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 User-Defined 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 $\mathrm{hubFrame}$ port of the Standard Tire Body component must be connected directly to the $\mathrm{hubFrame}$ port of a tire force/moment component, as shown in the diagram below.

Connections

 Name Description Color ${\mathrm{frame}}_{a}$ Frame connecting the tire to revolute joint, which is assumed to be rotating about the Y-axis Gray $\mathrm{hubFrame}$ Tire reference frame, which must be connected to a tire force/moment component White $\mathrm{SpinRate}$ Real output signal. The spin rate of tire (rads/sec). White $\mathrm{Inclination}$ Real output signal. The inclination angle of tire (rads). White $\mathrm{Rloaded}$ Real output signal. The loaded radius of tire (m). White $\mathrm{LongSlip}$ Real output signal. The longitudinal slip of tire (m). White $\mathrm{SlipAngle}$ Real output signal. The slip angle of tire (rads). White $\mathrm{TireVel}$ 3x1 output array. The velocity of tire center expressed in the inertial frame (m/s). White $\mathrm{Fz}$ Real output signal. The tire normal force (N). White

Tire Body Details

In the Standard Tire model, ${Z}_{\mathrm{ISO}}$ is assumed to be parallel with the inertial z-axis and the spin axis (${u}_{\mathrm{Spin}}$) is assumed to be about the positive y-axis of ${\mathrm{frame}}_{a}$ of the Standard Tire model (that is, the leftmost multibody frame on the Standard Tire icon). The  ${X}_{\mathrm{ISO}}$ and  ${Y}_{\mathrm{ISO}}$ axes are calculated using the following vector algebra equations:

${X}_{\mathrm{ISO}}$ =  ,  = ${Z}_{\mathrm{ISO}}$ x ${X}_{\mathrm{ISO}}$

The inclination angle γ, which is needed to compute the tire forces and moments resulting from camber effects, is computed as:

γ = arcsin ( (${Y}_{\mathrm{ISO}}$ x ${u}_{\mathrm{Spin}}$) · ${X}_{\mathrm{ISO}}$ )

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 ${u}_{\mathrm{CP}}$ is then obtained as:

${u}_{\mathrm{CP}}$= ${u}_{\mathrm{Spin}}$ x ${X}_{\mathrm{ISO}}$

The distance from C to the road plane along ${u}_{\mathrm{CP}}$ gives the loaded radius of the tire, ${R}_{\mathrm{loaded}}$:

${R}_{\mathrm{loaded}}$ =

where rC·${Z}_{\mathrm{ISO}}$ 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 wC is assumed to be the vector sum of the yaw rate vector wZ, the inclination rate wX, and the spin rate vector Ω uS.  The spin rate of the tire,  Ω, can be defined as:

Ω =  (wC - wZ - wX) · ${u}_{\mathrm{Spin}}$

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 ${X}_{\mathrm{ISO}}$ and ${Y}_{\mathrm{ISO}}$ 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 ${Y}_{\mathrm{ISO}}$ 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 ${\theta }_{slip}$ Calc parameter of the Standard Tire.

The longitudinal slip, ε, is calculated as follows:

ε =  $\frac{\mathrm{Ω}{R}_{\mathrm{eff}}-\mathrm{VxC}}{\left|\mathrm{VxC}\right|}$

where VxC, the ${X}_{\mathrm{ISO}}$ component of the translational velocity of C,  is the forward speed of the tire. The effective tire radius ${R}_{\mathrm{eff}}$ is defined as the forward speed divided by the spin rate, for a free-rolling wheel. Thus, the longitudinal slip is zero for a free-rolling 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, ${R}_{\mathrm{eff}}$ : the unloaded tire radius (${R}_{\mathrm{unloaded}}$), the loaded tire radius (${R}_{\mathrm{loaded}}$), 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:

$\mathrm{MapleSim}/\mathrm{Tire}{R}_{\mathrm{eff}}$ =   ${R}_{\mathrm{unloaded}}$ − 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 time-varying value of the tire deflection:

r = ${R}_{\mathrm{unloaded}}$ −  | rCP | = ${R}_{\mathrm{unloaded}}$ −  ${R}_{\mathrm{loaded}}$

where Runloaded is the unloaded radius of the tire and rCP is the vector from frame C to P.

