Independent Suspension - Mapped

Mapped independent suspension

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Libraries:
Vehicle Dynamics Blockset / Suspension

Description

The Independent Suspension - Mapped block implements a mapped independent suspension for multiple axles with multiple wheels per axle. You can use the block to model suspension geometry, compliance, and damping effects from measured or simulated suspension response data.

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 Element	Setting
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

Parameter	Setting
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

Wheel	Axle	Wheel Number
Front left	Front	1
Front right	Front	2
Rear left	Rear	1
Rear right	Rear	2

Suspension Compliance and Damping

The block uses a lookup table that relates the vertical damping and compliance to the suspension height, suspension height rate of change, and steering angle. You can calibrate the wheel force lookup table so that steering angle changes from the nominal center position generate a force that increases the vehicle height.

The block implements these equations.

$$\begin{array}{l}{F}_{wzlooku{p}_a}=f(z_{v_{a,t}}-z_{w_{a,t}},{\dot{z}}_{v_{a,t}}-{\dot{z}}_{w_{a,t}},{\delta }_{stee{r}_{a,t}})\\ \\ F_{w{z}_{a,t}}=F_{wzlooku{p}_a}+F_{zasw{y}_{a,t}}\end{array}$$

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.

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

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

$$\begin{array}{l}{x}_{w_{a,t}}=x_{v_{a,t}}\\ y_{w_{a,t}}=y_{v_{a,t}}\\ {\dot{x}}_{w_{a,t}}={\dot{x}}_{v_{a,t}}\\ {\dot{y}}_{w_{a,t}}={\dot{y}}_{v_{a,t}}\end{array}$$

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

Anti-Sway Bar

Optionally, use the Anti-sway axle enable by axle, AntiSwayEnByAxl parameter to implement an anti-sway bar force, F_zaswya,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.

Calculation	Equation
Anti-sway bar angular deflection for a given axle and wheel, Δϴa,t	$\begin{array}{l}{\theta }_{0a}={\tan }^{-1}\left(\frac{z_0}{r}\right)\\ \Delta {\theta }_{a,t}={\tan }^{-1}\left(\frac{r\tan {\theta }_{0a}-z_{w_{a,t}}+z_{v_{a,t}}}{r}\right)\end{array}$
Anti-sway bar twist angle, ϴa	${\theta }_a=-{\tan }^{-1}\left(\frac{r\tan {\theta }_{0a}-z_{w_{a,1}}+z_{v_{a,1}}}{r}\right)-{\tan }^{-1}\left(\frac{r\tan {\theta }_{0a}-z_{w_{a,2}}+z_{v_{a,2}}}{r}\right)$
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	$\begin{array}{l}{F}_{zasw{y}_{a,1}}=\left(\frac{{\tau }_a}{r}\right)\cos \left({\theta }_{0a}-{\tan }^{-1}\left(\frac{r\tan {\theta }_{0a}-z_{w_{a,1}}+z_{v_{a,1}}}{r}\right)\right)\\ F_{zasw{y}_{a,2}}=\left(\frac{{\tau }_a}{r}\right)\cos \left({\theta }_{0a}-{\tan }^{-1}\left(\frac{r\tan {\theta }_{0a}-z_{w_{a,2}}+z_{v_{a,2}}}{r}\right)\right)\end{array}$

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, the block uses a lookup table, G_alookup, that is a function of the suspension height and steering angle.

$$\left[\begin{array}{ccc}{\xi }_{a,t}& {\eta }_{a,t}& {\zeta }_{a,t}\end{array}\right]=G_{alookup}f(z_{w_{a,t}}-z_{v_{a,t}},{\delta }_{stee{r}_{a,t}})$$

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
δsteera,t	Steering angle input for axle a, wheel t
zva,t	Vehicle displacement at axle a, wheel t, along inertial-fixed z-axis
zwa,t	Wheel displacement at axle a, wheel t, along inertial-fixed z-axis

Steering Angles

Optionally, you can input steering angles for the wheels. To calculate the steering angles for the wheels, the block offsets the input steering angles as a function of the suspension height. For the calculation, the block uses a lookup table, G_alookup, that is a function of the suspension position and steering angle.

${\delta }_{whlstee{r}_{a,t}}={\delta }_{stee{r}_{a,t}}+G_{alookup}f\left(z_{w_{a,t}}-z_{v_{a,t}},{\delta }_{stee{r}_{a,t}}\right)$

The equation uses these variables.

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, and wheel, t.

