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# Independent Suspension - Double Wishbone

Double wishbone independent suspension

Libraries:
Vehicle Dynamics Blockset / Suspension

## Description

The Independent Suspension - Double Wishbone block implements an independent double wishbone suspension for multiple axles with multiple wheels per axle.

The block models the suspension compliance, damping, and geometric effects as functions of the relative positions and velocities of the vehicle and wheel carrier with axle-specific compliance and damping parameters. Using the suspension compliance and damping, the block calculates the suspension force on the vehicle and wheel. The block uses the Z-down coordinate system (defined in SAE J670). This table describes the settings you can specify for each suspension element.

Suspension ElementSetting

Axle

• Multiple wheels

• An anti-sway bar for axles with two wheels

• Suspension parameters

Wheel

• Steering angles

The block contains energy-storing spring elements and energy-dissipating damper elements. It does not contain energy-storing mass elements. The block assumes that the vehicle (sprung) and wheel (unsprung) blocks connected to the block store the mass-related suspension energy.

This table summarizes the block parameter settings for a vehicle with:

• Two axles

• Two wheels per axle

• Steering angle input for both wheels on the front axle

• An anti-sway bar on the front axle

ParameterSetting
Number of axles, NumAxl

2

Number of wheels by axle, NumWhlsByAxl

[2 2]

Steered axle enable by axle, StrgEnByAxl

[1 0]

Anti-sway axle enable by axle, AntiSwayEnByAxl

[1 0]

The block uses the wheel number, t, to index the input and output signals. This table summarizes the wheel, axle, and corresponding wheel number for a vehicle with:

• Two axles

• Two wheels per axle

WheelAxleWheel Number
Front leftFront1
Front rightFront2
Rear leftRear1
Rear rightRear2

### Suspension Compliance and Damping

The block uses a linear spring and damper to model the vertical dynamic effects of the suspension system. Using the relative positions and velocities of the vehicle and wheel carrier, the block calculates the vertical suspension forces on the wheel and vehicle. The block uses a linear equation that relates the vertical damping and compliance to the suspension height, suspension height rate of change, and absolute value of the steering angles.

The block implements this equation.

${F}_{w{z}_{a,t}}={F}_{z{0}_{a}}+{k}_{{z}_{a}}\left({z}_{{v}_{a,t}}-{z}_{{w}_{a,t}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+c\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}}\right)+{F}_{zhsto{p}_{a,t}}+{F}_{zasw{y}_{a,t}}$

The damping coefficient, c, depends on the Enable active damping parameter setting.

Enable active damping Setting

Damping
off

Constant, c = cza

on

Lookup table that is a function of active damper duty cycle and actuator velocity

$c=f\left(duty,\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}}\right)\right)$

The block assumes that the suspension elements have no mass. Therefore, the suspension forces and moments applied to the vehicle are equal to the suspension forces and moments applied to the wheel.

$\begin{array}{l}{F}_{v{x}_{a,t}}={F}_{w{x}_{a,t}}\\ {F}_{v{y}_{a,t}}={F}_{w{y}_{a,t}}\\ {F}_{v{z}_{a,t}}=-{F}_{w{z}_{a,t}}\\ \\ {M}_{v{x}_{a,t}}={M}_{w{x}_{a,t}}+{F}_{w{y}_{a,t}}\left(R{e}_{w{y}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{y}_{a,t}}={M}_{w{y}_{a,t}}+{F}_{w{x}_{a,t}}\left(R{e}_{w{x}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{z}_{a,t}}={M}_{w{z}_{a,t}}\end{array}$

The block sets the wheel positions and velocities equal to the vehicle lateral and longitudinal positions and velocities.

$\begin{array}{l}{x}_{{w}_{a,t}}={x}_{{v}_{a,t}}\\ {y}_{{w}_{a,t}}={y}_{{v}_{a,t}}\\ {\stackrel{˙}{x}}_{{w}_{a,t}}={\stackrel{˙}{x}}_{{v}_{a,t}}\\ {\stackrel{˙}{y}}_{{w}_{a,t}}={\stackrel{˙}{y}}_{{v}_{a,t}}\end{array}$

The equations use these variables.

