Dugoff Wheel 2DOF

Dugoff Wheel 2DOF wheel with disc, drum, or mapped brake

Since R2023a

Libraries:
Vehicle Dynamics Blockset / Wheels and Tires

Description

The Dugoff Wheel 2DOF block implements a simplified tire with lateral and longitudinal slip capability based on the H. Dugoff 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 degrees-of-freedom (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 tire-road 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 analysis, 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.

You can set your own user-defined model parameter values or use a built-in tire model.

If you install the Extended Tire Features for Vehicle Dynamics Blockset support package, you can click the Plot steady state force, moment response button to generate these plots:

Lateral force [N] vs Slip angle [rad]
Self-aligning moment [Nm] vs Slip angle [rad]
Longitudinal force [N] vs Longitudinal slip []
Longitudinal force [N] vs Lateral force [N]

With the support package, you can also import tire parameter values defined in the Dugoff Wheel 2DOF block to a tireModel object or export tire parameter values from a tireModel object to the Dugoff Wheel 2DOF block. For more information, see tireModel.get and set.

Use the Tire type parameter to either input tire parameter values or select a fitted tire parameter set.

Goal Action

Goal	Action
Input user-defined tire parameter values.	Update the block parameters with user-defined parameter values: Set Tire type to `User defined`. In the Wheel and Tire Parameters section, input user-defined values. Click Apply.
Select one of the built-in Dugoff tire models to drive the lateral and longitudinal calculations. [Link].	Update the applicable block parameters with values from a built-in tire model: Set Tire type to the tire that you want to implement. Options include: `Light passenger car 205/60R15` `Light passenger car 245/60R16` `Mid-size passenger car 235/45R18` `Performance car 225/40R19` `SUV 265/50R20` `Light truck 275/65R18` `Commercial truck 295/75R22.5` Click Update block with applicable tire values. In the Wheel and Tire Parameters section, the block updates the applicable Longitudinal and Lateral parameters. Click Apply.

Input user-defined tire parameter values.

Update the block parameters with user-defined parameter values:

Set Tire type to User defined.
In the Wheel and Tire Parameters section, input user-defined values.
Click Apply.

Select one of the built-in Dugoff tire models to drive the lateral and longitudinal calculations. [Link].

Update the applicable block parameters with values from a built-in tire model:

Set Tire type to the tire that you want to implement. Options include:
- Light passenger car 205/60R15
- Light passenger car 245/60R16
- Mid-size passenger car 235/45R18
- Performance car 225/40R19
- SUV 265/50R20
- Light truck 275/65R18
- Commercial truck 295/75R22.5
Click Update block with applicable tire values. In the Wheel and Tire Parameters section, the block updates the applicable Longitudinal and Lateral parameters.
Click Apply.

Use the Model slip type parameter to select slip type.

Action	Model Slip Type Setting
Calculate longitudinal and lateral forces under nominal slip conditions	`Nominal slip`
Calculate longitudinal and lateral forces with additional correction factors for a more accurate response at higher slip values	`Extended slip`

Use the Brake Type parameter to select the brake.

Action	Brake Type Setting
No braking	`None`
Implement brake that converts the brake cylinder pressure into a braking force	`Disc`
Implement simplex drum brake that converts the applied force and brake geometry into a net braking torque	`Drum`
Implement lookup table that is a function of the wheel speed and applied brake pressure	`Mapped`

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

Setting	Block Implementation
`None`	None
`Pressure and velocity`	Method in Stepwise Coastdown Methodology for Measuring Tire Rolling Resistance. The rolling resistance is a function of tire pressure, normal force, and velocity.
`ISO 28580`	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`	Magic formula equations from 4.E70 in Tire and Vehicle Dynamics. The magic formula is an empirical equation based on fitting coefficients.
`Mapped torque`	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
`None`	Block passes the applied chassis forces directly through to the rolling resistance and longitudinal force calculations.
`Mapped stiffness and damping`	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.
`External deflection`	The block uses the defined sidewall deflection directly in the effective radius calculation.

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 tire-road 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 first-order dynamics. The rolling resistance has a time constant parameterized in terms of a relaxation length.

$T_{d} (s) = \frac{1}{\frac{L_{e}}{| ω | R_{e}} s + 1} (F_{x} R_{e} + M_{y})$

To calculate the rolling resistance torque, you can specify one of these Rolling Resistance parameters.

Setting	Block Implementation
`None`	Block sets rolling resistance, `M_y`, to zero.
`Pressure and velocity`	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: $M_{y} = R_{e} {a + b \| V_{x} \| + c V_{x}^{2}} {F_{z}^{β} p_{i}^{α}} \tanh (4 V_{x})$
`ISO 28580`	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: $M_{y} = R_{e} (\frac{F_{z} C_{r}}{1 + K_{t} (T_{a m b} - T_{m e a s})} - F_{p l}) \tanh (ω)$
`Magic Formula`	Block calculates the rolling resistance, `M_y`, using the Magic Formula equations from 4.E70 in Tire and Vehicle Dynamics. The magic formula is an empirical equation based on fitting coefficients.
`Mapped torque`	For the rolling resistance, `M_y`, the block uses a lookup table that is a function of the normal force and spin axis longitudinal velocity.

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 lock-up condition, the block implements these friction and dynamic models.

Equation	Lock-Up Condition	Friction Model	Dynamic Model
$\begin{array}{l} ω \neq 0 \\ or \\ T_{S} < \| T_{i} + T_{f} - ω b \| \end{array}$	Unlocked	$\begin{array}{l} T_{f} = T_{k}, \\ where \\ T_{k} = F_{c} R_{e f f} μ_{k} \tanh [4 (- ω_{d})] \\ T_{s} = F_{c} R_{e f f} μ_{s} \\ R_{e f f} = \frac{2 (R_{o}^{3} - R_{i}^{3})}{3 (R_{o}^{2} - R_{i}^{2})} \end{array}$	$\dot{ω} J = - ω b + T_{i} + T_{o}$
$\begin{array}{l} ω = 0 \\ and \\ T_{S} \geq \| T_{i} + T_{f} - ω b \| \end{array}$	Locked	$T_{f} = T_{s}$	$ω = 0$

The equations use these variables.

Variable	Value
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
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
Nominal Slip	$\begin{array}{l} F_{x} = \frac{C_{k} κ}{1 - κ} f (z), \\ where \\ f (z) = {\begin{cases} z (2 - z) when z < 1 \\ 1 when z \geq 1 \end{cases} \\ z = \frac{μ F_{z} (1 - κ)}{2 \sqrt{{(C_{k} κ)}^{2} + {(C_{α} \tan (α))}^{2}}} \end{array}$
Extended	$\begin{array}{l} F_{x} = \frac{C_{k} κ}{1 - κ} f (z) g_{x}, \\ where \\ f (z) = {\begin{cases} z (2 - z) when z < 1 \\ 1 when z \geq 1 \end{cases} \\ z = \frac{μ F_{z} (1 - κ)}{2 \sqrt{{(C_{k} κ)}^{2} + {(C_{α} \tan (α))}^{2}}} \\ g_{x} = (g_{x 1} + g_{x 2} μ) κ^{2} - (g_{x 3} + g_{x 4} μ) κ + g_{x 5} \end{array}$
Friction coefficient	$\begin{array}{l} μ = μ_{0} (1 - A_{s} V_{s}), \\ where \\ V_{s} = u \sqrt{κ^{2} + \tan^{2} (α)} \end{array}$

The equations use these variables.

