Abstract Tire model is required in order to study vehicle [621666]


Abstract — Tire model is required in order to study vehicle
dynamic behavior for designing cont rol system such as electronic
stability control. In this study, a Magic Formula tire model was
implemented using Matlab Simulink bl ock diagram. Tire modeling is
the first step to investigate vehicle handling stability. The model was developed based on a set of mathem atical equations. The model was
developed based on pure slip and combine slip condition. The feasibility of Magic Formula block diagram is validated via simulation software which is Cars im. For the validation procedure,
Double Lane Change (DLC) test was used. The output forces and moments from Carsim are compared with the development model. The validation results are discussed. Once the model was validated, the Magic Formula model will be use as a subsystem for vehicle
stability control.

Keywords — Carsim, Magic Formula, Simulink, Tire Slip
I. INTRODUCTION
OMMONLY , tire 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 the lateral force with the pneumatic trail produced
by the deformation of rubber tire [1].

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-ma il: [anonimizat]).
Pakharuddin Mohd Samin, is with the Department of Automotive
Engineering, Faculty of Mechanical Engineering, Universiti Teknologi
Malaysia (UTM), 81300, Skudai, Johor, Malaysia (e-mail:
[anonimizat]).
Hishamuddin Jamaluddin, is with the Department of System Dynamics and
Control, Faculty of Mechanical Engi neering, Universiti Teknologi Malaysia
(UTM), 81300, Skudai, Johor, Malaysia (e-mail: [anonimizat]).
Roslan Abd Rahman, is with the Department of System Dynamics and
Control, Faculty of Mechanical Engi neering, Universiti Teknologi Malaysia
(UTM), 81300, Skudai, Johor, Malays ia (e-mail: [anonimizat]).
Syabillah Sulaiman, is with the De partment of Automotive Engineering,
Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM),
81300, Skudai, Johor, Malaysia (e-m ail: [anonimizat]).

Combination slip conditions are represented in Magic
Formula mathematical formulae. Magic Formula tire is able to
handle combine slip condition. This model provides an
accurate behavior of tire mechanics based on real experimental test data. 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 [2]. In this study, vertical wheel load input are kept constant but the vehicle slip ratio and slip angle are varied.
The objective of this paper is to implement a tire model
based on Magic Formula mathematical equation [3], [4]. The 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 are shown in section II. While in the section III, the modeling block is introduced and validated with Carsim. For the purpose of tire forces and moment analysis, section IV discusses the results obtained from the proposed model. Some researchers are working on several tire models such as UniTire Model [2], Dugoff’s Tire Model [4], and ot hers. Different tire model used
different approach and equati ons and of course different
model parameters
.
This tire model will be used for vehicle dynamic analysis
and control. The forces and moments that are generated by the tire are monitored in order to enhance vehicle stability. A control algorithm required to control the tire forces and moments to be maintained at its adhesion limit to ensure vehicle handing stability.

II.
MATHEMATICAL MODELING
A. Slip Ratio and Slip Angle
Slip ratio, κ defines the ratio between slip velocity and
vehicle velocity. The expression represents the slip ratio is

vv v
vehiclevehicle wheel
 . ( 1 )
Slip ratio exist when vehicle is under braking and accelerating
condition.
Slip angle, α, defines the angle between the directions of
tire to the velocity vector of the vehicle [5]. The value is Modeling and Validation of Magic
Formula Tire Model
Mohammad Safwan Burhaumudin, Pakharuddin Mohd Samin, Hishamuddin Jamaluddin,
Roslan Abd Rahman, Syabillah Sulaiman
C
International Conference on Automotive, Mechanical and Materials Engineering (ICAMME'2012) Penang (Malaysia) May 19-20, 2012
113

typically small under steady cornering. However, the value
suddenly changes in critical driving condition. Fig. 1
illustrates a tire moving along the velocity vector, v, at a
sideslip angle, α. The tire is steered by the angle δ. If the angle
between the velocity vector, v, and the vehicle x-axis is shown
by β, then slip angle, α, is defined as

 . ( 2 )

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

Fig. 1 Tire slip angle.