 Fwza,t, Mwza,t Suspension force and moment applied to the wheel on axle a, wheel t along wheel-fixed z-axis Fwxa,t, Mwxa,t Suspension force and moment applied to the wheel on axle a, wheel t along wheel-fixed x-axis Fwya,t, Mwya,t Suspension force and moment applied to the wheel on axle a, wheel t along wheel-fixed y-axis Fvza,t, Mvza,t Suspension force and moment applied to the vehicle on axle a, wheel t along wheel-fixed z-axis Fvxa,t, Mvxa,t Suspension force and moment applied to the vehicle on axle a, wheel t along wheel-fixed x-axis Fvya,t, Mvya,t Suspension force and moment applied to the vehicle on axle a, wheel t along wheel-fixed y-axis Fz0a Vertical suspension spring preload force applied to the wheels on axle a kza Vertical spring constant applied to wheels on axle a kwaz Wheel and axle interface compliance constant mhsteera Steering angle to vertical force slope applied at wheel carrier for wheels on axle a δsteera,t Steering angle input for axle a, wheel t cza Vertical damping constant applied to wheels on axle a cwaz Wheel and axle interface damping constant Rewa,t Effective wheel radius for axle a, wheel t Fzhstopa,t Vertical hardstop force at axle a, wheel t, along the inertial-fixed z-axis Fzaswya,t Vertical anti-sway force at axle a, wheel t, along the inertial-fixed z-axis Fwaz0 Wheel and axle interface compliance constant zva,t, żva,t Vehicle displacement and velocity at axle a, wheel t, along the inertial-fixed z-axis zwa,t, żwa,t Wheel displacement and velocity at axle a, wheel t, along the inertial-fixed z-axis xva,t, ẋva,t Vehicle displacement and velocity at axle a, wheel t, along the inertial-fixed z-axis xwa,t, ẋwa,t Wheel displacement and velocity at axle a, wheel t, along the inertial-fixed z-axis yva,t, ẏva,t Vehicle displacement and velocity at axle a, wheel t, along the inertial-fixed y-axis ywa,t, ẏwa,t Wheel displacement and velocity at axle a, wheel t, along the inertial-fixed y-axis Ha,t Suspension height at axle a, wheel t Rewa,t Effective wheel radius at axle a, wheel t

### Hardstop Forces

The hardstop feedback force, Fzhstopa,t, that the block applies depends on whether the suspension is compressing or extending. The block applies the force:

• In compression, when the suspension is compressed more than the maximum distance specified by the Suspension maximum height, Hmax parameter

• In extension, when the suspension extension is greater than maximum extension specified by the Suspension maximum height, Hmax parameter

To calculate the force, the block uses a stiffness based on a hyperbolic tangent and exponential scaling.

### Anti-Sway Bar

Optionally, use the parameter to implement an anti-sway bar force, Fzaswya,t, for axles that have two wheels. This figure shows how the anti-sway bar transmits torque between two independent suspension wheels on a shared axle. Each independent suspension applies a torque to the anti-sway bar via a radius arm that extends from the anti-sway bar back to the independent suspension connection point.

To calculate the sway bar force, the block implements these equations.

CalculationEquation

Anti-sway bar angular deflection for a given axle and wheel, Δϴa,t

$\begin{array}{l}{\theta }_{0a}={\mathrm{tan}}^{-1}\left(\frac{{z}_{0}}{r}\right)\\ \Delta {\theta }_{a,t}={\mathrm{tan}}^{-1}\left(\frac{r\mathrm{tan}{\theta }_{0a}-{z}_{{w}_{a,t}}+{z}_{{v}_{a,t}}}{r}\right)\end{array}$

Anti-sway bar twist angle, ϴa

${\theta }_{a}=-{\mathrm{tan}}^{-1}\left(\frac{r\mathrm{tan}{\theta }_{0a}-{z}_{{w}_{a,1}}+{z}_{{v}_{a,1}}}{r}\right)-{\mathrm{tan}}^{-1}\left(\frac{r\mathrm{tan}{\theta }_{0a}-{z}_{{w}_{a,2}}+{z}_{{v}_{a,2}}}{r}\right)$

Anti-sway bar torque, τa

${\tau }_{a}={k}_{a}{\theta }_{a}$

Anti-sway bar forces applied to the wheel on axle a, wheel t along wheel-fixed z-axis

$\begin{array}{l}{F}_{zasw{y}_{a,1}}=\left(\frac{{\tau }_{a}}{r}\right)\mathrm{cos}\left({\theta }_{0a}-{\mathrm{tan}}^{-1}\left(\frac{r\mathrm{tan}{\theta }_{0a}-{z}_{{w}_{a,1}}+{z}_{{v}_{a,1}}}{r}\right)\right)\\ {F}_{zasw{y}_{a,2}}=\left(\frac{{\tau }_{a}}{r}\right)\mathrm{cos}\left({\theta }_{0a}-{\mathrm{tan}}^{-1}\left(\frac{r\mathrm{tan}{\theta }_{0a}-{z}_{{w}_{a,2}}+{z}_{{v}_{a,2}}}{r}\right)\right)\end{array}$

The equations and figure use these variables.

 τa Anti-sway bar torque θ Anti-sway bar twist angle θ0a Initial anti-sway bar twist angle Δϴa,t Anti-sway bar angular deflection at axle a, wheel t r Anti-sway bar arm radius z0 Vertical distance from anti-sway bar connection point to anti-sway bar centerline Fzswaya,t Anti-sway bar force applied to the wheel on axle a, wheel t along wheel-fixed z-axis zva,t Vehicle displacement at axle a, wheel t, along the inertial-fixed z-axis zwa,t Wheel displacement at axle a, wheel t, along the inertial-fixed z-axis

### Camber, Caster, and Toe Angles

To calculate the camber, caster, and toe angles, block uses linear functions of the suspension height and steering angle.

The equations use these variables.

