Fiala Wheel 2DOF
Fiala wheel 2DOF wheel with disc, drum, or mapped brake
 Library:
Vehicle Dynamics Blockset / Wheels and Tires
Description
The Fiala Wheel 2DOF block implements a simplified tire with lateral and longitudinal slip capability based on the E. Fiala model^{[1]}. The block uses a translational friction model to calculate the forces and moments during combined longitudinal and lateral slip, requiring fewer parameters than the Combined Slip Wheel 2DOF block. If you do not have the tire coefficients needed by the Magic Formula, consider using this block for studies that do not involve extensive nonlinear combined lateral slip or lateral dynamics. If your study does require nonlinear combined slip or lateral dynamics, consider using the Combined Slip Wheel 2DOF block.
The block determines the wheel rotation rate, vertical motion, and forces and moments in all six degreesoffreedom (DOFs) based on the driveline torque, brake pressure, road height, wheel camber angle, and inflation pressure. You can use this block for these types of analyses:
Driveline and vehicle simulations that require low frequency tireroad and braking forces for vehicle acceleration, braking, and wheel rolling resistance calculations with minimal tire parameters.
Wheel interaction with an idealized road surface.
Ride and handling maneuvers for vehicles undergoing mild combined slip. For this analysis, you can connect the block to driveline and chassis components such as differentials, suspension, and vehicle body systems.
Yaw stability. For this analyses, you can connect this block to more detailed braking system models.
Tire stiffness and unsprung mass interactions with ground variations, load transfer, or chassis motion using the block vertical DOF.
The block integrates rotational wheel, vertical mass, and braking dynamics models. For the slipdependent tire forces and moments, the block implements the Fiala tire model.
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 

To calculate the rolling resistance torque, specify one of these Rolling Resistance parameters.
Setting  Block Implementation 

 None 
 Method in Stepwise Coastdown Methodology for Measuring Tire Rolling Resistance. The rolling resistance is a function of tire pressure, normal force, and velocity. 
 Method specified in ISO 28580:2018, Passenger car, truck and bus tyre rolling resistance measurement method — Single point test and correlation of measurement results. 
 Magic formula equations from 4.E70 in Tire and Vehicle Dynamics. The magic formula is an empirical equation based on fitting coefficients. 
 Lookup table that is a function of the normal force and spin axis longitudinal velocity. 
To calculate vertical motion, specify one of these Vertical Motion parameters.
Setting  Block Implementation 

 Block passes the applied chassis forces directly through to the rolling resistance and longitudinal force calculations. 
 Vertical motion depends on wheel stiffness and damping. Stiffness is a function of tire sidewall displacement and pressure. Damping is a function of tire sidewall velocity and pressure. 
Rotational Wheel Dynamics
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
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.
$${T}_{d}(s)=\frac{1}{\frac{{L}_{e}}{\left\omega \right{R}_{e}}s+1}$$
To calculate the rolling resistance torque, you can specify one of these Rolling Resistance parameters.
Setting  Block Implementation 

 Block sets rolling resistance,

 Block uses the method in SAE Stepwise Coastdown Methodology for Measuring Tire Rolling Resistance. The rolling resistance is a function of tire pressure, normal force, and velocity. Specifically, $${M}_{y}={R}_{e}\{a+b\left{V}_{x}\right+c{V}_{x}{}^{2}\}\left\{{F}_{z}{}^{\beta}{p}_{i}{}^{\alpha}\right\}\mathrm{tanh}\left(4{V}_{x}\right)$$

 Block uses the method specified in ISO 28580:2018, Passenger car, truck and bus tyre rolling resistance measurement method — Single point test and correlation of measurement results. The method accounts for normal load, parasitic loss, and thermal corrections from test conditions. Specifically, $${M}_{y}={R}_{e}(\frac{{F}_{z}{C}_{r}}{1+{K}_{t}({T}_{amb}{T}_{meas})}{F}_{pl})\mathrm{tanh}(\omega )$$ 
 Block calculates the rolling resistance,

 For the rolling resistance,

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  Velocityindependent 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 
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 
C_{r}  Rolling resistance constant 
T_{amb}  Ambient temperature 
T_{meas}  Measured temperature for rolling resistance constant 
F_{pl}  Parasitic force loss 
K_{t}  Thermal correction factor 
ɑ  Tire pressure exponent 
β  Normal force exponent 
p_{i}  Tire pressure 
μ_{s}  Coefficient of static friction 
μ_{k}  Coefficient of kinetic friction 
Longitudinal Force
The block implements the longitudinal force as a function of wheel slip relative to the road surface using these equations.
Calculation  Equation 

Critical slip  $$\kappa {\text{'}}_{Critical}=\left\frac{\mu {F}_{z}}{2{C}_{\kappa}}\right$$ 
Longitudinal force  $${F}_{x}=\{\begin{array}{c}{C}_{k}\kappa \text{'}\text{when}\left{\kappa}^{\text{'}}\right\le {\kappa}^{\text{'}}{}_{Critical}\\ \text{tanh}(4\kappa \text{'})\left(\mu \left{F}_{z}\right\left\frac{{\left(\mu {F}_{z}\right)}^{2}}{4\kappa \text{'}{C}_{\kappa}}\right\right)\text{when}\left{\kappa}^{\text{'}}\right{\kappa}^{\text{'}}{}_{Critical}\end{array}$$ 
Friction coefficient  $\mu =({\mu}_{s}({\mu}_{s}{\mu}_{k})\text{}{\kappa}_{k\alpha}){\lambda}_{\mu}$ 
Slip coefficient  ${\kappa}_{k\alpha}=\sqrt{\kappa {\text{'}}^{2}+{\text{tan}}^{2}(\alpha \text{'})}$ 
The equations use these variables.
κ'  Slip state 
F_{x}  Longitudinal force acting on axle along tirefixed xaxis, 
C_{κ}  Longitudinal stiffness 
F_{z}  Vertical contact patch normal force along tirefixed zaxis, 
μ  Friction coefficient 
μ_{s}  Coefficient of static friction 
μ_{k}  Coefficient of kinetic friction 
κ_{ka}  Comprehensive slip coefficient 
α'  Slip angle state 
λ_{μ}  Friction scaling 
Lateral Force
The block implements the lateral force as a function of wheel slip angle state using these equations.
Calculation  Equation 

