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International Internatio nal Journal of Autom Automotive otive Technolog Technologyy , Vol. 9, No. 6, pp. 687−693 (2008) DOI 10.1007/s12239−008−0081−y

Copyright Copyri ght © 200 2008 8 KS KSAE AE 1229−9138/2008/043−06

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS J. KIM* Department of Vehicle Dynamics Research Team, Hankook Tire Co., LTD., R&D Center, 23-1 Jang-dong, Yuseong-gu, Daejeon 305-725, Korea (Received 28 December 2007; Revised 23 June 2008)

−In this article, the analysis methods for vehicle handling performance are studied. Using simple models, dynamic characteristic parameters such as yaw, natural frequency, and the damping coefficient of a vehicle can be theoretically formulated. Here, the vehicle is simplified by a bicycle (single-track) model, and the tire is modeled by an equivalent cornering stiffness and first order lag. From the experimental road data, the tire model parameters (equivalent cornering stiffness and time lag constant) are extracted. These parameters are then inserted into the theoretically formulated equations of dynamic characteristic parameters. For the purpose of validating the efficiency of the suggested methods, experimental road tests (where the cars have different handling performances) are performed. The results show that vehicle handling performance can be sufficien sufficiently tly represent represented ed by the suggeste suggested d dynamic dynamic characte characteristic ristic paramete parameters. rs. So, it is concluded concluded that that the proposed proposed method method has practical use for the development of new cars or for the comparison of similar cars since the evaluations of the vehicle handling performance can be efficiently determined by the suggested dynamic characteristic parameters. ABSTRACT

KEY WORDS :

Bicycle model, First order lag tire model, Equivalent cornering stiffness, Damping coefficient

1. INTRODUCTION

The purpose of this article is to study the possibility of using vehicle lateral and yaw dynamics for the analysis of vehicle handling performance (i.e., based on the simple bicyclee model). bicycl model). For this, dynamic dynamic charact characteristic eristic paramete parameters rs such as yaw natural frequency and the damping coefficient of car are theoretically formulated using the approaches of Sakai and Satoh (1995). Specifically, the vehicle is simplified by bicycle (single-track) model, and actual tire and suspension characteristics are merged into one force generator per axle that is modeled by an equivalent cornering stiffness and first order lag. In this paper, new approaches for obtaining dynamic characteristic parameters from experimental road test data are introduced. For this, the equivalent cornering stiffness and first order lag are extracted from road test data, which fundamentally allows more accurate compared with those methods that use indoor tire behavior machines (such as Flat-trac). The reason is that the tire has a smaller inertia, and its behavior is influenced by road roughness, the antilock braking system (ABS) operation, tire pressure and tread variations, carcass radial non-uniformities, as well as vehicle roll, pitch, and vertical motions. Therefore, tire motions tend to be a lot more oscillatory and contain higher frequency components. Separating the effects of these disturbances from those of tire and road interactions is difficult. The extracted equivalent cornering stiffness and first order lag parameter are then inserted into the theoretically formulated equations to determine dynamic characteristic parameters.

Handling analysis is one of the most attractive themes in automobile engineering. Here, many suspension and tire designers want to know how their design parameters contribute to handling characteristics, and test drivers want to know what the sensations that they feel are derived from. A simplified bicycle model, which is also known as the single-track model, is convenient for the theoretical analysis of vehicle dynamics. Due to this majority of single-track model, automobile engineers can easily understand the handling characteristics of a car. In the literature, several different methods have been introduced. For example, Sakai and Satoh (1995) used complex cornering stiffness, which is the same as the equivalent cornering stiffness when the phase lead time is taken into account. From this approach, the natural frequency and damping coefficient of the plane motion are theoretically formulated. Kusaka et al . (1995) examined a simpler bicycle model for the purpose of investigating the influence of tire transient characteristics on vehicle handling properties. Mimuro et al. (1990) developed the four parameter evaluation methods of lateral transient response that also come from the bicycle model. This technique allows a rapid comparison of handling characteristics of different tire and vehicle combinations by condensing the response data to be displaye displayed d into a rhombus form. [email protected] hankookti ooktire.co re.com m *Corresponding author. e-mail: [email protected] 687

688

J. KIM

In order to validate the efficiency of the suggested methods, experimental road tests for cars having different handling performances are performed. The results show that vehicle handling performance can be sufficiently represented by the suggested dynamic characteristic parameters.

2. LATERAL VEHICLE DYNAMICS In order to formulate the dynamic characteristic parameters theoretically, a first order lag tire model and simple bicycle vehicle model are considered. 2.1. First Order Lag Tire Model The lateral force generated at the contact point between the road and tire is transferred to the wheel. This mechanism can be modeled with an effective spring and damper, as shown in Figure 1. The contact patch (including tread, carcass, and belt of the tire) is connected to the wheel by an effective spring (k y) and damper (d y). Here, the axis η (c) is fixed at the center of wheel to describe the lateral motion of the contact patch. The equation of motion of the contact patch can be written as ·· – θ · 2 η ) + d η · + k η = F (1) m ( η ( c)

(c)

s

(c)

( c)

y

y

( c)

y

(c )

y

(c)

y

(c )

( c)

· η ( c )⎞ F y = C α ⎛ α – ------⎝ V x ⎠

(4)

where C α is the cornering stiffness of the tire. By combining equations (1) and (4), the equation of motion of the tire contact patch can be obtained by ·· – θ · 2 η ) + d η · + k η = C ( α – η · / V ) . (5) m ( η (c)

(c)

s

(c )

y

(c)

y

(c)

α

( c)

x

In general, m(c) and d y are very small compared to k y. This allows equation (5) to be simplified as · (6) k y η ( c) = C α ( α – η ( c ) / V x ) = F y . From equation (6), the following relationship can be obtained. · · y = k η F y ( c ) .

