Integration of Magic Formula Tire Model With Vehicle Handling Model

AbstractA tire model is required in order to study the dynamic

behavior of a vehicle when designing control system such as electronic stability control. In this study, Magic Formula tire simulator model was implemented using Matlab Simulink block diagram. Tire modeling is the first step needed where vehicle handling stability needs to be investigated. The model was implemented based on a set of mathematical equations. The tire model is integrated with a 3 degree of freedom (3-DOF) vehicle handling model was also implemented. This handling model was developed based on Newtons second law equation of motion. Responses of interest for handling model are yaw motion, roll motion and pitch motion. Simulink blocks that represent tire responses are simplified into 3 subsystems that generated longitudinal force, Fx, lateral force, Fy, and aligning moment, Mz. The responses of every tire are observed. The four tire responses are needed to compute vehicle handling response. The implemented models was validated using simulation software, Carsim. For the validation procedure, Double Lane Change (DLC) test was used. The output variables generated from the implemented model are compared with Carsim output variables. The validation results are discussed. Once the models have been validated, the Magic Formula and vehicle handling model were used as a subsystem for designing a controller for vehicle handling stability.

KeywordsCarsim, Matlab Simulink, Magic Formula model,

Handling model.

Mohammad Safwan Burhaumudin, is with the Department of Automotive Engineering, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81300, Skudai, Johor, Malaysia (corresponding author phone: +60137147460; e-mail: [email protected]).

Pakharuddin Mohd Samin, is with the Department of Automotive Engineering, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81300, Skudai, Johor, Malaysia (e-mail: [email protected]).

Hishamuddin Jamaluddin, is with the Department of System Dynamics and Control, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81300, Skudai, Johor, Malaysia (e-mail: [email protected]).

Roslan Abd Rahman, is with the Department of System Dynamics and Control, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81300, Skudai, Johor, Malaysia (e-mail: [email protected]).

Syabillah Sulaiman, is with the Department of Automotive Engineering, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM), 81300, Skudai, Johor, Malaysia (e-mail: [email protected]).

I. INTRODUCTION IRE forces and moments are generated when there is friction between tire and road surface. The forces and

moments produced are crucial parameters that influence vehicle handling. Effective tire model for handling simulation should comprise of two basic elements; lateral tire force and longitudinal tire force which depend on slip angle and slip ratio respectively. Aligning moment is computed by multiplying lateral force with pneumatic trail produced by the deformation of rubber tire [1].

Combination slip conditions are represented in Magic Formula mathematical formulae. Thus, enable it to handle combined slip condition. This model provides an accurate behavior of tire mechanics based on real experimental test data [2]. Vertical wheel load is one of the input variables that influence the production of tire forces and moments. Vertical wheel load will contribute to different contact patch area and tire deformation [3]. In this study, vertical wheel load, slip ratio and slip angle input are obtained from vehicle response generated by Carsim.

The vehicle handling model employed consists of 3-DOF for vehicle body and 4-DOF for vehicle tires [4]. In vehicle body, the vehicle experience motion along the longitudinal x-axis and the lateral y-axis, and the angular motions of yaw around the vertical z-axis. The diagram representing a 3-DOF vehicle handling model [5] is shows in Fig.1.

Fig. 1 Vehicle handling model [5]

The roll and pitch motions depend on lateral and

longitudinal acceleration respectively. Since only the vehicle body undergoes roll and pitch, so sprung mass has to be taken into consideration instead of total vehicle mass.

Integration of Magic Formula Tire Model with Vehicle Handling Model

Mohammad Safwan Burhaumudin, Pakharuddin Mohd Samin, Hishamuddin Jamaluddin, Roslan Abd Rahman and Syabillah Sulaiman

T

International Journal of Research in Engineering and Technology (IJRET) Vol. 1, No. 3, 2012 ISSN 2277 4378

139

The objective of this paper is to implement a tire model based on Magic Formula mathematical equation [6], [7] and vehicle handling model based on Newtons second law of motion [4],[8]. The Magic Formula equations consist of several coefficients to be tuned so that the output trend is similar to the experimental test data. The details of the equations for Magic Formula and vehicle handling model are shown in section II. While in the section III, the modeling blocks are introduced and validated with Carsim. For the purpose of tire forces, moments and vehicle body response analysis, section IV discusses the results obtained from the proposed models. Some researchers are working on several tire models such as UniTire Model [3], Dugoffs Tire Model [7], and others. Different tire model used different approach and equations and thus different model parameters.

This tire model and vehicle handling model will be used for vehicle dynamic analysis and control. The forces and moments that are generated by the tire will affect the vehicle body stability. A control algorithm is required to control vehicle state to enhance vehicle handling stability. A particular control algorithm with appropriate actuator must be well defined.