Critical slip angle  $\alpha {\text{'}}_{Critical}=\text{atan}(\frac{3\mu \left{F}_{z}\right}{{C}_{a}})$ 
Lateral force  $$\begin{array}{l}{F}_{y}=\{\begin{array}{c}\text{tanh}(4\alpha \text{'})\mu \left{F}_{z}\right\text{when}\left\alpha \text{'}\right\alpha {\text{'}}_{Critical}\\ \mathrm{tanh}\left(4\alpha \text{'}\right)\mu \left{F}_{z}\right\left(1{\xi}^{3}\right)+\gamma {C}_{\gamma}\text{when}\left\alpha \text{'}\right\le \alpha {\text{'}}_{Critical}\end{array}\\ \\ \xi =1\frac{{C}_{a}\left\text{tan}(\alpha \text{'})\right}{3\mu \left{F}_{z}\right}\end{array}$$ 
The equations use these variables.
α'  Slip angle state 
F_{y}  Lateral force acting on axle along tirefixed yaxis, 
F_{z}  Vertical contact patch normal force along tirefixed zaxis 
C_{ɣ}  Camber stiffness 
C_{α}  Lateral stiffness per slip angle 
μ  Friction coefficient 
Vertical Dynamics
For the vertical dynamics, the block implements these equations.
Calculation  Equation 

Vertical response  $\ddot{z}m={F}_{ztire}+mgFz$ 
Tire normal force  ${F}_{ztire}={\rho}_{z}kb\dot{z}$ 
Vertical sidewall deflection  ${\rho}_{z}={z}_{gnd}z,z\ge 0$ 
The equations use these variables.
z  Tire deflection along tirefixed zaxis 
z_{gnd}  Ground displacement along tirefixed zaxis 
F_{ztire}  Tire normal force along tirefixed zaxis 
F_{z}  Vertical force acting on axle along tirefixed zaxis 
ρ_{z}  Vertical sidewall deflection along tirefixed zaxis 
k  Vertical sidewall stiffness 
b  Vertical sidewall damping 
Overturning, Aligning, and Scaling
This table summarizes the overturning, aligning, and scaling implementation.
Calculation  Implementation 

Overturning moment  The Fiala model does not define an overturning moment. The block implements this equation, requiring minimal parameters. ${M}_{x}={F}_{y}{R}_{e}\text{cos}(\gamma )$ 
Aligning moment  The block implements the aligning moment as a combination of yaw rate damping and slip angle state. $$\begin{array}{l}{M}_{z}=\{\begin{array}{c}\dot{\psi}{b}_{{M}_{z}}\text{when}\left\alpha \text{'}\right\alpha {\text{'}}_{Critical}\\ \mathrm{tanh}\left(4{\alpha}^{\text{'}}\right)w\mu \left{\text{F}}_{z}\right\left(1\xi \right){\xi}^{3}+\dot{\psi}{b}_{{M}_{z}}\text{when}\left\alpha \text{'}\right\le \alpha {\text{'}}_{Critical}\end{array}\\ \\ \xi =1\frac{{C}_{a}\left\text{tan}(\alpha \text{'})\right}{3\mu \left{F}_{z}\right}\end{array}$$ 
Friction scaling  To vary the coefficient of friction, use the

The equations use these variables.
M_{x}  Overturning moment acting on axle about tirefixed xaxis 
M_{z}  Aligning moment acting on axle about tirefixed zaxis 
R_{e}  Effective contact patch to wheel carrier radial distance 
ɣ  Camber angle 
k  Vertical sidewall stiffness 
b  Vertical sidewall damping 
$\dot{\psi}$  Tire angular velocity about the tirefixed zaxis (yaw rate) 
w  Tire width 
α'  Slip angle state 
b_{Mz}  Linear yaw rate resistance 
F_{y}  Lateral force acting on axle along tirefixed yaxis 
C_{ɣ}  Camber stiffness 
C_{α}  Lateral stiffness per slip angle 
μ  Friction coefficient 
F_{z}  Vertical contact patch normal force along tirefixed zaxis 
Tire and Wheel Coordinate Systems
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}
Brakes
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.
Ports
Input
Output
Parameters
Model Examples
References
[1] Fiala, E. "Seitenkrafte am Rollenden Luftreifen." VDI Zeitschrift, V.D.I.. Vol 96, 1954.
[2] Highway Tire Committee. Stepwise Coastdown Methodology for Measuring Tire Rolling Resistance. Standard J2452_199906. Warrendale, PA: SAE International, June 1999.
[3] ISO 28580:2018. Passenger car, truck and bus tyre rolling resistance measurement method — Single point test and correlation of measurement results. ISO (International Organization for Standardization), 2018.
[4] Pacejka, H. B. Tire and Vehicle Dynamics. 3rd ed. Oxford, UK: SAE and ButterworthHeinemann, 2012.
Extended Capabilities
Version History
Introduced in R2019aSee Also
Combined Slip Wheel 2DOF  Combined Slip Wheel 2DOF CPI  Combined Slip Wheel 2DOF STI  Longitudinal Wheel
^{1} Reprinted with permission Copyright © 2008 SAE International. Further distribution of this material is not permitted without prior permission from SAE.