(7)

By combining equations (4) and (7), the first order lag tire model that considers the lateral force and steady-state slip angle can be obtained by · y + F = C α τ F y α

(8)

where

y

· where m(c) is the mass of the contact patch, θ s is the steering rate of the wheel, and F y is the lateral force at the tire contact patch. The lateral force at the wheel, F y(w) can be expressed as · + k η = F − m ( η ·· – θ · 2 η ) . (2) F = d η y ( w )

small slip angle area, F y can be expressed as

s

(c)

Equation (2) shows the phase delay between the lateral force of contact patch and the lateral force of wheel, which · is the comes from the inertia of the tire contact patch. η (c ) lateral velocity difference between the wheel and tire contact patch. This represents the direction between the wheel and tire contact patch. The tire slip angle, which is defined as the angle between the wheel heading direction (ξ -axis) and the traveling direction of the tire contact patch, can be written as · η (c) ------− (3) α = α V x · where α is steady-state slip angle when η is zero. In a

C α -. τ = --------k y V x

(9)

In equation (9), τ is the time constant of the first order lag tire model. Also, from equation (8), the transfer function of the lateral force of the tire can be obtained by F y ( s ) C α ------------ = -------------. α ( s ) 1 + τ s

(10)

2.2. Vehicle Model An actual vehicle model with a single track, which describes the lateral and yaw dynamics of a 2-axle and 1-rigid body ground vehicle, is shown in Figure 2(a). The equation of motion for this model can be expressed by mV [ s β ( s ) + r ( s ) ] = F y 1 ( s ) + F y 2 ( s )

(c)

Figure 1. Transient characteristics of the tire lateral force.

Figure 2. Two DOF (degree of freedom) of the (a) actual and (b) equivalent bicycle model.

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS

Table 1. Vehicle model parameters. Symbol

Description

a a y b F y1 F y2 I z l m r V β δ s

distance from front wheels to c.g. lateral acceleration distance from rear wheels to c.g. front axle lateral force rear axle lateral force yaw moment of inertia wheelbase mass of the vehicle vehicle yaw rate vehicle speed vehicle sideslip angle Wheel steer angle

sI z ( s ) = aF y 1 ( s )− bF y 2 ( s )

aC α 1 ( s ) – b C α 2 ( s ) * * mV s + C α 1 ( s ) + C α 2 ( s ) mV + ------------------------------------------- β ( s ) V *

2

C α 1 ( s ) δ s ( s )

=

2

*

r(s )

aC α 1 ( s ) δ s ( s ) *

.

(15)

Using the first order lag of the lateral tire force model that * is previously formulated in equation (10), C α i ( s ) can be written as C α i * C α i ( s ) = --------------- , i=1, 2 1 + τ i s

(11)

(12)

where ξ i represents the change in tire slip angle induced by the suspension and the steering compliance. Also, δ s ( s ) is the wheel steer angle, which can be obtained by dividing the steering wheel angle by the overall steering gear ratio. The actual bicycle model of Figure 2(a) can be simplified by the equivalent vehicle model of Figure 2(b). In order to this, pseudo slip angles are defined by

(16)

where τ i represents the time constant of the first order lag model. Also, C αi represents the equivalent cornering stiffness under steady-state conditions. The complex cornering stiffness is represented by a steady-state equivalent cornering stiffness and a time lag constant. Equation (16) can be simplified with the help of a Taylor expansion series around s=0, such as C α i * C α i ( s ) = --------------- ≈ C α i ( 1 − τ i s ) . 1 + τ i s

(17)

By substituting equation (17) into (15), the characteristic equation of the vehicle can be obtained; it is A2 2 A 1 + A 1 ′ s + ------------------- s + ------------------- =0 ′ + A 0 A 0 A 0 + A 0 ′

(18)

where A 0 = mV I z 2

l C α 1 C α 2 τ 1 τ 2 2 2 A 0 ′ = ------------------------------ − m ( a C α 1 τ 1 + b C α 2 τ 2 ) V

− I z ( C α 1 τ 1 + C α 2 τ 2 ) A 1 = m ( a C α 1 + b C α 2 ) + I z ( C α 1 + C α 2 ) 2

2

2

2

l l A 1 ′ = − ⎛ --- C α 2 – am V ⎞ C α 1 τ 1 − ⎛ --- C α 1 – bm V ⎞ C α 2 τ 2 ⎝ V ⎠ ⎝ V ⎠

a α ( s ) = δ s ( s )− ---- r ( s )− β ( s ) V * 1

b α ( s ) = ---- r ( s )− β ( s ) V

*

*

a α 1 ( s ) = δ s ( s ) + ξ 1 − ---- r ( s ) −β ( s ) V

* 2

*

a C α 1 ( s ) + b C α 2 ( s ) * * aC α 1 ( s ) – b C α 2 ( s ) I z s + ---------------------------------------------V

where the vehicle model parameters are explained in Table 1. For the purpose of investigating the influence of the transient tire characteristics on vehicle maneuverability, the equation of motion of vehicle is expressed as a Laplace transform. In a small slip angle area, the tire slip angle is given by

b α 2 ( s ) = ξ 2 + ---- r ( s ) −β ( s ) V

689

2

(13)

l A 2 = − m V( a C α 1 – bC α 2 ) + --- C α 1 C α 2 . V

Using equation (13), the lateral force in a small slip angle area can by obtained by

Equation (18) can be expressed with the dynamic characteristic parameters of the system as

F i ( s ) = C α i ( s ) α i ( s ) , i=1, 2

s +2 ζω n s + ω n =0

*

*

2

(14)

where C α i ( s ) is the complex cornering stiffness and i=1, 2 (which refers to the front and rear, respectively). Both ξ i and the tire characteristic in the vehicle model can be expressed by the complex cornering stiffness. The mathematical definition of the complex cornering stiffness is the transfer function of the lateral force against the pseudo slip angle. By substituting equation (14) for the lateral force of the tires in equation (11), the equivalent vehicle model can be obtain by *

2

(19)

where A2 2 2 = C w ω n 0 ω n = ------------------ A 0 + A 0 ′

(19a)

A1 + A1 ′ ζω n = --------------------------- = C ζ C w ζ 0 ω n 0 2 ( A0 + A0 ′ )

(19b)

l C α 1 C α 2 ⎧ 1 m ⎛ b a ⎫ -----2 + ---2- -------- – --------⎞ ⎬ ω = -------------------⎨ mI z ⎩ V l ⎝ C α 1 C α 2⎠ ⎭

(19c)

2

2 n0

690

J. KIM

2

2

1 C α 1 + C α 2 a C α 1 + b C α 2⎞ + --------------------------------- . ζ 0 ω n 0 = ------- ⎛ -------------------⎠ 2 V ⎝ m I z