Variable	Value
F_x	Longitudinal force acting on axle along tire-fixed x-axis
C_κ	Longitudinal stiffness
C_α	Lateral stiffness per slip angle
k	Longitudinal slip ratio of tires
F_z	Vertical contact patch normal force along tire-fixed z-axis
u	Velocity component in the wheel plane
μ	Maximum friction coefficient
μ₀	Maximum friction scaling coefficient
A_s	Friction reduction factor
V_s	Friction reduction magnitude
α	Side slip angle of tires
g_x	Longitudinal correction factor
g_x1	Longitudinal squared slip correction factor
g_x2	Longitudinal squared slip friction correction factor
g_x3	Longitudinal linear slip correction factor
g_x4	Longitudinal linear slip friction correction factor
g_x5	Longitudinal offset correction factor

Lateral Force

The block implements the lateral force as a function of wheel slip angle state using these equations.

Calculation	Equation
Nominal Slip	$\begin{array}{l} F_{y} = \frac{C_{α} \tan (α)}{1 - κ} f (z) + γ C_{γ}, \\ where \\ f (z) = {\begin{cases} z (2 - z) when z < 1 \\ 1 when z \geq 1 \end{cases} \\ z = \frac{μ F_{z} (1 - κ)}{2 \sqrt{{(C_{k} κ)}^{2} + {(C_{α} \tan (α))}^{2}}} \end{array}$
Extended Slip	$\begin{array}{l} F_{y} = \frac{C_{α} \tan (α)}{1 - κ} f (z) g_{y} + γ C_{γ}, \\ where \\ f (z) = {\begin{cases} z (2 - z) when z < 1 \\ 1 when z \geq 1 \end{cases} \\ z = \frac{μ F_{z} (1 - κ)}{2 \sqrt{{(C_{k} κ)}^{2} + {(C_{α} \tan (α))}^{2}}} \\ g_{y} = (μ + g_{y 1}) \tan (α) + g_{y 2} \end{array}$
Friction Coefficient	$\begin{array}{l} μ = μ_{0} (1 - A_{s} V_{s}), \\ where \\ V_{s} = u \sqrt{κ^{2} + \tan^{2} (α)} \end{array}$

The equations use these variables.

Variable	Value
α	Side slip angle of tires
F_y	Lateral force acting on axle along tire-fixed y-axis
F_z	Vertical contact patch normal force along tire-fixed z-axis
ɣ	Camber angle
C_ɣ	Camber stiffness
C_α	Lateral stiffness per angle slip
C_k	Longitudinal stiffness
k	Longitudinal slip ratio of tires
u	Velocity component in the wheel plane
μ	Maximum friction coefficient
μ₀	Maximum friction scaling coefficient
V_s	Friction reduction magnitude
A_s	Friction reduction factor
g_y	Lateral correction factor
g_y1	Lateral maximum friction correction factor
g_y2	Lateral offset correction factor

Vertical Dynamics

The block implements these equations for the vertical dynamics.

Calculation	Equation
Vertical response	$\ddot{z} m = F_{z t i r e} + m g - F z$
Tire normal force	$F_{z t i r e} = ρ_{z} k - b \dot{z}$
Vertical sidewall deflection	$ρ_{z} = z_{g n d} - z, z \geq 0$

The equations use these variables.

Variable	Value
z	Tire deflection along tire-fixed z-axis
z_gnd	Ground displacement along tire-fixed z-axis
F_ztire	Tire normal force along tire-fixed z-axis
F_z	Vertical force acting on axle along tire-fixed z-axis
ρ_z	Vertical sidewall deflection along tire-fixed z-axis
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 Dugoff model does not define an overturning moment. The block implements this equation, requiring minimal parameters. $M_{x} = F_{y} R_{e} cos (γ)$
Aligning moment	The block implements the aligning moment as a combination of yaw rate damping and slip angle state. $\begin{array}{l} M_{z} = {\begin{matrix} \dot{ψ} b_{M_{z}} when \| α' \| > α'_{C r i t i c a l} \\ \tanh (4 α^{'}) w μ \| F_{z} \| (1 - ξ) ξ^{3} + \dot{ψ} b_{M_{z}} when \| α' \| \leq α'_{C r i t i c a l} \end{matrix} \\ ξ = 1 - \frac{C_{a} \| tan (α') \|}{3 μ \| F_{z} \|} \end{array}$
Friction scaling	To vary the coefficient of friction, use the ScaleFctr input port.

Calculation

Implementation

Overturning moment

The Dugoff model does not define an overturning moment. The block implements this equation, requiring minimal parameters.

$M_{x} = F_{y} R_{e} cos (γ)$

Aligning moment

The block implements the aligning moment as a combination of yaw rate damping and slip angle state.

$\begin{array}{l} M_{z} = {\begin{matrix} \dot{ψ} b_{M_{z}} when | α' | > α'_{C r i t i c a l} \\ \tanh (4 α^{'}) w μ | F_{z} | (1 - ξ) ξ^{3} + \dot{ψ} b_{M_{z}} when | α' | \leq α'_{C r i t i c a l} \end{matrix} \\ ξ = 1 - \frac{C_{a} | tan (α') |}{3 μ | F_{z} |} \end{array}$

Friction scaling

To vary the coefficient of friction, use the ScaleFctr input port.

The equations use these variables.

Variable	Value
M_x	Overturning moment acting on axle about tire-fixed x-axis
M_z	Aligning moment acting on axle about tire-fixed z-axis
R_e	Effective contact patch to wheel carrier radial distance
ɣ	Camber angle
k	Vertical sidewall stiffness
b	Vertical sidewall damping
$\dot{ψ}$	Tire angular velocity about the tire-fixed z-axis (yaw rate)
w	Tire width
α'	Slip angle state
b_Mz	Linear yaw rate resistance
F_y	Lateral force acting on axle along tire-fixed y-axis
C_ɣ	Camber stiffness
C_α	Lateral stiffness per slip angle
μ	Friction coefficient
F_z	Vertical contact patch normal force along tire-fixed z-axis

Tire and Wheel Coordinate Systems

To resolve the forces and moments, the block uses the Z-Up 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.

Z-Up Orientation¹

Z-Up tire and wheel coordinate systems showing wheel plane and road plane

Brakes

Disc

If you specify the Brake Type parameter as Disc, the block implements a disc brake. This figure shows the side and front views of a disc brake.

Front and side view of disc brake, showing pad, disc, and caliper

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.

$T = {\begin{matrix} \frac{μ P π B_{a}^{2} R_{m} N_{p a d s}}{4} when N \neq 0 \\ \frac{μ_{s t a t i c} P π B_{a}^{2} R_{m} N_{p a d s}}{4} when N = 0 \end{matrix}$

$R m = \frac{R o + R i}{2}$

The equations use these variables.

Variable	Value
T	Brake torque
P	Applied brake pressure
N	Wheel speed
N_pads	Number of brake pads in disc brake assembly
μ_static	Disc pad-rotor coefficient of static friction
μ	Disc pad-rotor 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

Drum

If you specify the Brake Type parameter as Drum, the block implements a static (steady-state) simplex drum brake. A simplex drum brake consists of a single two-sided 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.

$\begin{array}{l} T_{r s h o e} = (\frac{π μ c r (\cos θ_{2} - \cos θ_{1}) B_{a}^{2}}{2 μ (2 r (\cos θ_{2} - \cos θ_{1}) + a (\cos^{2} θ_{2} - \cos^{2} θ_{1})) + a (2 θ_{1} - 2 θ_{2} + \sin 2 θ_{2} - \sin 2 θ_{1})}) P \\ T_{l s h o e} = (\frac{π μ c r (\cos θ_{2} - \cos θ_{1}) B_{a}^{2}}{- 2 μ (2 r (\cos θ_{2} - \cos θ_{1}) + a (\cos^{2} θ_{2} - \cos^{2} θ_{1})) + a (2 θ_{1} - 2 θ_{2} + \sin 2 θ_{2} - \sin 2 θ_{1})}) P \end{array}$

$T = {\begin{matrix} T_{r s h o e} + T_{l s h o e} when N \neq 0 \\ (T_{r s h o e} + T_{l s h o e}) \frac{μ_{s t a t i c}}{μ} when N = 0 \end{matrix}$

Side view of drum brake

The equations use these variables.