Calculation	Equation
Dissipated power, Psuspa,t	$$P_{sus{p}_{a,t}}=F_{wzlooku{p}_a}\left({\dot{z}}_{v_{a,t}}-{\dot{z}}_{w_{a,t}},{\dot{z}}_{v_{a,t}}-{\dot{z}}_{w_{a,t}},{\delta }_{stee{r}_{a,t}}\right)$$
Absorbed energy, Esuspa,t	$$E_{sus{p}_{a,t}}=F_{wzlooku{p}_a}\left({\dot{z}}_{v_{a,t}}-{\dot{z}}_{w_{a,t}},{\dot{z}}_{v_{a,t}}-{\dot{z}}_{w_{a,t}},{\delta }_{stee{r}_{a,t}}\right)$$
Suspension height, Ha,t	$$H_{a,t}=-\left(z_{v_{a,t}}-z_{w_{a,t}}-\mathrm{median}(f\_susp\_dz\_bp)\right)$$
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
f_susp_dz_bp	Vertical axis suspension height breakpoints
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

Examples

Kinematics and Compliance Virtual Test Laboratory Reference Application

Generate optimized suspension parameters for the vehicle dynamics mapped suspension blocks.

• Open Script

Ports

Input

WhlPz — Wheel z-axis displacement

array

Wheel displacement, z_w, 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].

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

  Wheel	Array Element	Axle	Wheel Number
  Front left	WhlPz(1,1)	1	1
  Front right	WhlPz(1,2)	1	2
  Rear left	WhlPz(1,3)	2	1
  Rear right	WhlPz(1,4)	2	2

WhlRe — Wheel effective radius

array

Effective wheel radius, Re_w, 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].

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

  Wheel	Array Element	Axle	Wheel Number
  Front left	WhlRe(1,1)	1	1
  Front right	WhlRe(1,2)	1	2
  Rear left	WhlRe(1,3)	2	1
  Rear right	WhlRe(1,4)	2	2

WhlVz — Wheel z-axis velocity

array

Wheel velocity, ż_w, 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].

  $$\mathrm{WhlVz}={\dot{z}}_w=\left[\begin{array}{cccc}{\dot{z}}_{w_{1,1}}& {\dot{z}}_{w_{1,2}}& {\dot{z}}_{w_{2,1}}& {\dot{z}}_{w_{2,2}}\end{array}\right]$$

  Wheel	Array Element	Axle	Wheel Number
  Front left	WhlVz(1,1)	1	1
  Front right	WhlVz(1,2)	1	2
  Rear left	WhlVz(1,3)	2	1
  Rear right	WhlVz(1,4)	2	2

WhlFx — Longitudinal wheel force on vehicle

array

Longitudinal wheel force applied to vehicle, F_wx, 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].

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

  Wheel	Array Element	Axle	Wheel Number
  Front left	WhlFx(1,1)	1	1
  Front right	WhlFx(1,2)	1	2
  Rear left	WhlFx(1,3)	2	1
  Rear right	WhlFx(1,4)	2	2

WhlFy — Lateral wheel force on vehicle

array

Lateral wheel force applied to vehicle, F_wy, 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].

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

  Wheel	Array Element	Axle	Wheel Number
  Front left	WhlFy(1,1)	1	1
  Front right	WhlFy(1,2)	1	2
  Rear left	WhlFy(1.3)	2	1
  Rear right	WhlFy(1,4)	2	2

WhlM — Suspension moment on wheel

array

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.

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

  Wheel	Array Element	Axle	Wheel Number	Moment Axis
  Front left	WhlM(1,1)	1	1	Inertial-fixed x-axis (longitudinal)
  Front right	WhlM(1,2)	1	2
  Rear left	WhlM(1,3)	2	1
  Rear right	WhlM(1,4)	2	2
  Front left	WhlM(2,1)	1	1	Inertial-fixed y-axis (lateral)
  Front right	WhlM(2,2)	1	2
  Rear left	WhlM(2,3)	2	1
  Rear right	WhlM(2,4)	2	2
  Front left	WhlM(3,1)	1	1	Inertial-fixed z-axis (vertical)
  Front right	WhlM(3,2)	1	2
  Rear left	WhlM(3,3)	2	1
  Rear right	WhlM(3,4)	2	2

VehP — Vehicle displacement

array

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, x_v, along the inertial-fixed x-axis

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

• VehP(3,...) — Vehicle displacement from wheel, z_v, 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.

