Longitudinal Wheel

Longitudinal wheel with disc, drum, or mapped brake

Libraries:
Powertrain Blockset / Drivetrain / Wheels
Vehicle Dynamics Blockset / Wheels and Tires

Description

The Longitudinal Wheel block implements the longitudinal behavior of an ideal wheel. You can specify the longitudinal force and rolling resistance calculation method, and brake type. Use the block in driveline and longitudinal vehicle simulations where low frequency tire-road and braking forces are required to determine vehicle acceleration, braking, and wheel-rolling resistance. For example, you can use the block to determine the torque and power requirements for a specified drive cycle or braking event. The block is not suitable for applications that require combined lateral slip.

There are four types of Longitudinal Wheel blocks. Each block implements a different brake type.

Block Name	Brake Type Setting	Brake Implementation
Longitudinal Wheel - No Brake	`None`	None
Longitudinal Wheel - Disc Brake	`Disc`	Brake that converts the brake cylinder pressure into a braking force.
Longitudinal Wheel - Drum Brake	`Drum`	Simplex drum brake that converts the applied force and brake geometry into a net braking torque.
Longitudinal Wheel - Mapped Brake	`Mapped`	Lookup table that is a function of the wheel speed and applied brake pressure.

The block models longitudinal force as a function of wheel slip relative to the road surface. To calculate the longitudinal force, specify one of these Longitudinal Force parameters.

Setting	Block Implementation
`Magic Formula constant value`	Magic Formula with constant coefficient for stiffness, shape, peak, and curvature.
`Magic Formula pure longitudinal slip`	Magic Formula with load-dependent coefficients that implement equations 4.E9 through 4.E18 in Tire and Vehicle Dynamics.
`Mapped force`	Lookup table that is a function of the normal force and wheel slip ratio.

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.

If	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.

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 disc outer radius
R_i	Annular disc 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

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

Longitudinal Force

To model the Longitudinal Wheel block longitudinal forces, you can use the Magic Formula. The model provides a steady-state tire characteristic function F_x = f(κ, F_z), the longitudinal force F_x on the tire, based on:

Vertical load F_z
Wheel slip κ

Image of tire with applied vertical and longitudinal forces

The Magic Formula model uses these variables.

Ω	Wheel angular velocity
r_w	Wheel radius
V_x	Wheel hub longitudinal velocity
r_wΩ	Tire tread longitudinal velocity
V_sx = r_wΩ – V_x	Wheel slip velocity
κ = V_sx/\|V_x\|	Wheel slip
F_z, F_z0	Vertical load and nominal vertical load on tire
F_x = f(κ, F_z)	Longitudinal force exerted on the tire at the contact point. Also a characteristic function f of the tire.

Magic Formula Constant Value

If you set Longitudinal Force to Magic Formula constant value, the block implements the Magic Formula as a specific form of the tire characteristic function, characterized by four dimensionless coefficients (B, C, D, E), or stiffness, shape, peak, and curvature:

$F_{x} = f (κ, F_{z}) = F_{z} D sin (C \tan^{- 1} [{B κ - E [B κ - \tan^{- 1} (B κ)]}])$

The slope of f at κ = 0 is BCD·F_z.

The coefficients are based on empirical tire data. These values are typical sets of constant Magic Formula coefficients for common road conditions.

Surface	B	C	D	E
Dry tarmac	10	1.9	1	0.97
Wet tarmac	12	2.3	0.82	1
Snow	5	2	0.3	1
Ice	4	2	0.1	1

Magic Formula Pure Longitudinal Slip

If you set Longitudinal Force to Magic Formula pure longitudinal slip, the block implements a more general Magic Formula using dimensionless coefficients that are functions of the tire load. The block implements the longitudinal force equations in Chapter 4 of Tire and Vehicle Dynamics, including 4.E9 through 4.E18:

$\begin{array}{l} F_{x 0} = D_{x} sin (C_{x} \tan^{- 1} [{B_{x} κ_{x} - E_{x} [B_{x} κ_{x} - \tan^{- 1} (B_{x} κ_{x})]}]) + S_{Vx} \\ where: \\ κ_{x} = κ + S_{H x} \\ C_{x} = p_{C x 1} λ_{C x} \\ D_{x} = μ_{x} F_{z} ς_{1} \\ μ_{x} = (p_{D x 1} + p_{D x 2} d f_{z}) (1 + p_{p x 3} d p_{i} + p_{p x 4} d p_{i}^{2}) (1 - p_{D x 3} γ^{2}) λ^{*}_{μ x} \\ E_{x} = (p_{E x 1} + p_{E x 2} d f_{z} + p_{E x 3} d f_{z}^{2}) [1 - p_{E x 4} sgn (κ_{x})] λ_{E x} \\ K_{x κ} = F_{z} (p_{K x 1} + p_{K x 2} d f_{z}) exp (p_{Kx3} d f_{z}) (1 + p_{p x 1} d p_{i} + p_{p x 2} d p_{i}^{2}) \\ B_{x} = K_{x κ} / (C_{x} D_{x} + ε_{x}) \\ S_{H x} = p_{H x 1} + p_{H x 2} d f_{z} \\ S_{V x} = F_{z} \cdot (p_{V x 1} + p_{V x 2} d f_{z}) λ_{V x} λ^{'}_{μ x} ς_{1} \end{array}$

S_Hx and S_Vx represent offsets to the slip and longitudinal force in the force-slip function, or horizontal and vertical offsets if the function is plotted as a curve. μ_x is the longitudinal load-dependent friction coefficient. ε_x is a small number inserted to prevent division by zero as F_z approaches zero.

Vertical Dynamics

If you select no vertical degrees-of-freedom by setting Vertical Motion to None, the block passes the applied chassis forces directly through to the rolling resistance and longitudinal force calculations.

If you set Vertical Motion to Mapped stiffness and damping, the 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.

$F z t i r e (z, \dot{z}, P_{t i r e}) = F_{z k} (z, P_{t i r e}) + F_{z b} (\dot{z}, P_{t i r e})$

The block determines the vertical response using this differential equation.

$\ddot{z} m = F z t i r e - F_{z} - m g$

When you disable the vertical degree-of-freedom, the input normal force from the vehicle passes directly to the longitudinal and rolling force calculations.

$\begin{array}{l} \ddot{z} = \dot{z} = m = 0 \\ F z t i r e = m g \end{array}$

The block uses the wheel-fixed frame to resolve the vertical forces.

Figure of inclined plan with tires and body-fixed, wheel-fixed, and inertial frames

The equations use these variables.

Fztire	Tire normal force along the wheel-fixed z-axis
m	Axle mass
F_zk	Tire normal force due to wheel stiffness along the wheel-fixed z-axis
F_zb	Tire normal force due to wheel damping along the wheel-fixed z-axis
F_z	Suspension or vehicle normal force along the wheel-fixed z-axis
P_Tire	Tire pressure
$z, \dot{z}, \ddot{z}$	Tire displacement, velocity, and acceleration, respectively, along the wheel-fixed z-axis

Power Accounting

For the power accounting, the block implements these equations.

Bus Signal			Description	Equations
`PwrInfo`	`PwrTrnsfrd` — Power transferred between blocks Positive signals indicate flow into block Negative signals indicate flow out of block	`PwrRoad`	Tractive power applied from the axle	$P_{r o a d} = F_{x} V_{x}$
		`PwrAxlTrq`	External torque applied by the axle to the wheel	$P_{T} = T ω$
		`PwrFz`	Vertical force applied to the wheel by the vehicle or suspension	$P_{F z} = F_{z} \dot{z}$
	`PwrNotTrnsfrd` — Power crossing the block boundary, but not transferred Positive signals indicate an input Negative signals indicate a loss	`PwrSlip`	Tractive power loss	$P_{κ} = F_{x} V_{x} + (- F_{c p} R_{e} + M_{y}) ω$
		`PwrMyRoll`	Rolling resistance power	$P_{M y} = M_{y} ω$
		`PwrMyBrk`	Braking power	$P_{b r k} = M_{b r k} ω$
		`PwrMyb`	Rolling viscous damping loss	$P_{b} = - b ω^{2}$
		`PwrFzDamp`	Vertical damping power	$P_{F z b} = F_{z b} \dot{z}$
	`PwrStored` — Stored energy rate of change Positive signals indicate an increase Negative signals indicate a decrease	`PwrStoredzdot`	Rate of change of vertical kinetic energy	$P_{\dot{z}} = m \ddot{z} \dot{z}$
		`PwrStoredq`	Rate of change of rotational kinetic energy	$P_{ω} = I_{y y} \dot{ω} ω$
		`PwrStoredFsFzSprng`	Rate of change of stored sidewall potential energy	$P_{F z k} = F_{z k} {\dot{z}}_{x}$
		`PwrStoredGrvty`	Rate of change of gravitational potential energy	$P_{g} = - m g \dot{Z}$

The equations use these variables.