B. Magic Formula
Magic Formula [3], [4] deve loped by Pacejka has been
widely used to calculate stead y-state tire force and moment
characteristics. The semi-empiri cal 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 [6]. The form ulae 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 Formul a introduce coefficient S
v 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 is shown in Fig. 2.

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

Fig. 2 Original sine version of the Magic Formula graph.

The full set of equation of Magic Formula [6] that
represents pure and combine slip condition are defined in
equation (3) to (16). In this study, 3 outputs are considered which are longitudinal force, lateral force and aligning moment.
Longitudinal force for pure slip, F
xo, consists of coefficients
B, C, D, E and Sv. The subscript x represents condition along
x-axis. Slip ratio, κ, is the input of Fxo as given by

    
.arctan arctan sin
SVxx Bx x Bx Ex x Bx Cx Dx Fxo
      (3)

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, α, is the input to Fyo as given by

      
.arctan arctan sin
SVyy By y By Ey y By Cy Dy Fyo
      (4)

Aligning moment for pure slip, M zo, is the product of
pneumatic trail, to with lateral force, Fyo, as given by

 . Mzro Fyo to Mzo  ( 5 )

The second term on the right side of equation (5) represents moment occurs due to camber angle. Pneumatic trail, t
o
consist of coefficients B, C, D and E as given by

    . arctan arctan cos    tBt t Bt Et t Bt Ct Dt to    (6)

Aligning moment due to camber angle, Mzro, as given by

  . arctan cos rBr Cr Dr Mzro ( 7 )

Longitudinal force for combined slip, Fx, is the product of
factor Gα with pure longitudinal force, Fxo, as given by

. Fxo Gx Fx ( 8 )

Factor Gα represents increment or d ecrement factor when slip
angle is introduced in the presence of slip ratio as given by

International Conference on Automotive, Mechanical and Materials Engineering (ICAMME'2012) Penang (Malaysia) May 19-20, 2012
114

 G o x
S Bx S BxEx S BxCx Gx  
  /arctanarctan cos



 (9)
and
 




SHx Bx SHx BxEx SHx BxCx G o x
 
 arctanarctan cos (10)

Lateral force for combined slip, Fy, is the product of factor
Gκ with the pure lateral force, Fyo, as given by

. SVy Fyo Gy Fy     ( 1 1 )

Factor Gκ represents increment or d ecrement factor when slip
ratio is introduced in presence of slip angle as given by

 Go y
S By S ByEy S By
Cy Gy  
  /arctanarctan cos 






 (12)
and
  







SHy By SHy ByEy SHy By
Cy Go y
 
 arctanarctan cos (13)

Aligning moment for combined slip, Mz, as shown by

     . , arctan cos 'Fxs eq r Br Cr Dr Fyt Mz        ( 1 4 )

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

         eq t Bt eq t Bt Et eq t Bt Ct Dt t , arctan, , arctan cos    (15)
and
Ro s 1 . 0  ( 1 6 )
III. SIMULATION AND VALIDATION
A. Simulation
All the equations in section II are converted into Matlab
Simulink block diagram. The final simplified model is shown
in Fig. 3. The inputs to the subsystem are vertical load, Fz,
road friction coefficient, μ, slip ratio, κ, and slip angle, α.
Camber angle is set to zero degree for simplicity.

Fig. 3 Modeling of Magic Fo rmula in Matlab Simulink. The slip ratio and slip angle that are generated under
Double Lane Change (DLC) maneuver at 120 km/h are shown in Fig. 4. The values of slip ratio and slip angle generated by Carsim are fed into Matlab Simulink as inputs variable.

Fig. 4 Slip ratio and slip angle during Double Lane Change (DLC)
Maneuver.