Variable	Value
T	Brake torque
P	Applied brake pressure
N	Wheel speed
μ_static	Disc pad-rotor coefficient of static friction
μ	Disc pad-rotor 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
Θ₁	Angle from shoe hinge pin center to start of brake pad material on shoe
Θ₂	Angle from shoe hinge pin center to end of brake pad material on shoe

Mapped

If you specify the Brake Type parameter as Mapped, the block uses a lookup table to determine the brake torque.

$T = {\begin{matrix} f_{b r a k e} (P, N) when N \neq 0 \\ (\frac{μ_{s t a t i c}}{μ}) f_{b r a k e} (P, N) when N = 0 \end{matrix}$

The equations use these variables.

Variable	Value
T	Brake torque
$f_{b r a k e}^{} (P, N)$	Brake torque lookup table
P	Applied brake pressure
N	Wheel speed
μ_static	Friction coefficient of drum pad-face interface under static conditions
μ	Friction coefficient of disc pad-rotor interface

The lookup table for the brake torque, $f_{b r a k e}^{} (P, N)$ , 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.

Plot of brake torque as a function of wheel speed and applied brake pressure

Examples

Double Lane Change Reference Application

Simulate a full vehicle dynamics model undergoing a double lane change maneuver standard ISO 3888-2. Use for vehicle dynamics ride and handling analysis and chassis controls development, including yaw stability and lateral acceleration limits.

Open Script

Ports

Input

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BrkPrs — Brake pressure
scalar | N-by-`1` vector

Brake pressure, in Pa.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Dependencies

To enable this port, set the Brake Type parameter, to one of these types:

Disc
Drum
Mapped

AxlTrq — Axle torque
scalar | N-by-`1` vector

Axle torque, T_a, about wheel spin axis, in N·m.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Vx — Longitudinal velocity
scalar | N-by-`1` vector

Axle longitudinal velocity, V_x, along tire-fixed x-axis, in m/s.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Vy — Lateral velocity
scalar | N-by-`1` vector

Axle lateral velocity, V_y, along tire-fixed y-axis, in m/s.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Camber — Inclination angle
scalar | N-by-`1` vector

Camber angle, ɣ, or inclination angle, ε, in rad.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

YawRate — Tire angular velocity
scalar | N-by-`1` vector

Tire angular velocity, r, about the tire-fixed z-axis (yaw rate), in rad/s.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Prs — Tire inflation pressure
scalar | N-by-`1` vector

Tire inflation pressure, p_i, in Pa.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Gnd — Ground displacement
scalar | N-by-`1` vector

Ground displacement along tire-fixed z-axis, in m. Positive input produces wheel lift.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Fext — Axle force applied to tire
scalar | N-by-`1` vector

Axle force applied to tire, F_ext, along vehicle-fixed z-axis (positive input compresses the tire), in N.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Dependencies

To enable this parameter, set Vertical Motion to None or Mapped stiffness and damping.

RadialDeflct — Tire radial deflection
scalar | N-by-`1` vector

Tire radial deflection, RadialDeflct. This value is used directly in the effective radius calculation.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Dependencies

To enable this port, set Vertical Motion to External Deflection.

ScaleFctrs — Scale factor
scalar | N-by-`1` vector

Scale factor to account for variations in the coefficient of friction.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Output

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Info — Block data
bus

Block data, returned as a bus signal containing these block values.

Signal	Description	Units
`AxlTrq`	Axle torque about wheel-fixed y-axis	N·m
`Omega`	Wheel angular velocity about wheel-fixed y-axis	rad/s
`Fx`	Longitudinal vehicle force along tire-fixed x-axis	N
`Fy`	Lateral vehicle force along tire-fixed y-axis	N
`Fz`	Vertical vehicle force along tire-fixed z-axis	N
`Mx`	Overturning moment about tire-fixed x-axis	N·m
`My`	Rolling resistance torque about tire-fixed y-axis	N·m
`Mz`	Aligning moment about tire-fixed z-axis	N·m
`Vx`	Vehicle longitudinal velocity along tire-fixed x-axis	m/s
`Vy`	Vehicle lateral velocity along tire-fixed y-axis	m/s
`Re`	Loaded effective radius	m
`Kappa`	Longitudinal slip ratio	NA
`Alpha`	Side slip angle	rad
`a`	Contact patch half length	m
`b`	Contact patch half width	m
`Gamma`	Camber angle	rad
`psidot`	Tire angular velocity about the tire-fixed z-axis (yaw rate)	rad/s
`BrkTrq`	Brake torque about the vehicle-fixed y-axis	N·m
`BrkPrs`	Brake pressure	Pa
`z`	Axle vertical displacement along tire-fixed z-axis	m
`zdot`	Axle vertical velocity along tire-fixed z-axis	m/s
`Gnd`	Ground displacement along tire-fixed z-axis (positive input produces wheel lift)	m
`GndFz`	Vertical sidewall force on ground along tire-fixed z-axis	N
`Prs`	Tire inflation pressure	Pa
`WhlTrq`	Wheel torque	N·m
`RL`	Loaded radius	m
`RadialDeflct`	Tire radial deflection	m

Omega — Wheel angular velocity
scalar | N-by-`1` vector

Wheel angular velocity, ω, about wheel-fixed y-axis, in rad/s.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Fx — Longitudinal axle force
scalar | N-by-`1` vector

Longitudinal force acting on axle, F_x, along tire-fixed x-axis, in N. Positive force acts to move the vehicle forward.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Fy — Lateral axle force
scalar | N-by-`1` vector

Lateral force acting on axle, F_y, along tire-fixed y-axis, in N.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Fz — Vertical axle force
scalar | N-by-`1` vector

Vertical force acting on axle, F_z, along tire-fixed z-axis, in N.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Mx — Overturning moment
scalar | N-by-`1` vector

Longitudinal moment acting on axle, M_x, about tire-fixed x-axis, in N·m.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

My — Rolling resistive moment
scalar | N-by-`1` vector

Lateral moment acting on axle, M_y, about tire-fixed y-axis, in N·m.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Mz — Aligning moment
scalar | N-by-`1` vector

Vertical moment acting on axle, M_z, about tire-fixed z-axis, in N·m.

Vector is the number of wheels, N, by 1. If you provide a scalar value, the block assumes that number of wheels is one.

Parameters

expand all

Block Options

Tire type — Select type
`User defined` (default) | `Light passenger car 205/60R15` | `Light passenger car 245/60R16` | `Mid-size passenger car 235/45R18` | `Performance car 225/40R19` | `SUV 265/50R20` | `Light truck 275/65R18` | `Commercial truck 295/75R22.5`

Use the Tire type parameter to either input tire parameter values or select a fitted tire parameter set.

Goal Action

Goal	Action
Input user-defined tire parameter values.	Update the block parameters with user-defined parameter values: Set Tire type to `User defined`. In the Wheel and Tire Parameters section, input user-defined values. Click Apply.
Select one of the built-in Dugoff tire models to drive the lateral and longitudinal calculations. [Link].	Update the applicable block parameters with values from a built-in tire model: Set Tire type to the tire that you want to implement. Options include: `Light passenger car 205/60R15` `Light passenger car 245/60R16` `Mid-size passenger car 235/45R18` `Performance car 225/40R19` `SUV 265/50R20` `Light truck 275/65R18` `Commercial truck 295/75R22.5` Click Update block with applicable tire values. In the Wheel and Tire Parameters section, the block updates the applicable Longitudinal and Lateral parameters. Click Apply.

