Three-Axle Vehicle Body

R2026b

Vehicle body with three axles

Since R2026a

• 

Three-Axle Vehicle Body block

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Libraries:
Simscape / Driveline / Tires & Vehicles

Description

The Three-Axle Vehicle Body block represents a three-axle vehicle in longitudinal motion. You can specify the number of wheels for each axle. The block accounts for body mass, aerodynamic drag, road incline, pitch and heave, and weight distribution between axles due to acceleration and road profile. The vehicle mass and center of gravity remain constant.

Vehicle Model

The vehicle axles are parallel and form a plane. The local frame moves with the center of gravity. The x-axis is tangential to the road surface in the direction of forward motion, while the y-axis points to the center of gravity.

If the vehicle travels on an inclined road, the normal direction, y, is not parallel to gravity but is always perpendicular to the axle-longitudinal plane. θ is the relative pitch angle between the road inclination angle, β, and the angle of the vehicle. p = θ+β, where p is the pitch angle, measured against the X-axis of the world frame.

Equations

The block balances force in the x and y-directions, such that

$$\begin{array}{l}m{\ddot{x}}_{CG}=0=F_{road}-F_{drag}-mg\sin (\beta )-m\dot{v}\\ m{\ddot{y}}_{CG}=\left(F_f+F_m+F_r\right)-mg\cos (\beta )-m\dot{v}\beta \end{array}$$

where:

• m is the mass of the vehicle.

• x_CG and y_CG are the positions of the center of gravity along the x-axis and y-axis, respectively.

• F_road is the force of the road acting on the vehicle in the x-direction.

• F_drag is the force due to drag acting on the vehicle in the x-direction.

• g is the gravitational acceleration.

• v is the vehicle velocity tangential to the road surface.

• F_f, F_m, and F_r are the forces applied to the front, middle, and rear suspension in the y-direction, respectively.

The block defines the horizontal locations of the center of gravity (CG), middle axis, and rear axle with respect to the location of the front axle. The figure shows the vehicle body with and without suspension deformation.

The block balances torque about the CG, such that

$${\tau }_{CG}=\ddot{I}p=L_{CG}{F}_f-\left(L_m-L_{CG}\right)F_m-\left(L_r-L_{CG}\right)F_r+\left(y_c-s\right)F_{road},$$

where:

• I is the pitch moment of inertia of the vehicle.

• L_CG is the distance from the front axle to the CG along the x-axis.

• L_m is the distance from the front axle to the middle axle along the x-axis.

• L_r is the distance from the front axle to the rear axle along the x-axis.

• y_c is the height of the CG with no deformation.

• s is the change in position of the CG due to suspension deformation, such that s = y_c-y_CG.

The block defines the forces at the front, middle, and rear suspensions, such that

$$\begin{array}{l}{F}_f=F_{f,suspension}{(}_{s_f}{F}_{f,hardstop}(s_f,{\dot{s}}_f)\\ F_m=F_{m,suspension}{(}_{s_m}{F}_{m,hardstop}(s_m,{\dot{s}}_m)\\ F_r=F_{r,suspension}{(}_{s_r}{F}_{r,hardstop}(s_r,{\dot{s}}_r)\end{array}$$

where:

• F_f,suspension, F_m,suspension, F_r,suspension are the forces transmitted by the front, middle, and rear suspension, respectively.

• F_f,hardstop, F_m,hardstop, and F_r,hardstop are the forces applied by the hard stops at the front, middle, and rear suspensions, respectively.

The figure illustrates the geometric relationships.

The block leverages the small angle and rigid body assumptions to simplify calculations, such that

$$\begin{array}{l}\theta =\frac{s_r-s_f}{L_r}\\ \frac{s_r-s_m}{L_r-L_m}=\frac{s_r-s_f}{L_r}\end{array}$$

The block calculates the height change of the CG due to deformation, such that

$$s=s_r-\theta \left(L_r-L_{CG}\right).$$

When you select the Enable hard stops parameter, the block uses a subcomponent implementation of the Translational Hard Stop block, where the setting of the Hard stop model parameter is Stiffness and damping applied smoothly through transition region, damped rebound.

Ports

Input

W — Headwind speed, m/s

physical signal

Physical signal input port associated with the headwind speed, in m/s.

Programmatic Use

Port: wind

CD — Drag coefficient, unitless

physical signal

Physical signal input port associated with the drag coefficient.

Dependencies

To enable this port, select Variable Drag Coefficient.

Programmatic Use

Port: CD

β — Road incline angle, rad

physical signal

Physical signal input port associated with the road incline angle, β, in rad.

Programmatic Use

Port: beta

Output

V — Longitudinal velocity, m/s

physical signal

Physical signal output port associated with the longitudinal velocity of the vehicle, in m/s.

