Combined slip 2DOF wheel with disc, drum, or mapped brake
Vehicle Dynamics Blockset / Wheels and Tires
The Combined Slip Wheel 2DOF block implements the longitudinal and lateral behavior of a wheel characterized by the Magic Formula^{[1] and [2]}. You can import your own tire data or use fitted tire data sets provided by the Global Center for Automotive Performance Simulation (GCAPS). Use the block in driveline and vehicle simulations where low frequency tireroad and braking forces are required to determine vehicle acceleration, braking, and wheelrolling resistance. The block is suitable for applications that require combined lateral slip, for example, in lateral motion and yaw stability studies.
Based on the driveline torque, brake pressure, road height, wheel camber angle, and inflation pressure, the block determines the wheel rotation rate, vertical motion, forces, and moments in all six degrees of freedom (DOF). Use the vertical DOF to study tiresuspension resonances from road profiles or chassis motion.
Use the Tire type parameter to select the source of the tire data.
Goal  Action 

Implement the Magic Formula using empirical equations^{1 and 2}. The equations use fitting coefficients that correspond to the block parameters.  Update the block parameters with fitting coefficients from a file:

Implement fitted tire data sets provided by the Global Center for Automotive Performance Simulation (GCAPS).  Update the applicable block parameters with GCAPS fitted tire data:

Use the Brake Type parameter to select the brake.
Goal  Brake Type Setting 

No braking 

Implement brake that converts the brake cylinder pressure into a braking force 

Implement simplex drum brake that converts the applied force and brake geometry into a net braking torque 

Implement lookup table that is a function of the wheel speed and applied brake pressure 

The block calculates the inertial response of the wheel subject to:
Axle losses
Brake and drive torque
Tire rolling resistance
Ground contact through the tireroad interface
To implement the Magic Formula, the block uses these equations.
Calculation  Equations 

Longitudinal force  Tire and Vehicle Dynamics^{[2]} equations 4.E9 through 4.E57 
Lateral force  pure sideslip  Tire and Vehicle Dynamics^{[2]} equations 4.E19 through 4.E30 
Lateral force  combined slip  Tire and Vehicle Dynamics^{[2]} equations 4.E58 through 4.E67 
Vertical dynamics  Tire and Vehicle Dynamics^{[2]} equations 4.E68, 4.E1, 4.E2a, and 4.E2b 
Overturning couple  Tire and Vehicle Dynamics^{[2]} equation 4.E69 
Rolling resistance 

Aligning moment  Tire and Vehicle Dynamics^{[2]} equation 4.E31 through 4.E49 
Aligning torque  combined slip  Tire and Vehicle Dynamics^{[2]} equation 4.E71 through 4.E78 
The input torque is the summation of the applied axle torque, braking torque, and moment arising from the combined tire torque.
$${T}_{i}={T}_{a}{T}_{b}+{T}_{d}$$
For the moment arising from the combined tire torque, the block implements tractive wheel forces and rolling resistance with firstorder dynamics. The rolling resistance has a time constant parameterized in terms of a relaxation length.
$${T}_{d}(s)=\frac{1}{\frac{\left\omega \right{R}_{e}}{{L}_{e}}s+1}({F}_{x}\text{}{R}_{e}+{M}_{y})$$
If the brakes are enabled, the block determines the braking locked or unlocked condition based on an idealized dry clutch friction model. Based on the lockup condition, the block implements these friction and dynamic models.
If  Lockup Condition  Friction Model  Dynamic Model 