Input user-defined tire parameter values.

Update the block parameters with user-defined parameter values:

Set Tire type to User defined.
In the Wheel and Tire Parameters section, input user-defined values.
Click Apply.

Select one of the built-in Dugoff tire models to drive the lateral and longitudinal calculations. [Link].

Update the applicable block parameters with values from a built-in tire model:

Set Tire type to the tire that you want to implement. Options include:
- Light passenger car 205/60R15
- Light passenger car 245/60R16
- Mid-size passenger car 235/45R18
- Performance car 225/40R19
- SUV 265/50R20
- Light truck 275/65R18
- Commercial truck 295/75R22.5
Click Update block with applicable tire values. In the Wheel and Tire Parameters section, the block updates the applicable Longitudinal and Lateral parameters.
Click Apply.

Model slip type — Model slip type
`Nominal slip` | `Extended slip`

Use the Model slip type parameter to select slip type.

Action	Model Slip Type Setting
Calculate longitudinal and lateral forces under nominal slip conditions	`Nominal slip`
Calculate longitudinal and lateral forces with additional correction factors for a more accurate response at higher slip values	`Extended slip`

Dependencies

Setting Model slip type to Extended slip enables these parameters:

Setting	Parameters Enabled
`Extended slip`	Longitudinal squared slip correction factor, gx1 Longitudinal squared slip friction correction factor, gx2 Longitudinal linear slip correction factor, gx3 Longitudinal linear slip friction correction factor, gx4 Longitudinal offset correction factor, gx5 Lateral maximum friction correction factor, gy1 Lateral offset correction factor, gy2

Brake type — Brake type
`None` | `Disc` | `Drum` | `Mapped`

Use the Brake Type parameter to select the brake.

Action	Brake Type Setting
No braking	`None`
Implement brake that converts the brake cylinder pressure into a braking force	`Disc`
Implement simplex drum brake that converts the applied force and brake geometry into a net braking torque	`Drum`
Implement lookup table that is a function of the wheel speed and applied brake pressure	`Mapped`

Rolling Resistance — Rolling resistance torque
`None` (default) | `Pressure and velocity` | `ISO 28580` | `Magic Formula` | `Mapped torque`

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

Setting	Block Implementation
`None`	None
`Pressure and velocity`	Method in Stepwise Coastdown Methodology for Measuring Tire Rolling Resistance. The rolling resistance is a function of tire pressure, normal force, and velocity.
`ISO 28580`	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`	Magic formula equations from 4.E70 in Tire and Vehicle Dynamics. The magic formula is an empirical equation based on fitting coefficients.
`Mapped torque`	Lookup table that is a function of the normal force and spin axis longitudinal velocity.

Dependencies

Each Rolling Resistance setting enables additional parameters.

Setting	Parameters Enabled
`Pressure and velocity`	Velocity independent force coefficient, aMy Linear velocity force component, bMy Quadratic velocity force component, cMy Tire pressure exponent, alphaMy Normal force exponent, betaMy
`ISO 28580`	Parasitic losses force, Fpl Rolling resistance constant, Cr Thermal correction factor, Kt Measured temperature, Tmeas Parasitic losses force, Fpl Ambient temperature, Tamb
`Magic Formula`	Rolling resistance torque coefficient, QSY Longitudinal force rolling resistance coefficient, QSY2 Linear rotational speed rolling resistance coefficient, QSY3 Quartic rotational speed rolling resistance coefficient, QSY4 Camber squared rolling resistance torque, QSY5 Load based camber squared rolling resistance torque, QSY6 Normal load rolling resistance coefficient, QSY7 Pressure load rolling resistance coefficient, QSY8 Rolling resistance scaling factor, lam_My
`Mapped torque`	Spin axis velocity breakpoints, VxMy Normal force breakpoints, FzMy Rolling resistance torque map, MyMap

Vertical Motion — Vertical motion model
`None` (default) | `Mapped stiffness and damping` | `External deflection`

To calculate vertical motion, specify one of these Vertical Motion parameters.

Setting	Block Implementation
`None`	Block passes the applied chassis forces directly through to the rolling resistance and longitudinal force calculations.
`Mapped stiffness and damping`	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.
`External deflection`	The block uses the defined sidewall deflection directly in the effective radius calculation.

Dependencies

Setting Vertical Motion to Mapped stiffness and damping enables these parameters:

Setting	Parameters Enabled
`Mapped stiffness and damping`	Wheel mass, MASS Initial tire displacement, zo Initial velocity, zdoto Initial wheel vertical velocity (wheel fixed frame), zdoto Vertical deflection breakpoints, zFz Pressure breakpoints, pFz Force due to deflection, Fzz Vertical velocity breakpoints, zdotFz Force due to velocity, Fzzdot

Longitudinal and Lateral

Longitudinal stiffness, Ckappa — Longitudinal stiffness
`1e7` (default) | scalar | N-by-`1` vector

Longitudinal stiffness, C_κ, specified as a scalar or N-by-1 vector, in N. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Lateral stiffness per slip angle, Calpha — Lateral stiffness
`4.5e4` (default) | scalar | N-by-`1` vector

Lateral stiffness per slip angle, C_α, specified as a scalar or N-by-1 vector, in N/rad. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Camber stiffness, Cgamma — Camber stiffness
`1e3` (default) | scalar | N-by-`1` vector

Camber stiffness, C_ɣ, specified as a scalar or N-by-1 vector, in N/rad. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Maximum friction scaling coefficient, mu0 — Maximum friction scaling coefficient
`0.8` (default) | scalar | N-by-`1` vector

Maximum friction scaling coefficient, μ₀, specified as a scalar or N-by-1 vector, dimensionless. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Friction reduction factor, As — Friction reduction factor
`0.01` (default) | scalar | N-by-`1` vector

Friction reduction factor, A_s, specified as a scalar or N-by-1 vector, dimensionless. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Longitudinal relaxation length, Lrelx — Longitudinal relaxation length
`0.05` (default) | scalar | N-by-`1` vector

Longitudinal relaxation length, L_relx, specified as a scalar or N-by-1 vector, in m. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Lateral relaxation length, Lrely — Lateral relaxation length
`0.15` (default) | scalar | N-by-`1` vector

Lateral relaxation length, L_rely, specified as a scalar or N-by-1 vector, in m/rad. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other longitudinal and lateral parameters.

N is the number of wheels and must match the input signal dimensions.

Extended slip

Lateral offset correction factor, gy2 — Lateral offset correction factor
`1.5` (default) | scalar

Lateral offset correction factor, g_y2, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Lateral maximum friction correction factor, gy1 — Lateral maximum friction correction factor
`-1.6` (default) | scalar

Lateral maximum friction correction factor, g_y1, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Longitudinal offset correction factor, gx5 — Longitudinal offset correction factor
`1.5` (default) | scalar

Longitudinal offset correction factor, g_x5, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Longitudinal linear slip friction correction factor, gx4 — Longitudinal linear slip friction correction factor
`-0.75` (default) | scalar

Longitudinal linear slip friction correction factor, g_x4, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Longitudinal linear slip correction factor, gx3 — Longitudinal linear slip correction factor
`1.63` (default) | scalar

Longitudinal linear slip correction factor, g_x3, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Longitudinal squared slip friction correction factor, gx2 — Longitudinal squared slip friction correction factor
`-0.75` (default) | scalar

Longitudinal squared slip friction correction factor, g_x2, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Longitudinal squared slip correction factor, gx1 — Longitudinal squared slip correction factor
`1.14` (default) | scalar

Longitudinal squared slip correction factor, g_x1, dimensionless.

