International Journal of Automotive Technology, Vol. 9, No. 6, pp. 687−693 (2008)DOI 10.1007/s12239−008−0081−y

Copyright © 2008 KSAE1229−9138/2008/043−06

687

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)

ABSTRACT−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 theoreticallyformulated. Here, the vehicle is simplified by a bicycle (single-track) model, and the tire is modeled by an equivalent corneringstiffness and first order lag. From the experimental road data, the tire model parameters (equivalent cornering stiffness andtime lag constant) are extracted. These parameters are then inserted into the theoretically formulated equations of dynamiccharacteristic 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 canbe sufficiently represented by the suggested dynamic characteristic parameters. So, it is concluded that the proposed methodhas practical use for the development of new cars or for the comparison of similar cars since the evaluations of the vehiclehandling performance can be efficiently determined by the suggested dynamic characteristic parameters.

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

1. INTRODUCTION

Handling analysis is one of the most attractive themes inautomobile engineering. Here, many suspension and tiredesigners want to know how their design parameters con-tribute to handling characteristics, and test drivers want toknow what the sensations that they feel are derived from. Asimplified bicycle model, which is also known as thesingle-track model, is convenient for the theoretical analysisof vehicle dynamics. Due to this majority of single-trackmodel, automobile engineers can easily understand thehandling characteristics of a car.

In the literature, several different methods have beenintroduced. For example, Sakai and Satoh (1995) usedcomplex cornering stiffness, which is the same as the equi-valent cornering stiffness when the phase lead time is takeninto account. From this approach, the natural frequency anddamping coefficient of the plane motion are theoreticallyformulated. Kusaka et al. (1995) examined a simpler bicyclemodel for the purpose of investigating the influence of tiretransient characteristics on vehicle handling properties.Mimuro et al. (1990) developed the four parameter evalu-ation methods of lateral transient response that also comefrom the bicycle model. This technique allows a rapidcomparison of handling characteristics of different tire andvehicle combinations by condensing the response data tobe displayed into a rhombus form.

The purpose of this article is to study the possibility ofusing vehicle lateral and yaw dynamics for the analysis ofvehicle handling performance (i.e., based on the simplebicycle model). For this, dynamic characteristic parameterssuch as yaw natural frequency and the damping coefficientof car are theoretically formulated using the approaches ofSakai and Satoh (1995). Specifically, the vehicle is simpli-fied by bicycle (single-track) model, and actual tire andsuspension characteristics are merged into one force gene-rator per axle that is modeled by an equivalent corneringstiffness and first order lag.

In this paper, new approaches for obtaining dynamiccharacteristic parameters from experimental road test dataare introduced. For this, the equivalent cornering stiffnessand first order lag are extracted from road test data, whichfundamentally allows more accurate compared with thosemethods that use indoor tire behavior machines (such asFlat-trac). The reason is that the tire has a smaller inertia,and its behavior is influenced by road roughness, the anti-lock braking system (ABS) operation, tire pressure andtread variations, carcass radial non-uniformities, as well asvehicle roll, pitch, and vertical motions. Therefore, tiremotions tend to be a lot more oscillatory and contain higherfrequency components. Separating the effects of thesedisturbances from those of tire and road interactions isdifficult. The extracted equivalent cornering stiffness andfirst order lag parameter are then inserted into the theoreti-cally formulated equations to determine dynamic charac-teristic parameters.*Corresponding author. e-mail: jskim@hankooktire.com

688 J. KIM

In order to validate the efficiency of the suggestedmethods, experimental road tests for cars having differenthandling performances are performed. The results showthat vehicle handling performance can be sufficiently repre-sented by the suggested dynamic characteristic parameters.

2. LATERAL VEHICLE DYNAMICS

In order to formulate the dynamic characteristic parameterstheoretically, a first order lag tire model and simple bicyclevehicle model are considered.