(19d)

1 C w = -------------------------------------------------------------------------------------------------------------------------------------------2 2 2 1 C α 1 τ 1 + C α 2 τ 2 a C α 1 τ 1 + b C α 2 τ 2 l C α 1 C α 2 τ 1 τ 2⎞ 1+ --- ⎛ – -------------------------------– ------------------------------------------- + -----------------------------⎠ V ⎝ m I z mV I z

( l C α 2 – ma V ) C α 1 τ 1 + ( l C α 1 + mb V ) C α 2 τ 2 ----------------------------------------------------------C ζ =1 − --------------------------------------------2 2 V [ ( I z + a m ) C α 1 + ( I z + b m ) C α 2 ] 2

2

2

2

From equation (19), the natural frequency and damping coefficient of the plane motion can be obtained. Figure 3 shows an example of the calculation of ω n and ζω n (m=1239 kg, a=1.06 m, b=1.63 m, V =100 kph, I z =2140.7 kg·m2). It can be observed that the time lag of the lateral force affects ω n and ζω n remarkably. 2.3. Identification of the Axle Force Characteristics In order to identify the lateral force characteristics, the motion of two DOF is considered. By simply solving for F y1 and F y2 in equation (11), the axle lateral forces can be directly obtained, where the result is · ma y b + I z r (20) F y 1 = ---------------------l · ma y a + I z r (21) F y 2 = ---------------------l · where a y = V ( β + r ) is the lateral acceleration, and l =a+b is the wheelbase. The pseudo slip angles related to the lateral forces can be obtained from equation (13). For this, three data points ( r , β , δ s) are directly measured by the sensors. If the vehicle motion in the horizontal road plane is determined accurately, the lateral forces and slip angles are determined as well. The lateral forces are plotted as functions of the pseudo slip angles in Figure 4, where the non-linear form of the pneumatic tire behavior is recognized. The lateral forces and pseudo slip angles are applied to the most commonly

Figure 3. Effects of τ 1 and τ 2 on the dynamic characteristics of the vehicle.

Figure 4. Lateral force generation at the front and rear axle. used empirical model, the so-called “Magic formula” (Bakker and Pacejka, 1989). For clarity, the simplest form has been adopted in the following: F yi = − K 3 i sin[ K 2 i tan { K 1 i α i − –1

K 4 i K 1 i α i − tan

–1

( K 1 i α i ) )}]

(22)

where K 3i represents the peak value, if present, and the product K 1 K i 2 K i 3i equals the gradient of the function at zero slip angle, which is the equivalent cornering stiffness, C αi , in equation (17). From C α1 and C α2 , the understeer coefficient can be obtained by 1 W W K us = --- ⎛ -------1- – --------2 ⎞ ⎝ g C α 1 C α 2 ⎠

(23)

where W 1 and W 2 are the normal load on the front and rear axle (Wong, 2001). As explained in Figure 1, the tires do not instantly produce lateral forces as a result of the slip angle delay. This phenomenon is mathematically represented using the time constant τ in equation (8). Thus, τ is the time required to attain 63.2% of the final steady-state value for the step response. In the case of rolling tire, the tire has to roll some distance before the steady-state lateral force is fully developed, where this distance is called the relaxation length. By multiplying τ in equation (9) by the speed V x, the relaxation length can easily be obtained. Loeb et al . (1990) report that the length a value of 1/2 up to 1 full revolution of the tire (0.9 up to 1.8 m with a 0.3 m rolling radius). Figure 4 shows the cross plot of the axle lateral force versus slip angle. A counterclockwise hysteresis loop can be observed, which represents the energy accumulation between the slip angle and lateral force. This phenomenon is closely related to the time delay between the slip angle and lateral force. If a clockwise hysteresis loop occurs, it is usually related to the energy loss between the slip angle and lateral force. Since the Magic formula of equation (22) does not include the transient characteristics of the lateral force,

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS

691

Table 2. Test conditions and subjective assessment results (Tire size: 205/55R16, Rim: 6.5J). Test car

Front driving (Curb+2 persons)

Figure 5. Rear axle force (a) before reducing hysteresis and (b) after reducing hysteresis for τ 2=0.02 seconds. modeling of this effect is required. Firstly, transient parameters (such as the time delay) are identified by reduction of the hysteresis between lateral force and slip angle in Figure 4. Then, the function parameters (such as the Magic formula coefficient) of equation (22) are identified. To reduce the hysteresis, a first order filtering of equation (8) is used. The slip angle data in Figure 4 are driven through this filter, and then plotted against the lateral axle force data. If the right time lag constant τ is determined, the hysteresis area will be minimized. For a specific value of τ that minimizes the hysteresis area, an iterative routine based on a golden section search is used (Belegundu and Chandrupatla, 1999). Using an equivalent cornering stiffness that is estimated by the Magic formula and a time lag constant determined by a first order filter, the complex cornering stiffness of equation (17) can be determined. Figure 5(a) shows the rear axle force generation during a high speed maneuver, while Figure 5(b) shows the results of the hysteresis reduction, resulting in a time lag constant of 0.02 second. It should be noted that the hysteresis area is significantly reduced, which implies that the axle lateral forces can be approximated with a single-valued function. The coefficients of the Magic formula are determined by fitting the reduced hysteresis area data of Figure 5(b) with equation (22).

Test

Test conditions

Rating

ID

Inflation of pressure (kPa)

Stability

Front

Rear

T1

103.4

206.8

9.75

T2

206.8

206.8

7

T3

206.8

137.9

3.75

T4

206.8

103.4

3.6

T1, the inflation pressure of the front tires is set to 103.4 kPa, while the inflation pressure of the rear tires is set to 206.8 kPa. This makes the car understeer more. On the other hand, in test T4, the inflation pressure of the rear tires is set to 103.4 kPa, while the inflation pressure of the front tires is set to 206.8 kPa. This makes the car oversteer more. With regard to the handling performance of car, the stability of the yaw and lateral motions of the car is subjectively evaluated (Kim et al ., 2006). Among the four test conditions, there are big differences in the stability of the car, as shown in Table 2. As a decimal digit rating, larger values mean better stability. When the evaluator feels that handling is “very poor”, it is awarded about 2 points. If the evaluator feels that handling is “excellent”, 10 points are awarded. In the results, the smaller the inflation pressure of the front tires, the better the stability. The reason for this can be inferred from the fact that the tendency to understeer

3. APPLICATION OF THE VEHICLE HANDLING ANALYSIS In this section, the proposed methods are applied to the vehicle handling analysis. Both a subjective assessment of the driver and objective measurement are performed for four sets of tests, as shown in Table 2. The car is frontdriving, and has a specified distribution of the front (60%) and rear (40%) weight. The tire size is 205/55R17V, and the inflation pressures of the tires are varied. The original inflation pressure of the tires for the car is 206.8 kPa. In test

Figure 6. Measured signals during lane change maneuvers.