Dependencies

To enable this parameter, set Tire type to Extended slip.

Rolling Resistance

Pressure and Velocity

Velocity independent force coefficient, aMy — Velocity-independent force coefficient
`84e-4` (default) | scalar

Velocity-independent force coefficient, a, dimensionless.

Dependencies

To enable this parameter, set Rolling Resistance to Pressure and velocity.

Linear velocity force component, bMy — Linear velocity force component
`6.2e-4` (default) | scalar

Linear velocity force component, b, in s/m.

Dependencies

To enable this parameter, set Rolling Resistance to Pressure and velocity.

Quadratic velocity force component, cMy — Quadratic velocity force component
`1.6e-4` (default) | scalar

Quadratic velocity force component, c, in s^2/m^2.

Dependencies

To enable this parameter, set Rolling Resistance to Pressure and velocity.

Tire pressure exponent, alphaMy — Tire pressure exponent
`-0.003` (default) | scalar

Tire pressure exponent, ɑ, dimensionless.

Dependencies

To enable this parameter, set Rolling Resistance to Pressure and velocity.

Normal force exponent, betaMy — Normal force exponent
`0.97` (default) | scalar

Normal force exponent, β, dimensionless.

Dependencies

To enable this parameter, set Rolling Resistance to Pressure and velocity.

ISO 28580

Parasitic losses force, Fpl — Parasitic force loss
`10` (default) | scalar

Parasitic force loss, F_pl, in N.

Dependencies

To enable this parameter, set Rolling Resistance to ISO 28580.

Rolling resistance constant, Cr — Rolling resistance constant
`1e-3` (default) | scalar

Rolling resistance constant, C_r, in N/kN. ISO 28580 specifies the rolling resistance unit as one newton of tractive resistance for every kilonewtons of normal load.

Dependencies

To enable this parameter, set Rolling Resistance to ISO 28580.

Thermal correction factor, Kt — Thermal correction factor
`0.008` (default) | scalar

Thermal correction factor, K_t, in 1/degC.

Dependencies

To enable this parameter, set Rolling Resistance to ISO 28580.

Measured temperature, Tmeas — Temperature during testing
`298.15` (default) | scalar

Measured ambient temperature, T_meas, near tire during tire testing, in K.

Dependencies

To enable this parameter, set Rolling Resistance to ISO 28580.

Ambient temperature, Tamb — Temperature in application environment
`298.15` (default) | scalar

Measured ambient temperature, T_amb, near tire in application environment, in K. For example, the measured ambient temperature is the ambient temperature near the tire when the vehicle is on the road.

Dependencies

To enable this parameter, set Rolling Resistance to ISO 28580.

Input ambient temperature — Option to input ambient temperature
`off` (default) | on

Select to create input port Tamb to input the measured ambient temperature.

The measured ambient temperature, T_amb, is the temperature near tire in application environment, in K. For example, the measured ambient temperature is the ambient temperature near the tire when the vehicle is on the road.

Dependencies

To enable this parameter, set Rolling Resistance to ISO 28580.

Magic Formula

Rolling resistance torque coefficient, QSY1 — Torque coefficient
`0.007` (default) | scalar

Rolling resistance torque coefficient, dimensionless.

Dependencies

To enable this parameter, set Rolling Resistance to Magic Formula.

Longitudinal force rolling resistance coefficient, QSY2 — Force resistance coefficient
`0` (default) | scalar

Longitudinal force rolling resistance coefficient, dimensionless.

Dependencies

To enable this parameter, set Rolling Resistance to Magic Formula.

Linear rotational speed rolling resistance coefficient, QSY3 — Linear speed coefficient
`0.0015` (default) | scalar

Linear rotational speed rolling resistance coefficient, dimensionless.

Dependencies

To enable this parameter, set Rolling Resistance to Magic Formula.

Quartic rotational speed rolling resistance coefficient, QSY4 — Quartic speed coefficient
`8.5e-05` (default) | scalar

Quartic rotational speed rolling resistance coefficient, dimensionless.

Dependencies

To enable this parameter, set Rolling Resistance to Magic Formula.

Camber squared rolling resistance torque, QSY5 — Camber resistance torque
`0` (default) | scalar

Camber squared rolling resistance torque, in 1/rad^2.

Dependencies

To enable this parameter, set Rolling Resistance to Magic Formula.

Load based camber squared rolling resistance torque, QSY6 — Load resistance torque
`0` (default) | scalar

Load based camber squared rolling resistance torque, in 1/rad^2.

Dependencies

To enable this parameter, set Rolling Resistance to Magic Formula.

Normal load rolling resistance coefficient, QSY7 — Normal resistance coefficient
`0.9` (default) | scalar

Normal load rolling resistance coefficient, dimensionless.

Dependencies

To enable this parameter, set Rolling Resistance to Magic Formula.

Pressure load rolling resistance coefficient, QSY8 — Pressure resistance coefficient
`-0.4` (default) | scalar

Pressure load rolling resistance coefficient, dimensionless.

Dependencies

To enable this parameter, set Rolling Resistance to Magic Formula.

Rolling resistance scaling factor, lam_My — Scaling factor
`1` (default) | scalar

Rolling resistance scaling factor, dimensionless.

Dependencies

To enable this parameter, set Rolling Resistance to Magic Formula.

Mapped

Spin axis velocity breakpoints, VxMy — Spin axis velocity breakpoints
`-20:1:20` (default) | vector

Spin axis velocity breakpoints, in m/s.

Dependencies

To enable this parameter, set Rolling Resistance to Mapped torque.

Normal force breakpoints, FzMy — Normal force breakpoints
`0:200:1e4` (default) | vector

Normal force breakpoints, in N.

Dependencies

To enable this parameter, set Rolling Resistance to Mapped torque.

Rolling resistance torque map, MyMap — Rolling resistance torque map
array

Rolling resistance torque versus axle speed and normal force, in N·m.

Dependencies

To enable this parameter, set Rolling Resistance to Mapped torque.

Aligning

Tire nominal section width, WIDTH — Tire nominal section width
scalar

Tire nominal section width, WIDTH, in m.

Linear yaw rate resistance, bMz — Linear yaw rate resistance
`0` | scalar

Linear yaw rate resistance, b_Mz, in N·m·s/rad.

Brake

Static friction coefficient, mu_static — Static friction coefficient
`0.3` (default) | scalar | N-by-`1` vector

Static friction coefficient, specified as a scalar or N-by-1 vector, dimensionless. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Disc, Drum, or Mapped.

Kinetic friction coefficient, mu_kinetic — Kinetic friction
`0.2` (default) | scalar | N-by-`1` vector

Kinematic friction coefficient, specified as a scalar or N-by-1 vector, dimensionless. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Disc, Drum, or Mapped.

Disc

Disc brake actuator bore, disc_abore — Bore distance
`0.05` (default) | scalar | N-by-`1` vector

Disc brake actuator bore, specified as a scalar or N-by-1 vector, in m. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Disc.

Brake pad mean radius, Rm — Radius
`0.177` (default) | scalar | N-by-`1` vector

Brake pad mean radius, specified as a scalar or N-by-1 vector, in m. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Disc.

Number of brake pads, num_pads — Number of brake pads
`2` (default) | scalar | N-by-`1` vector

Number of brake pads, specified as a scalar or N-by-1 vector, dimensionless. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Disc.

Drum

Drum brake actuator bore, disc_abore — Bore distance
`0.0508` (default) | scalar | N-by-`1` vector

Drum brake actuator bore, specified as a scalar or N-by-1 vector, in m. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other brake parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Brake Type to Drum.