2.1. First Order Lag Tire ModelThe lateral force generated at the contact point between theroad and tire is transferred to the wheel. This mechanismcan be modeled with an effective spring and damper, asshown in Figure 1. The contact patch (including tread,carcass, and belt of the tire) is connected to the wheel by aneffective spring (ky) and damper (dy). Here, the axis η(c) isfixed at the center of wheel to describe the lateral motion ofthe contact patch. The equation of motion of the contactpatch can be written as

(1)

where m(c) is the mass of the contact patch, is thesteering rate of the wheel, and Fy is the lateral force at thetire contact patch. The lateral force at the wheel, Fy(w) can beexpressed as

. (2)

Equation (2) shows the phase delay between the lateralforce of contact patch and the lateral force of wheel, whichcomes from the inertia of the tire contact patch. is thelateral velocity difference between the wheel and tirecontact patch. This represents the direction between thewheel and tire contact patch. The tire slip angle, which isdefined as the angle between the wheel heading direction(ξ-axis) and the traveling direction of the tire contact patch,can be written as

(3)

where α is steady-state slip angle when is zero. In a

small slip angle area, Fy can be expressed as

(4)

where Cα is the cornering stiffness of the tire. By combin-ing equations (1) and (4), the equation of motion of the tirecontact patch can be obtained by

. (5)

In general, m(c) and dy are very small compared to ky. Thisallows equation (5) to be simplified as

. (6)

From equation (6), the following relationship can beobtained.

. (7)

By combining equations (4) and (7), the first order lag tiremodel that considers the lateral force and steady-state slipangle can be obtained by

(8)

where

. (9)

In equation (9), τ is the time constant of the first orderlag tire model. Also, from equation (8), the transfer func-tion of the lateral force of the tire can be obtained by

. (10)

2.2. Vehicle ModelAn actual vehicle model with a single track, which describesthe lateral and yaw dynamics of a 2-axle and 1-rigid bodyground vehicle, is shown in Figure 2(a). The equation ofmotion for this model can be expressed by

m c( ) η·· c( ) θs2· η c( )–( )+dyη· c( )+kyη c( )=Fy

θ· s

Fy w( )=dyη· c( )+kyη c( )=Fy−m c( ) η·· c( ) θs2· η c( )–( )

η· c( )

α = α−η· c( )

Vx

-------

η· c( )

Fy=Cα α η· c( )

Vx

-------–⎝ ⎠⎛ ⎞

m c( ) η·· c( ) θs2· η c( )–( )+dyη· c( )+kyη c( )=Cα α η· c( )/Vx–( )

kyη c( )=Cα α η· c( )/Vx–( )=Fy

F· y=kyη· c( )

τF· y+Fy=Cαα

τ=Cα

kyVx

----------

Fy s( )α s( )------------=

Cα

1 τs+--------------

mV sβ s( )+r s( )[ ]=Fy1 s( )+Fy2 s( )

Figure 1. Transient characteristics of the tire lateral force.Figure 2. Two DOF (degree of freedom) of the (a) actualand (b) equivalent bicycle model.

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS 689

(11)

where the vehicle model parameters are explained in Table1. For the purpose of investigating the influence of thetransient tire characteristics on vehicle maneuverability, theequation of motion of vehicle is expressed as a Laplacetransform.

In a small slip angle area, the tire slip angle is given by

(12)

where ξi represents the change in tire slip angle induced bythe suspension and the steering compliance. Also, isthe wheel steer angle, which can be obtained by dividingthe steering wheel angle by the overall steering gear ratio.