II. MATHEMATICAL MODELING

A. Vertical Wheel Load Vehicle weight transfer will cause vertical load, Fzij, varies,

with the index i indicating front (f) or rear (r) tires and j indicating left (l) or right (r) tires. The equations represent vertical loads, Fzij, at each tire are

lhma

lc

tha

mmgF xy

rzfl

=

21

/ (1)

and

lhma

lb

tha

mmgF xy

rzrl

=

21

/ (2)

Where, left side tire using plus operation, +, and right side

tire using minus operation, -, m denotes the overall vehicle mass, g the gravitational acceleration, h the vehicle mass center of gravity, t the vehicle track width, b and c are the distance between the center of gravity and the front and the rear axle respectively, ax and ay are the accelerations in x-axis and y-axis respectively, and l the vehicle wheelbase.

B. Slip Ratio and Slip Angle Slip ratio, , defines the ratio between slip velocity and

vehicle velocity. The expression represents the slip ratio is

. v

vvvehicle

vehiclewheel = (3)

Slip ratio exists when vehicle is under braking or

accelerating conditions.

Slip angle, , defines the angle between the directions of tire to the velocity vector of the vehicle [9]. The value is typically small under steady cornering. However, the value may change suddenly in critical driving condition. Fig. 2 illustrates a tire moving [9] along the velocity vector, v, at a sideslip angle, . The tire is steered by the angle . If the angle between the velocity vector, v, and the vehicle x-axis is shown by , then slip angle, , is defined as.

= (4)

In Matlab Simulink simulation the slip angle generated by Carsim software was used. The slip angle produced by Carsim yielded from the movement of vehicle under Double Lane Change (DLC) maneuver.

Fig. 2 Tire slip angle [9].

C. Magic Formula Magic Formula [6], [7] developed by Pacejka has been

widely used to calculate steady-state tire force and moment characteristics. The semi-empirical tire model combined slip situation from a physical view point. The pneumatic trail is introduced as a basis to calculate this moment about the vertical axis [2]. The formulae consist of coefficient B, C, D, and E that determine the trend of force and moment similar to the actual test data. To avoid symmetrical asymptote about the origin, the Magic Formula incorporate coefficient Sv and Sh. This offset from the origin occurs due to the presence of camber angle. Table 1 shows the coefficients that govern the Magic Formula equation. The general form of graph produced by Magic Formula [2] is shown in Fig. 3.

TABLE I COEFFICIENT IN MAGIC FORMULA

Symbol Quantity

B stiffness factor

C shape factor

D peak value

E curvature factor

Sh horizontal shift

Sv vertical shift


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Fig. 3 Original sine version of the Magic Formula graph [2].

The full set of equation of Magic Formula [2] that

represents pure and combine slip condition are defined in equations (5) to (13). In this study, 3 outputs are considered which are longitudinal force, lateral force and aligning moment.Longitudinal force for pure slip, Fxo, consists of coefficients B, C, D, E and Sv. The subscript x represents condition along x-axis. Slip ratio, , is the input of Fxo as given by

( )( ){ }[ ].

arctanarctansin

SVx

xBxxBxE xxBxC xDxF xo+

= (5)

Lateral force for pure slip, Fyo consists of coefficients B, C, D, E and Sv. The subscript y represents condition along y-axis. Slip angle, , is the input to Fyo as given by

( )( ){ }[ ].

arctanarctansin

SVy

yB yyB yE yyB yC yD yF yo

+

= (6)

Aligning moment for pure slip, Mzo, is the product of

pneumatic trail, to, with lateral force, Fyo, as given by .yoFotM zo = (7)

Where pneumatic trail, to, responding towards slip angle, ,

and consists of coefficients B, C, D and E given by ( )( ){ }[ ].arctanarctancos tBttBtEttBtCtDtto = (8) Where subscript t represents condition occur due to existing

of trail.Longitudinal force for combined slip, Fx, is given by .F xoGxF x = (9) Where Gx represents positive factor when slip angle, ,

was introduced in the presence of slip ratio, , and Fxo is the longitudinal force for pure slip.Lateral force for combined slip, Fy, is given by

.F yoG yF y = (10)

Where Gy represent positive factor when slip ratio, , is introduced in the presence of slip angle, , and Fyo is the lateral force for pure slip.