ω	Wheel angular velocity
b	Linear velocity force component
F_x	Longitudinal force developed by the tire road interface due to slip
F_cp	Tire slip force at contact patch
F_z	Vehicle normal force
F_zb	Tire normal force due to wheel damping
F_zk	Tire normal force due to wheel stiffness
I_yy	Wheel rotational inertia
M_brk	Braking moment
M_y	Rolling resistance torque
R_e	Effective tire radius while under load and for a given pressure
T	Axle torque applied on wheel
V_x	Longitudinal axle velocity
$z, \dot{z}, \ddot{z}$	Tire displacement, velocity, and acceleration, respectively
ω	Wheel angular velocity
$\dot{Z}$	Vehicle vertical velocity along the vehicle-fixed z-axis

Ports

Input

expand all

BrkPrs — Brake pressure
`scalar`

Brake pressure, in Pa.

Dependencies

To enable this port, for the Brake Type parameter, specify one of these types:

Disc
Drum
Mapped

AxlTrq — Axle torque
`scalar`

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

Vx — Velocity
`scalar`

Axle longitudinal velocity along vehicle(body)-fixed x-axis, in m/s.

Fz — Normal force
`scalar`

Absolute value of suspension or vehicle normal force along body-fixed z-axis, in N.

Gnd — Ground displacement
`scalar`

Ground displacement, Grndz, along negative wheel-fixed z-axis, in m.

Figure of inclined plan with tires and body-fixed, wheel-fixed, and inertial frames

Dependencies

To create Gnd:

Set Vertical Motion to Mapped stiffness and damping.
On the Vertical pane, select Input ground displacement.

lam_mux — Friction scaling factor
`scalar`

Longitudinal friction scaling factor, dimensionless.

Dependencies

To enable this port, select Input friction scale factor.

TirePrs — Tire pressure
`scalar`

Tire pressure, in Pa.

Dependencies

To enable this port:

Set one of these parameters:
- Longitudinal Force to Magic Formula pure longitudinal slip.
- Rolling Resistance to Pressure and velocity or Magic Formula.
- Vertical Motion to Mapped stiffness and damping.
On the Wheel Dynamics pane, select Input tire pressure.

Tamb — Ambient temperature
`scalar`

Ambient temperature, T_amb, in K.

The 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.

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

Dependencies

To enable this port:

Set Rolling Resistance to ISO 28580.
On the Rolling Resistance pane, select to Input ambient temperature.

Output

expand all

Info — Bus signal
bus

Bus signal containing these block calculations.