692

J. KIM

Figure 7. Estimation results of the (a) equivalent cornering stiffness and (b) time lag constant: , front; , rear. is increased due to a reduction of the inflation pressure of the front tires. All of the tests are carried out on dry asphalt. In the measurement, the signals are pre-filtered with a 5 Hz cutoff frequency, while the data acquisition is performed at a sampling rate of 100 Hz. In order to get enough data to describe the axle cornering characteristics, a lane change maneuver is chosen. The measurement is repeated at a constant speed of 33.3 m/s. Figure 6 shows measurement signals obtained during lane change maneuvers. Using these signals, all of the dynamic characteristic parameters are obtained. Figure 7(a) shows the results of the estimation of the equivalent cornering stiffness, C α1 and C α2 , by adapting the Magic formula of equation (22). All of the values of C α1 are smaller than C α 2, which means that understeering occurred for all test sets. Also, C α2 decreases as the inflation of the rear tires is decreased. Figure 7(b) shows the results of the estimation of the time lag constant, τ 1 and τ 2 obtained by adapting a first order filter. The results show that τ 1 is larger than τ 2. Also, both τ 1 and τ 2 increases as the inflation pressure of the tire is decreased. Figure 8 shows the vehicle dynamic characteristics, ω n and ζω n , obtained by substituting the estimated equivalent cornering stiffness and time lag constant into equations (19a) and (19b), respectively. In considering the time lag constant, both ω n and ζω n must be changed considerably. Figure 9 shows the correlation between dynamic characteristic parameters and the stability performance of the car. C α1 and C α2 show a low correlation, while K us shows a high correlation. τ 1 and τ 2 show the highest correlation of all parameters. τ 1 has a positive correlation while τ 2 has a negative correlation. This means that the stability of car can

Figure 8. Numerical results of the vehicle dynamic parameters.

Figure 9. Correlation between the stability of the car and the vehicle dynamic parameter.

Figure 10. Stability of the vehicle response. be improved by setting τ 1 larger than τ 2, which increases the damping coefficient, ζω n. Using ω n from equation (19a)

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS

instead of ω n0 from equation (19c), the correlation value can be improved from 0.80 to 0.84. Similarly, the correlation with the stability of the car can be improved from 0.72 to 0.89 using ζω n from equation (19b) instead of ζω n0 from equation (19d). These two results show that the stability of car can be analyzed more accurately by considering both the equivalent cornering stiffness and time lag constant when evaluating the dynamic characteristic parameter of car. Figure 10 shows the measured vehicle responses, which are the cross plot between the yaw rate and sideslip angle of the car (Wu et al ., 2007). It can be observed that T1 is more stable than T4. This kind of characteristic can be easily identified and effectively quantified by the formulated dynamic characteristic parameters.

4. CONCLUSION A simple 2 DOF vehicle model combined with a transient representation of the axle data is shown in order to analyze the handling performance of the car. From the model, the vehicle dynamic parameters are formulated, and these are estimated from the actual vehicle responses during lane change maneuvers. In order to validate the suggested method, a correlation study between the subjective stability, which is one of the handling performance capacities of the car, and the estimated vehicle dynamics parameter is performed. The following points summarize the conclusions of this study. (1) The bicycle model, which is the simplest available in describing the handling properties of car, is shown to give reliable results. (2) The effect of the time lag constant of the axle lateral force on the stability of the car is examined, which shows the time lag constant plays an important role in the handling performance of car. (3) By considering the effect of the time lag constant on the vehicle dynamic parameters, the stability of car can be analyzed more accurately. By using the proposed analysis technique, it will be possible to analyze the handling performance of the car

693

from the experimental or simulation data more effectively. This will allow for a reduction of the time and money spent on shortening the development time of new vehicles and tires, which is very critical in today’s market.

−This research is supported by the

ACKNOWLEDGEMENT

R&D centre of HANKOOK Tyre Co., Ltd., under the KONTROLTechnology program. The author also thanks the Vehicle Dynamics Research Team for providing field test data and valuable suggestions.

REFERENCES Bakker, E. and Pacejka, H. B. (1989). A new tyre models with an application in vehicle dynamics studies. SAE Paper No. 890087. Belegundu, A. D. and Chandrupatla, T. R. (1999). Optimization Concepts and Applications in Engineering. Prentice-Hall. New Jersey. Kim, J., Kim, Y. T. and Yoon, Y. S. (2006). Analysis of transient maneuvers for objectifying evaluation of vehicle stability. Trans. Korean Society of Automotive Engineers 14, 1, 167−175. Kusaka, K., Higuchi, M., Shibusawa, K., Muraoka, N. and Tsukagoshi, M. (1995). Transient tire model for handling analysis and prediction. Proc.IPC-8, 2, 75−80. Loeb, J. S., Guenther, D. A. and Chen, H. F. (1990). Lateral stiffness, cornering stiffness and relaxation length of the pneumatic tire. SAE Paper No. 900129. Mimuro, T., Ohsaki, M., Yasunaga, H. and Satoh, K. (1990). Four parameter evaluation method of lateral transient response. SAE Paper No. 901734, 51−60. Sakai, H. and Satoh, Y. (1995). A theoretical study of vehicle dynamics behavior with complex cornering stiffness. Proc. IPC-8, 2, 63−68. Wong, J. Y. (2001). Theory of Ground Vehicles. John Wiley & Sons. New York. Wu, J., Tang, H., Li, S. and Zheng, S. B. (2007). Integrated control system design of active front wheel steering and four wheel torque to improve vehicle handling and stability. Int. J. Automotive Technology 8, 3, 299−308.