Programmatic Use

Port: V

NF — Front axle normal force, N

physical signal

Physical signal output port associated with the front axle normal force, in N.

Programmatic Use

Port: NF

NM — Middle axle normal force, N

physical signal

Physical signal output port associated with the middle axle normal force, in N.

Programmatic Use

Port: NM

NR — Rear axle normal force, N

physical signal

Physical signal output port associated with the rear axle normal force, in N.

Programmatic Use

Port: NR

Conserving

H — Horizontal motion

mechanical translational

Mechanical translational conserving port associated with the horizontal motion of the vehicle.

Programmatic Use

Port: H

Parameters

Main

Mass — Vehicle mass

5000 kg (default) | positive scalar

Mass of the vehicle.

Programmatic Use

Parameter: mass

Number of wheels per axle [front, middle, rear] — Wheel count

[2, 4, 4] (default) | three-element vector

Number of wheels on the front, middle, and rear axles. From left to right, the elements represent the number of wheels for the front, middle, and rear axles. For example, if the input is the array [2,4,4], then the front axle has two wheels, and the middle and rear axles have four wheels each.

Programmatic Use

Parameter: n_w

Horizontal distance from front axle to CG — Center of gravity location

3.0 m (default) | nonnegative scalar

Location of the center of gravity with respect to the front axle, LC_CG.

Programmatic Use

Parameter: L_cg

Horizontal distance from front axle to middle axle — Middle axle location

4.0 m (default) | positive scalar

Location of the middle axle with respect to the front axle, L_m.

Programmatic Use

Parameter: L_m

Horizontal distance from front axle to rear axle — Rear axle location

5.0 m (default) | positive scalar

Location of the rear axle with respect to the front axle, L_r.

Programmatic Use

Parameter: L_r

CG height with no suspension deformation — Distance between CG and ground

1.5 m (default) | scalar

Distance, y_c, between the center of gravity of the vehicle and the ground.

Programmatic Use

Parameter: y_c

Gravitational acceleration — Gravitational acceleration

9.81 m/s^2 (default) | nonnegative scalar

Acceleration due to gravitational force acting at the center of gravity of the vehicle.

Programmatic Use

Parameter: g

Negative normal force warning — Option to generate warning for negative normal force

off (default) | on

Option to generate a warning for negative normal force.

Programmatic Use

Parameter:	liftoff_warning
Values:	"true" | "false"

Input filtering time constant — Time constant for input filtering

1e-6 s (default) | positive scalar

Time constant to filter inputs.

Programmatic Use

Parameter: tau

Drag

Frontal area — Effective cross-sectional area

8 m^2 (default) | nonnegative scalar

Effective cross-sectional area, A, presented by the vehicle in longitudinal motion. The block uses this value to calculate the aerodynamic drag force on the vehicle.

Programmatic Use

Parameter: area

Variable drag coefficient — Variable drag option

off (default) | on

Whether the drag coefficient is variable. Enable this parameter to expose port CD.

Programmatic Use

Parameter:	Cd_var
Values:	"true" | "false"

Drag coefficient — Aerodynamic drag coefficient

0.6 (default) | nonnegative scalar

Aerodynamic drag coefficient, C_D. The block uses this value to calculate the aerodynamic drag force on the vehicle.

Dependencies

To enable this parameter, clear the Variable Drag Coefficient parameter.

Programmatic Use

Parameter: Cd

Air density — Ambient air density

1.18 kg/m^3 (default) | nonnegative scalar

Density of the air that surrounds the vehicle.

Programmatic Use

Parameter: rho

Pitch

Pitch moment of inertia — Pitch moment of inertia

15000 kg*m^2 (default) | positive scalar

Pitch moment of inertia of the vehicle, I.

Programmatic Use

Parameter: inertia

Suspension model — Suspension parameterization method

Linear (default) | By table lookup

Parameterization method to use to model the suspension. The setting determines how the block calculates F_f,suspension, F_m,suspension, and F_r,suspension. When you select Linear, you specify the front, middle, and rear stiffness and damping coefficients by using scalar values. When you select By table lookup, you specify the stiffness and damping forces and suspension deformation values as vectors.