 ξa,t Camber angle of wheel on axle a, wheel t ηa,t Caster angle of wheel on axle a, wheel t ζa,t Toe angle of wheel on axle a, wheel t ξ0a, η0a, ζ0a Nominal suspension axle a camber, caster, and toe angles, respectively, at zero steering angle mhcambera, mhcastera, mhtoea Camber, caster, and toe angles, respectively, versus suspension height slope for axle a mcambersteera, mcastersteera, mtoesteera Camber, caster, and toe angles, respectively, versus steering angle slope for axle a mhsteera Steering angle versus vertical force slope for axle a δsteera,t Steering angle input for axle a, wheel t zva,t Vehicle displacement at axle a, wheel t, along the inertial-fixed z-axis zwa,t Wheel displacement at axle a, wheel t, along the inertial-fixed z-axis

### Steering Angles

Optionally, use the Steered axle enable by axle, StrgEnByAxl parameter to input steering angles for the wheels. To calculate the steering angles for the wheels, the block offsets the input steering angles with a linear function of the suspension height.

${\delta }_{whlstee{r}_{a,t}}={\delta }_{stee{r}_{a,t}}+{m}_{hto{e}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{toestee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|$

The equation uses these variables.

 mtoesteera Axle a toe angle versus steering angle slope mhsteera Axle a steering angle versus vertical force slope mhtoea Axle a toe angle versus suspension height slope δwhlsteera,t Wheel steering angle for axle a, wheel t δsteera,t Steering angle input for axle a, wheel t zva,t Vehicle displacement at axle a, wheel t, along the inertial-fixed z-axis zwa,t Wheel displacement at axle a, wheel t, along the inertial-fixed z-axis

### Power and Energy

The block calculates these suspension characteristics for each axle, a, wheel, t.

CalculationEquation

Dissipated power, Psuspa,t

${P}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)$

Absorbed energy, Esuspa,t

${E}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)$

Suspension height, Ha,t

${H}_{a,t}=-\left({z}_{{v}_{a,t}}-{z}_{{w}_{a,t}}+\frac{{F}_{z{0}_{a}}}{{k}_{{z}_{a}}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)$

Distance from wheel carrier center to tire/road interface

${z}_{wt{r}_{a,t}}=R{e}_{{w}_{a,t}}+{H}_{a,t}$

The equations use these variables.

 mhsteera Steering angle to vertical force slope applied at wheel carrier for wheels on axle a δsteera,t Steering angle input for axle a, wheel t Rewa,t Axle a, wheel t effective wheel radius from wheel carrier center to tire/road interface Fz0a Vertical suspension spring preload force applied to the wheels on axle a zwtra,t Distance from wheel carrier center to tire/road interface, along the inertial-fixed z-axis zva,t, żva,t Vehicle displacement and velocity at axle a, wheel t, along the inertial-fixed z-axis zwa,t, żwa,t Wheel displacement and velocity at axle a, wheel t, along the inertial-fixed z-axis

## Ports

### Input

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Wheel displacement, zw, along wheel-fixed z-axis, in m. Array dimensions are 1 by the total number of wheels on the vehicle.

For example, for a two-axle vehicle with two wheels per axle, the WhlPz:

• Signal array dimensions are [1x4].

$\mathrm{WhlPz}={z}_{w}=\left[\begin{array}{cccc}{z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel Number
Front leftWhlPz(1,1)11
Front rightWhlPz(1,2)12
Rear leftWhlPz(1,3)21
Rear rightWhlPz(1,4)22

Effective wheel radius, Rew, in m. Array dimensions are 1 by the total number of wheels on the vehicle.

For example, for a two-axle vehicle with two wheels per axle, the WhlRe:

• Signal array dimensions are [1x4].

$\mathrm{Whl}\mathrm{Re}=R{e}_{w}=\left[\begin{array}{cccc}R{e}_{{w}_{1,1}}& R{e}_{{w}_{1,2}}& R{e}_{{w}_{2,1}}& R{e}_{{w}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel Number
Front leftWhlRe(1,1)11
Front rightWhlRe(1,2)12
Rear leftWhlRe(1,3)21
Rear rightWhlRe(1,4)22

Wheel velocity, żw, along wheel-fixed z-axis, in m. Array dimensions are 1 by the total number of wheels on the vehicle.

For example, for a two-axle vehicle with two wheels per axle, the WhlVz:

• Signal array dimensions are [1x4].

$\mathrm{WhlVz}={\stackrel{˙}{z}}_{w}=\left[\begin{array}{cccc}{\stackrel{˙}{z}}_{{w}_{1,1}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel Number
Front leftWhlVz(1,1)11
Front rightWhlVz(1,2)12
Rear leftWhlVz(1,3)21
Rear rightWhlVz(1,4)22

Longitudinal wheel force applied to vehicle, Fwx, along the inertial-fixed x-axis. Array dimensions are 1 by the total number of wheels on the vehicle.

For example, for a two-axle vehicle with two wheels per axle, the WhlFx:

• Signal array dimensions are [1x4].