  $$\mathrm{VehP}=\left[\begin{array}{c}{x}_v\\ y_v\\ z_v\end{array}\right]=\left[\begin{array}{cccc}{x}_v{}_{{}_{1,1}}& x_v{}_{{}_{1,2}}& x_v{}_{{}_{2,1}}& x_v{}_{{}_{2,2}}\\ y_v{}_{{}_{1,1}}& y_v{}_{{}_{1,2}}& y_v{}_{{}_{2,1}}& y_v{}_{{}_{2,2}}\\ z_v{}_{{}_{1,1}}& z_v{}_{{}_{1,2}}& z_v{}_{{}_{2,1}}& z_v{}_{{}_{2,2}}\end{array}\right]$$

  Wheel	Array Element	Axle	Wheel Number	Axis
  Front left	VehP(1,1)	1	1	Inertial-fixed x-axis
  Front right	VehP(1,2)	1	2
  Rear left	VehP(1,3)	2	1
  Rear right	VehP(1,4)	2	2
  Front left	VehP(2,1)	1	1	Inertial-fixed y-axis
  Front right	VehP(2,2)	1	2
  Rear left	VehP(2,3)	2	1
  Rear right	VehP(2,4)	2	2
  Front left	VehP(3,1)	1	1	inertial-fixed z-axis
  Front right	VehP(3,2)	1	2
  Rear left	VehP(3,3)	2	1
  Rear right	VehP(3,4)	2	2

VehV — Vehicle velocity

array

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, x_v, along the inertial-fixed x-axis

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

• VehV(3,...) — Vehicle velocity at wheel, z_v, 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.

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

  Wheel	Array Element	Axle	Wheel Number	Axis
  Front left	VehV(1,1)	1	1	Inertial-fixed x-axis
  Front right	VehV(1,2)	1	2
  Rear left	VehV(1,3)	2	1
  Rear right	VehV(1,4)	2	2
  Front left	VehV(2,1)	1	1	Inertial-fixed y-axis
  Front right	VehV(2,2)	1	2
  Rear left	VehV(2,3)	2	1
  Rear right	VehV(2,4)	2	2
  Front left	VehV(3,1)	1	1	Inertial-fixed z-axis
  Front right	VehV(3,2)	1	2
  Rear left	VehV(3,3)	2	1
  Rear right	VehV(3,4)	2	2

StrgAng — Steering angle, optional

array

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.

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

  Wheel	Array Element	Axle	Wheel Number
  Front left	StrgAng(1,1)	1	1
  Front right	StrgAng(1,2)	1	2

Dependencies

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

Output

Info — Bus signal

bus

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)

  Wheel	Array Element	Axle	Wheel Number
  Front left	(1,1)	1	1
  Front right	(1,2)	1	2
  Rear left	(1,3)	2	1
  Rear right	(1,4)	2	2

• 3D array signal (3-by-4)

  Wheel	Array Element	Axle	Wheel Number
  Front left	(1,1)	1	1
  Front right	(1,2)	1	2
  Rear left	(1,3)	2	1
  Rear right	(1,4)	2	2
  Front left	(2,1)	1	1
  Front right	(2,2)	1	2
  Rear left	(2,3)	2	1
  Rear right	(2,4)	2	2
  Front left	(3,1)	1	1
  Front right	(3,2)	1	2
  Rear left	(3,3)	2	1
  Rear right	(3,4)	2	2

Signal	Description	Array Signal	Variable	Units
Camber	Wheel angles according to the axle and wheel location.	1D	$$\mathrm{WhlAng}[1,\mathrm{...}]=\xi =\left[{\xi }_{a,t}\right]$$	rad
Caster			$$\mathrm{WhlAng}[2,\mathrm{...}]=\eta =\left[{\eta }_{a,t}\right]$$
Toe			$$\mathrm{WhlAng}[3,\mathrm{...}]=\zeta =\left[{\zeta }_{a,t}\right]$$
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: $$\mathrm{WhlV}=\left[\begin{array}{c}{\dot{x}}_w\\ {\dot{y}}_w\\ {\dot{z}}_w\end{array}\right]=\left[\begin{array}{cccc}{\dot{x}}_{w_{1,1}}& {\dot{x}}_{w_{1,2}}& {\dot{x}}_{w_{2,1}}& {\dot{x}}_{w_{2,2}}\\ {\dot{y}}_{w_{{}_{1,1}}}& {\dot{y}}_{w_{1,2}}& {\dot{y}}_{w_{2,1}}& {\dot{y}}_{w_{2,2}}\\ {\dot{z}}_{w_{{}_{1,1}}}& {\dot{z}}_{w_{1,2}}& {\dot{z}}_{w_{2,1}}& {\dot{z}}_{w_{2,2}}\end{array}\right]$$	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

