bounce and relative eccentricity, as follows:

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Y ¼ ½€zs zus  zs zg  zur zur  zus 41 T

ð23Þ

The controllable input indicated by Eq. (21) is actuator force of the active suspension:

U ¼ ½F a 

ð24Þ

Defining the exogenous disturbances W as:

W ¼ ½z_ g F r Z 21 T

ð25Þ

Then the system matrix can be derived:

2

m1 s cs 6 1 6 6 6 0 A¼6 6 0 6 6 1 4 mus ct

m1 s ks 0

0 0

0 0

m1 s cs 1

0 0

0

0

0

0

1

m1 s

7 7 7 7 7 7 7 7 5

0

0

0

1

1

m1 us ks

0

m1 us km

m1 us c t

0

0

m1 us kt

m1 ur km

0

m1 ur ct

0 2

3

66

3

2 2

6 0 7 7 6 7 6 6 0 7 7 B¼6 6 0 7 7 6 7 6 5 4 m1 us 0

6 6 ; C¼6 4

1 1 m1 s cs ms ks 0 0 ms c s 0

3

0

1

0 0

0

0

0

1 0

0

07 7 7 05

0

0

0 1

0

0

3 m1 s 6 0 7 7 6 ; D¼6 7 4 0 5 2

46

0

6 6 6 6 ; G¼6 6 6 6 4 41

0 0 1 0 0

0

0 7 7 7 0 7 7 0 7 7 7 5 m1 us

1 m1 ur c t mur

61

3 2

0 0

3

60 07 7 6 ; H¼6 7 40 05 0 0

42

62

The cost function of the controller is selected as:

Z J¼

1

Z ½YT QY þ UT RUdt ¼

0

1

0



X

T 

U

Qd

Nd

NTd

Rd



X U

 ð26Þ

dt

Where the weight matrices are assigned:

Q ¼ diag½ a1

a2

a3

a4 

ð27Þ

R ¼ ½b

ð28Þ

Where,

Q d ¼ CT QC; Rd ¼ DT QD þ R; Nd ¼ CT QD Controller input is expressed as:

U ¼ KX

ð29Þ

The feedback gain matrix is obtained from Eq. (30). T T K ¼ R1 d ðB S þ Nd Þ

ð30Þ

Where S can be obtained by the Riccati equation: T T AT S þ SA  ðSB þ Nd ÞR1 d ðB S þ Nd Þ þ Dd ¼ 0

ð31Þ _

The Kalman estimator is used for state estimation of the system state X: _ _ _ _ X ¼ A X þBU þ FðY  C X DUÞ

ð32Þ

Gain matrix in Kalman estimator is generated by Eq. (33):

 R  1 F ¼ ðPCT þ NÞ

ð33Þ

 and R  is obtained by solving Riccati equation: In Eq. (33), N

 ¼ Rn þ HNn þ NT HT þ HQ HT R n n

ð34Þ

   ¼ V Q HT þ N n N n

ð35Þ

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In Eq. (35),

Q n ¼ EðWWT Þ; Rn ¼ EðvvT Þ; Nn ¼ EðWvT Þ Where v is the nominal measurement noise in Kalman estimator. Finally, the governing equations of control force of active suspension can be obtained by Eq. (36): _

U ¼ K X

ð36Þ

3.2. A test case As mentioned before, the unbalanced electromagnetic excitation of IWM is directly related to the relative eccentricity and motor current demand. If the eccentricity is fixed, the higher driving force the vehicle required, the higher magnitude of unbalanced radial force is generated. In this paper, an acceleration process is introduced as a test case and utilized for the proposed analysis. In this case, the transient relative eccentricity e is calculated by Eq. (20). The desired speed of vehicle, average output torque of driven motor and vehicle speed error are shown as Fig. 10. This test case is a dynamic process which combines the road roughness excitation and time-varying driving torque requirement in vehicle while rouses transient coupling motions between unbalanced electromagnetic excitation and vertical kinematics and dynamics in IWM vehicle system. 3.3. Optimal parameters design It is well known that different design and control parameters in active suspension could result in diverse performance. As for the active suspension used in IWM-EVs, the priority control purpose is to alleviate the negative vibration effect caused by electromechanical coupling. On the other hand, the active suspension system is hoped to maintain its inherent friendly characteristics to vehicle dynamic performance. It is implied that the optimization of vehicle suspension in this case is a typical nonlinear and multi-objective procedures in which not only a balance between the ride comfort and the road holding, but also negative electromechanical coupling attenuation in both response ranges of time and frequency domains should be considered together. A multi-objective nonlinear optimization problem with m goals and multiple constraints is generated:

 F fitness ðxÞ ¼

bestff 1 ðxÞ; f 2 ðxÞ;    f m ðxÞg; x 2 K;

Pbestn < 1:5  Pbestn1

w;

Pbestn P 1:5  Pbestn1

ð37Þ

Where Pbestn is the optimal position of the external solution set when the optimization progress is iterating to n steps; x is the optimization variable; w is defined as a penalty function to search the optimal solution more efficiently, its dimension is consistent with the fitness function; K is constraint condition of the design variable:

K ¼ fx 2 Rn g

ð38Þ

Where Rn is the design variable field. 3.3.1. Objective functions In previous studies, most of the objective functions in the optimization of active suspension control was focused on the balance of dynamic performance in vehicle system [35]. In this paper, however, the major concerns are the emergence of unbalanced electromagnetic excitation and vibration issues caused by it. Therefore, the purposes of the optimization approach are to reduce the vertical component of unbalanced electromagnetic excitation and decouple the negative effects of electromechanical coupling on vehicle system as well as preserve dynamical advantages of active suspension in vehicle.

Fig. 10. Acceleration test case: (a) vehicle speed error; (b) average output torque of driven motor.

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The Root Mean Square (RMS) responses of vertical acceleration (€zs ), deflection in suspension (zus-zs) and tire bounce (zg-zur) are evaluated in order to improve the ride comfort and the road handling. They are considered as the first objective function. As mentioned before, the relative eccentricity of driven motor is the only influencing factor to increase the amplitude of unbalanced electromagnetic excitation when the current requirement is defined. Therefore, RMS value of e in the vehicle dynamics equation (which is Eq. (20)) is also considered as objective function in the optimization approach. Furthermore, in the weak damping IWM system, electromagnetic excitation of driven motor, structural kinematic of IWM system and vehicle system dynamics are highly electromechanical-coupled due to the compliance of hub and motor bearing. One of the optimal solutions is to reduce the sensitivity of vehicle system to electromagnetic excitation by active suspension system. Hoping to be partially electromechanical decoupled from the system level. On the other side, the time domain evaluation is applied to limited behaviors. Compared to time domain analysis, frequency response analysis is a widely used technology to evaluate the ride comfort and the road holding in vehicle, especially for the ride comfort assessment. For an IWM-EV, driving speed range is 0–100 km/h, the corresponding unbalanced vertical force excitation frequency in driven motor is from 0 Hz to 140 Hz. Therefore, this paper adopts the areas and maximum values within this particular frequency transfer characteristics range for quantifying the negative vibration ffects. More specifically, the transfer characteristics of dynamics responses in vehicle vertical direction (which is €zs and zg  zur ) to unbalanced electromagnetic excitation are selected and quantified as electromechanical coupling target and subsequently classified into objective 3. These objective functions are normalized by RMS values of vehicle in passive suspension in the same test condition in order to generate reasonable optimal results. They are described in details as follows: Objective 1: F 1 ðxÞ ¼ RMSð€zs Þ þ RMSðzus  zs Þ þ RMSðzg  zur Þ Objective 2: F 2 ðxÞ ¼ RMSðeÞ Objective 3:

F 3 ðxÞ ¼ Sðtransfer ratio€zs !F r Z Þ þ Maxðtransfer ratio€zs !F r Z Þ þ Sðtransfer ratioðzg zur Þ!F r Z Þ þ Maxðtransfer ratioðzg zur Þ!F r Z Þ