The normal force (${F}_{z}$) of the tire is computed from the tire stiffness (${K}_{\mathrm{stiffness}}$), damping (${K}_{\mathrm{damping}}$), deflection of the tire (ρ), and tire center speed along the ${Z}_{\mathrm{ISO}}$ axis ($\mathrm{VzC}$) as follows:

${F}_{z}={K}_{\mathrm{stiffness}}\mathrm{ρ}-{K}_{\mathrm{damping}}\mathrm{VzC}$

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 first-order ordinary differential equations:

=     −

=     −

where Blong and Blat are the longitudinal and lateral relaxation lengths, respectively. For steady-state conditions, the above equations reduce to the previous expressions for longitudinal slip and slip angle presented above.  Note that Vy is the ${Y}_{\mathrm{ISO}}$ 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 first-order 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, time-varying 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 drop-down list and filling in numeric values for [Fz0, R0, LSkappa, LSalpha, PTx1, PTx2, PTx3, PTy1, PTy2, PKy3].

Fz0:  nominal value of normal tire force

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 ${K}_{\mathrm{stiffness}}$ 304000 $\frac{N}{m}$ Tire stiffness Kspring ${K}_{\mathrm{damper}}$ 500 $\frac{\mathrm{Ns}}{m}$ Tire damping Kdamper ${r}_{unloaⅆⅇⅆ}$ $0.355$ $m$ Unloaded radius Rtire ${\theta }_{slip}$ Calc ISO Standard - Determines how the slip angle of the tire is calculated SlipAngle ${r}_{ⅇff}$ Calc Loaded Radius - Determines how the effective rolling radius of the tire is calculated EffRollRad ${r}_{ⅇff}$ 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 ${r}_{ⅇff}$ Calc = Pacejka, otherwise the parameter is not shown. ReffPacejka Time Lag (TL) None - The way in which which time-lag 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 $\mathrm{kg}$ The mass of the tire Mass $\stackrel{}{\left[I\right]}$ $\left[\begin{array}{ccc}\frac{78}{100}& 0& 0\\ 0& \frac{156}{100}& 0\\ 0& 0& \frac{78}{100}\end{array}\right]$ $\mathrm{kg}\cdot {m}^{2}$ The inertia matrix for the tire, expressed in the $\mathrm{hubFrame}$ Inertia ${\mathrm{Type}}_{v}$ $\mathrm{Inboard}$ - Indicates whether the initial velocity is expressed in the inboard (gray) or outboard (white) frame. VelType ${\mathrm{Type}}_{\mathrm{θ}}$ $\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$ - Indicates the sequence of body-fixed rotations used to describe the initial orientation of the center of mass frame. For example, $\left[1,2,3\right]$ refers to sequential rotations about the x, then y, and then z axis (123 - Euler angles) RotType ${\mathrm{Type}}_{\mathrm{ω}}$ $\mathrm{Euler}$ - 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 ${\mathrm{IC}}_{r,v}$ Ignore - Indicates whether MapleSim ignores, tries to enforce, or strictly enforces the translational initial conditions MechTranTree ${\stackrel{&conjugate0;}{r}}_{0}$ $\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ $m$ Initial displacement of the center of mass frame at the start of the simulation. These values are expressed along the x-, y- and z-axis of the inboard (gray) frame respectively InitPos ${\stackrel{&conjugate0;}{v}}_{0}$ $\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ $\frac{m}{s}$ Initial velocity of the center of mass frame at the start of the simulation. These values are expressed along the x-, y- and z-axis of the inboard (gray) frame respectively InitVel ${\mathrm{IC}}_{\mathrm{θ},\mathrm{ω}}$ Ignore - Indicates whether MapleSim ignores, tries to enforce, or strictly enforces the rotational initial conditions MechRotTree ${\stackrel{&conjugate0;}{\theta }}_{0}$ $\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ $\mathrm{rad}$ Initial rotation of the center of mass frame at the start of the simulation, based on the ${\mathrm{Type}}_{\mathrm{\theta }}$ parameter values InitAng ${\stackrel{&conjugate0;}{\omega }}_{0}$ $\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ $\frac{\mathrm{rad}}{s}$ Initial velocity of the center of mass frame at the start of the simulation, based on the ${\mathrm{Type}}_{\mathrm{ω}}$ 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 Low-speed and High-speed Calculations",  SAE Technical Paper Series, (950311), 1995.