VehF — Suspension force on vehicle

array

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.

  $$\mathrm{VehF}=F_v=\left[\begin{array}{cccc}{F}_v{}_{x_{1,1}}& F_v{}_{x_{1,2}}& F_v{}_{x_{2,1}}& F_v{}_{x_{2,2}}\\ F_v{}_{y_{1,1}}& F_v{}_{y_{1,2}}& F_v{}_{y_{2,1}}& F_v{}_{y_{2,2}}\\ F_v{}_{z_{1,1}}& F_v{}_{z_{1,2}}& F_v{}_{z_{2,1}}& F_v{}_{z_{2,2}}\end{array}\right]$$

  Wheel	Array Element	Axle	Wheel Number	Force Axis
  Front left	VehF(1,1)	1	1	Inertial-fixed x-axis (longitudinal)
  Front right	VehF(1,2)	1	2
  Rear left	VehF(1,3)	2	1
  Rear right	VehF(1,4)	2	2
  Front left	VehF(2,1)	1	1	Inertial-fixed y-axis (lateral)
  Front right	VehF(2,2)	1	2
  Rear left	VehF(2,3)	2	1
  Rear right	VehF(2,4)	2	2
  Front left	VehF(3,1)	1	1	Inertial-fixed z-axis (vertical)
  Front right	VehF(3,2)	1	2
  Rear left	VehF(3,3)	2	1
  Rear right	VehF(3,4)	2	2

VehM — Suspension moment on vehicle

array

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 Element	Axle	Wheel Number	Moment Axis
  VehM(1,1)	1	1	Inertial-fixed x-axis (longitudinal)
  VehM(1,2)	1	2
  VehM(1,3)	2	1
  VehM(1,4)	2	2
  VehM(2,1)	1	1	Inertial-fixed y-axis (lateral)
  VehM(2,2)	1	2
  VehM(2,3)	2	1
  VehM(2,4)	2	2
  VehM(3,1)	1	1	Inertial-fixed z-axis (vertical)
  VehM(3,2)	1	2
  VehM(3,3)	2	1
  VehM(3,4)	2	2

WhlF — Suspension force on wheel

array

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.

  $$\mathrm{WhlF}=F_w=\left[\begin{array}{cccc}{F}_w{}_{x_{1,1}}& F_w{}_{x_{1,2}}& F_w{}_{x_{2,1}}& F_w{}_{x_{2,2}}\\ F_w{}_{y_{1,1}}& F_w{}_{y_{1,2}}& F_w{}_{y_{2,1}}& F_w{}_{y_{2,2}}\\ F_w{}_{z_{1,1}}& F_w{}_{z_{1,2}}& F_w{}_{z_{2,1}}& F_w{}_{z_{2,2}}\end{array}\right]$$

  Wheel	Array Element	Axle	Wheel Number	Force Axis
  Front left	WhlF(1,1)	1	1	Inertial-fixed x-axis (longitudinal)
  Front right	WhlF(1,2)	1	2
  Rear left	WhlF(1,3)	2	1
  Rear right	WhlF(1,4)	2	2
  Front left	WhlF(2,1)	1	1	Inertial-fixed y-axis (lateral)
  Front right	WhlF(2,2)	1	2
  Rear left	WhlF(2,3)	2	1
  Rear right	WhlF(2,4)	2	2
  Front left	WhlF(3,1)	1	1	Inertial-fixed z-axis (vertical)
  Front right	WhlF(3,2)	1	2
  Rear left	WhlF(3,3)	2	1
  Rear right	WhlF(3,4)	2	2

WhlV — Wheel velocity

array

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.

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

  Wheel	Array Element	Axle	Wheel Number	Force Axis
  Front left	WhlV(1,1)	1	1	Inertial-fixed x-axis (longitudinal)
  Front right	WhlV(1,2)	1	2
  Rear left	WhlV(1,3)	2	1
  Rear right	WhlV(1,4)	2	2
  Front left	WhlV(2,1)	1	1	Inertial-fixed y-axis (lateral)
  Front right	WhlV(2,2)	1	2
  Rear left	WhlV(2,3)	2	1
  Rear right	WhlV(2,4)	2	2
  Front left	WhlV(3,1)	1	1	Inertial-fixed z-axis (vertical)
  Front right	WhlV(3,2)	1	2
  Rear left	WhlV(3,3)	2	1
  Rear right	WhlV(3,4)	2	2

WhlAng — Wheel camber, caster, toe angles

array

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.