Shoe pin to drum center distance, drum_a — Shoe pin to drum center distance
`0.123` (default) | scalar

Shoe pin to drum center distance, in m.

Dependencies

To enable this parameter, set Brake Type to Drum.

Shoe pin center to force application point distance, drum_c — Shoe pin center to force application point distance
`0.212` (default) | scalar

Shoe pin center to force application point distance, in m.

Dependencies

To enable this parameter, set Brake Type to Drum.

Drum internal radius, drum_r — Drum internal radius
`0.15` (default) | scalar

Drum internal radius, in m.

Dependencies

To enable this parameter, set Brake Type to Drum.

Shoe pin to pad start angle, drum_theta1 — Shoe pin to pad start angle
`0` (default) | scalar

Shoe pin to pad start angle, in deg.

Dependencies

To enable this parameter, set Brake Type to Drum.

Shoe pin to pad end angle, drum_theta2 — Shoe pin to pad end angle
`126` (default) | scalar

Shoe pin to pad end angle, in deg.

Dependencies

To enable this parameter, set Brake Type to Drum.

Mapped

Brake actuator pressure breakpoints, brake_p_bpt — Brake actuator pressure breakpoints
vector

Brake actuator pressure breakpoints, in bar.

Dependencies

To enable this parameter, set Brake Type to Mapped.

Wheel speed breakpoints, brake_n_bpt — Wheel speed breakpoints
vector

Wheel speed breakpoints, in rpm.

Dependencies

To enable this parameter, set Brake Type to Mapped.

Brake torque map, f_brake_t — Lookup table for brake torque
array

The lookup table for the brake torque, $f_{b r a k e}^{} (P, N)$ , 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.

Plot showing brake torque as a function of wheel speed and applied brake pressure

Dependencies

To enable this parameter, set Brake Type to Mapped.

Wheel

Rotational damping, br — Rotational damping
scalar | N-by-`1` vector

Rotational damping, specified as a scalar or N-by-1 vector, in N·m·s/rad. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other rotational parameters.

N is the number of wheels and must match the input signal dimensions.

Tire rotational inertia (rolling axis), IYY — Tire rotational inertia
scalar | N-by-`1` vector

Tire rotational inertia (rolling axis), specified as a scalar or N-by-1 vector, in kg·m². If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other rotational parameters.

N is the number of wheels and must match the input signal dimensions.

Initial wheel rotational velocity, omegao — Initial wheel rotational velocity
scalar | N-by-`1` vector

Initial wheel rotational velocity, specified as a scalar or N-by-1 vector, in rad/s. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other rotational parameters.

N is the number of wheels and must match the input signal dimensions.

Tire unloaded radius, UNLOADED_RADIUS — Tire unloaded radius
`0.309384029954441` (default) | scalar

Tire unloaded radius, in m.

Vertical

Tire mass, MASS — Tire mass
`9.46491996974568` (default) | scalar | N-by-`1` vector

Tire mass, specified as a scalar or N-by-1 vector, in kg. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other vertical parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Vertical Motion to Mapped stiffness and damping.

Initial tire displacement, zo — Initial tire displacement
`0` (default) | scalar | N-by-`1` vector

Initial tire displacement, specified as a scalar or N-by-1 vector, in m. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other vertical parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Vertical Motion to Mapped stiffness and damping.

Initial wheel vertical velocity (wheel fixed frame), zdoto — Initial wheel vertical velocity
`0` (default) | scalar | N-by-`1` vector

Initial wheel vertical velocity, specified as a scalar or N-by-1 vector, in m/s. If you specify a scalar, the block uses that value for all wheels. If you specify a vector, you must specify vectors for the other vertical parameters.

N is the number of wheels and must match the input signal dimensions.

Dependencies

To enable this parameter, set Vertical Motion to Mapped stiffness and damping.

Gravitational acceleration, GRAVITY — Gravitational acceleration
`-9.81` (default) | scalar

Gravitational acceleration, in m/s^2.

Dependencies

To enable this parameter, set Vertical Motion to Mapped stiffness and damping.

Mapped Stiffness and Damping

Vertical deflection breakpoints, zFz — Vertical deflection breakpoints
`[0 .01 .1]` (default) | vector

Vector of sidewall deflection breakpoints corresponding to the force table, in m.

Dependencies

To enable this parameter, set Vertical Motion to Mapped stiffness and damping.

Pressure breakpoints, pFz — Pressure breakpoints
`[10000 1000000]` (default) | vector

Vector of pressure data points corresponding to the force table, in Pa.

Dependencies

To enable this parameter, set Vertical Motion to Mapped stiffness and damping.

Force due to deflection, Fzz — Force due to deflection
`[0 1e3 1e4; 0 1e4 1e5]` (default) | vector

Force due to sidewall deflection and pressure along wheel-fixed z-axis, in N.

Dependencies

To enable this parameter, set Vertical Motion to Mapped stiffness and damping.

Vertical velocity breakpoints, zdotFz — Vertical velocity breakpoints
`[-20 0 20]` (default) | scalar

Vector of sidewall velocity breakpoints corresponding to the force due to velocity table, in m.

Dependencies

To enable this parameter, set Vertical Motion to Mapped stiffness and damping.

Force due to velocity, Fzzdot — Force due to velocity
`[500 0 -500;250 0 -250]` (default) | array

Force due to sidewall velocity and pressure along wheel-fixed z-axis, in N.

Dependencies

To enable this parameter, set Vertical Motion to Mapped stiffness and damping.

Simulation Setup

Maximum normal force, FZMAX — Maximum normal force
`10000` (default) | scalar

Maximum normal force, in N. Used with all vertical force calculations.

Minimum normal force, FZMIN — Minimum normal force
`0` (default) | scalar

Minimum normal force, in N. Used with all vertical force calculations.

Maximum pressure, PRESMAX — Maximum pressure
`1003118` (default) | scalar

Maximum pressure, PRESMAX, in Pa.

Minimum pressure, PRESMIN — Minimum pressure
`9982` (default) | scalar

Minimum pressure, PRESMIN, in Pa.

Max allowable slip ratio (absolute), KPUMAX — Max allowable slip ratio
`0.999` (default) | scalar

Max allowable slip ratio (absolute), KPUMAX, dimensionless.

Minimum allowable slip ratio (absolute), KPUMIN — Minimum allowable slip ratio
`-0.999` (default) | scalar

Minimum allowable slip ratio (absolute), KPUMIN, dimensionless.

Max allowable slip angle (absolute), ALPMAX — Max allowable slip angle
`1.5708` (default) | scalar

Max allowable slip angle (absolute), ALPMAX, in rad.

Minimum allowable slip angle (absolute), ALPMIN — Minimum allowable slip angle
`-1.5708` (default) | scalar

Minimum allowable slip angle (absolute), ALPMIN, in rad.

Maximum allowable camber angle, CAMMAX — Maximum allowable camber angle
`0.173` | scalar

Maximum allowable camber angle CAMMAX, in rad.

Minimum allowable camber angle, CAMMIN — Minimum allowable camber angle
`-0.173` | scalar

Minimum allowable camber angle, CAMMIN, in rad.

Minimum ambient temperature, TMIN — Minimum ambient temperature
`0` (default) | scalar

Minimum ambient temperature, T_MIN, in K.

Dependencies

To enable this parameter, set Rolling Resistance to ISO 28580.

Maximum ambient temperature, TMAX — Maximum ambient temperature
`400` (default) | scalar

Maximum ambient temperature, T_MAX, in K.

Dependencies

To enable this parameter, set Rolling Resistance to ISO 28580.