The actual bicycle model of Figure 2(a) can be simpli-fied by the equivalent vehicle model of Figure 2(b). Inorder to this, pseudo slip angles are defined by

(13)

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

, i=1, 2 (14)

where 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 beexpressed by the complex cornering stiffness. The mathe-matical definition of the complex cornering stiffness is thetransfer function of the lateral force against the pseudo slipangle. By substituting equation (14) for the lateral force ofthe tires in equation (11), the equivalent vehicle model canbe obtain by

. (15)

Using the first order lag of the lateral tire force model thatis previously formulated in equation (10), can bewritten as

, i=1, 2 (16)

where τ i represents the time constant of the first order lagmodel. Also, Cαi represents the equivalent cornering stiff-ness under steady-state conditions. The complex corneringstiffness is represented by a steady-state equivalent corner-ing stiffness and a time lag constant. Equation (16) can besimplified with the help of a Taylor expansion series arounds=0, such as

. (17)

By substituting equation (17) into (15), the characteristicequation of the vehicle can be obtained; it is

(18)

where

.

Equation (18) can be expressed with the dynamic charac-teristic parameters of the system as

(19)

where

(19a)

(19b)

(19c)

sIz s( )=aFy1 s( )−bFy2 s( )

α1 s( )=δs s( )+ξ1−aV----r s( )−β s( )

α2 s( )=ξ2+bV----r s( )−β s( )

δs s( )

α1* s( )=δs s( )− a

V----r s( )−β s( )

α2* s( )= b

V----r s( )−β s( )

Fi s( )=Cαi* s( )αi

* s( )

Cαi* s( )

mVs+Cα1* s( )+Cα2

* s( ) mV+aCα1

* s( ) b– Cα2* s( )

V--------------------------------------------

aCα1* s( ) b– Cα2

* s( ) Izs+a2Cα1

* s( )+b2Cα2* s( )

V----------------------------------------------

β s( )r s( )

=Cα1

* s( )δs s( )

aCα1* s( )δs s( )

Cαi* s( )

Cα i* s( )= Cα i

1 τis+---------------

Cα i* s( )= Cα i

1 τis+--------------- Cα i 1−τis( )≈

s2+A1 A1′+A0 A0′+-------------------s+

A2

A0 A0′+-------------------=0

A0=mVIz

A0′=l2Cα1Cα2τ1τ2

V------------------------------−m a2Cα1τ1 b2Cα2τ2+( )

−Iz Cα1τ1 Cα2τ2+( )

A1=m a2Cα1 b2Cα2+( )+Iz Cα1 Cα2+( )

A1′=−l2

V---Cα2 amV–⎝ ⎠⎛ ⎞Cα1τ1−

l2

V---Cα1 bmV–⎝ ⎠⎛ ⎞Cα2τ2

A2=−mV aCα1 bCα2–( )+ l2

V---Cα1Cα2

s2+2ζωns+ωn2=0

ωn2=

A2

A0 A0′+-------------------=Cwωn0

2

ζωn=A1 A1′+

2 A0 A0′+( )---------------------------=CζCwζ0ωn0

ωn02 =

l2Cα1Cα2

mIz

-------------------- 1V2----- m

l2----+ b

Cα1

-------- aCα2

--------–⎝ ⎠⎛ ⎞

⎩ ⎭⎨ ⎬⎧ ⎫

Table 1. Vehicle model parameters.

Symbol Descriptiona distance from front wheels to c.g.ay lateral accelerationb distance from rear wheels to c.g.Fy1 front axle lateral forceFy2 rear axle lateral forceIz yaw moment of inertial wheelbase m mass of the vehicler vehicle yaw rateV vehicle speedβ vehicle sideslip angleδ s Wheel steer angle

690 J. KIM

. (19d)

From equation (19), the natural frequency and dampingcoefficient of the plane motion can be obtained. Figure 3shows an example of the calculation of ωn and ζωn

(m=1239 kg, a=1.06 m, b=1.63 m, V=100 kph, Iz=2140.7kg·m2). It can be observed that the time lag of the lateralforce affects ωn and ζωn remarkably.

2.3. Identification of the Axle Force CharacteristicsIn order to identify the lateral force characteristics, themotion of two DOF is considered. By simply solving forFy1 and Fy2 in equation (11), the axle lateral forces can bedirectly obtained, where the result is

(20)

(21)

where is the lateral acceleration, and l=a+bis the wheelbase.

