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  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 

Tire Body Details

In the Standard Tire model,  is assumed to be parallel with the inertial z-axis and the spin axis () is assumed to be about the positive y-axis 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 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,  : 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 time-varying 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 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  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

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

References

4 C. Schmitke, K. Morency, and J. McPhee, "Using Graph Theory and Symbolic Computing to Generate Efficient Models for Multi-body Vehicle Dynamics", IMechE Journal of Multi-body Dynamics, Vol. 222, pp.339-352, 2008.

See Also

Linear, Calspan,  Fiala, Pacejka, User-Defined