Standard Tire
Tire body used to determine the normal force and kinematic parameters required by the tire force and moment components
The Standard Tire component internally calculates the kinematic tire parameters and the tire normal force required by the Linear, Calspan, Fiala, and Pacejka force and moment components. For the UserDefined Tire component, these parameters are exported through the signal ports at the bottom of the Standard Tire component. The geometry is assumed to be a thin circular disk, which is common in automotive applications.
Note: The port of the Standard Tire Body component must be connected directly to the port of a tire force/moment component, as shown in the diagram below.
Connections
Name

Description

Color


Frame connecting the tire to revolute joint, which is assumed to be rotating about the Yaxis

Gray


Tire reference frame, which must be connected to a tire force/moment component

White


Real output signal. The spin rate of tire (rads/sec).

White


Real output signal. The inclination angle of tire (rads).

White


Real output signal. The loaded radius of tire (m).

White


Real output signal. The longitudinal slip of tire (m).

White


Real output signal. The slip angle of tire (rads).

White


3x1 output array. The velocity of tire center expressed in the inertial frame (m/s).

White


Real output signal. The tire normal force (N).

White



Tire Body Details
In the Standard Tire model, is assumed to be parallel with the inertial zaxis and the spin axis () is assumed to be about the positive yaxis of of the Standard Tire model (that is, the leftmost multibody frame on the Standard Tire icon). The and axes are calculated using the following vector algebra equations:
= , = x
The inclination angle γ, which is needed to compute the tire forces and moments resulting from camber effects, is computed as:
γ = arcsin ( ( x ) · )
The position vector from the tire center, C, to the point of contact, P, between the tire and ground is needed to determine the moments about C resulting from tire forces acting at P. To compute this vector, the tire is represented by a thin circular disk with a radius equal to the unloaded tire radius. The unit vector is then obtained as:
= x
The distance from C to the road plane along gives the loaded radius of the tire, :
=
where rC· gives the height of the tire center above the road plane, with rC being the global position of the tire center.
The angular velocity of the tire ωC is assumed to be the vector sum of the yaw rate vector ωZ, the inclination rate ωX, and the spin rate vector Ω uS. The spin rate of the tire, Ω, can be defined as:
Ω = (ωC  ωZ  ωX) ·
The slip angle α is also needed to compute tire forces and moments. The MapleSim Tire Library uses the ISO standard definition to compute α, which represents the angle from the direction of the tire velocity to the direction of the tire heading:
α = − arctan ( )
where VxC and VyP are the and components of the translational velocity of C and P, respectively. If the inclination angle is not varying too rapidly, you can obtain an approximation of the slip angle by using the alternate expression:
α = − arctan ( )
where VyC is the component of the velocity of the tire center C. Since the velocity of C is easier to compute than the velocity of P, this alternate expression may lead to faster simulations without affecting modeling fidelity. To select this option, choose Tire Center, instead of the default ISO Standard option for the Calc parameter of the Standard Tire.
The longitudinal slip, ε, is calculated as follows:
ε =
where VxC, the component of the translational velocity of C, is the forward speed of the tire. The effective tire radius is defined as the forward speed divided by the spin rate, for a freerolling wheel. Thus, the longitudinal slip is zero for a freerolling wheel. During braking, the wheel spins slower than the free rolling condition and longitudinal slip is negative. During acceleration, the longitudinal slip is positive.
The MapleSim Tire Library offers three options for calculating the effective rolling radius, : the unloaded tire radius (), the loaded tire radius (), and a more complex formula suggested by Pacejka[1]that computes an effective rolling radius that lies between the values of the loaded and unloaded radii:
= − DRe arctan(BRe ) + FRe
where:
Fz0: nominal value of normal tire force
BRe, Dre, Fre: tire parameters determined from experiments
and is the tire deflection at the nominal value of the normal tire force Fz0, and ρ is the timevarying value of the tire deflection:
ρ = −  rCP  = −
where Runloaded is the unloaded radius of the tire and rCP is the vector from frame C to P.
The normal force () of the tire is computed from the tire stiffness (), damping (), deflection of the tire (ρ), and tire center speed along the axis () as follows:
Tire transients can be added through the inclusion of the "relaxation lengths" proposed by Bernard and Clover [2]. In this approach, the previous definitions of longitudinal slip (ε) and lateral slip angle (α) are replaced by the firstorder ordinary differential equations:
= −
= −
where Blong and Blat are the longitudinal and lateral relaxation lengths, respectively. For steadystate conditions, the above equations reduce to the previous expressions for longitudinal slip and slip angle presented above. Note that Vy is the component of the translational velocity of either P or C, depending upon which slip angle calculation is used (ISO Standard versus Tire Center).
Not only do the firstorder ordinary differential equations allow for tire transients to be included in the model, they also allow for the simulation of maneuvers involving zero forward speeds, for which the previous definitions would fail due to a division by zero. You can include the above formulas for constant values of relaxation length by setting the Time Lag parameter of the Standard Tire component to Constant and specifying numeric values for [Blong, Blat].
Alternatively, you can use a more complex model in which the deforming tire carcass in the contact patch is represented as a "stretched string." With this model, timevarying values of Blong and Blat are computed and combined with the differential equations for the tire transients.
There are many stretched string models in the literature [1]; the MapleSim Tire Library uses a model proposed by the developers of MSC.Adams [3]. This model is accessed by selecting the StretchedStringA (SSA) option from the Time Lag dropdown list and filling in numeric values for [Fz0, R0, LSkappa, LSalpha, PTx1, PTx2, PTx3, PTy1, PTy2, PKy3].
Fz0: nominal value of normal tire force
R0: radius of unloaded tire
L___: scale factors
P___: coefficients of magic formula type model
Complete definitions of the StretchedStringA parameters, and expressions for the computed values of Blong and Blat, are given in Reference [3].
Parameters
Symbol

