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.
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Abstract— Tire model is required in order to study vehicle
dynamic behavior for designing control system such as electronic
stability control In this study, a Magic Formula tire model was
implemented using Matlab Simulink block diagram Tire modeling is
the first step to investigate vehicle handling stability The model was
developed based on a set of mathematical 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 Carsim 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-mail: safwan_burhaumudin@yahoo.com)
Pakharuddin Mohd Samin, is with the Department of Automotive
Engineering, Faculty of Mechanical Engineering, Universiti Teknologi
Malaysia (UTM), 81300, Skudai, Johor, Malaysia (e-mail:
pakhar@fkm.utm.my)
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: hishamj@fkm.utm.my)
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: roslan@fkm.utm.my)
Syabillah Sulaiman, is with the Department of Automotive Engineering,
Faculty of Mechanical Engineering, Universiti Teknologi Malaysia (UTM),
81300, Skudai, Johor, Malaysia (e-mail: syabillahsulaiman@gmail.com)
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 others Different tire model used different approach and equations 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
v v
v
vehicle
vehicle wheel
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
Trang 2typically 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] 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 [6] 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 introduce coefficient S v and S h 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
C OEFFICIENT I N M AGIC F ORMULA
Symbol Quantity
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 S v The subscript x represents condition along x-axis Slip ratio, κ, is the input of F xo as given by
.
arctan arctan
sin
SVx
x Bx x
Bx
E x x Bx
C x Dx
F xo
Lateral force for pure slip, F yo consists of coefficients B, C,
D, E and S v The subscript y represents condition along y-axis Slip angle, α, is the input to F yo as given by
.
arctan arctan
sin
SVy
y
B y y
B y
E y y
B y
C y
D y
F yo
Aligning moment for pure slip, M zo, is the product of
pneumatic trail, t o with lateral force, F yo, as given by
to F yo M zro.
M zo (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
Dt
Aligning moment due to camber angle, M zro, as given by
Dr
Longitudinal force for combined slip, F x, is the product of
factor G α with pure longitudinal force, F xo, as given by
.
F xo
G x
F x (8)
Factor G α represents increment or decrement factor when slip angle is introduced in the presence of slip ratio as given by
Trang 3
S
B x S
Bx
E x S Bx
C x
arctan arctan
and
S Hx
B x
S Hx
B x
E x
S Hx
B x
C x
Lateral force for combined slip, F y, is the product of factor
G κ with the pure lateral force, F yo, as given by
.
SVy
F yo
G y
F y (11)
Factor G κ represents increment or decrement factor when slip
ratio is introduced in presence of slip angle as given by
S
B y S
B y
E y S
B y
C y
arctan arctan
and
S Hy
B y
S Hy
B y
E y
S Hy
B y
C y
Aligning moment for combined slip, M z, as shown by
t F y' DrcosCrarctanBr r,eq s F x.
The aligning moment, Mz, 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
Ct Btt eq Et Btt eq Btt eq
Dt
and
Ro
s 0 1 (16)
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, F z,
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 Formula 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 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 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 for 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 2 Section 3
Fz
μ
α
к
Fxo, Fx Fyo, Fy Mzo, Mz
Trang 4Fig 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 suspension 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 computed 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 the 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
Trang 5From 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 the Magic Formula tire can be
used to represent actual tire dynamic behavior under any
maneuver
ACKNOWLEDGMENT 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 Department of Automotive, Faculty of
Mechanical Engineering, Universiti Teknologi Malaysia
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International Symposium on Intelligent Information Technology and
Security Informatics, pp 280-284, 2 April 2010
[2] N Xu, D Lu, 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, 12 August 2011
[3] E Bakker, L Nyborg, 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