Programmatic Use

Parameter:	suspension_model
Values:	"sdl.enum.linearTLU.linear" | "sdl.enum.linearTLU.TLU"

Front suspension stiffness — Front suspension stiffness

1e5 N/m (default) | positive scalar

Suspension stiffness at the front axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Programmatic Use

Parameter: k_f

Front suspension damping — Front suspension damping

1e4 N*s/m (default) | positive scalar

Suspension damping at the front axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Programmatic Use

Parameter: b_f

Middle suspension stiffness — Middle suspension stiffness

1e5 N/m (default) | positive scalar

Suspension stiffness at the middle axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Programmatic Use

Parameter: k_m

Middle suspension damping — Middle suspension damping

1e4 N*s/m (default) | positive scalar

Suspension damping at the middle axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Programmatic Use

Parameter: b_m

Rear suspension stiffness — Rear suspension stiffness

1e5 N/m (default) | positive scalar

Suspension stiffness at the rear axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Programmatic Use

Parameter: k_r

Rear suspension damping — Rear suspension damping

1e4 N*s/m (default) | positive scalar

Suspension damping at the rear axle.

Dependencies

To enable this parameter, set the Suspension model parameter to Linear.

Programmatic Use

Parameter: b_r

Front suspension stiffness force vector — Front suspension stiffness force

[-200000, -100000, 0, 100000, 200000] N (default) | vector

Stiffness force at the front suspension. Each element in the vector corresponds to the elements in the Front suspension deformation vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: k_f_TLU

Front suspension deformation vector — Front suspension deformation

[-.4, -.2, 0, .2, .4] m (default) | vector

Front suspension deformation for a given stiffness force. Each element in the vector corresponds to the elements in the Front suspension stiffness vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: x_f_TLU

Front suspension damping force vector — Front suspension damping force

[-10000, -5000, 0, 5000, 10000] N (default) | vector

Damping force at the front suspension for a given velocity. Each element in the vector corresponds to the elements in the Front suspension velocity vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: b_f_TLU

Front suspension velocity vector — Front suspension velocity

[-.4, -.2, 0, .2, .4] m/s (default) | vector

Velocity of the front suspension. Each element in the vector corresponds to the elements in the Front suspension damping force vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: v_f_TLU

Middle suspension stiffness force vector — Middle suspension stiffness force

[-200000, -100000, 0, 100000, 200000] N (default) | vector

Stiffness force at the middle suspension. Each element in the vector corresponds to the elements in the Middle suspension deformation vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: k_m_TLU

Middle suspension deformation vector — Middle suspension deformation

[-.4, -.2, 0, .2, .4] m (default) | vector

Middle suspension deformation for a given stiffness force. Each element in the vector corresponds to the elements in the Middle suspension stiffness vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: x_m_TLU

Middle suspension damping force vector — Middle suspension damping force

[-10000, -5000, 0, 5000, 10000] N (default) | vector

Damping force at the middle suspension for a given velocity. Each element in the vector corresponds to the elements in the Middle suspension velocity vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: b_m_TLU

Middle suspension velocity vector — Middle suspension velocity

[-.4, -.2, 0, .2, .4] m/s (default) | vector

Velocity of the middle suspension. Each element in the vector corresponds to the elements in the Middle suspension damping force vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: v_m_TLU

Rear suspension stiffness force vector — Rear suspension stiffness

[-200000, -100000, 0, 100000, 200000] N (default) | vector

Stiffness force at the rear suspension. Each element in the vector corresponds to the elements in the Rear suspension deformation vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: k_r_TLU

Rear suspension deformation vector — Rear suspension deformation

[-.4, -.2, 0, .2, .4] m (default) | vector

Rear suspension deformation for a given stiffness force. Each element in the vector corresponds to the elements in the Rear suspension stiffness vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: x_r_TLU

Rear suspension damping force vector — Rear suspension damping force

[-10000, -5000, 0, 5000, 10000] N (default) | vector

Damping force at the rear suspension for a given velocity. Each element in the vector corresponds to the elements in the Rear suspension velocity vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: b_r_TLU

Rear suspension velocity vector — Rear suspension velocity

[-.4, -.2, 0, .2, .4] m/s (default) | vector

Velocity of the rear suspension. Each element in the vector corresponds to the elements in the Rear suspension damping force vector parameter.

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter: v_r_TLU

Interpolation method — Interpolation method

Linear (default) | Smooth

Interpolation method to use to approximate the output value when the input value is between two consecutive grid points. You can select:

• Linear — Select this default option to get the best performance. Provide at least two values per dimension.

• Smooth — Select this option to produce a continuous curve with continuous first-order derivatives. Provide at least three values per dimension.

For more information on interpolation algorithms, see PS Lookup Table (1D).

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter:	interp_method
Values:	"simscape.enum.interpolation.linear" | "simscape.enum.interpolation.smooth"

Extrapolation method — Extrapolation method

Linear (default) | Nearest | Error

Extrapolation method to use to determine the output value when the input value is outside the specified operating range. You can select:

• Linear — Produces a curve with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.

• Nearest — Produces an extrapolation that does not go above the highest point in the data or below the lowest point in the data.