The pseudo slip angles related to the lateral forces can beobtained from equation (13). For this, three data points (r,β, δs) are directly measured by the sensors. If the vehiclemotion in the horizontal road plane is determined accurate-ly, the lateral forces and slip angles are determined as well.

The lateral forces are plotted as functions of the pseudoslip angles in Figure 4, where the non-linear form of thepneumatic tire behavior is recognized. The lateral forcesand pseudo slip angles are applied to the most commonly

used empirical model, the so-called “Magic formula”(Bakker and Pacejka, 1989). For clarity, the simplest formhas been adopted in the following:

(22)

where K3i represents the peak value, if present, and theproduct K1iK2iK3i equals the gradient of the function at zeroslip angle, which is the equivalent cornering stiffness, Cαi,in equation (17). From Cα1 and Cα2, the understeer coeffi-cient can be obtained by

(23)

where W1 and W2 are the normal load on the front and rearaxle (Wong, 2001).

As explained in Figure 1, the tires do not instantlyproduce lateral forces as a result of the slip angle delay.This phenomenon is mathematically represented using thetime constant τ in equation (8). Thus, τ is the time requiredto attain 63.2% of the final steady-state value for the stepresponse. In the case of rolling tire, the tire has to roll somedistance before the steady-state lateral force is fullydeveloped, where this distance is called the relaxationlength. By multiplying τ in equation (9) by the speed Vx,the relaxation length can easily be obtained. Loeb et al.(1990) report that the length a value of 1/2 up to 1 fullrevolution of the tire (0.9 up to 1.8 m with a 0.3 m rollingradius).

Figure 4 shows the cross plot of the axle lateral forceversus slip angle. A counterclockwise hysteresis loop canbe observed, which represents the energy accumulationbetween the slip angle and lateral force. This phenomenonis closely related to the time delay between the slip angleand lateral force. If a clockwise hysteresis loop occurs, it isusually related to the energy loss between the slip angleand lateral force.

Since the Magic formula of equation (22) does notinclude the transient characteristics of the lateral force,

ζ0ωn0=1

2V------- Cα1+Cα2

m-------------------- a2Cα1 b2Cα2+

Iz

---------------------------------+⎝ ⎠⎛ ⎞

Cw= 1

1+ 1V--- Cα1τ1 Cα2τ2+

m--------------------------------– a2Cα1τ1 b2Cα2τ2+

Iz

-------------------------------------------– l2Cα1Cα2τ1τ2

mVIz

------------------------------+⎝ ⎠⎛ ⎞

--------------------------------------------------------------------------------------------------------------------------------------------

Cζ=1− l2Cα2 maV2–( )Cα1τ1 l2Cα1 mbV2+( )Cα2τ2+V Iz a2m+( )Cα1+ Iz b2m+( )Cα2[ ]

----------------------------------------------------------------------------------------------------------

Fy1=mayb+Izr·

l-----------------------

Fy2=maya+Izr·

l-----------------------

ay=V β· r+( )

Fyi=−K3 isin[K2itan 1– {K1iαi−K4iK1 iαi−tan 1– K1 iαi( ))}]

Kus=1g--- W1

Cα1

-------- W2

Cα2

--------–⎝ ⎠⎛ ⎞

Figure 3. Effects of τ1 and τ2 on the dynamic characteri-stics of the vehicle.

Figure 4. Lateral force generation at the front and rear axle.

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS 691

modeling of this effect is required. Firstly, transient para-meters (such as the time delay) are identified by reductionof the hysteresis between lateral force and slip angle inFigure 4. Then, the function parameters (such as the Magicformula coefficient) of equation (22) are identified. Toreduce the hysteresis, a first order filtering of equation (8)is used. The slip angle data in Figure 4 are driven throughthis filter, and then plotted against the lateral axle forcedata. If the right time lag constant τ is determined, thehysteresis area will be minimized. For a specific value of τthat minimizes the hysteresis area, an iterative routinebased on a golden section search is used (Belegundu andChandrupatla, 1999).