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Copyright Copyri ght © 200 2008 8 KS KSAE AE 1229−9138/2008/043−06

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS J. KIM* Department of Vehicle Dynamics Research Team, Hankook Tire Co., LTD., R&D Center, 23-1 Jang-dong, Yuseong-gu, Daejeon 305-725, Korea (Received 28 December 2007; Revised 23 June 2008)

−In this article, the analysis methods for vehicle handling performance are studied. Using simple models, dynamic characteristic parameters such as yaw, natural frequency, and the damping coefficient of a vehicle can be theoretically formulated. Here, the vehicle is simplified by a bicycle (single-track) model, and the tire is modeled by an equivalent cornering stiffness and first order lag. From the experimental road data, the tire model parameters (equivalent cornering stiffness and time lag constant) are extracted. These parameters are then inserted into the theoretically formulated equations of dynamic characteristic parameters. For the purpose of validating the efficiency of the suggested methods, experimental road tests (where the cars have different handling performances) are performed. The results show that vehicle handling performance can be sufficien sufficiently tly represent represented ed by the suggeste suggested d dynamic dynamic characte characteristic ristic paramete parameters. rs. So, it is concluded concluded that that the proposed proposed method method has practical use for the development of new cars or for the comparison of similar cars since the evaluations of the vehicle handling performance can be efficiently determined by the suggested dynamic characteristic parameters. ABSTRACT

KEY WORDS :

Bicycle model, First order lag tire model, Equivalent cornering stiffness, Damping coefficient

1. INTRODUCTION

The purpose of this article is to study the possibility of using vehicle lateral and yaw dynamics for the analysis of vehicle handling performance (i.e., based on the simple bicyclee model). bicycl model). For this, dynamic dynamic charact characteristic eristic paramete parameters rs such as yaw natural frequency and the damping coefficient of car are theoretically formulated using the approaches of Sakai and Satoh (1995). Specifically, the vehicle is simplified by bicycle (single-track) model, and actual tire and suspension characteristics are merged into one force generator per axle that is modeled by an equivalent cornering stiffness and first order lag. In this paper, new approaches for obtaining dynamic characteristic parameters from experimental road test data are introduced. For this, the equivalent cornering stiffness and first order lag are extracted from road test data, which fundamentally allows more accurate compared with those methods that use indoor tire behavior machines (such as Flat-trac). The reason is that the tire has a smaller inertia, and its behavior is influenced by road roughness, the antilock braking system (ABS) operation, tire pressure and tread variations, carcass radial non-uniformities, as well as vehicle roll, pitch, and vertical motions. Therefore, tire motions tend to be a lot more oscillatory and contain higher frequency components. Separating the effects of these disturbances from those of tire and road interactions is difficult. The extracted equivalent cornering stiffness and first order lag parameter are then inserted into the theoretically formulated equations to determine dynamic characteristic parameters.

Handling analysis is one of the most attractive themes in automobile engineering. Here, many suspension and tire designers want to know how their design parameters contribute to handling characteristics, and test drivers want to know what the sensations that they feel are derived from. A simplified bicycle model, which is also known as the single-track model, is convenient for the theoretical analysis of vehicle dynamics. Due to this majority of single-track model, automobile engineers can easily understand the handling characteristics of a car. In the literature, several different methods have been introduced. For example, Sakai and Satoh (1995) used complex cornering stiffness, which is the same as the equivalent cornering stiffness when the phase lead time is taken into account. From this approach, the natural frequency and damping coefficient of the plane motion are theoretically formulated. Kusaka et al . (1995) examined a simpler bicycle model for the purpose of investigating the influence of tire transient characteristics on vehicle handling properties. Mimuro et al. (1990) developed the four parameter evaluation methods of lateral transient response that also come from the bicycle model. This technique allows a rapid comparison of handling characteristics of different tire and vehicle combinations by condensing the response data to be displaye displayed d into a rhombus form. [email protected] hankookti ooktire.co re.com m *Corresponding author. e-mail: [email protected] 687

688

J. KIM

In order to validate the efficiency of the suggested methods, experimental road tests for cars having different handling performances are performed. The results show that vehicle handling performance can be sufficiently represented by the suggested dynamic characteristic parameters.

2. LATERAL VEHICLE DYNAMICS In order to formulate the dynamic characteristic parameters theoretically, a first order lag tire model and simple bicycle vehicle model are considered. 2.1. First Order Lag Tire Model The lateral force generated at the contact point between the road and tire is transferred to the wheel. This mechanism can be modeled with an effective spring and damper, as shown in Figure 1. The contact patch (including tread, carcass, and belt of the tire) is connected to the wheel by an effective spring (k y) and damper (d y). Here, the axis η (c) is fixed at the center of wheel to describe the lateral motion of the contact patch. The equation of motion of the contact patch can be written as ·· – θ · 2 η ) + d η · + k η = F (1) m ( η ( c)

(c)

s

(c)

( c)

y

y

( c)

y

(c )

y

(c)

y

(c )

( c)

· η ( c )⎞ F y = C α ⎛ α – ------⎝ V x ⎠

(4)

where C α is the cornering stiffness of the tire. By combining equations (1) and (4), the equation of motion of the tire contact patch can be obtained by ·· – θ · 2 η ) + d η · + k η = C ( α – η · / V ) . (5) m ( η (c)

(c)

s

(c )

y

(c)

y

(c)

α

( c)

x

In general, m(c) and d y are very small compared to k y. This allows equation (5) to be simplified as · (6) k y η ( c) = C α ( α – η ( c ) / V x ) = F y . From equation (6), the following relationship can be obtained. · · y = k η F y ( c ) .