$\begin{array}{l}\omega \ne 0\\ \text{or}\\ {T}_{S}<\left{T}_{i}+{T}_{f}\omega b\right\end{array}$  Unlocked  $$\begin{array}{l}{T}_{f}={T}_{k}\\ \text{where,}\\ {T}_{k}={F}_{c}{R}_{eff}{\mu}_{k}\mathrm{tanh}\left[4\left({\omega}_{d}\right)\right]\\ {T}_{s}={F}_{c}{R}_{eff}{\mu}_{s}\\ {R}_{eff}=\frac{2({R}_{o}{}^{3}{R}_{i}{}^{3})}{3({R}_{o}{}^{2}{R}_{i}{}^{2})}\end{array}$$  $$\dot{\omega}J=\omega b+{T}_{i}+{T}_{o}$$ 
$\begin{array}{l}\omega =0\\ \text{and}\\ {T}_{S}\ge \left{T}_{i}+{T}_{f}\omega b\right\end{array}$  Locked  $${T}_{f}={T}_{s}$$  $$\omega =0$$ 
The equations use these variables.
ω  Wheel angular velocity 
a  Velocity independent force component 
b  Linear velocity force component 
c  Quadratic velocity force component 
L_{e}  Tire relaxation length 
J  Moment of inertia 
M_{y}  Rolling resistance torque 
T_{a}  Applied axle torque about wheel spin axis 
T_{b}  Braking torque 
T_{d}  Combined tire torque 
T_{f}  Frictional torque 
T_{i}  Net input torque 
T_{k}  Kinetic frictional torque 
T_{o}  Net output torque 
T_{s}  Static frictional torque 
F_{c}  Applied clutch force 
F_{x}  Longitudinal force developed by the tire road interface due to slip 
R_{eff}  Effective clutch radius 
R_{o}  Annular disk outer radius 
R_{i}  Annular disk inner radius 
R_{e}  Effective tire radius while under load and for a given pressure 
V_{x}  Longitudinal axle velocity 
F_{z}  Vehicle normal force 
ɑ  Tire pressure exponent 
β  Normal force exponent 
p_{i}  Tire pressure 
μ_{s}  Coefficient of static friction 
μ_{k}  Coefficient of kinetic friction 
To resolve the forces and moments, the block uses the ZUp orientation of the tire and wheel coordinate systems.
Tire coordinate system axes (X_{T}, Y_{T}, Z_{T}) are fixed in a reference frame attached to the tire. The origin is at the tire contact with the ground.
Wheel coordinate system axes (X_{W}, Y_{W}, Z_{W}) are fixed in a reference frame attached to the wheel. The origin is at the wheel center.
ZUp Orientation^{[1]}
If you specify the Brake Type parameter Disc
,
the block implements a disc brake. This figure shows the side and front views of a disc brake.
A disc brake converts brake cylinder pressure from the brake cylinder into force. The disc brake applies the force at the brake pad mean radius.
The block uses these equations to calculate brake torque for the disc brake.
$T=\{\begin{array}{c}\frac{\mu P\pi {B}_{a}{}^{2}{R}_{m}{N}_{pads}}{4}\text{when}N\ne 0\\ \frac{{\mu}_{static}P\pi {B}_{a}{}^{2}{R}_{m}{N}_{pads}}{4}\text{when}N=0\end{array}$
$$Rm=\frac{Ro+Ri}{2}$$
The equations use these variables.
T  Brake torque 
P  Applied brake pressure 
N  Wheel speed 
N_{pads}  Number of brake pads in disc brake assembly 
μ_{static}  Disc padrotor coefficient of static friction 
μ  Disc padrotor coefficient of kinetic friction 
B_{a}  Brake actuator bore diameter 
R_{m}  Mean radius of brake pad force application on brake rotor 
R_{o}  Outer radius of brake pad 
R_{i}  Inner radius of brake pad 
If you specify the Brake Type parameter Drum
,
the block implements a static (steadystate) simplex drum brake. A simplex drum brake
consists of a single twosided hydraulic actuator and two brake shoes. The brake shoes
do not share a common hinge pin.