• Error — Issues an error if the input signal is outside the range of the table. Select this option when you want your data to be within the table range.

For more information on extrapolation algorithms, see PS Lookup Table (1D).

Dependencies

To enable this parameter, set the Suspension model parameter to By table lookup.

Programmatic Use

Parameter:	extrap_method
Values:	"simscape.enum.extrapolation.linear" | "simscape.enum.extrapolation.nearest" | "simscape.enum.extrapolation.error"

Initialize at pitch and heave equilibrium — Option to initialize block at pitch and heave equilibrium

on (default) | off

Whether to initialize the block at pitch and heave equilibrium.

Programmatic Use

Parameter:	init_at_equilibrium
Values:	"true" | "false"

Initial pitch angle — Initial pitch angle

0 deg (default) | scalar

Pitch angle of the vehicle at the start of simulation.

Dependencies

To enable this parameter, clear the Initialize at pitch and heave equilibrium parameter.

Programmatic Use

Parameter: pitch_init

Initial pitch angle rate — Initial pitch rate

0 deg/s (default) | scalar

Rate of change of the pitch at the start of simulation.

Dependencies

To enable this parameter, clear the Initialize at pitch and heave equilibrium parameter.

Programmatic Use

Parameter: pitch_rate_init

Initial CG height — Initial height of center of gravity

1.5 m (default) | scalar

Height of the CG at the start of the simulation.

Programmatic Use

Parameter: y_init

Initial CG vertical velocity — Initial velocity of center of gravity

0 m/s (default) | scalar

Vertical velocity of the CG at the start of the simulation.

Programmatic Use

Parameter: vy_init

Pitch Hard Stops

Enable hard stops — Option to enable hard stops

off (default) | on

Whether to enable hard stops. The setting determines how the block calculates F_f,hardstop, F_m,hardstop, and F_r,hardstop.

Programmatic Use

Parameter:	enable_hard_stop
Values:	"true" | "false"

Front upper bound — Gap between front slider and front upper bound

0.25 m (default) | scalar

Gap between the front axle slider and the front upper bound. Specify the direction with respect to the local coordinate system, with the slider located on the origin.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: upper_bound_f

Front lower bound — Gap between front slider and lower bound

-0.25 m (default) | scalar

Gap between the front axle slider and the front lower bound. Specify the direction with respect to the local coordinate system, with the slider located on the origin.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: lower_bound_f

Front contact stiffness — Front contact stiffness

1e8 N/m (default) | nonnegative scalar

Stiffness of the front suspension at the upper and lower bounds.

Programmatic Use

Parameter: stop_k_f

Front contact damping — Front contact damping

1500 N*s/m (default) | nonnegative scalar

Damping of the front suspension at the upper and lower bounds.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: stop_b_f

Middle upper bound — Gap between middle slider and upper bound

0.25 m (default) | scalar

Gap between the middle axle slider and the middle upper bound. Specify the direction with respect to the local coordinate system, with the slider located on the origin.

Programmatic Use

Parameter: upper_bound_m

Middle lower bound — Gap between middle slider and lower bound

-0.25 m (default) | scalar

Gap between the middle axle slider and the middle upper bound. Specify the direction with respect to the local coordinate system, with the slider located on the origin.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: lower_bound_m

Middle contact stiffness — Middle contact stiffness

1e8 N/m (default) | nonnegative scalar

Stiffness of the middle suspension at the upper and lower bounds.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: stop_k_m

Middle contact damping — Middle contact damping

1500 N*s/m (default) | nonnegative scalar

Damping of the front suspension at the upper and lower bounds.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: stop_b_m

Rear upper bound — Rear upper bound

0.25 m (default) | scalar

Gap between the rear axle slider and the rear upper bound. Specify the direction with respect to the local coordinate system, with the slider located on the origin.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: upper_bound_r

Rear lower bound — Rear lower bound

-0.25 m (default) | scalar

Gap between the rear axle slider and the rear upper bound. Specify the direction with respect to the local coordinate system, with the slider located on the origin.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: lower_bound_r

Rear contact stiffness — Rear contact stiffness

1e8 N/m (default) | nonnegative scalar

Stiffness of the rear suspension at the upper and lower bounds.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: stop_k_r

Rear contact damping — Rear contact damping

1500 N*s/m (default) | nonnegative scalar

Damping of the rear suspension at the upper and lower bounds.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: stop_b_r

Transition region — Region to start or stop motion

1 mm (default) | scalar

Region where the force ramps between 0 and the full value. To learn more, see Threshold Parameter Tuning.

Dependencies

To enable this parameter, select the Enable hard stops parameter.

Programmatic Use

Parameter: transition

Extended Capabilities

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

Version History

Introduced in R2026a