ð39Þ

Here, S represents area, Max represents maximum value. 3.3.2. Variables and constraints In this paper, the weight matrices in LQG controller are considered as the design variables in multi-objective optimization of active suspension system. Besides, the stiffness (ks) and damping (cs) in the suspension system are also considered as the mechanical design variables. This means that the conventional mechanical design variables and control parameters in LQR controller are considered together in the proposed optimization, which makes the search of the optimal solution set more efficient. Therefore, the design variables are described as:

x ¼ ½a1 ;    ; a4 ; b; ks ; cs 

ð40Þ

In the stiffness and the damping variables, the stiffness is limited in a range between 0.75 time of the reference stiffness and 1.5 times of the reference stiffness. In this way, the partial frequency of in electric power vehicle is controlled in between 1 Hz and 1.45 Hz. The damping in suspension system is limited from 0.25 time to 1.5 time of the reference damping. All these are considered as the constraints in multi-objective optimization formulation. Therefore, the constraints are described as follows:

s:t: ks 2 ½0:75; 1:5  ksðpassiveÞ cs 2 ½0:25; 1:5  csðpassiveÞ a1 ;    ; a4 2 ½1; 1e10 ; b 2 ½1; 1e10  3.3.3. Parameters of the particle swarm optimization The experienced method for parameters’ selection in active suspension system is a time consuming and complex process which is not always accurate and available. The traditional algorithm transforms the multi-objective problem into the singleobjective problem by means of weighting coefficient, however, the choice of different weight coefficients will give rise to different results which are usually trapped into local optimum. In recent years, many optimization methods are developed to search optimal parameters for active suspension systems and improve global convergence. For example, Zhang [36] developed Simulated Annealing (SA) algorithm to optimize a half-car suspension system. Tung et al. [37,38] used Particle Swarm Optimization (PSO) method to search the optimal control parameters for exponential decay controller. Do [39] proposed a Genetic Algorithm (GA) method to optimize H-inf/LPV integrated controller parameters for semi-active suspension system. Nariman et al. [40] developed a Multi-objective Uniform-diversity Genetic Algorithm (MUGA) to achieve a comprehensive

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multi-objective optimization in vehicle dynamic control. The algorithm was further applied to the probabilistic uncertainty [41], which validated the robustness of the optimization algorithm. Dynamic control in IWM-EVs is a multi-objective coordination control issue in which a compromise between various considerations should be achieved. According to the aforementioned optimization methods, intelligent methodological 0optimization such as GA and PSO is essential instead of traditional ones to achieve the global optimal results. Here, PSO method is employed to search Pareto solution set and obtain the optimal vertical dynamics control for IWM-EVs. The parameters in the PSO are selected as follows: population size: 500; the inertia coefficient W changes linearly with the iteration process with an initial value 0.7 and 0.4 to the end; the weighting factors R1 and R2 change randomly between 0 and 1; the constraint factor a = 0.5; the iteration step: 1800. Fig. 11 is the flow chart and the steps in details can be described as follows: (a) The particle swarm velocity VEL[i] in the current iteration step is calculated by the follow equation:

VEL½i ¼ W  VEL½i þ R1  ðPbest½i  POP½iÞ þ R2  ðGbest½h  POP½iÞ

ð41Þ

Where W is the inertia factor; R1 and R2 are weighting factors, which represent the weights of individual optimum and global optimum respectively; Pbest[i] is the individual optimal location of the current particle; Gbest[h] is the global optimal location of the current particle; POP[i] is the particle current location. (b) The location of the particle swarm is updated according to the following equation:

POP½i ¼ POP½i þ a  VEL½i

ð42Þ

Where a is the constraint factor, if the particle flies out of the set boundary (constraint), the particle stays on the boundary and change the flight direction. (c) The fitness function of the location updated particle POP[i]is calculated. (d) The particles in the Archive by the adaptive mesh method is updated, and then the global optimal location Gbest is searched by roulette. (e) The new individual optimal location Pbest is compared with elder one, if dominate, the new Pbest is set, otherwise the elder is remained, if the two solution does not dominate each other, then randomly one of the two is chose as Pbest. (f) If the condition is reached, the iteration is terminated, otherwise, repeat (a). 3.4. Optimization iteration process Fig. 12 illustrates the iteration progress of PSO. It can be seen that the group distribution of three objectives tend to stabilization after 600 generations although some fluctuations are observed after about 1300 generations. Fig. 12(b) shows the number of external solution set tends to stabilization after about 1600 generations, in which contains 138 optimal solutions. This means elite particles for each generation have emerged. It can be also found from Fig. 12(c) that the solution set is distributed evenly in the solution space, which satisfies global optimal solution characteristics. Moreover, the terminal condition is satisfied at 1800 generations and PSO search finished. The optimized parameters are partly listed in Table 2. 4. Results and discussion The optimal solution 108 is selected and utilized as active control (b) for active suspension system with electromechanical coupling is considered in IWM-EVs to analyze and demonstrate dynamic performances in vehicle. A comparison between optimized approaches in active suspension system is made. The reference one is conducted by the same PSO parameters and approach without objective functions specific formulated for electromechanical coupling parameters in optimization process (i.e. only objective 1 is evaluated). The result Pareto solution set is adopted as active control (a). For the fairness of comparison, design parameters stiffness (ks) and damping (cs) active control (a) remain consistent with optimal solution 108. 4.1. Objective function for dynamic performance in vehicle Simulation results in vehicle dynamic responses by passive suspension, active control method (a) and active control method (b) in the test case are shown in Fig. 13 and Table 3. It can be noted that the acceleration of sprung mass and suspension deflection are reduced successively via active control methods. It is seen that RMS value of the sprung mass acceleration are decreased from 0.1621 m2/s to 0.1100 m2/s in active control (a) and 0.1596 m2/s in active control (b) respectively, a reduction of 32.14% and 1.54% respectively and the RMS value of the suspension deflection are decreased from 1.2218 mm to 1.1565 mm in active control (a) and 0.6728 mm in active control (b) respectively, a reduction of 5.34% and 44.93% respectively. This implies that a promotion for the ride comfort is achieved. It is also pointed out that the active control (b) does not

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Fig. 11. The flow chart of PSO.

improve the vehicle dynamics over the full range of the time domain, which specific manifested as that the peak value of the sprung mass response at certain interval (T = 2.75–3.25, for example) is greater than passive and active control (a). This is due to the adjustment of the active control (b) for the multi-objective balancing. A sacrifice in vertical acceleration is made to attenuate the unbalance radial force in motor. For this issue, we will discuss further with regard to the other two objective functions of active control (b) later.

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Fig. 12. The iteration process: (a) the mean value of fitness function; (b) the number of external solution set; (c) Pareto set distribution.

Table 2 Original and optimized parameters. Suspension parameters

a1

a2

a3

a4

b

ks

cs

*

– 9.2E+3 4.6E4 4.2E+6 4.8E+6 1.7E+6

– 3.3E+8 3.5E+9 1.5E+1 1.2E+1 1.3E+1

– 5.8E+1 1.1E+4 7.1E2 4.9E2 4.1E2

– 8.5E8 4.7E3 4.4E3 4.5E3 5.8E3

– 1.3E+0 4.5E+0 1.8E+2 1.6E+2 1.1E+3

1.0 0.61 1.39 1.25 1.21 1.04

1.0 0.59 0.72 0.52 0.52 0.50

Passive suspension Optimal solution 1 Optimal solution 2 ... * Optimal solution 108 ...