$\mathrm{WhlFx}={F}_{wx}=\left[\begin{array}{cccc}{F}_{w{x}_{1,1}}& {F}_{w{x}_{1,2}}& {F}_{w{x}_{2,1}}& {F}_{w{x}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel Number
Front leftWhlFx(1,1)11
Front rightWhlFx(1,2)12
Rear leftWhlFx(1,3)21
Rear rightWhlFx(1,4)22

Lateral wheel force applied to vehicle, Fwy, along the inertial-fixed y-axis. Array dimensions are 1 by the total number of wheels on the vehicle.

For example, for a two-axle vehicle with two wheels per axle, the WhlFy:

• Signal array dimensions are [1x4].

$\mathrm{WhlFy}={F}_{wy}=\left[\begin{array}{cccc}{F}_{w{y}_{1,1}}& {F}_{w{y}_{1,2}}& {F}_{w{y}_{2,1}}& {F}_{w{y}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel Number
Front leftWhlFy(1,1)11
Front rightWhlFy(1,2)12
Rear leftWhlFy(1.3)21
Rear rightWhlFy(1,4)22

Longitudinal, lateral, and vertical suspension moments at axle a, wheel t, applied to the wheel at the axle wheel carrier reference coordinate, in N·m. Input array dimensions are 3 by the number of wheels on the vehicle.

• WhlM(1,...) — Suspension moment applied to the wheel about the inertial-fixed x-axis (longitudinal)

• WhlM(2,...) — Suspension moment applied to the wheel about the inertial-fixed y-axis (lateral)

• WhlM(3,...) — Suspension moment applied to the wheel about the inertial-fixed z-axis (vertical)

For example, for a two-axle vehicle with two wheels per axle, the WhlM:

• Signal dimensions are [3x4].

• Signal contains suspension moments applied to four wheels according to their axle and wheel locations.

$\mathrm{WhlM}={M}_{w}=\left[\begin{array}{cccc}{M}_{w{x}_{1,1}}& {M}_{w{x}_{1,2}}& {M}_{w{x}_{2,1}}& {M}_{w{x}_{2,2}}\\ {M}_{w{y}_{1,1}}& {M}_{w{y}_{1,2}}& {M}_{w{y}_{2,1}}& {M}_{w{y}_{2,2}}\\ {M}_{w{z}_{1,1}}& {M}_{w{z}_{1,2}}& {M}_{w{z}_{2,1}}& {M}_{w{z}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel NumberMoment Axis
Front leftWhlM(1,1)11Inertial-fixed x-axis (longitudinal)
Front rightWhlM(1,2)12
Rear leftWhlM(1,3)21
Rear rightWhlM(1,4)22
Front leftWhlM(2,1)11Inertial-fixed y-axis (lateral)
Front rightWhlM(2,2)12
Rear leftWhlM(2,3)21
Rear rightWhlM(2,4)22
Front leftWhlM(3,1)11Inertial-fixed z-axis (vertical)
Front rightWhlM(3,2)12
Rear leftWhlM(3,3)21
Rear rightWhlM(3,4)22

Vehicle displacement from axle a, wheel t along inertial-fixed coordinate system, in m. Input array dimensions are 3 by the number of wheels on the vehicle.

• VehP(1,...) — Vehicle displacement from wheel, xv, along the inertial-fixed x-axis

• VehP(2,...) — Vehicle displacement from wheel, yv, along the inertial-fixed y-axis

• VehP(3,...) — Vehicle displacement from wheel, zv, along the inertial-fixed z-axis

For example, for a two-axle vehicle with two wheels per axle, the VehP:

• Signal dimensions are [3x4].

• Signal contains four displacements according to their axle and wheel locations.

$\mathrm{VehP}=\left[\begin{array}{c}{x}_{v}\\ {y}_{v}\\ {z}_{v}\end{array}\right]=\left[\begin{array}{cccc}{x}_{v}{}_{{}_{1,1}}& {x}_{v}{}_{{}_{1,2}}& {x}_{v}{}_{{}_{2,1}}& {x}_{v}{}_{{}_{2,2}}\\ {y}_{v}{}_{{}_{1,1}}& {y}_{v}{}_{{}_{1,2}}& {y}_{v}{}_{{}_{2,1}}& {y}_{v}{}_{{}_{2,2}}\\ {z}_{v}{}_{{}_{1,1}}& {z}_{v}{}_{{}_{1,2}}& {z}_{v}{}_{{}_{2,1}}& {z}_{v}{}_{{}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel NumberAxis
Front leftVehP(1,1)11Inertial-fixed x-axis
Front rightVehP(1,2)12
Rear leftVehP(1,3)21
Rear rightVehP(1,4)22
Front leftVehP(2,1)11Inertial-fixed y-axis
Front rightVehP(2,2)12
Rear leftVehP(2,3)21
Rear rightVehP(2,4)22
Front leftVehP(3,1)11inertial-fixed z-axis
Front rightVehP(3,2)12
Rear leftVehP(3,3)21
Rear rightVehP(3,4)22

Vehicle velocity at axle a, wheel t along inertial-fixed coordinate system, in m. Input array dimensions are 3 by the number of wheels on the vehicle.