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

  Wheel	Array Element	Axle	Wheel Number	Angle
  Front left	WhlAng(1,1)	1	1	Camber
  Front right	WhlAng(1,2)	1	2
  Rear left	WhlAng(1,3)	2	1
  Rear right	WhlAng(1,4)	2	2
  Front left	WhlAng(2,1)	1	1	Caster
  Front right	WhlAng(2,2)	1	2
  Rear left	WhlAng(2,3)	2	1
  Rear right	WhlAng(2,4)	2	2
  Front left	WhlAng(3,1)	1	1	Toe
  Front right	WhlF(3,2)	1	2
  Rear left	WhlF(3,3)	2	1
  Rear right	WhlF(3,4)	2	2

Parameters

Axles

Number of axles, NumAxl — Number of axles

2 (default) | scalar

Number of axles, N_a, dimensionless.

Number of wheels by axle, NumWhlsByAxl — Number of wheels per axle

[2 2] (default) | vector

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

Steered axle enable by axle, StrgEnByAxl — Boolean vector to enable axle steering

[1 0] (default) | vector

Boolean vector that enables axle steering, En_steer, dimensionless. Vector is 1 by the number of vehicle axles, N_a. 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.

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

  Wheel	Array Element	Axle	Wheel Number
  Front left	StrgAng(1,1)	1	1
  Front right	StrgAng(1,2)	1	2

Anti-sway axle enable by axle, AntiSwayEnByAxl — Boolean vector to enable axle anti-sway

[0 0] (default) | vector

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, N_a.

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

Mapped

Axle breakpoints, f_susp_axl_bp — Breakpoints

[1 2] (default) | 1-by-P array

Axle breakpoints, dimensionless.

Vertical axis suspension height breakpoints, f_susp_dz_bp — Breakpoints

1-by-M array

Vertical axis suspension height breakpoints, in m.

Vertical axis suspension height velocity breakpoints, f_susp_dzdot_bp — Breakpoints

1-by-N array

Vertical axis suspension height velocity breakpoints, in m/s.

Vertical axis suspension force and moment responses, f_susp_fmz — Output array

zeros(31,31,61,2,4) (default) | M-by-N-by-O-by-P-by-4 array

Array of output values as a function of:

• Vertical suspension height, M

• Vertical suspension height velocity, N

• Steering angle, O

• Axle, P

• 4 output types

  • 1 — Vertical force, in N

  • 2 — User-defined

  • 3 — Stored energy, in J

  • 4 — Absorbed power, in W

The array dimensions must match the breakpoint dimensions

Suspension geometry responses, f_susp_geom — Suspension geometry responses

zeros(31,61,2,3) (default) | M-by-O-by-P-by-3 array

Array of geometric suspension values as a function of:

• Vertical suspension height, M

• Steering angle, O

• Axle, P

• 3 output types

  • 1 — Camber angle, in rad

  • 2 — Caster angle, in rad

  • 3 — Toe angle, in rad

The array dimensions must match the breakpoint dimensions

Steering angle breakpoints, f_susp_strgdelta_bp — Steering angle breakpoints

1-by-O array

Steering angle breakpoints, in rad.

Anti-Sway

Anti-sway arm radius, AntiSwayR — Anti-sway arm radius

0.2 (default) | scalar | vector

Anti-sway arm radius, r, in m.

Vector is 1 by the number of vehicle axles, N_a. 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, AntiSwayNtrlAng — Anti-sway arm neutral angle

0.5236 (default) | scalar | vector

Anti-sway arm neutral angle, θ_0a, at nominal suspension height, in rad.

Vector is 1 by the number of vehicle axles, N_a. 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 torsion spring constant, AntiSwayTrsK — Anti-sway torsion spring constant

5.7296e+03 (default) | scalar | vector

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

Vector is 1 by the number of vehicle axles, N_a. 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.

Extended Capabilities

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

Version History

Introduced in R2018a

R2022b: Parameter name change from NumTracksByAxl to NumWhlsByAxl

The Number of tracks by axle, NumTracksByAxl parameter is renamed to Number of wheels by axle, NumWhlsByAxl.

The block uses the number of wheels per axle to index the input and output block signals.