Plotting

Install Extended Tire Features — Option to install Extended Tire Features for Vehicle Dynamics Blockset™ support package
button

Click Install Extended Tire Features to install the Extended Tire Features for Vehicle Dynamics Blockset support package. With the support package, you can plot steady-state force and moment tire responses from the Dugoff Wheel 2DOF Block Parameters dialog box.

Plot steady state force, moment response — Option to plot steady-state force and moment tire responses
button

Click Plot steady state force, moment response to generate these plots:

Lateral force [N] vs Slip angle [rad]
Self-aligning moment [Nm] vs Slip angle [rad]
Longitudinal force [N] vs Longitudinal slip []
Longitudinal force [N] vs Lateral force [N]

Dependencies

To enable this parameter, click Install Extended Tire Features.

References

[1] Bhoraskar, A. and P. Sakthivel. "A Review and a Comparison of Dugoff and Modified Dugoff Formula with Magic Formula." 2017 International Conference on Nascent Technologies in Engineering (ICNTE)(2017): 1–4. https://doi.org/10.1109/ICNTE.2017.7947898.

[2] Highway Tire Committee. Stepwise Coastdown Methodology for Measuring Tire Rolling Resistance. Standard J2452_199906. Warrendale, PA: SAE International, June 1999.

[3] International Organization for Standardization. Passenger car, truck and bus tyre rolling resistance measurement method — Single point test and correlation of measurement results. ISO 28580: 2018. https://www.iso.org/standard/67531.html.

[4] Pacejka, H. B. Tire and Vehicle Dynamics, 3rd ed. Oxford, UK: SAE and Butterworth-Heinemann, 2012.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2023a

expand all

R2024b: New Tire type option

Starting in R2024b, you can specify Tire type to use the new built-in tire model, Light passenger car 245/60R16.

R2024b: New Vertical Motion option

Starting in R2024b, specifying Vertical Motion as External Deflection enables the RadialDeflct port. Use this port to define the sidewall deflection when interfacing with a tire in cases where forces are not provided.

R2024b: New signals added to the Info port bus

Starting in R2024b, the Info port bus contains these new signals.

Signal	Description
`RadialDeflct`	Tire radial deflection
`RL`	Loaded radius
`WhlTrq`	Wheel torque

R2024b: Renamed block parameters

Starting in R2024b, these parameters have been renamed.

Old Name	New Name
Wheel width	Tire nominal section width
Unloaded radius	Tire unloaded radius
Initial rotational velocity	Initial wheel rotational velocity
Wheel mass	Tire mass
Rotational inertia	Tire rotational inertia

R2024a: Block parameter update

Starting in R2024a, the options Nominal slip and Extended slip are available for the new parameter Model slip type instead of Tire type.

These options are now available for Tire type:

User defined
Light passenger car 205/60R15
Mid-size passenger car 235/45R18
Performance car 225/40R19
SUV 265/50R20
Light truck 275/65R18
Commercial truck 295/75R22.5

R2024a: Plot steady-state force and moment responses

If you have the Extended Tire Features for Vehicle Dynamics Blockset support package installed, you can use the new Plot steady state force, moment response button to generate plots.

Dugoff Wheel 2DOF

Description

Rotational Wheel Dynamics

Longitudinal Force

Lateral Force

Vertical Dynamics

Overturning, Aligning, and Scaling

Tire and Wheel Coordinate Systems

Brakes

Examples

Double Lane Change Reference Application

Ports

Input

BrkPrs — Brake pressure scalar | N-by-1 vector

Dependencies

AxlTrq — Axle torque scalar | N-by-1 vector

Vx — Longitudinal velocity scalar | N-by-1 vector

Vy — Lateral velocity scalar | N-by-1 vector

Camber — Inclination angle scalar | N-by-1 vector

YawRate — Tire angular velocity scalar | N-by-1 vector

Prs — Tire inflation pressure scalar | N-by-1 vector

Gnd — Ground displacement scalar | N-by-1 vector

Fext — Axle force applied to tire scalar | N-by-1 vector

Dependencies

RadialDeflct — Tire radial deflection scalar | N-by-1 vector

Dependencies

ScaleFctrs — Scale factor scalar | N-by-1 vector

Output

Info — Block data bus

Omega — Wheel angular velocity scalar | N-by-1 vector

Fx — Longitudinal axle force scalar | N-by-1 vector

Fy — Lateral axle force scalar | N-by-1 vector

Fz — Vertical axle force scalar | N-by-1 vector

Mx — Overturning moment scalar | N-by-1 vector

My — Rolling resistive moment scalar | N-by-1 vector

Mz — Aligning moment scalar | N-by-1 vector

Parameters

Block Options

Tire type — Select type User defined (default) | Light passenger car 205/60R15 | Light passenger car 245/60R16 | Mid-size passenger car 235/45R18 | Performance car 225/40R19 | SUV 265/50R20 | Light truck 275/65R18 | Commercial truck 295/75R22.5

Model slip type — Model slip type Nominal slip | Extended slip

Dependencies

Brake type — Brake type None | Disc | Drum | Mapped

Rolling Resistance — Rolling resistance torque None (default) | Pressure and velocity | ISO 28580 | Magic Formula | Mapped torque

Dependencies

Vertical Motion — Vertical motion model None (default) | Mapped stiffness and damping | External deflection

Dependencies

Longitudinal and Lateral

Longitudinal stiffness, Ckappa — Longitudinal stiffness 1e7 (default) | scalar | N-by-1 vector

Lateral stiffness per slip angle, Calpha — Lateral stiffness 4.5e4 (default) | scalar | N-by-1 vector

Camber stiffness, Cgamma — Camber stiffness 1e3 (default) | scalar | N-by-1 vector

Maximum friction scaling coefficient, mu0 — Maximum friction scaling coefficient 0.8 (default) | scalar | N-by-1 vector

Friction reduction factor, As — Friction reduction factor 0.01 (default) | scalar | N-by-1 vector

Longitudinal relaxation length, Lrelx — Longitudinal relaxation length 0.05 (default) | scalar | N-by-1 vector

Lateral relaxation length, Lrely — Lateral relaxation length 0.15 (default) | scalar | N-by-1 vector

Lateral offset correction factor, gy2 — Lateral offset correction factor 1.5 (default) | scalar

Dependencies

Lateral maximum friction correction factor, gy1 — Lateral maximum friction correction factor -1.6 (default) | scalar

Dependencies

Longitudinal offset correction factor, gx5 — Longitudinal offset correction factor 1.5 (default) | scalar

Dependencies

Longitudinal linear slip friction correction factor, gx4 — Longitudinal linear slip friction correction factor -0.75 (default) | scalar

Dependencies

Longitudinal linear slip correction factor, gx3 — Longitudinal linear slip correction factor 1.63 (default) | scalar

Dependencies

Longitudinal squared slip friction correction factor, gx2 — Longitudinal squared slip friction correction factor -0.75 (default) | scalar

Dependencies

Longitudinal squared slip correction factor, gx1 — Longitudinal squared slip correction factor 1.14 (default) | scalar

Dependencies

Rolling Resistance

Velocity independent force coefficient, aMy — Velocity-independent force coefficient 84e-4 (default) | scalar

Dependencies

Linear velocity force component, bMy — Linear velocity force component 6.2e-4 (default) | scalar

Dependencies

Quadratic velocity force component, cMy — Quadratic velocity force component 1.6e-4 (default) | scalar

Dependencies

Tire pressure exponent, alphaMy — Tire pressure exponent -0.003 (default) | scalar