Signal			Description	Units
`AxlTrq`			Axle torque about body-fixed y-axis	N·m
`Omega`			Wheel angular velocity about body-fixed y-axis	rad/s
`Omegadot`			Wheel angular acceleration about body-fixed y-axis	rad/s^2
`Fx`			Longitudinal vehicle force along body-fixed x-axis	N
`Fz`			Vertical vehicle force along body-fixed z-axis	N
`Fzb`			Tire normal force due to wheel damping along the wheel-fixed z-axis	N
`Fzk`			Tire normal force due to wheel stiffness along the wheel-fixed z-axis	N
`My`			Rolling resistance torque about body-fixed y-axis	N·m
`Myb`			Rolling resistance torque due to damping about body-fixed y-axis	N·m
`Kappa`			Slip ratio	NA
`Vx`			Vehicle longitudinal velocity along body-fixed x-axis	m/s
`Re`			Wheel effective radius along wheel-fixed z-axis	m
`BrkTrq`			Brake torque about body-fixed y-axis	N·m
`BrkPrs`			Brake pressure	Pa
`z`			Wheel vertical deflection along wheel-fixed z-axis	m
`zdot`			Wheel vertical velocity along wheel-fixed z-axis	m/s
`zddot`			Wheel vertical acceleration along wheel-fixed z-axis	m/s^2
`Gndz`			Ground displacement along negative of wheel-fixed z-axis (positive input produces wheel lift)	m
`GndFz`			Vertical wheel force on ground along negative of wheel-fixed z-axis	N
`TirePrs`			Tire pressure	Pa
`Fpatch`			Tractive force (Fx considering relaxation effects). Use for vehicle transient maneuvers.	N
`PwrInfo`	`PwrTrnsfrd`	`PwrRoad`	External torque applied by the axle to the wheel	W
		`PwrAxlTrq`	Vertical force applied to the wheel by the vehicle or suspension	W
		`PwrFz`	Tractive power loss	W
	`PwrNotTrnsfrd`	`PwrSlip`	Rolling resistance power	W
		`PwrMyRoll`	Braking power	W
		`PwrMyBrk`	Rolling viscous damping loss	W
		`PwrMyb`	Vertical damping power	W
		`PwrFzDamp`	Rate of change of vertical kinetic energy	W
	`PwrStored`	`PwrStoredzdot`	Rate of change of rotational kinetic energy	W
		`PwrStoredq`	Rate of change of stored sidewall potential energy	W
		`PwrStoredFsFzSprng`	Rate of change of gravitational potential energy	W
		`PwrStoredGrvty`	Tractive power applied from the axle	W

Fx — Longitudinal axle force
`scalar`

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

Omega — Wheel angular velocity
`scalar`

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

z — Wheel vertical deflection
`scalar`

Wheel vertical deflection along wheel-fixed z-axis, in m.

Dependencies

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

zdot — Wheel vertical velocity
`scalar`

Wheel vertical velocity along wheel-fixed z-axis, in m/s.

Dependencies

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

Parameters

expand all

Block Options

Longitudinal Force — Select type
`Magic Formula constant value` (default) | `Magic Formula pure longitudinal slip` | `Mapped force`

The block models longitudinal force as a function of wheel slip relative to the road surface. To calculate the longitudinal force, specify one of these Longitudinal Force parameters.

Setting	Block Implementation
`Magic Formula constant value`	Magic Formula with constant coefficient for stiffness, shape, peak, and curvature.
`Magic Formula pure longitudinal slip`	Magic Formula with load-dependent coefficients that implement equations 4.E9 through 4.E18 in Tire and Vehicle Dynamics.
`Mapped force`	Lookup table that is a function of the normal force and wheel slip ratio.

Dependencies

Selecting Enables These Parameters

Selecting	Enables These Parameters
`Magic Formula constant value`	Pure longitudinal peak factor, Dx Pure longitudinal shape factor, Cx Pure longitudinal stiffness factor, Bx Pure longitudinal curvature factor, Ex
`Magic Formula pure longitudinal slip`	Cfx shape factor, PCX1 Longitudinal friction at nominal normal load, PDX1 Frictional variation with load, PDX2 Frictional variation with camber, PDX3 Longitudinal curvature at nominal normal load, PEX1 Variation of curvature factor with load, PEX2 Variation of curvature factor with square of load, PEX3 Longitudinal curvature factor with slip, PEX4 Longitudinal slip stiffness at nominal normal load, PKX1 Variation of slip stiffness with load, PKX2 Slip stiffness exponent factor, PKX3 Horizontal shift in slip ratio at nominal normal load, PHX1 Variation of horizontal slip ratio with load, PHX2 Vertical shift in load at nominal normal load, PVX1 Variation of vertical shift with load, PVX2 Linear variation of longitudinal slip stiffness with tire pressure, PPX1 Quadratic variation of longitudinal slip stiffness with tire pressure, PPX2 Linear variation of peak longitudinal friction with tire pressure, PPX3 Quadratic variation of peak longitudinal friction with tire pressure, PPX4 Linear variation of longitudinal slip stiffness with tire pressure, PPX1 Slip speed decay function scaling factor, lam_muV Brake slip stiffness scaling factor, lam_Kxkappa Longitudinal shape scaling factor, lam_Cx Longitudinal curvature scaling factor, lam_Ex Longitudinal horizontal shift scaling factor, lam_Hx Longitudinal vertical shift scaling factor, lam_Vx
`Mapped force`	Slip ratio breakpoints, kappaFx Normal force breakpoints, FzFx Longitudinal force map, FxMap

