Vehicle Body

Two-axle vehicle with longitudinal dynamics and motion and adjustable mass, geometry, and drag properties

• Library:

• Simscape / Driveline / Tires & Vehicles

• 

Description

The Vehicle Body block represents a two-axle vehicle body in longitudinal motion. The vehicle can have the same or a different number of wheels on each axle. For example, two wheels on the front axle and one wheel on the rear axle. The vehicle wheels are assumed identical in size. The vehicle can also have a center of gravity (CG) that is at or below the plane of travel.

The block accounts for body mass, aerodynamic drag, road incline, and weight distribution between axles due to acceleration and road profile. Optionally include pitch and suspension dynamics. The vehicle does not move vertically relative to the ground.

The block has an option to include an externally-defined mass and an externally-defined inertia. The mass, inertia, and center of gravity of the vehicle body can vary over the course of simulation in response to system changes.

Model

The vehicle axles are parallel and form a plane. The longitudinal, x, direction lies in this plane and perpendicular to the axles. If the vehicle is traveling on an incline slope, β, the normal, z, direction is not parallel to gravity but is always perpendicular to the axle-longitudinal plane.

This figure and table define the vehicle motion model variables.

Vehicle Dynamics and Motion

Vehicle Model Variables

Symbol	Description
g	Gravitational acceleration
β	Incline angle
m	Mass of the vehicle
h	Height of vehicle center of gravity (CG) above the ground
a, b	Distance of front and rear axles, respectively, from the normal projection point of vehicle CG onto the common axle plane
Vx	Velocity of the vehicle. When Vx > 0, the vehicle moves forward. When Vx < 0, the vehicle moves backward.
Vw	Wind speed. When Vw > 0, the wind is headwind. When Vw < 0, the wind is tailwind.
n	Number of wheels on each axle
Fxf, Fxr	Longitudinal forces on each wheel at the front and rear ground contact points, respectively
Fzf, Fzr	Normal load forces on each wheel at the front and rear ground contact points, respectively
A	Effective frontal vehicle cross-sectional area
Cd	Aerodynamic drag coefficient
ρ	Mass density of air
Fd	Aerodynamic drag force

Equations

Vehicle Dynamics

The vehicle motion is a result of the net effect of all the forces and torques acting on it. The longitudinal tire forces push the vehicle forward or backward. The weight mg of the vehicle acts through its center of gravity (CG). Depending on the incline angle, the weight pulls the vehicle to the ground and pulls it either backward or forward. Whether the vehicle travels forward or backward, aerodynamic drag slows it down. For simplicity, the drag is assumed to act through the CG.

$$m{\dot{V}}_x=F_x-\text{}{F}_d-mg\cdot \sin \beta$$

$$F_x=n(F_{xf}+F_{xr})$$

$$F_d=\frac{1}{2}{C}_d\rho A{(V_x+V_w)}^2\cdot \mathrm{sgn}(V_x+V_w)$$

Zero normal acceleration and zero pitch torque determine the normal force on each front and rear wheel.

$$F_{zf}=\frac{-h(F_d+mg\sin \beta +m{\dot{V}}_x)+b\cdot mg\cos \beta }{n(a+b)}$$

$$F_{zr}=\frac{+h(F_d+mg\sin \beta +m{\dot{V}}_x)+a\cdot mg\cos \beta }{n(a+b)}$$

The wheel normal forces satisfy $$F_{zf}+F_{zr}=mg\frac{\cos \beta }{n}$$.

If you include an externally-defined mass or inertia, the equations are shifted by the weighted value of the input.

Pitch dynamics

Pitch acceleration depends on three torque components and the inertia of the vehicle:

$$\alpha =\frac{(f\cdot h)+(F_{zf}a)-(F_{zr}b)}{J}$$

Where:

• ɑ is the pitch acceleration.

• f is the longitudinal force.

• h is the height of the center of gravity when measured parallel to the z-axis.

• J is the inertia.

If you choose a linear model for suspension stiffness and damping, the block uses small angle approximation for pitch calculations. If you choose a table lookup model, the block uses the vectors that you specify calculate pitch dynamics. For hard stop equations, see Translational Hard Stop.

Variables

Use the Variables tab to set the priority and initial target values for the block variables before simulating. For more information, see Set Priority and Initial Target for Block Variables.

Limitations and Assumptions

The Vehicle Body block lets you model only longitudinal dynamics, parallel to the ground and oriented along the direction of motion. The vehicle is assumed to be in pitch and normal equilibrium. The block does not model pitch or vertical movement. As such, the equations assume that the wheels never lose contact. This constraint can result in negative normal forces.

Ports

Input

W — Headwind speed, m/s

scalar

Physical signal input port for headwind speed.

beta — Road incline angle, rad

scalar

Physical signal input port for road incline angle.

CG — Center of gravity, m

two-element vector

Physical signal input port for the center of gravity, in m, of the externally-defined mass relative to the CG of the vehicle body.

Dependencies

This port is visible only when the Externally-defined additional mass parameter in the Main settings is set to On.