(7)

By combining equations (4) and (7), the first order lag tire model that considers the lateral force and steady-state slip angle can be obtained by · y + F = C α τ F y α

(8)

where

y

· where m(c) is the mass of the contact patch, θ s is the steering rate of the wheel, and F y is the lateral force at the tire contact patch. The lateral force at the wheel, F y(w) can be expressed as · + k η = F − m ( η ·· – θ · 2 η ) . (2) F = d η y ( w )

small slip angle area, F y can be expressed as

s

(c)

Equation (2) shows the phase delay between the lateral force of contact patch and the lateral force of wheel, which · is the comes from the inertia of the tire contact patch. η (c ) lateral velocity difference between the wheel and tire contact patch. This represents the direction between the wheel and tire contact patch. The tire slip angle, which is defined as the angle between the wheel heading direction (ξ -axis) and the traveling direction of the tire contact patch, can be written as · η (c) ------− (3) α = α V x · where α is steady-state slip angle when η is zero. In a

C α -. τ = --------k y V x

(9)

In equation (9), τ is the time constant of the first order lag tire model. Also, from equation (8), the transfer function of the lateral force of the tire can be obtained by F y ( s ) C α ------------ = -------------. α ( s ) 1 + τ s

(10)

2.2. Vehicle Model An actual vehicle model with a single track, which describes the lateral and yaw dynamics of a 2-axle and 1-rigid body ground vehicle, is shown in Figure 2(a). The equation of motion for this model can be expressed by mV [ s β ( s ) + r ( s ) ] = F y 1 ( s ) + F y 2 ( s )

(c)

Figure 1. Transient characteristics of the tire lateral force.

Figure 2. Two DOF (degree of freedom) of the (a) actual and (b) equivalent bicycle model.

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS

Table 1. Vehicle model parameters. Symbol

Description

a a y b F y1 F y2 I z l m r V β δ s

distance from front wheels to c.g. lateral acceleration distance from rear wheels to c.g. front axle lateral force rear axle lateral force yaw moment of inertia wheelbase mass of the vehicle vehicle yaw rate vehicle speed vehicle sideslip angle Wheel steer angle

sI z ( s ) = aF y 1 ( s )− bF y 2 ( s )

aC α 1 ( s ) – b C α 2 ( s ) * * mV s + C α 1 ( s ) + C α 2 ( s ) mV + ------------------------------------------- β ( s ) V *

2

C α 1 ( s ) δ s ( s )

=

2

*

r(s )

aC α 1 ( s ) δ s ( s ) *

.

(15)

Using the first order lag of the lateral tire force model that * is previously formulated in equation (10), C α i ( s ) can be written as C α i * C α i ( s ) = --------------- , i=1, 2 1 + τ i s

(11)

(12)

where ξ i represents the change in tire slip angle induced by the suspension and the steering compliance. Also, δ s ( s ) is the wheel steer angle, which can be obtained by dividing the steering wheel angle by the overall steering gear ratio. The actual bicycle model of Figure 2(a) can be simplified by the equivalent vehicle model of Figure 2(b). In order to this, pseudo slip angles are defined by

(16)

where τ i represents the time constant of the first order lag model. Also, C αi represents the equivalent cornering stiffness under steady-state conditions. The complex cornering stiffness is represented by a steady-state equivalent cornering stiffness and a time lag constant. Equation (16) can be simplified with the help of a Taylor expansion series around s=0, such as C α i * C α i ( s ) = --------------- ≈ C α i ( 1 − τ i s ) . 1 + τ i s

(17)

By substituting equation (17) into (15), the characteristic equation of the vehicle can be obtained; it is A2 2 A 1 + A 1 ′ s + ------------------- s + ------------------- =0 ′ + A 0 A 0 A 0 + A 0 ′

(18)

where A 0 = mV I z 2

l C α 1 C α 2 τ 1 τ 2 2 2 A 0 ′ = ------------------------------ − m ( a C α 1 τ 1 + b C α 2 τ 2 ) V

− I z ( C α 1 τ 1 + C α 2 τ 2 ) A 1 = m ( a C α 1 + b C α 2 ) + I z ( C α 1 + C α 2 ) 2

2

2

2

l l A 1 ′ = − ⎛ --- C α 2 – am V ⎞ C α 1 τ 1 − ⎛ --- C α 1 – bm V ⎞ C α 2 τ 2 ⎝ V ⎠ ⎝ V ⎠

a α ( s ) = δ s ( s )− ---- r ( s )− β ( s ) V * 1

b α ( s ) = ---- r ( s )− β ( s ) V

*

*

a α 1 ( s ) = δ s ( s ) + ξ 1 − ---- r ( s ) −β ( s ) V

* 2

*

a C α 1 ( s ) + b C α 2 ( s ) * * aC α 1 ( s ) – b C α 2 ( s ) I z s + ---------------------------------------------V

where the vehicle model parameters are explained in Table 1. For the purpose of investigating the influence of the transient tire characteristics on vehicle maneuverability, the equation of motion of vehicle is expressed as a Laplace transform. In a small slip angle area, the tire slip angle is given by

b α 2 ( s ) = ξ 2 + ---- r ( s ) −β ( s ) V

689

2

(13)

l A 2 = − m V( a C α 1 – bC α 2 ) + --- C α 1 C α 2 . V

Using equation (13), the lateral force in a small slip angle area can by obtained by

Equation (18) can be expressed with the dynamic characteristic parameters of the system as

F i ( s ) = C α i ( s ) α i ( s ) , i=1, 2

s +2 ζω n s + ω n =0

*

*

2

(14)

where C α i ( s ) is the complex cornering stiffness and i=1, 2 (which refers to the front and rear, respectively). Both ξ i and the tire characteristic in the vehicle model can be expressed by the complex cornering stiffness. The mathematical definition of the complex cornering stiffness is the transfer function of the lateral force against the pseudo slip angle. By substituting equation (14) for the lateral force of the tires in equation (11), the equivalent vehicle model can be obtain by *

2

(19)

where A2 2 2 = C w ω n 0 ω n = ------------------ A 0 + A 0 ′

(19a)

A1 + A1 ′ ζω n = --------------------------- = C ζ C w ζ 0 ω n 0 2 ( A0 + A0 ′ )

(19b)

l C α 1 C α 2 ⎧ 1 m ⎛ b a ⎫ -----2 + ---2- -------- – --------⎞ ⎬ ω = -------------------⎨ mI z ⎩ V l ⎝ C α 1 C α 2⎠ ⎭

(19c)

2

2 n0

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J. KIM

2

2

1 C α 1 + C α 2 a C α 1 + b C α 2⎞ + --------------------------------- . ζ 0 ω n 0 = ------- ⎛ -------------------⎠ 2 V ⎝ m I z

(19d)