Aligning moment for combined slip, Mz, is given by ( ) .F xsF ytM z += (11)

The aligning moment, Mz, is the summation of moments occurs due to lateral force and longitudinal force. The offset location where the force acting in y-axis is given by

( )( ){ }[ ] eqtBteqtBtEteqtBtCtDtt ,arctan,,arctancos = (12)

Where t,eq is equivalent slip angle incorporating the effect

of on the composite slip and

( )Ros 1.0= (13) Where Ro is the wheel radius.

D. Vehicle Handling Model The vehicle experiences motion along the longitudinal x-

axis and the lateral y-axis, and the angular motions of yaw around the vertical z-axis. Roll and pitch motions depend on lateral and longitudinal accelerations respectively. The equation of roll and pitch motions are

( ) ( )J

khmahmsx

ssyss g ++

=

(14)

and ( ) ( )

Jkhmahm

sy

ssxss g ++=

(15)

Where hs is the height of sprung mass at the center of gravity, g is the gravitational acceleration, is roll acceleration , is pitch acceleration and k , , k and are the stiffness and damping constants for roll and pitch respectively. The moment of inertia of the sprung mass about x-axis and y-axis are denoted by Jsx and Jsy respectively.

Angular motion of yaw dependents on the tire forces Fxij and Fyij as well as the self-aligning moment, denoted by Mzij acting on each tire. The equation represent yaw motion is

]

sinsincoscos

sin2

sin2

22cos

2cos

2[1

MMMMFLFLFLFL

FLFLFwFw

FwFwFwFwJ

zrrzrlzfrzfl

xfrfxflfyfrfyflf

yrrryrlryfrf

yflf

xrrr

xrlr

xfrf

xflf

z

++++

++

+

+=

(16)

Where w denoted the track width, Jz the moment of inertia

around z-axis, the vehicle steer angle and, Lf and Lr are the distance between center of gravity and front axle and rear axle respectively.


141

III. SIMULATION AND VALIDATION A. Simulation All the equations in section II are converted into Matlab

Simulink block diagram. The final simplified model for Magic Formula tire model is shown in Fig. 4. The inputs to the subsystem are vertical load, Fz, road friction coefficient, , slip ratio, , and slip angle, . Camber angle is set to zero degree for simplicity.

Fig. 4 Modeling of Magic Formula in Matlab Simulink.

The final simplified vehicle handling model is shown in Fig. 5. The inputs to the subsystem are vehicle mass, m, height of center of gravity, h, steering angle, , and, tire forces, Fx/y, and moments, Mz.

Fig. 5 Modeling of vehicle handling model

Combination of tire model and handling model is shown in Fig. 6. Tire model was duplicated to represent each tire. However, tire responses are different at every tire due to input variations.

The slip ratio and slip angle that were generated under Double Lane Change (DLC) maneuver at 120 km/h are shown in Fig. 7. The values of slip ratio and slip angle generated by Carsim were fed into Matlab Simulink as inputs variable.

Fig. 6 Integration of tire model and handling model

Fig. 7 Slip ratio and slip angle during Double Lane Change (DLC)

Maneuver.

B. Validation For validation purpose, Carsim software was used to

simulate a vehicle during Double Lane Change (DLC) maneuver. The default build-in testing module was used for the Carsim simulation.

The command window of Carsim testing simulation illustrated in Fig. 8 is divided into 3 sections. In the command window, vehicle type and testing procedure are selected from section 1. The vehicle specification chosen was E-Class, Sedan and the testing procedure was Double Lane Change (DLC) at 120 km/h maneuver. Once the vehicle type and testing procedure have been chosen, the solver is selected in section 2. The display results and animation are chosen in section 3. The animation of vehicle maneuver for Double Lane Change (DLC) is shown in Fig. 9.

The left image in Fig. 9 shows the vehicle about to change lane while the right image shows the vehicle at the end of the maneuver.


142

Fig. 8 Carsim command window for simulate vehicle maneuver.

Fig. 9 Double Lane Change maneuver.

IV. RESULTS AND DISCUSSIONS During vehicle travelling on the designated course, the

vehicle response will generate vertical force, slip angle and slip ratio. Those variables generated by Carsim are fed into Matlab Simulink Magic Formula block diagram to produce the forces and moments. These forces and moments were used as the input for the vehicle handling model. The trend of the graphs for tire forces and moments are compared and the results are shown in Figs.10 to 12.

Fig. 10 shows the longitudinal force generated at the tire. The trend of the graph produced by Matlab Simulink is quite similar to Carsim except when the gradient is increasing and decreasing. These happen due to some assumption made by Magic Formula tire model but not in Carsim. Aerodynamic effect, gear shifting effect and kinematics of vehicle suspension are some factors that are neglected by Magic Formula tire model. Additional vehicle model is required to compensate the situation will be considered in future work.