B. Validation
For validation purpose, Carsim software was used to
simulate a vehicle moving in Double Lane Change (DLC)
maneuver. The default build-in testing module was used for the Carsim simulation.
The command window of Carsim testing simulation is
illustrated in Fig. 5 can be divided into 3 sections. In the command window, vehicle type and testing procedure were selected from section 1. The ve hicle 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 was selected in section 2. The results and animation was chosen section 3. Vehicle maneuver for Double Lane Change (DLC) is shown in Fig. 6.

Fig. 5 Carsim command window fo r simulate vehicle maneuver.

The left image in Fig. 6 shows the vehicle about to
change lane while the right image shows the vehicle at the end of the maneuver.
Section 1 Section 3 Section 2

Fz
μ
α
к
Fxo, Fx
Fyo, Fy Mzo, Mz
International Conference on Automotive, Mechanical and Materials Engineering (ICAMME'2012) Penang (Malaysia) May 19-20, 2012
115

Fig. 6 Double Lane Change Maneuver.
IV. RESULTS AND DISCUSSIONS
During vehicle travelling on the designated course, the
vehicle response will generate slip angle and slip ratio. Those
slip generated by Carsim is fed into Matlab Simulink Magic Formula block diagram. The trends of the graph are compared and the results are shown in Figs.7 to 9.
Fig. 7 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 factors that are neglected by Magic Formula tire model but in Carsim are considered. Aerodynamic effect , gear shifting effect and
kinematics of vehicle suspensi on are some factors that are
neglected by Magic Formula tire model. Additional vehicle model is required to compensate the situation, which is will considered in future work.

Fig. 7 Longitudinal force com puted from Matlab Simulink and
Carsim.

The lateral force shown in Fig. 8 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 during the maneuver.

Fig. 8 Lateral force computed from Matlab Simulink and Carsim.

Aligning moment shown in Fig. 9 has inverse trend to
lateral force. This moment exists due to the deformation of tire prevailing at 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. Distance offset from th e origin of the tire creates
pneumatic trail that contribute to the generation of aligning
moment
.

Fig. 9 Aligning moment computed from Matlab Simulink and
Carsim.

IV. CONCLUSION
Modeling of Magic Formula tire model in Matlab Simulink
was developed in order to initiate a future project in vehicle
stability control. Tire model is a paramount subsystem affecting vehicle dynamic behavior. The developed tire model was validated using Carsim software.
The Magic Formula tire was validated based on Double
Lane Change (DLC) testing method maneuver at 120 km/h using Carsim. During maneuver, tire slip ratio and slip angle were generated. The slip ratio and slip angle yielded from that maneuver were used as the input variable to the Magic Formula tire developed using Matlab Simulink. The longitudinal force, lateral force and aligning moment produced during that course maneuver were compared.

International Conference on Automotive, Mechanical and Materials Engineering (ICAMME'2012) Penang (Malaysia) May 19-20, 2012
116

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,
gear shifting effect and kinematics of vehicle suspension effect being ignored in the model.
The validation result shows th e Magic Formula tire can be
used to represent actual tire dynamic behavior under any maneuver.
A
CKNOWLEDGMENT
The author wish to thank the Ministry of Higher
Education (MOHE) and Universiti Teknologi Malaysia (UTM) for providing the research facilities and support, especially all staff’s of Depart ment of Automotive, Faculty of
Mechanical Engineering, Un iversiti Teknologi Malaysia.
R
EFERENCES
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[2] N. Xu, D. Lu, S. Ran, “A Pred icted 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, 12 August 2011.
[3] E. Bakker, L. Nybor g, and H.B. Pacejka, “Tyre Modelling for Use in
Vehicle Dynamics Studies,” SAE Paper 870421, pp. 1-15, 1987.
[4] R. Rajamani, Vehicle Dynamics and Control, New York: Springer, 2006,
ch. 13.
[5] Jazar, G. Nakhaie, Vehicle Dynamic Theory and Application, New
York: Springer, 2008, pp. 600-605.
[6] H.B. Pacejka, Tire and Vehicle Dynamics, SAE International and
Elsevier, 2005, ch. 4.
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