Using an equivalent cornering stiffness that is estimatedby the Magic formula and a time lag constant determinedby a first order filter, the complex cornering stiffness ofequation (17) can be determined. Figure 5(a) shows therear axle force generation during a high speed maneuver,while Figure 5(b) shows the results of the hysteresisreduction, resulting in a time lag constant of 0.02 second. Itshould be noted that the hysteresis area is significantlyreduced, which implies that the axle lateral forces can beapproximated with a single-valued function. The coefficientsof the Magic formula are determined by fitting the reducedhysteresis area data of Figure 5(b) with equation (22).

3. APPLICATION OF THE VEHICLE HANDLING ANALYSIS

In this section, the proposed methods are applied to thevehicle handling analysis. Both a subjective assessment ofthe driver and objective measurement are performed forfour sets of tests, as shown in Table 2. The car is front-driving, and has a specified distribution of the front (60%)and rear (40%) weight. The tire size is 205/55R17V, andthe inflation pressures of the tires are varied. The originalinflation pressure of the tires for the car is 206.8 kPa. In test

T1, the inflation pressure of the front tires is set to 103.4kPa, while the inflation pressure of the rear tires is set to206.8 kPa. This makes the car understeer more. On theother hand, in test T4, the inflation pressure of the rear tiresis set to 103.4 kPa, while the inflation pressure of the fronttires is set to 206.8 kPa. This makes the car oversteer more.With regard to the handling performance of car, the stabilityof the yaw and lateral motions of the car is subjectivelyevaluated (Kim et al., 2006). Among the four test condi-tions, there are big differences in the stability of the car, asshown in Table 2. As a decimal digit rating, larger valuesmean better stability. When the evaluator feels that hand-ling is “very poor”, it is awarded about 2 points. If theevaluator feels that handling is “excellent”, 10 points areawarded. In the results, the smaller the inflation pressure ofthe front tires, the better the stability. The reason for thiscan be inferred from the fact that the tendency to understeer

Figure 5. Rear axle force (a) before reducing hysteresis and(b) after reducing hysteresis for τ2=0.02 seconds.

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

Test car Test Test conditions Rating

Front driving

(Curb+2 persons)

ID Inflation of pressure (kPa) Stability

Front RearT1 103.4 206.8 9.75T2 206.8 206.8 7T3 206.8 137.9 3.75T4 206.8 103.4 3.6

Figure 6. Measured signals during lane change maneuvers.

692 J. KIM

is increased due to a reduction of the inflation pressure ofthe front tires.

All of the tests are carried out on dry asphalt. In themeasurement, the signals are pre-filtered with a 5 Hz cut-off frequency, while the data acquisition is performed at asampling rate of 100 Hz. In order to get enough data todescribe the axle cornering characteristics, a lane changemaneuver is chosen. The measurement is repeated at aconstant speed of 33.3 m/s.

Figure 6 shows measurement signals obtained duringlane change maneuvers. Using these signals, all of thedynamic characteristic parameters are obtained.

Figure 7(a) shows the results of the estimation of theequivalent cornering stiffness, Cα1 and Cα2, by adapting theMagic formula of equation (22). All of the values of Cα1 aresmaller than Cα2, which means that understeering occurredfor all test sets. Also, Cα2 decreases as the inflation of therear tires is decreased.

Figure 7(b) shows the results of the estimation of thetime lag constant, τ1 and τ2 obtained by adapting a firstorder filter. The results show that τ1 is larger than τ2. Also,both τ1 and τ2 increases as the inflation pressure of the tireis decreased.

Figure 8 shows the vehicle dynamic characteristics, ωn

and ζωn, obtained by substituting the estimated equivalentcornering stiffness and time lag constant into equations(19a) and (19b), respectively. In considering the time lagconstant, both ωn and ζωn must be changed considerably.