M — Mass, kg

scalar

Physical signal input port for the mass, in kg, of the externally-defined mass.

Dependencies

This port is visible only when the Externally-defined additional mass parameter in the Main settings is set to On.

J — External moment of inertia, kg*m^2

scalar

Physical signal input port for the moment of inertia, in kg*m^2, of the externally-defined mass.

Dependencies

This port is visible only when the Externally-defined additional mass parameter in the Main settings and the Pitch dynamics parameter in the Pitch settings are both set to On.

Output

V — Longitudinal velocity, m/s

scalar

Physical signal output port for vehicle longitudinal velocity in the vehicle reference frame.

NF — Front axle normal force, N

scalar

Physical signal output port for normal force on the front axle. Wheel forces are considered positive if acting downwards.

NR — Rear axle normal force, N

scalar

Physical signal output port for normal force on the rear axle. Wheel forces are considered positive if acting downwards.

Conserving

H — Horizontal motion

mechanical translational

Conserving port associated with the horizontal motion of the vehicle body. Connect tire traction motion to this port.

Parameters

Main

Mass — Vehicle mass

1200 kg (default)

Mass of the vehicle.

Number of wheels per axle — Wheel count

2 (default) | scalar number or a two-element array

Wheel counts on the front and rear axles. If the input is a scalar number, the wheel counts of the front and rear axles are assumed the same. For example, if the input is 2, then the front and rear axles are each assumed to have two wheels.

If the input is a two-element array, the first number is the front-axle wheel count and the second number the rear-axle wheel count. For example, if the input is the array [2,1], then the front axle is assumed to have two wheels and the rear axle one wheel.

Horizontal distance from CG to front axle — CG-to-front axle distance

1.4 m (default)

Horizontal distance, a, from the center of gravity to the front wheel axle of the vehicle.

Horizontal distance from CG to rear axle — CG-to-rear axle distance

1.6 m (default)

Horizontal distance, b, from the center of gravity to the rear wheel axle of the vehicle.

CG height above ground — CG-to-ground distance

0.5 m (default)

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

Gravitational acceleration — Gravitational acceleration

9.81 m/s^2 (default)

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

Externally-defined additional mass — Variable mass option

Off (default) | On

Option to include as time- or event-variant mass that affects the mass and CG of the vehicle body.

Dependencies

The CG, M, and J ports are visible only when this parameter is set to On. Exposure of the J port also requires that you select On for the Pitch dynamics parameter in the Pitch settings.

Negative normal force warning — Warning

Off (default) | On

To see a warning when the block calculates negative values for normal force during simulation, select On. The block can generate negative forces because the equations assume that the wheels never lose contact.

Drag

Frontal area — Effective cross-sectional area

3 m^2 (default)

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.

Drag coefficient — Aerodynamic drag coefficient

0.4 (default)

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

Air density — Ambient air density

1.18 kg/m^3 (default)

Density of the air that surrounds the vehicle.

Pitch

The table shows how the visibility of some parameters depends on the option that you choose for other parameters. To learn how to read the table, see Parameter Dependencies.

Pitch Parameter Dependencies

Pitch
Pitch dynamics — Choose Off or On
Off	On
	Pitch moment of inertia
	Suspension model — Choose Linear or By table lookup
	Linear	By table lookup
	Front suspension stiffness	Front suspension stiffness force vector
	Front suspension damping	Front suspension deformation vector
	Rear suspension stiffness	Front suspension damping force vector
	Rear suspension damping	Front suspension velocity vector
		Rear suspension stiffness force vector
		Rear suspension deformation vector
		Rear suspension damping force vector
		Rear suspension velocity vector
		Interpolation method
		Extrapolation method
	Hard stop — Choose Off or On
	Off	On
		Front upper bound
		Front lower bound
		Front contact stiffness
		Front contact damping
		Rear upper bound
		Rear lower bound
		Rear contact stiffness
		Rear contact damping
	Initial pitch
	Initial pitch rate

Pitch dynamics — Pitch dynamics modelling

Off (default) | On

Option to model pitch dynamics.

Dependencies

Selecting On for this parameter enables parameters for:

• Suspension model

• Hard-stop model

• Initial pitch conditions

The J port is visible only when this parameter is set to On. Exposure of the J port also requires that you select On for the Externally-defined additional mass parameter in the Main settings.

For more information, see the Pitch Parameter Dependencies table.

Pitch moment of inertia — Pitch moment of inertia

4000 kg/m^2 (default) | positive scalar

Moment of inertia for pitch calculations.

Dependencies

This parameter is enabled when you set Pitch dynamics to On.

For more information, see the Pitch Parameter Dependencies table.

Suspension model — Suspension parameterization method

Linear (default) | Table lookup

Parameterization method for suspension modeling.

You parameterize the suspension using stiffness and damping data. To specify the data using scalar values, select Linear. To specify the data using vector values, select By table lookup.

Dependencies

This parameter is enabled when you set Pitch dynamics to On.

For more information, see the Pitch Parameter Dependencies table.