On the other hand, the RMS value of tire bounce fallen from 0.5998 mm to 0.4553 mm by adopting active control method (b). However, RMS value of tire bounce increases from 0.5998 mm to 0.6236 mm by adopting active control (a), an obvious deterioration is seen in Fig. 13(c) and Table 3. It is seen that active control method (a) pays more attention to the acceleration reduction of sprung mass, so that the redundant vibration energy of the actuator is transmitted to the wheel. Meanwhile, in the optimization process for active control (a), the electromechanical coupling is not considered, even if its negative effects has a greater impact on the motion of unsprung mass as discussed in Section 2.4.3. In general, both two active suspension systems exhibit obvious dynamical advantages in vehicle vertical dynamics after parameters and controllers are optimized. 4.2. Objective function for transient electromechanical parameters The dynamic changes of relative eccentricity and corresponding unbalanced electromagnetic excitation in driven motor under test case are depicted in Fig. 14(a), (b) and Fig. 15(a). It can be observed that the continuous high torque demand around 0.75 s to 2.5 s (which shown in Fig. 10(b)) induces a shock responses on motor relative eccentricity in passive suspension system, in which the peak value reaches to 0.48. At this moment, the vertical unbalanced electromagnetic excitation which is influenced by a variety of factors such as vehicle driven current requirements and kinematic response of IWM system, reaches 1275 N in peak as shown in Fig. 15(a). On the other hand, it can be coincidentally noted in Fig. 13 that the

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Fig. 13. Comparisons of vehicle responses in the test case: (a) the vertical acceleration; (b) the suspension deflection; (c) the tire bounce.

Table 3 The RMS value of the vehicle responses under test case. Dynamics response

Vertical acceleration (m/s2) Suspension deflection (mm) Tire bounce (mm)

Passive

0.1621 1.2218 0.5998

Active suspension

Changed

control (a)

control (b)

0.1100 1.1565 0.6236

0.1596 0.6728 0.4553

"45.1% ;41.8% ;27.0%

acceleration of sprung mass, the suspension deflection and the tire bounce in vehicle system are raised substantially in the very same interval where the driving torque starts to rise. It can be deduced that the magnet gap deformation is initially caused by road surface roughness excitation and is electromechanical coupled with unbalanced radial forces of motor, will eventually result in a deterioration in vehicle dynamics. For active control method (b) with targeted optimal function, it is noted that the peak value of the relative eccentricity falls to 0.38 and the corresponding motor vertical unbalanced excitation is 838 N, has a reduction of 20.83% and 34.27% respectively. Both low and high frequency components of vertical unbalanced excitation are attenuated effectively as shown in Fig. 15(b), which are contributed to vibration reduction of vehicle sprung mass and unsprung mass. Moreover, Fig. 14(c) also illustrates the comparison of the actuator forces of active suspension system between different active control methods. It can be seen that more frequent output force in actuator is generated by the active control method (b) to coordinate the relative motion of stator and rotor when relative eccentricity in motor is increased. The simulation results verify that active control method (b) can maintain the relative eccentricity of motor in a reasonable interval by outputting compensated active actuator force, sequentially preventing further enlargement of air-gap deformation and unbalanced electromagnetic excitation which enables the system self-coordinate the relative motion of stator and rotor, adjusting the fluctuation axis and makeing the fluctuation more concentrate. All of these are benefit to attenuate the RMS value of unbalanced electromagnetic excitation as well as weaken its influence on vertical dynamic performance in vehicle. The specific RMS results are listed in Table 4.

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Fig. 14. Comparisons of relative eccentricity and actuator force in the test case: (a) the dynamic change of relative eccentricity in passive suspension system; (b) the dynamic changes of relative eccentricity in active suspension system; (c) the actuator forces.

Fig. 15. Comparisons of unbalanced electromagnetic forces in the test case: (a) vertical unbalanced excitation in motor; (b) FFT of vertical unbalanced excitation in motor.

Table 4 The RMS value of the electromechanical parameters. Mechanical/ electromagnetic results

Motor relative eccentricity Vertical motor unbalanced radial force (N)

Passive

0.0911 65.56

Active suspension

Changed

control(a)

control(b)

0.1049 68.91

0.0817 53.45

;22.1% ;22.4%

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Fig. 16. Vibration transfer characteristics of vehicle to the unbalanced electromagnetic excitation: (a) the vertical acceleration; (b) the tire bounce.