• VehV(1,...) — Vehicle velocity at wheel, xv, along the inertial-fixed x-axis

• VehV(2,...) — Vehicle velocity at wheel, yv, along the inertial-fixed y-axis

• VehV(3,...) — Vehicle velocity at wheel, zv, along the inertial-fixed z-axis

For example, for a two-axle vehicle with two wheels per axle, the VehV:

• Signal dimensions are [3x4].

• Signal contains 4 velocities according to their axle and wheel locations.

$\mathrm{VehV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{v}\\ {\stackrel{˙}{y}}_{v}\\ {\stackrel{˙}{z}}_{v}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{v}_{1,1}}& {\stackrel{˙}{x}}_{{v}_{1,2}}& {\stackrel{˙}{x}}_{{v}_{2,1}}& {\stackrel{˙}{x}}_{{v}_{2,2}}\\ {\stackrel{˙}{y}}_{{v}_{1,1}}& {\stackrel{˙}{y}}_{{v}_{1,2}}& {\stackrel{˙}{y}}_{{v}_{2,1}}& {\stackrel{˙}{y}}_{{v}_{2,2}}\\ {\stackrel{˙}{z}}_{{v}_{1,1}}& {\stackrel{˙}{z}}_{{v}_{1,2}}& {\stackrel{˙}{z}}_{{v}_{2,1}}& {\stackrel{˙}{z}}_{{v}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel NumberAxis
Front leftVehV(1,1)11Inertial-fixed x-axis
Front rightVehV(1,2)12
Rear leftVehV(1,3)21
Rear rightVehV(1,4)22
Front leftVehV(2,1)11Inertial-fixed y-axis
Front rightVehV(2,2)12
Rear leftVehV(2,3)21
Rear rightVehV(2,4)22
Front leftVehV(3,1)11Inertial-fixed z-axis
Front rightVehV(3,2)12
Rear leftVehV(3,3)21
Rear rightVehV(3,4)22

Optional steering angle for each wheel, δ. Input array dimensions are 1 by the number of steered wheels.

For example, for a two-axle vehicle with two wheels per axle, you can input steering angles for both wheels on the first axle.

• To enable the StrgAng port, set Steered axle enable by axle, StrgEnByAxl to [1 0]. The input signal array dimensions are [1x2].

• The StrgAng signal contains two steering angles according to their axle and wheel locations.

$\mathrm{StrgAng}={\delta }_{steer}=\left[\begin{array}{cc}{\delta }_{stee{r}_{1,1}}& {\delta }_{stee{r}_{1,2}}\end{array}\right]$

WheelArray ElementAxleWheel Number
Front leftStrgAng(1,1)11
Front rightStrgAng(1,2)12

#### Dependencies

To enable the port StrgAng, set an element of the Steered axle enable by axle, StrgEnByAxl vector to 1.

### Output

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Bus signal containing block values. The signals are arrays that depend on the wheel location.

For example, these are the indices for a two-axle, two-wheel vehicle. The total number of wheels is four.

• 1D array signal (1-by-4)

WheelArray ElementAxleWheel Number
Front left(1,1)11
Front right(1,2)12
Rear left(1,3)21
Rear right(1,4)22

• 3D array signal (3-by-4)

WheelArray ElementAxleWheel Number
Front left(1,1)11
Front right(1,2)12
Rear left(1,3)21
Rear right(1,4)22
Front left(2,1)11
Front right(2,2)12
Rear left(2,3)21
Rear right(2,4)22
Front left(3,1)11
Front right(3,2)12
Rear left(3,3)21
Rear right(3,4)22

SignalDescriptionArray SignalVariableUnits
Camber

Wheel angles according to the axle and wheel location.

1D

$\mathrm{WhlAng}\left[1,...\right]=\xi =\left[{\xi }_{a,t}\right]$

rad

Caster

$\mathrm{WhlAng}\left[2,...\right]=\eta =\left[{\eta }_{a,t}\right]$

Toe

$\mathrm{WhlAng}\left[3,...\right]=\zeta =\left[{\zeta }_{a,t}\right]$

Height

Suspension height

1D

H

m

Power

Suspension power dissipation

1D

Psusp

W

Energy

Suspension absorbed energy

1D

Esusp

J

VehF

Suspension forces applied to the vehicle

3D

For a two-axle, two wheels per axle vehicle:

$\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]$

N

VehM

Suspension moments applied to vehicle

3D

For a two-axle, two wheels per axle vehicle:

$\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]$

N·m

WhlF

Suspension force applied to wheel

3D

For a two-axle, two wheels per axle vehicle:

$\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]$

N

WhlP

Wheel displacement

3D

For a two-axle, two wheels per axle vehicle:

$\mathrm{WhlP}=\left[\begin{array}{c}{x}_{w}\\ {y}_{w}\\ {z}_{w}\end{array}\right]=\left[\begin{array}{cccc}{x}_{w}{}_{{}_{1,1}}& {x}_{w}{}_{{}_{1,2}}& {x}_{w}{}_{{}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{w}{}_{{}_{1,1}}& {y}_{w}{}_{{}_{1,2}}& {y}_{w}{}_{{}_{2,1}}& {y}_{w}{}_{{y}_{2,2}}\\ {z}_{wtr}{}_{{}_{1,1}}& {z}_{wtr}{}_{{}_{1,2}}& {z}_{wtr}{}_{{}_{2,1}}& {z}_{wt{r}_{2,2}}\end{array}\right]$

m

WhlV

Wheel velocity

3D

For a two-axle, two wheels per axle vehicle:

$\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{w}\\ {\stackrel{˙}{y}}_{w}\\ {\stackrel{˙}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{w}_{1,1}}& {\stackrel{˙}{x}}_{{w}_{1,2}}& {\stackrel{˙}{x}}_{{w}_{2,1}}& {\stackrel{˙}{x}}_{{w}_{2,2}}\\ {\stackrel{˙}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{y}}_{{w}_{1,2}}& {\stackrel{˙}{y}}_{{w}_{2,1}}& {\stackrel{˙}{y}}_{{w}_{2,2}}\\ {\stackrel{˙}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]$

m/s

WhlAng

Wheel camber, caster, toe angles

3D

For a two-axle, two wheels per axle vehicle:

$\mathrm{WhlAng}=\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]=\left[\begin{array}{cccc}{\xi }_{1,1}& {\xi }_{1,2}& {\xi }_{2,1}& {\xi }_{2,2}\\ {\eta }_{1,1}& {\eta }_{1,2}& {\eta }_{2,1}& {\eta }_{2,2}\\ {\zeta }_{1,1}& {\zeta }_{1,2}& {\zeta }_{2,1}& {\zeta }_{2,2}\end{array}\right]$

rad

Longitudinal, lateral, and vertical suspension force at axle a, wheel t, applied to the vehicle at the suspension connection point, in N. Array dimensions are 3 by the number of wheels on the vehicle.

• VehF(1,...) — Suspension force applied to vehicle along the inertial-fixed x-axis (longitudinal)

• VehF(2,...) — Suspension force applied to vehicle along the inertial-fixed y-axis (lateral)

• VehF(3,...) — Suspension force applied to vehicle along the inertial-fixed z-axis (vertical)

For example, for a two-axle vehicle with two wheels per axle, the VehF:

• Signal dimensions are [3x4].

• Signal contains suspension forces applied to the vehicle according to the axle and wheel locations.

$\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel NumberForce Axis
Front leftVehF(1,1)11Inertial-fixed x-axis (longitudinal)
Front rightVehF(1,2)12
Rear leftVehF(1,3)21
Rear rightVehF(1,4)22
Front leftVehF(2,1)11Inertial-fixed y-axis (lateral)
Front rightVehF(2,2)12
Rear leftVehF(2,3)21
Rear rightVehF(2,4)22
Front leftVehF(3,1)11Inertial-fixed z-axis (vertical)
Front rightVehF(3,2)12
Rear leftVehF(3,3)21
Rear rightVehF(3,4)22

Longitudinal, lateral, and vertical suspension moment at axle a, wheel t, applied to the vehicle at the suspension connection point, in N·m. Array dimensions are 3 by the number of wheels on the vehicle.

• VehM(1,...) — Suspension moment applied to the vehicle about the inertial-fixed x-axis (longitudinal)

• VehM(2,...) — Suspension moment applied to the vehicle about the inertial-fixed y-axis (lateral)

• VehM(3,...) — Suspension moment applied to the vehicle about the inertial-fixed z-axis (vertical)

For example, for a two-axle vehicle with two wheels per axle, the VehM:

• Signal dimensions are [3x4].

• Signal contains suspension moments applied to vehicle according to the axle and wheel locations.

$\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]$

Array ElementAxleWheel NumberMoment Axis
VehM(1,1)11Inertial-fixed x-axis (longitudinal)
VehM(1,2)12
VehM(1,3)21
VehM(1,4)22
VehM(2,1)11Inertial-fixed y-axis (lateral)
VehM(2,2)12
VehM(2,3)21
VehM(2,4)22
VehM(3,1)11Inertial-fixed z-axis (vertical)
VehM(3,2)12
VehM(3,3)21
VehM(3,4)22

Longitudinal, lateral, and vertical suspension forces at axle a, wheel t, applied to the wheel at the axle wheel carrier reference coordinate, in N. Array dimensions are 3 by the number of wheels on the vehicle.

• WhlF(1,...) — Suspension force on wheel along the inertial-fixed x-axis (longitudinal)

• WhlF(2,...) — Suspension force on wheel along the inertial-fixed y-axis (lateral)

• WhlF(3,...) — Suspension force on wheel along the inertial-fixed z-axis (vertical)

For example, for a two-axle vehicle with two wheels per axle, the WhlF:

• Signal dimensions are [3x4].

• Signal contains wheel forces applied to the vehicle according to the axle and wheel locations.