Dependencies

Normal force exponent, betaMy — Normal force exponent 0.97 (default) | scalar

Dependencies

Parasitic losses force, Fpl — Parasitic force loss 10 (default) | scalar

BrkPrs — Brake pressure
scalar | N-by-`1` vector

AxlTrq — Axle torque
scalar | N-by-`1` vector

Vx — Longitudinal velocity
scalar | N-by-`1` vector

Vy — Lateral velocity
scalar | N-by-`1` vector

Camber — Inclination angle
scalar | N-by-`1` vector

YawRate — Tire angular velocity
scalar | N-by-`1` vector

Prs — Tire inflation pressure
scalar | N-by-`1` vector

Gnd — Ground displacement
scalar | N-by-`1` vector

Fext — Axle force applied to tire
scalar | N-by-`1` vector

RadialDeflct — Tire radial deflection
scalar | N-by-`1` vector

ScaleFctrs — Scale factor
scalar | N-by-`1` vector

Info — Block data
bus

Omega — Wheel angular velocity
scalar | N-by-`1` vector

Fx — Longitudinal axle force
scalar | N-by-`1` vector

Fy — Lateral axle force
scalar | N-by-`1` vector

Fz — Vertical axle force
scalar | N-by-`1` vector

Mx — Overturning moment
scalar | N-by-`1` vector

My — Rolling resistive moment
scalar | N-by-`1` vector

Mz — Aligning moment
scalar | N-by-`1` vector

Tire type — Select type
`User defined` (default) | `Light passenger car 205/60R15` | `Light passenger car 245/60R16` | `Mid-size passenger car 235/45R18` | `Performance car 225/40R19` | `SUV 265/50R20` | `Light truck 275/65R18` | `Commercial truck 295/75R22.5`

Model slip type — Model slip type
`Nominal slip` | `Extended slip`

Brake type — Brake type
`None` | `Disc` | `Drum` | `Mapped`

Rolling Resistance — Rolling resistance torque
`None` (default) | `Pressure and velocity` | `ISO 28580` | `Magic Formula` | `Mapped torque`

Vertical Motion — Vertical motion model
`None` (default) | `Mapped stiffness and damping` | `External deflection`

Longitudinal stiffness, Ckappa — Longitudinal stiffness
`1e7` (default) | scalar | N-by-`1` vector

Lateral stiffness per slip angle, Calpha — Lateral stiffness
`4.5e4` (default) | scalar | N-by-`1` vector

Camber stiffness, Cgamma — Camber stiffness
`1e3` (default) | scalar | N-by-`1` vector

Maximum friction scaling coefficient, mu0 — Maximum friction scaling coefficient
`0.8` (default) | scalar | N-by-`1` vector

Friction reduction factor, As — Friction reduction factor
`0.01` (default) | scalar | N-by-`1` vector

Longitudinal relaxation length, Lrelx — Longitudinal relaxation length
`0.05` (default) | scalar | N-by-`1` vector

Lateral relaxation length, Lrely — Lateral relaxation length
`0.15` (default) | scalar | N-by-`1` vector

Lateral offset correction factor, gy2 — Lateral offset correction factor
`1.5` (default) | scalar

Lateral maximum friction correction factor, gy1 — Lateral maximum friction correction factor
`-1.6` (default) | scalar

Longitudinal offset correction factor, gx5 — Longitudinal offset correction factor
`1.5` (default) | scalar

Longitudinal linear slip friction correction factor, gx4 — Longitudinal linear slip friction correction factor
`-0.75` (default) | scalar

Longitudinal linear slip correction factor, gx3 — Longitudinal linear slip correction factor
`1.63` (default) | scalar

Longitudinal squared slip friction correction factor, gx2 — Longitudinal squared slip friction correction factor
`-0.75` (default) | scalar

Longitudinal squared slip correction factor, gx1 — Longitudinal squared slip correction factor
`1.14` (default) | scalar

Velocity independent force coefficient, aMy — Velocity-independent force coefficient
`84e-4` (default) | scalar

Linear velocity force component, bMy — Linear velocity force component
`6.2e-4` (default) | scalar

Quadratic velocity force component, cMy — Quadratic velocity force component
`1.6e-4` (default) | scalar

Tire pressure exponent, alphaMy — Tire pressure exponent
`-0.003` (default) | scalar

Normal force exponent, betaMy — Normal force exponent
`0.97` (default) | scalar

Parasitic losses force, Fpl — Parasitic force loss
`10` (default) | scalar

Rolling resistance constant, Cr — Rolling resistance constant
`1e-3` (default) | scalar

Thermal correction factor, Kt — Thermal correction factor
`0.008` (default) | scalar

Measured temperature, Tmeas — Temperature during testing
`298.15` (default) | scalar

Ambient temperature, Tamb — Temperature in application environment
`298.15` (default) | scalar

Input ambient temperature — Option to input ambient temperature
`off` (default) | on

Rolling resistance torque coefficient, QSY1 — Torque coefficient
`0.007` (default) | scalar

Longitudinal force rolling resistance coefficient, QSY2 — Force resistance coefficient
`0` (default) | scalar

Linear rotational speed rolling resistance coefficient, QSY3 — Linear speed coefficient
`0.0015` (default) | scalar

Quartic rotational speed rolling resistance coefficient, QSY4 — Quartic speed coefficient
`8.5e-05` (default) | scalar

Camber squared rolling resistance torque, QSY5 — Camber resistance torque
`0` (default) | scalar

Load based camber squared rolling resistance torque, QSY6 — Load resistance torque
`0` (default) | scalar

Normal load rolling resistance coefficient, QSY7 — Normal resistance coefficient
`0.9` (default) | scalar

Pressure load rolling resistance coefficient, QSY8 — Pressure resistance coefficient
`-0.4` (default) | scalar

Rolling resistance scaling factor, lam_My — Scaling factor
`1` (default) | scalar

Spin axis velocity breakpoints, VxMy — Spin axis velocity breakpoints
`-20:1:20` (default) | vector

Normal force breakpoints, FzMy — Normal force breakpoints
`0:200:1e4` (default) | vector

Rolling resistance torque map, MyMap — Rolling resistance torque map
array

Tire nominal section width, WIDTH — Tire nominal section width
scalar

Linear yaw rate resistance, bMz — Linear yaw rate resistance
`0` | scalar

Static friction coefficient, mu_static — Static friction coefficient
`0.3` (default) | scalar | N-by-`1` vector

Kinetic friction coefficient, mu_kinetic — Kinetic friction
`0.2` (default) | scalar | N-by-`1` vector

Disc brake actuator bore, disc_abore — Bore distance
`0.05` (default) | scalar | N-by-`1` vector

Brake pad mean radius, Rm — Radius
`0.177` (default) | scalar | N-by-`1` vector

Number of brake pads, num_pads — Number of brake pads
`2` (default) | scalar | N-by-`1` vector

Drum brake actuator bore, disc_abore — Bore distance
`0.0508` (default) | scalar | N-by-`1` vector

Shoe pin to drum center distance, drum_a — Shoe pin to drum center distance
`0.123` (default) | scalar

Shoe pin center to force application point distance, drum_c — Shoe pin center to force application point distance
`0.212` (default) | scalar

Drum internal radius, drum_r — Drum internal radius
`0.15` (default) | scalar

Shoe pin to pad start angle, drum_theta1 — Shoe pin to pad start angle
`0` (default) | scalar

Shoe pin to pad end angle, drum_theta2 — Shoe pin to pad end angle
`126` (default) | scalar

Brake actuator pressure breakpoints, brake_p_bpt — Brake actuator pressure breakpoints
vector

Wheel speed breakpoints, brake_n_bpt — Wheel speed breakpoints
vector

Brake torque map, f_brake_t — Lookup table for brake torque
array

Rotational damping, br — Rotational damping
scalar | N-by-`1` vector