1 C w = -------------------------------------------------------------------------------------------------------------------------------------------2 2 2 1 C α 1 τ 1 + C α 2 τ 2 a C α 1 τ 1 + b C α 2 τ 2 l C α 1 C α 2 τ 1 τ 2⎞ 1+ --- ⎛ – -------------------------------– ------------------------------------------- + -----------------------------⎠ V ⎝ m I z mV I z

( l C α 2 – ma V ) C α 1 τ 1 + ( l C α 1 + mb V ) C α 2 τ 2 ----------------------------------------------------------C ζ =1 − --------------------------------------------2 2 V [ ( I z + a m ) C α 1 + ( I z + b m ) C α 2 ] 2

2

2

2

From equation (19), the natural frequency and damping coefficient of the plane motion can be obtained. Figure 3 shows an example of the calculation of ω n and ζω n (m=1239 kg, a=1.06 m, b=1.63 m, V =100 kph, I z =2140.7 kg·m2). It can be observed that the time lag of the lateral force affects ω n and ζω n remarkably. 2.3. Identification of the Axle Force Characteristics In order to identify the lateral force characteristics, the motion of two DOF is considered. By simply solving for F y1 and F y2 in equation (11), the axle lateral forces can be directly obtained, where the result is · ma y b + I z r (20) F y 1 = ---------------------l · ma y a + I z r (21) F y 2 = ---------------------l · where a y = V ( β + r ) is the lateral acceleration, and l =a+b is the wheelbase. The pseudo slip angles related to the lateral forces can be obtained from equation (13). For this, three data points ( r , β , δ s) are directly measured by the sensors. If the vehicle motion in the horizontal road plane is determined accurately, the lateral forces and slip angles are determined as well. The lateral forces are plotted as functions of the pseudo slip angles in Figure 4, where the non-linear form of the pneumatic tire behavior is recognized. The lateral forces and pseudo slip angles are applied to the most commonly

Figure 3. Effects of τ 1 and τ 2 on the dynamic characteristics of the vehicle.

Figure 4. Lateral force generation at the front and rear axle. used empirical model, the so-called “Magic formula” (Bakker and Pacejka, 1989). For clarity, the simplest form has been adopted in the following: F yi = − K 3 i sin[ K 2 i tan { K 1 i α i − –1

K 4 i K 1 i α i − tan

–1

( K 1 i α i ) )}]

(22)

where K 3i represents the peak value, if present, and the product K 1 K i 2 K i 3i equals the gradient of the function at zero slip angle, which is the equivalent cornering stiffness, C αi , in equation (17). From C α1 and C α2 , the understeer coefficient can be obtained by 1 W W K us = --- ⎛ -------1- – --------2 ⎞ ⎝ g C α 1 C α 2 ⎠

(23)

where W 1 and W 2 are the normal load on the front and rear axle (Wong, 2001). As explained in Figure 1, the tires do not instantly produce lateral forces as a result of the slip angle delay. This phenomenon is mathematically represented using the time constant τ in equation (8). Thus, τ is the time required to attain 63.2% of the final steady-state value for the step response. In the case of rolling tire, the tire has to roll some distance before the steady-state lateral force is fully developed, where this distance is called the relaxation length. By multiplying τ in equation (9) by the speed V x, the relaxation length can easily be obtained. Loeb et al . (1990) report that the length a value of 1/2 up to 1 full revolution of the tire (0.9 up to 1.8 m with a 0.3 m rolling radius). Figure 4 shows the cross plot of the axle lateral force versus slip angle. A counterclockwise hysteresis loop can be observed, which represents the energy accumulation between the slip angle and lateral force. This phenomenon is closely related to the time delay between the slip angle and lateral force. If a clockwise hysteresis loop occurs, it is usually related to the energy loss between the slip angle and lateral force. Since the Magic formula of equation (22) does not include the transient characteristics of the lateral force,

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS

691

Table 2. Test conditions and subjective assessment results (Tire size: 205/55R16, Rim: 6.5J). Test car

Front driving (Curb+2 persons)

Figure 5. Rear axle force (a) before reducing hysteresis and (b) after reducing hysteresis for τ 2=0.02 seconds. modeling of this effect is required. Firstly, transient parameters (such as the time delay) are identified by reduction of the hysteresis between lateral force and slip angle in Figure 4. Then, the function parameters (such as the Magic formula coefficient) of equation (22) are identified. To reduce the hysteresis, a first order filtering of equation (8) is used. The slip angle data in Figure 4 are driven through this filter, and then plotted against the lateral axle force data. If the right time lag constant τ is determined, the hysteresis area will be minimized. For a specific value of τ that minimizes the hysteresis area, an iterative routine based on a golden section search is used (Belegundu and Chandrupatla, 1999). Using an equivalent cornering stiffness that is estimated by the Magic formula and a time lag constant determined by a first order filter, the complex cornering stiffness of equation (17) can be determined. Figure 5(a) shows the rear axle force generation during a high speed maneuver, while Figure 5(b) shows the results of the hysteresis reduction, resulting in a time lag constant of 0.02 second. It should be noted that the hysteresis area is significantly reduced, which implies that the axle lateral forces can be approximated with a single-valued function. The coefficients of the Magic formula are determined by fitting the reduced hysteresis area data of Figure 5(b) with equation (22).

Test

Test conditions

Rating

ID

Inflation of pressure (kPa)

Stability

Front

Rear

T1

103.4

206.8

9.75

T2

206.8

206.8

7

T3

206.8

137.9

3.75

T4

206.8

103.4

3.6

T1, the inflation pressure of the front tires is set to 103.4 kPa, while the inflation pressure of the rear tires is set to 206.8 kPa. This makes the car understeer more. On the other hand, in test T4, the inflation pressure of the rear tires is set to 103.4 kPa, while the inflation pressure of the front tires is set to 206.8 kPa. This makes the car oversteer more. With regard to the handling performance of car, the stability of the yaw and lateral motions of the car is subjectively evaluated (Kim et al ., 2006). Among the four test conditions, there are big differences in the stability of the car, as shown in Table 2. As a decimal digit rating, larger values mean better stability. When the evaluator feels that handling is “very poor”, it is awarded about 2 points. If the evaluator feels that handling is “excellent”, 10 points are awarded. In the results, the smaller the inflation pressure of the front tires, the better the stability. The reason for this can be inferred from the fact that the tendency to understeer

3. APPLICATION OF THE VEHICLE HANDLING ANALYSIS In this section, the proposed methods are applied to the vehicle handling analysis. Both a subjective assessment of the driver and objective measurement are performed for four sets of tests, as shown in Table 2. The car is frontdriving, and has a specified distribution of the front (60%) and rear (40%) weight. The tire size is 205/55R17V, and the inflation pressures of the tires are varied. The original inflation pressure of the tires for the car is 206.8 kPa. In test

Figure 6. Measured signals during lane change maneuvers.