$\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel NumberForce Axis
Front leftWhlF(1,1)11Inertial-fixed x-axis (longitudinal)
Front rightWhlF(1,2)12
Rear leftWhlF(1,3)21
Rear rightWhlF(1,4)22
Front leftWhlF(2,1)11Inertial-fixed y-axis (lateral)
Front rightWhlF(2,2)12
Rear leftWhlF(2,3)21
Rear rightWhlF(2,4)22
Front leftWhlF(3,1)11Inertial-fixed z-axis (vertical)
Front rightWhlF(3,2)12
Rear leftWhlF(3,3)21
Rear rightWhlF(3,4)22

Longitudinal, lateral, and vertical wheel velocity at axle a, wheel t, in m/s. Array dimensions are 3 by the number of wheels on the vehicle.

• WhlV(1,...) — Wheel velocity along the inertial-fixed x-axis (longitudinal)

• WhlV(2,...) — Wheel velocity along the inertial-fixed y-axis (lateral)

• WhlV(3,...) — Wheel velocity along the inertial-fixed z-axis (vertical)

For example, for a two-axle vehicle with two wheels per axle, the WhlV:

• Signal dimensions are [3x4].

• Signal contains wheel forces applied to the vehicle according to the axle and wheel locations.

$\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{w}\\ {\stackrel{˙}{y}}_{w}\\ {\stackrel{˙}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{w}_{1,1}}& {\stackrel{˙}{x}}_{{w}_{1,2}}& {\stackrel{˙}{x}}_{{w}_{2,1}}& {\stackrel{˙}{x}}_{{w}_{2,2}}\\ {\stackrel{˙}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{y}}_{{w}_{1,2}}& {\stackrel{˙}{y}}_{{w}_{2,1}}& {\stackrel{˙}{y}}_{{w}_{2,2}}\\ {\stackrel{˙}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]$

WheelArray ElementAxleWheel NumberForce Axis
Front leftWhlV(1,1)11Inertial-fixed x-axis (longitudinal)
Front rightWhlV(1,2)12
Rear leftWhlV(1,3)21
Rear rightWhlV(1,4)22
Front leftWhlV(2,1)11Inertial-fixed y-axis (lateral)
Front rightWhlV(2,2)12
Rear leftWhlV(2,3)21
Rear rightWhlV(2,4)22
Front leftWhlV(3,1)11Inertial-fixed z-axis (vertical)
Front rightWhlV(3,2)12
Rear leftWhlV(3,3)21
Rear rightWhlV(3,4)22

Camber, caster, and toe angles at axle a, wheel t, in rad. Array dimensions are 3 by the number of wheels on the vehicle.

• WhlAng(1,...) — Camber angle

• WhlAng(2,...) — Caster angle

• WhlAng(3,...) — Toe angle

For example, for a two-axle vehicle with two wheels per axle, the WhlAng:

• Signal dimensions are [3x4].

• Signal contains angles according to the axle and wheel locations.

$\mathrm{WhlAng}=\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]=\left[\begin{array}{cccc}{\xi }_{1,1}& {\xi }_{1,2}& {\xi }_{2,1}& {\xi }_{2,2}\\ {\eta }_{1,1}& {\eta }_{1,2}& {\eta }_{2,1}& {\eta }_{2,2}\\ {\zeta }_{1,1}& {\zeta }_{1,2}& {\zeta }_{2,1}& {\zeta }_{2,2}\end{array}\right]$

WheelArray ElementAxleWheel NumberAngle
Front leftWhlAng(1,1)11

Camber

Front rightWhlAng(1,2)12
Rear leftWhlAng(1,3)21
Rear rightWhlAng(1,4)22
Front leftWhlAng(2,1)11

Caster

Front rightWhlAng(2,2)12
Rear leftWhlAng(2,3)21
Rear rightWhlAng(2,4)22
Front leftWhlAng(3,1)11

Toe

Front rightWhlF(3,2)12
Rear leftWhlF(3,3)21
Rear rightWhlF(3,4)22

## Parameters

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Include damping

#### Dependencies

Selecting this parameter creates:

Number of axles, Na, dimensionless.

Number of wheels per axle, Nta, dimensionless. Vector is 1 by the number of vehicle axles, Na. For example, [1,2] represents one wheel on axle one and two wheels on axle two.

Boolean vector that enables axle steering, Ensteer, dimensionless. Vector is 1 by the number of vehicle axles, Na. For example:

• [1 0] — For a two-axle vehicle, enables axle 1 steering and disables axle 2 steering

• [1 1] — For a two-axle vehicle, enables axle 1 and axle 2 steering

#### Dependencies

Setting any element of the Steered axle enable by axle, StrgEnByAxl vector to 1:

• Enables the port StrgAng.

• Enables these parameters:

• Toe angle vs steering angle slope, ToeStrgSlp

• Caster angle vs steering angle slope, CasterStrgSlp

• Camber angle vs steering angle slope, CamberStrgSlp

• Suspension height vs steering angle slope, StrgHgtSlp

For example, for a two-axle vehicle with two wheels per axle, you can input steering angles for both wheels on the first axle.

• To enable the StrgAng port, set Steered axle enable by axle, StrgEnByAxl to [1 0]. The input signal array dimensions are [1x2].

• The StrgAng signal contains two steering angles according to their axle and wheel locations.