Figure 9 shows the correlation between dynamic charac-teristic parameters and the stability performance of the car.Cα1 and Cα2 show a low correlation, while Kus shows a highcorrelation. τ1 and τ2 show the highest correlation of allparameters. τ1 has a positive correlation while τ2 has anegative correlation. This means that the stability of car can

be improved by setting τ1 larger than τ2, which increasesthe damping coefficient, ζωn. Using ωn from equation (19a)

Figure 7. Estimation results of the (a) equivalent corneringstiffness and (b) time lag constant: , front; , rear.

Figure 8. Numerical results of the vehicle dynamic para-meters.

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

Figure 10. Stability of the vehicle response.

ANALYSIS OF HANDLING PERFORMANCE BASED ON SIMPLIFIED LATERAL VEHICLE DYNAMICS 693

instead of ωn0 from equation (19c), the correlation valuecan be improved from 0.80 to 0.84. Similarly, the corre-lation with the stability of the car can be improved from0.72 to 0.89 using ζωn from equation (19b) instead of ζωn0

from equation (19d).These two results show that the stability of car can be

analyzed more accurately by considering both the equi-valent cornering stiffness and time lag constant whenevaluating the dynamic characteristic parameter of car.Figure 10 shows the measured vehicle responses, which arethe cross plot between the yaw rate and sideslip angle ofthe car (Wu et al., 2007). It can be observed that T1 is morestable than T4. This kind of characteristic can be easilyidentified and effectively quantified by the formulateddynamic characteristic parameters.

4. CONCLUSION

A simple 2 DOF vehicle model combined with a transientrepresentation of the axle data is shown in order to analyzethe handling performance of the car. From the model, thevehicle dynamic parameters are formulated, and these areestimated from the actual vehicle responses during lanechange maneuvers. In order to validate the suggested method,a correlation study between the subjective stability, whichis one of the handling performance capacities of the car,and the estimated vehicle dynamics parameter is performed.The following points summarize the conclusions of thisstudy.(1) The bicycle model, which is the simplest available in

describing the handling properties of car, is shown togive reliable results.

(2) The effect of the time lag constant of the axle lateralforce on the stability of the car is examined, whichshows the time lag constant plays an important role inthe handling performance of car.

(3) By considering the effect of the time lag constant on thevehicle dynamic parameters, the stability of car can beanalyzed more accurately.

By using the proposed analysis technique, it will bepossible to analyze the handling performance of the car

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

ACKNOWLEDGEMENT−This research is supported by theR&D centre of HANKOOK Tyre Co., Ltd., under the KONTROL-Technology program. The author also thanks the VehicleDynamics Research Team for providing field test data andvaluable suggestions.

REFERENCES

Bakker, E. and Pacejka, H. B. (1989). A new tyre modelswith an application in vehicle dynamics studies. SAEPaper 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 oftransient maneuvers for objectifying evaluation of vehiclestability. Trans. Korean Society of Automotive Engineers14, 1, 167−175.

Kusaka, K., Higuchi, M., Shibusawa, K., Muraoka, N. andTsukagoshi, M. (1995). Transient tire model for handlinganalysis and prediction. Proc.IPC-8, 2, 75−80.

Loeb, J. S., Guenther, D. A. and Chen, H. F. (1990). Lateralstiffness, cornering stiffness and relaxation length of thepneumatic tire. SAE Paper No. 900129.

Mimuro, T., Ohsaki, M., Yasunaga, H. and Satoh, K. (1990).Four parameter evaluation method of lateral transientresponse. SAE Paper No. 901734, 51−60.

Sakai, H. and Satoh, Y. (1995). A theoretical study of vehicledynamics 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). Integratedcontrol system design of active front wheel steering andfour wheel torque to improve vehicle handling andstability. Int. J. Automotive Technology 8, 3, 299−308.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.