692

J. KIM

Figure 7. Estimation results of the (a) equivalent cornering stiffness and (b) time lag constant: , front; , rear. is increased due to a reduction of the inflation pressure of the front tires. All of the tests are carried out on dry asphalt. In the measurement, the signals are pre-filtered with a 5 Hz cutoff frequency, while the data acquisition is performed at a sampling rate of 100 Hz. In order to get enough data to describe the axle cornering characteristics, a lane change maneuver is chosen. The measurement is repeated at a constant speed of 33.3 m/s. Figure 6 shows measurement signals obtained during lane change maneuvers. Using these signals, all of the dynamic characteristic parameters are obtained. Figure 7(a) shows the results of the estimation of the equivalent cornering stiffness, C α1 and C α2 , by adapting the Magic formula of equation (22). All of the values of C α1 are smaller than C α 2, which means that understeering occurred for all test sets. Also, C α2 decreases as the inflation of the rear tires is decreased. Figure 7(b) shows the results of the estimation of the time lag constant, τ 1 and τ 2 obtained by adapting a first order filter. The results show that τ 1 is larger than τ 2. Also, both τ 1 and τ 2 increases as the inflation pressure of the tire is decreased. Figure 8 shows the vehicle dynamic characteristics, ω n and ζω n , obtained by substituting the estimated equivalent cornering stiffness and time lag constant into equations (19a) and (19b), respectively. In considering the time lag constant, both ω n and ζω n must be changed considerably. Figure 9 shows the correlation between dynamic characteristic parameters and the stability performance of the car. C α1 and C α2 show a low correlation, while K us shows a high correlation. τ 1 and τ 2 show the highest correlation of all parameters. τ 1 has a positive correlation while τ 2 has a negative correlation. This means that the stability of car can

Figure 8. Numerical results of the vehicle dynamic parameters.

Figure 9. Correlation between the stability of the car and the vehicle dynamic parameter.

Figure 10. Stability of the vehicle response. be improved by setting τ 1 larger than τ 2, which increases the damping coefficient, ζω n. Using ω n from equation (19a)

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS

instead of ω n0 from equation (19c), the correlation value can be improved from 0.80 to 0.84. Similarly, the correlation with the stability of the car can be improved from 0.72 to 0.89 using ζω n from equation (19b) instead of ζω n0 from equation (19d). These two results show that the stability of car can be analyzed more accurately by considering both the equivalent cornering stiffness and time lag constant when evaluating the dynamic characteristic parameter of car. Figure 10 shows the measured vehicle responses, which are the cross plot between the yaw rate and sideslip angle of the car (Wu et al ., 2007). It can be observed that T1 is more stable than T4. This kind of characteristic can be easily identified and effectively quantified by the formulated dynamic characteristic parameters.

4. CONCLUSION A simple 2 DOF vehicle model combined with a transient representation of the axle data is shown in order to analyze the handling performance of the car. From the model, the vehicle dynamic parameters are formulated, and these are estimated from the actual vehicle responses during lane change maneuvers. In order to validate the suggested method, a correlation study between the subjective stability, which is one of the handling performance capacities of the car, and the estimated vehicle dynamics parameter is performed. The following points summarize the conclusions of this study. (1) The bicycle model, which is the simplest available in describing the handling properties of car, is shown to give reliable results. (2) The effect of the time lag constant of the axle lateral force on the stability of the car is examined, which shows the time lag constant plays an important role in the handling performance of car. (3) By considering the effect of the time lag constant on the vehicle dynamic parameters, the stability of car can be analyzed more accurately. By using the proposed analysis technique, it will be possible to analyze the handling performance of the car

693

from the experimental or simulation data more effectively. This will allow for a reduction of the time and money spent on shortening the development time of new vehicles and tires, which is very critical in today’s market.

−This research is supported by the

ACKNOWLEDGEMENT

R&D centre of HANKOOK Tyre Co., Ltd., under the KONTROLTechnology program. The author also thanks the Vehicle Dynamics Research Team for providing field test data and valuable suggestions.

REFERENCES Bakker, E. and Pacejka, H. B. (1989). A new tyre models with an application in vehicle dynamics studies. SAE Paper No. 890087. Belegundu, A. D. and Chandrupatla, T. R. (1999). Optimization Concepts and Applications in Engineering. Prentice-Hall. New Jersey. Kim, J., Kim, Y. T. and Yoon, Y. S. (2006). Analysis of transient maneuvers for objectifying evaluation of vehicle stability. Trans. Korean Society of Automotive Engineers 14, 1, 167−175. Kusaka, K., Higuchi, M., Shibusawa, K., Muraoka, N. and Tsukagoshi, M. (1995). Transient tire model for handling analysis and prediction. Proc.IPC-8, 2, 75−80. Loeb, J. S., Guenther, D. A. and Chen, H. F. (1990). Lateral stiffness, cornering stiffness and relaxation length of the pneumatic tire. SAE Paper No. 900129. Mimuro, T., Ohsaki, M., Yasunaga, H. and Satoh, K. (1990). Four parameter evaluation method of lateral transient response. SAE Paper No. 901734, 51−60. Sakai, H. and Satoh, Y. (1995). A theoretical study of vehicle dynamics behavior with complex cornering stiffness. Proc. IPC-8, 2, 63−68. Wong, J. Y. (2001). Theory of Ground Vehicles. John Wiley & Sons. New York. Wu, J., Tang, H., Li, S. and Zheng, S. B. (2007). Integrated control system design of active front wheel steering and four wheel torque to improve vehicle handling and stability. Int. J. Automotive Technology 8, 3, 299−308.

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