Magic Formula constant value

Pure longitudinal peak factor, Dx

Pure longitudinal shape factor, Cx

Pure longitudinal stiffness factor, Bx

Pure longitudinal curvature factor, Ex

Magic Formula pure longitudinal slip

Cfx shape factor, PCX1

Longitudinal friction at nominal normal load, PDX1

Frictional variation with load, PDX2

Frictional variation with camber, PDX3

Longitudinal curvature at nominal normal load, PEX1

Variation of curvature factor with load, PEX2

Variation of curvature factor with square of load, PEX3

Longitudinal curvature factor with slip, PEX4

Longitudinal slip stiffness at nominal normal load, PKX1

Variation of slip stiffness with load, PKX2

Slip stiffness exponent factor, PKX3

Horizontal shift in slip ratio at nominal normal load, PHX1

Variation of horizontal slip ratio with load, PHX2

Vertical shift in load at nominal normal load, PVX1

Variation of vertical shift with load, PVX2

Linear variation of longitudinal slip stiffness with tire pressure, PPX1

Quadratic variation of longitudinal slip stiffness with tire pressure, PPX2

Linear variation of peak longitudinal friction with tire pressure, PPX3

Quadratic variation of peak longitudinal friction with tire pressure, PPX4

Linear variation of longitudinal slip stiffness with tire pressure, PPX1

Slip speed decay function scaling factor, lam_muV

Brake slip stiffness scaling factor, lam_Kxkappa

Longitudinal shape scaling factor, lam_Cx

Longitudinal curvature scaling factor, lam_Ex

Longitudinal horizontal shift scaling factor, lam_Hx

Longitudinal vertical shift scaling factor, lam_Vx

Mapped force

Slip ratio breakpoints, kappaFx

Normal force breakpoints, FzFx

Longitudinal force map, FxMap

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

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

There are four types of Longitudinal Wheel blocks. Each block implements a different brake type.

Block Name	Brake Type Setting	Brake Implementation
Longitudinal Wheel - No Brake	`None`	None
Longitudinal Wheel - Disc Brake	`Disc`	Brake that converts the brake cylinder pressure into a braking force.
Longitudinal Wheel - Drum Brake	`Drum`	Simplex drum brake that converts the applied force and brake geometry into a net braking torque.
Longitudinal Wheel - Mapped Brake	`Mapped`	Lookup table that is a function of the wheel speed and applied brake pressure.

Vertical Motion — Select type
`None` (default) | `Mapped stiffness and damping`

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.

Selecting Enables These Parameters Creates These Output Ports

Selecting	Enables These Parameters	Creates These Output Ports
`Mapped stiffness and damping`	Wheel and unsprung mass, m Initial deflection, zo Initial velocity, zdoto Gravitational acceleration, g Vertical deflection breakpoints, zFz Pressure breakpoints, pFz Force due to deflection, Fzz Vertical velocity breakpoints, zdotFz Force due to velocity, Fzzdot Ground displacement, Gndz Input ground displacement	`z` `zdot`

Mapped stiffness and damping

Wheel and unsprung mass, m

Initial deflection, zo

Initial velocity, zdoto

Gravitational acceleration, g

Vertical deflection breakpoints, zFz

Pressure breakpoints, pFz

Force due to deflection, Fzz

Vertical velocity breakpoints, zdotFz

Force due to velocity, Fzzdot

Ground displacement, Gndz

Input ground displacement

z

zdot

Longitudinal scaling factor, lam_x — Friction scaling factor
`1` (default)

Longitudinal friction scaling factor, dimensionless.

Dependencies

To enable this parameter, clear Input friction scale factor.

Input friction scale factor — Selection
`Off` (default)

Create input port for longitudinal friction scaling factor.

Dependencies

Selecting this parameter:

Creates input port lam_mux.
Disables parameter Longitudinal scaling factor, lam_x.

Wheel Dynamics

Axle viscous damping coefficient, br — Damping
`0.001` (default) | `scalar`

Axle viscous damping coefficient, br, in N·m·s/rad.

Wheel inertia, Iyy — Inertia
`0.8` (default) | `scalar`

Wheel inertia, in kg·m^2.