4.3. Objective function for impact reduction on vehicle dynamics Fig. 16 is the transfer characteristics of dynamics responses in vertical direction to unbalanced electromagnetic excitation, which are selected and quantified as objective function 3 in the optimization process. As discussed in Section 2.4.3, the unbalanced radial force mainly excites three major vibration modals around the resonance frequencies of sprung and unsprung masses and gives rise to the vehicle resonance in three degrees of freedom around 1 Hz, 8 Hz and 65 Hz respectively. Of note, the proposed targeted multi-objective optimization method (b) attenuates the second and third resonance peaks of sprung mass, and reduces the sensitivity of the system sprung mass to the electromagnetic excitation in almost all frequency range. Moreover, the vibration energy distribution analysis in Section 2.4.3 indicates that the principal vibration modes in the second and the third order of the IWM system are mainly concentrated on unsprung masses. As a result, after taking this factor into consideration of optimization design, Fig. 16(b) also illustrates the active control (b) has a better second and third order vibration responses on unprung masses against unbalanced electromagnetic force than passive suspension system and active suspension system with active control (a). Generally speaking, the optimized active suspension system can achieve an overall better performance on both sprung and unsprung masses in vehicle. Less vehicle body acceleration and tire bounce are aroused by unbalanced electromagnetic force. The influence of unbalanced electromagnetic force on dynamics performance in vehicle is weaken by active suspension system with active control (b).

5. Conclusions The coupling between the electromagnetic excitation in motor and elastic structure in vehicle gives rise to electromechanical dynamic issues. The suffered weak damping system will deform the magnet gap in driving motor. The accompanying unbalanced electromagnetic excitation in vertical direction has an obvious influence on the ride comfort in IWM-EVs. The relative eccentricity is a key factor to cause the unbalanced electromagnetic excitation in IWM, and the amplitude and frequency of the unbalanced electromagnetic excitation are related to driving current requirements in electric vehicle. The consequent electromechanical coupling circle will aggravate vibration and noise in vehicle. As a result, the transient vibration problem becomes one of the key electromechanical dynamic issues in the IWM system. The negative influence of the electromechanical coupling effects on the ride comfort and the road holding should not be ignored. The proposed optimization method of active suspension (active control (b)) has a specific consideration for this issue. The simulate results show that the active suspension can maintain the relative eccentricity of motor in more reasonable interval by outputting targeted active control force, which attenuates the vertical component of unbalanced electromagnetic excitation rapidly, sequentially preventing further deterioration of the air-gap deformation in motor and enabling the system self-coordinate the relative motion of stator and rotor. Moreover, more comprehensive optimization process is conducted according not only in time domain but also in frequency domain. The frequency transfer characteristics of the system show that the proposed optimization method with targeted control criteria for the impact reduction on vehicle dynamics produces a better dynamic responses against unbalanced electromagnetic force, which means less vehicle body acceleration and tire bounce are aroused by unbalanced electromagnetic force. All of these are benefit to reduce the negative influence of electromechanical coupling effects on vertical dynamic performance in vehicle. In addition, thanks to the intelligent multi-objective optimization algorithm, the proposed optimization method preserves the dynamical advantages of active suspension system with a more balanced ride comfort and road holding responses.

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Acknowledgements The authors acknowledge the National Key Research and Development Program of China under Grant (2016YFB0100904) and Chongqing Science and Technology Commission under Grants (cstc2015jcyjBX0097, csts2015zdcy-ztzx30001) for financial support. Conflict of interest We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Appendix Nomenclature and value Name

Value

Structure parameters of SR-motor 210 0.5 lg 22 23 116 76 16.08 13.74 136 Nr 6 Parameters of vehicle model It 1.2 Re 0.269 ms 337.5 mus 37.5 mur 65 3,850,000 km 23,500 ks 250,000 kt 1450 cs ct 375 0.01 f0 64e-6 G0

Unit

Expression

mm mm degree degree mm mm mm mm

Rotor outer diameter Motor air-gap Stator pole curvature Rtator pole curvature Stator outer diameter Core length Height of stator yoke Height of rotor yoke Number of windings per phase The number of salient poles in rotor.

kgm2 m kg kg kg N/m N/m N/m Ns/m Ns/m Hz m3

Rotational inertia of the total wheel Wheel rolling radius Sprung mass of vehicle Mass of motor stator Mass of motor rotor Motor bearing stiffness Suspension stiffness Tire stiffness Suspension damping Tire damping low cutoff frequency Roughness coefficient

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