The simplex drum brake model uses the applied force and brake geometry to calculate a net torque for each brake shoe. The drum model assumes that the actuators and shoe geometry are symmetrical for both sides, allowing a single set of geometry and friction parameters to be used for both shoes.
The block implements equations that are derived from these equations in Fundamentals of Machine Elements.
$\begin{array}{l}{T}_{rshoe}=\left(\frac{\pi \mu cr(\mathrm{cos}{\theta}_{2}\mathrm{cos}{\theta}_{1}){B}_{a}{}^{2}}{2\mu (2r\left(\mathrm{cos}{\theta}_{2}\mathrm{cos}{\theta}_{1})+a\left({\mathrm{cos}}^{2}{\theta}_{2}{\mathrm{cos}}^{2}{\theta}_{1}\right)\right)+ar\left(2{\theta}_{1}2{\theta}_{2}+\mathrm{sin}2{\theta}_{2}\mathrm{sin}2{\theta}_{1}\right)}\right)P\\ \\ {T}_{lshoe}=\left(\frac{\pi \mu cr(\mathrm{cos}{\theta}_{2}\mathrm{cos}{\theta}_{1}){B}_{a}{}^{2}}{2\mu (2r\left(\mathrm{cos}{\theta}_{2}\mathrm{cos}{\theta}_{1})+a\left({\mathrm{cos}}^{2}{\theta}_{2}{\mathrm{cos}}^{2}{\theta}_{1}\right)\right)+ar\left(2{\theta}_{1}2{\theta}_{2}+\mathrm{sin}2{\theta}_{2}\mathrm{sin}2{\theta}_{1}\right)}\right)P\end{array}$
$T=\{\begin{array}{c}{T}_{rshoe}+{T}_{lshoe}\text{when}N\ne 0\\ ({T}_{rshoe}+{T}_{lshoe})\frac{{\mu}_{static}}{\mu}\text{when}N=0\end{array}$
The equations use these variables.
T  Brake torque 
P  Applied brake pressure 
N  Wheel speed 
μ_{static}  Disc padrotor coefficient of static friction 
μ  Disc padrotor coefficient of kinetic friction 
T_{rshoe}  Right shoe brake torque 
T_{lshoe}  Left shoe brake torque 
a  Distance from drum center to shoe hinge pin center 
c  Distance from shoe hinge pin center to brake actuator connection on brake shoe 
r  Drum internal radius 
B_{a}  Brake actuator bore diameter 
Θ_{1}  Angle from shoe hinge pin center to start of brake pad material on shoe 
Θ_{2}  Angle from shoe hinge pin center to end of brake pad material on shoe 
If you specify the Brake Type parameter
Mapped
, the block uses a lookup table to determine the brake
torque.
$T=\{\begin{array}{c}{f}_{brake}(P,N)\text{when}N\ne 0\\ \left(\frac{{\mu}_{static}}{\mu}\right){f}_{brake}(P,N)\text{when}N=0\end{array}$
The equations use these variables.
T  Brake torque 
${f}_{brake}^{}(P,N)$  Brake torque lookup table 
P  Applied brake pressure 
N  Wheel speed 
μ_{static}  Friction coefficient of drum padface interface under static conditions 
μ  Friction coefficient of disc padrotor interface 
The lookup table for the brake torque, ${f}_{brake}^{}(P,N)$, is a function of applied brake pressure and wheel speed, where:
T is brake torque, in N·m.
P is applied brake pressure, in bar.
N is wheel speed, in rpm.
[1] Besselink, Igo, Antoine J. M. Schmeitz, and Hans B. Pacejka, "An improved Magic Formula/Swift tyre model that can handle inflation pressure changes," Vehicle System Dynamics  International Journal of Vehicle Mechanics and Mobility 48, sup. 1 (2010): 337–52, https://doi.org/10.1080/00423111003748088.
[2] Pacejka, H. B. Tire and Vehicle Dynamics. 3rd ed. Oxford, United Kingdom: SAE and ButterworthHeinemann, 2012.
[3] Schmid, Steven R., Bernard J. Hamrock, and Bo O. Jacobson. Fundamentals of Machine Elements, SI Version. 3rd ed. Boca Raton: CRC Press, 2014.
Combined Slip Wheel 2DOF CPI  Combined Slip Wheel 2DOF STI  Fiala Wheel 2DOF  Longitudinal Wheel
^{[1]} Reprinted with permission Copyright © 2008 SAE International. Further distribution of this material is not permitted without prior permission from SAE.