Default

Units

Description

ID


304000


Tire stiffness

Kspring


500


Tire damping

Kdamper




Unloaded radius

Rtire

Calc

ISO Standard



Determines how the slip angle of the tire is calculated

SlipAngle

Calc

Loaded Radius



Determines how the effective rolling radius of the tire is calculated

EffRollRad

Params

[5900,8,0.24,0.01]



Parameters for the calculating the effective rolling radius. This option requires the use of the Pacejka tire. This option can be used only if Calc = Pacejka, otherwise the parameter is not shown.

ReffPacejka

Time Lag (TL)

None



The way in which which timelag will be used in the tire model

TimeLag

TL Const. Params

[Blong, Blat]



Parameters for constant time lag calculation. This option can be used only if Time Lag = Constant.

TLConstant

TL SSA Params

[5900,0.355,1,1,2.3657,1.4112,0.56626,2.1439,1.9829,0.90729]



Parameters for stretched string A time lag calculation. This option can be used only if Time Lag = Stretched String A.

TLSSA

m

28


The mass of the tire

Mass




The inertia matrix for the tire, expressed in the

Inertia





Indicates whether the initial velocity is expressed in the inboard (gray) or outboard (white) frame.

VelType





Indicates the sequence of bodyfixed rotations used to describe the initial orientation of the center of mass frame. For example, refers to sequential rotations about the x, then y, and then z axis (123  Euler angles)

RotType





Indicates whether the initial angular velocity is expressed in the inboard (gray) or outboard (white) frame. If Euler is selected, the initial angular velocities are assumed to be the direct derivatives of the Euler angles.

AngVelType


Ignore



Indicates whether MapleSim ignores, tries to enforce, or strictly enforces the translational initial conditions

MechTranTree




Initial displacement of the center of mass frame at the start of the simulation. These values are expressed along the x, y and zaxis of the inboard (gray) frame respectively

InitPos




Initial velocity of the center of mass frame at the start of the simulation. These values are expressed along the x, y and zaxis of the inboard (gray) frame respectively

InitVel


Ignore



Indicates whether MapleSim ignores, tries to enforce, or strictly enforces the rotational initial conditions

MechRotTree




Initial rotation of the center of mass frame at the start of the simulation, based on the parameter values

InitAng




Initial velocity of the center of mass frame at the start of the simulation, based on the parameter values

InitAngVel



References
1

H.B. Pacejka, Tire and Vehicle Dynamics, SAE International, 2002.

2

J.E. Bernard and C.L. Clover, "Tire Modelling for Lowspeed and Highspeed Calculations", SAE Technical Paper Series, (950311), 1995.

3

User's Manual, MSC.Adams 2005, Release 1. (MSC.Adams is a registered trademark of MSC.Software Corporation).

4 C. Schmitke, K. Morency, and J. McPhee, "Using Graph Theory and Symbolic Computing to Generate Efficient Models for Multibody Vehicle Dynamics", IMechE Journal of Multibody Dynamics, Vol. 222, pp.339352, 2008.
See Also
Linear, Calspan, Fiala, Pacejka, UserDefined