$\mathrm{StrgAng}={\delta }_{steer}=\left[\begin{array}{cc}{\delta }_{stee{r}_{1,1}}& {\delta }_{stee{r}_{1,2}}\end{array}\right]$

WheelArray ElementAxleWheel Number
Front leftStrgAng(1,1)11
Front rightStrgAng(1,2)12

Boolean vector that enables axle anti-sway for axle a, dimensionless. For example, [1 0] enables axle 1 anti-sway and disables axle 2 anti-sway. Vector is 1 by the number of vehicle axles, Na.

#### Dependencies

Setting an element of the Anti-sway axle enable by axle, AntiSwayEnByAxl vector to 1 creates these anti-sway parameters:

• Anti-sway arm radius, AntiSwayR

• Anti-sway arm neutral angle, AntiSwayNtrlAng

• Anti-sway torsion spring constant, AntiSwayTrsK

### Suspension

Compliance and Damping - Passive

Linear vertical spring constant for independent suspension wheels on axle a, kza, in N/m.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

Vertical preload spring force applied to the wheels on the axle at wheel carrier reference coordinates, Fz0a, in N. Positive preload forces:

• Cause the vehicle to lift.

• Point along the negative inertial-fixed z-axis.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

Linear vertical damping constant for independent suspension wheels on axle a, cza, in Ns/m.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

#### Dependencies

To create this parameter, clear .

Maximum suspension extension or minimum suspension compression height, Hmax, for axle a before the suspension reaches a hardstop, in m.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

Compliance and Damping - Active

Damping coefficient table as a function of active duty cycle and actuator compression velocity, in N·s/m. Each value specifies the damping for a specific combination of actuator duty cycle and velocity. The array dimensions must match the duty cycle, M, and actuator velocity, N, breakpoint vector dimensions.

#### Dependencies

To create this parameter, clear .

Damping actuator duty cycle breakpoints, dimensionless.

#### Dependencies

To create this parameter, clear .

Damping actuator velocity breakpoints, in m/s.

#### Dependencies

To create this parameter, clear .

Geometry

Nominal suspension toe angle at zero steering angle, ζ0a, in rad.

Roll steer angle versus suspension height, mhtoea, in rad/m.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

Toe angle versus steering angle slope, mtoesteera, dimensionless.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

#### Dependencies

To enable the port StrgAng, set an element of the Steered axle enable by axle, StrgEnByAxl vector to 1.

Nominal suspension caster angle at zero steering angle, η0a, in rad.

Caster angle versus suspension height, mhcastera, in rad/m.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

Caster angle versus steering angle slope, mcastersteera, dimensionless.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

#### Dependencies

To enable the port StrgAng, set an element of the Steered axle enable by axle, StrgEnByAxl vector to 1.

Nominal suspension camber angle at zero steering angle, ξ0a, in rad.

Camber angle versus suspension height, mhcambera, in rad/m.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

Camber angle versus steering angle slope, mcambersteera, dimensionless.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

#### Dependencies

To enable the port StrgAng, set an element of the Steered axle enable by axle, StrgEnByAxl vector to 1.

Steering angle to vertical force slope applied at suspension wheel carrier reference point, mhsteera, in m/rad.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

#### Dependencies

To enable the port StrgAng, set an element of the Steered axle enable by axle, StrgEnByAxl vector to 1.

### Anti-Sway

Anti-sway arm radius, r, in m.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

#### Dependencies

Setting an element of the Anti-sway axle enable by axle, AntiSwayEnByAxl vector to 1 creates these anti-sway parameters:

• Anti-sway arm radius, AntiSwayR

• Anti-sway arm neutral angle, AntiSwayNtrlAng

• Anti-sway torsion spring constant, AntiSwayTrsK

Anti-sway arm neutral angle, θ0a, at nominal suspension height, in rad.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

#### Dependencies

Setting an element of the Anti-sway axle enable by axle, AntiSwayEnByAxl vector to 1 creates these anti-sway parameters:

• Anti-sway arm radius, AntiSwayR

• Anti-sway arm neutral angle, AntiSwayNtrlAng

• Anti-sway torsion spring constant, AntiSwayTrsK

Anti-sway bar torsion spring constant, ka, in N·m/rad.

Vector is 1 by the number of vehicle axles, Na. If you provide a scalar value, the block uses that value for all axles.

#### Dependencies

Setting an element of the Anti-sway axle enable by axle, AntiSwayEnByAxl vector to 1 creates these anti-sway parameters:

• Anti-sway arm radius, AntiSwayR

• Anti-sway arm neutral angle, AntiSwayNtrlAng

• Anti-sway torsion spring constant, AntiSwayTrsK

## References

[1] Gillespie, Thomas. Fundamentals of Vehicle Dynamics. Warrendale, PA: Society of Automotive Engineers, 1992.

[2] Vehicle Dynamics Standards Committee. Vehicle Dynamics Terminology. SAE J670. Warrendale, PA: Society of Automotive Engineers, 2008.

[3] Technical Committee. Road vehicles — Vehicle dynamics and road-holding ability — Vocabulary. ISO 8855:2011. Geneva, Switzerland: International Organization for Standardization, 2011.

## Version History

Introduced in R2018a

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