Wheel initial angular velocity, omegao — Wheel speed
`0` (default) | `scalar`

Initial angular velocity of wheel, along body-fixed y-axis, in rad/s.

Relaxation length, Lrel — Relaxation length
`0.5` (default) | `scalar`

Wheel relaxation length, in m.

Loaded radius, Re — Loaded radius
`0.3` (default) | `scalar`

Loaded wheel radius, Re, in m.

Figure of inclined plan with tires and body-fixed, wheel-fixed, and inertial frames

Unloaded radius, UNLOADED_RADIUS — Unloaded radius
`0.4` (default) | `scalar`

Unloaded wheel radius, in m.

Dependencies

To create this parameter, set Rolling Resistance to Pressure and velocity or Magic Formula.

Nominal longitudinal speed, LONGVL — Speed
`16` (default) | `scalar`

Nominal longitudinal speed along body-fixed x-axis, in m/s.

Dependencies

To enable this parameter, set Longitudinal Force to Magic Formula pure longitudinal slip.

Nominal camber angle, gamma — Camber
`0` (default) | `scalar`

Nominal camber angle, in rad.

Dependencies

To enable this parameter, set either:

Longitudinal Force to Magic Formula pure longitudinal slip.
Rolling Resistance to Magic Formula.

Nominal pressure, NOMPRES — Pressure
`220000` (default) | `scalar`

Nominal pressure, in Pa.

Dependencies

To enable this parameter, set either:

Longitudinal Force to Magic Formula pure longitudinal slip.
Rolling Resistance to Magic Formula.

Pressure, press — Pressure
`220000` (default) | `scalar`

Pressure, in Pa.

Dependencies

To enable this parameter:

Set one of these:
- Longitudinal Force to Magic Formula pure longitudinal slip.
- Rolling Resistance to Pressure and velocity or Magic Formula.
- Vertical Motion to Mapped stiffness and damping.
On the Wheel Dynamics pane, clear Input tire pressure.

Longitudinal

Magic Formula Constant Value

Pure longitudinal peak factor, Dx — Factor
`1` (default) | `scalar`

Pure longitudinal peak factor, dimensionless.

The coefficients are based on empirical tire data. These values are typical sets of constant Magic Formula coefficients for common road conditions.

Surface	B	C	D	E
Dry tarmac	10	1.9	1	0.97
Wet tarmac	12	2.3	0.82	1
Snow	5	2	0.3	1
Ice	4	2	0.1	1

Dependencies

To create this parameter, select the Longitudinal Force parameter Magic Formula constant value.

Pure longitudinal shape factor, Cx — Factor
`1.65` (default) | `scalar`

Pure longitudinal shape factor, dimensionless.

The coefficients are based on empirical tire data. These values are typical sets of constant Magic Formula coefficients for common road conditions.

Surface	B	C	D	E
Dry tarmac	10	1.9	1	0.97
Wet tarmac	12	2.3	0.82	1
Snow	5	2	0.3	1
Ice	4	2	0.1	1

Dependencies

To create this parameter, select the Longitudinal Force parameter Magic Formula constant value.

Pure longitudinal stiffness factor, Bx — Factor
`10` (default) | `scalar`

Pure longitudinal stiffness factor, dimensionless.

The coefficients are based on empirical tire data. These values are typical sets of constant Magic Formula coefficients for common road conditions.

Surface	B	C	D	E
Dry tarmac	10	1.9	1	0.97
Wet tarmac	12	2.3	0.82	1
Snow	5	2	0.3	1
Ice	4	2	0.1	1

Dependencies

To create this parameter, select the Longitudinal Force parameter Magic Formula constant value.

Pure longitudinal curvature factor, Ex — Factor
`0.01` (default) | `scalar`

Pure longitudinal curvature factor, dimensionless.

The coefficients are based on empirical tire data. These values are typical sets of constant Magic Formula coefficients for common road conditions.

Surface	B	C	D	E
Dry tarmac	10	1.9	1	0.97
Wet tarmac	12	2.3	0.82	1
Snow	5	2	0.3	1
Ice	4	2	0.1	1

Dependencies

To create this parameter, select the Longitudinal Force parameter Magic Formula constant value.

Magic Formula Pure Longitudinal Slip