The lateral force shown in Fig. 11 has 2 peaks and a trough. The range of lateral force generated is from -4000 N to 3000 N. Maximum lateral force was generated due to increasing of slip angle when a vehicle changes the maneuver.

Fig. 10 Longitudinal force computed from Matlab Simulink and

Carsim.

Fig.11 Lateral force computed from Matlab Simulink and Carsim.

Aligning moment shown in Fig. 12 has inverse trend to

lateral force. This moment exists due to the deformation of tire during Double Lane Change (DLC) maneuver. Tire deformation will create concentrated point at the tire contact patch. This point is generally not at the center of the contact patch. The distance offset from the origin of the tire creates pneumatic trail that contribute to the generation of aligning moment.

Fig. 12 Aligning moment computed from Matlab Simulink and

Carsim.

Section 1 Section 3 Section 2


143

The tire responses are different at every wheel and are shown in Figs. 13 to 15. From these graphs, the responses of the right and left tires are quite similar at front or rear wheels. This happens due to slip angle generation at the left and right tires are almost similar with the front and rear wheels.

Fig. 13 Longitudinal force at every vehicle tire

Fig. 14 Lateral force at every vehicle tire

Fig. 15 Aligning moment at every vehicle tire

For handling model, output variables of interest are yaw

rate and roll angle. These output variables are shows in Figs. 16 and 17 and compared with the responses produced by Carsim. These 2 variables are crucial variables needed to be controlled for vehicle handling stability. Thus, the threshold

values for these variables should be well defined for a particular type of vehicle considered.

Fig. 16 Yaw rate of vehicle body

Fig. 17 Roll angle of vehicle body

IV. CONCLUSION Matlab Simulink model of Magic Formula tire model and

handling model was developed in order to initiate a future project in vehicle stability control. Tire model is a paramount subsystem affecting vehicle handling. The implemented model was validated using Carsim software.

The implemented model was validated based on Double Lane Change (DLC) testing method maneuvered at 120 km/h using Carsim. During maneuver, vertical load, slip angle and slip ratio were generated. Vertical load, slip ratio and slip angle yielded from the maneuver were used as the input variables to the Magic Formula tire. The longitudinal force, lateral force and aligning moment produced during that course maneuver were compared. The tire responses produces by Simulink model were used as the inputs for vehicle handling model. Responses from handling model were compared with Carsim.

From the validation results, the trends between Matlab Simulink and Carsim are similar with some difference in the magnitude. The difference arises due to aerodynamic effect,


144

ear shifting effect and kinematics of vehicle suspension effect being ignored.

ACKNOWLEDGMENT The authors wish to thank the Ministry of Higher

Education (MOHE) and Universiti Teknologi Malaysia (UTM) for providing the research facilities and support, especially all staffs of Department of Automotive, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia. This research is supported using a research grant, Vot No.78608.

REFERENCES [1] C. Long, and H. Chen, Comparative Study between the Magic Formula

and the Neural Network Tire Model Based on Genetic Algorithm, Third International Symposium on Intelligent Information Technology and Security Informatics, pp. 280-284, Jinggangshan, China, 2 April 2010.

[2] H.B. Pacejka, Tire and Vehicle Dynamics, Pennsylvania: SAE International and Elsevier, 2005 .

[3] N. Xu, D. Lu, and S. Ran, A Predicted Tire Model for Combined Tire Cornering and Braking Shear Forces Based on the Slip Condition, International Conference on Electronic & Mechanical Engineering and Information Technology, pp. 2073-2080, Heilongjiang, China, 12 August 2011.

[4] P.M. Samin. Hybrid Stability Augmentation System-Force Control of Semi-Active Suspension System with Magnetorheological Damper. Doctor Philosophy, Universiti Teknologi Malaysia, Skudai. 2010.

[5] M. Blundell, and D. Harty, Multibody Systems Approach to Vehicle Dynamics (1st ed.), Burlington: Elsevier Butterworth-Heinemann, 2004.

[6] E. Bakker, L. Nyborg, and H.B. Pacejka, Tyre Modelling for Use in Vehicle Dynamics Studies, SAE Paper 870421, pp. 1-15, 1987.

[7] R. Rajamani, Vehicle Dynamics and Control, New York: Springer, 2006, ch. 13.

[8] T.A. Wenzel, K.J Burnham, R.A. William, and M.V. Blundell, Closed-loop Driver/Vehicle Model for Automotive Control, International Conference on Systems Engineering, Las Vegas, Nevada, 16-18 August 2005.

[9] R.N. Jazar, Vehicle Dynamic Theory and Application, New York: Springer, 2008.


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Integration of Magic Formula Tire Model With Vehicle Handling Model

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