Front suspension stiffness — Front suspension stiffness

1e4 N/m (default) | positive scalar

Front suspension stiffness per axle.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to Linear

For more information, see the Pitch Parameter Dependencies table.

Front suspension damping — Front suspension damping

1e4N/(m/s) (default) | positive scalar

Front suspension damping per axle.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to Linear

For more information, see the Pitch Parameter Dependencies table.

Rear suspension stiffness — Rear suspension stiffness

1e4 N/m (default) | positive scalar

Rear suspension stiffness per axle.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to Linear

For more information, see the Pitch Parameter Dependencies table.

Rear suspension damping — Rear suspension damping

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

Rear suspension damping per axle.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to Linear

For more information, see the Pitch Parameter Dependencies table.

Front suspension stiffness force vector — Front suspension stiffness force

[-2000, -1000, 0, 1000, 2000] N (default) | increasing vector

Stiffness force of the front suspension. Specify the output values for the lookup table as a vector. The number of elements in the output vector must be the same as the number of elements in the input vector. The input vector parameter is the Front suspension deformation vector parameter.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to By table lookup

For more information, see the Pitch Parameter Dependencies table.

Front suspension deformation vector — Front suspension deformation

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

Deformation of the front suspension specified in terms of displacement. Specify the input values for the lookup table as a vector. The values of the elements in the vector must increase from left to right. The minimum number of elements in the vector depends on the interpolation method that you select. For linear interpolation, provide at least two elements. For smooth interpolation, provide at least three elements. The output vector parameter is the Front suspension stiffness force vector parameter.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to By table lookup

For more information, see the Pitch Parameter Dependencies table.

Front suspension damping force vector — Front suspension damping force

[-200, -100, 0, 100, 200] N (default) | increasing vector

Damping force of the front suspension. Specify the output values for the lookup table as a vector. The number of elements in the output vector must be the same as the number of elements in the input vector. The input vector parameter is the Front suspension velocity vector parameter.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to By table lookup

For more information, see the Pitch Parameter Dependencies table.

Front suspension velocity vector — Front suspension velocity

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

Velocity of the front suspension. Specify the input values for the lookup table as a vector. The values of the elements in the vector must increase from left to right. The minimum number of elements in the vector depends on the interpolation method that you select. For linear interpolation, provide at least two elements. For smooth interpolation, provide at least three elements. The output vector parameter is the Front suspension damping force vector parameter.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to By table lookup

For more information, see the Pitch Parameter Dependencies table.

Rear suspension stiffness force vector — Rear suspension stiffness force

[-2000, -1000, 0, 1000, 2000] N (default) | increasing vector

Stiffness force of the rear suspension. Specify the output values for the lookup table as a vector. The number of elements in the output vector must be the same as the number of elements in the input vector. The input vector parameter is the Rear suspension deformation vector parameter.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to By table lookup

For more information, see the Pitch Parameter Dependencies table.

Rear suspension deformation vector — Rear suspension deformation

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

Deformation of the rear suspension specified in terms of displacement. Specify the input values for the lookup table as a vector. The values of the elements in the vector must increase from left to right. The minimum number of elements in the vector depends on the interpolation method that you select. For linear interpolation, provide at least two elements. For smooth interpolation, provide at least three elements. The output vector parameter is the Rear suspension stiffness force vector parameter.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to By table lookup

For more information, see the Pitch Parameter Dependencies table.

Rear suspension damping force vector — Rear suspension damping force

[-200, -100, 0, 100, 200] N (default) | increasing vector

Damping force of the rear suspension. Specify the output values for the lookup table as a vector. The number of elements in the output vector must be the same as the number of elements in the input vector. The input vector parameter is the Rear suspension velocity vector parameter.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to By table lookup

For more information, see the Pitch Parameter Dependencies table.

Rear suspension velocity vector — Rear suspension velocity

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

Velocity of the rear suspension. Specify the input values for the lookup table as a vector. The values of the elements in the vector must increase from left to right. The minimum number of elements in the vector depends on the interpolation method that you select. For linear interpolation, provide at least two elements. For smooth interpolation, provide at least three elements. The output vector parameter is the Rear suspension damping force vector parameter.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to By table lookup

For more information, see the Pitch Parameter Dependencies table.

Interpolation method — Interpolation method

linear (default) | smooth

Interpolation methods for approximating the output value when the input value is between two consecutive grid points are:

• 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 the PS Lookup Table (1D) block reference page.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to By table lookup

For more information, see the Pitch Parameter Dependencies table.

Extrapolation method — Extrapolation method

linear (default) | smooth

Extrapolation method for determining the output value when the input value is outside the range specified in the argument list are:

• 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.

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

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Suspension model to By table lookup

For more information, see the Pitch Parameter Dependencies table.

Hard stop — Hard-stop modeling option

Off (default) | On

Option to model hardstops for the front and rear suspension.

Dependencies

This parameter is enabled when you set the Pitch dynamics to On:

Selecting On for this parameter enables parameters for the front and rear hard stops:

• Upper bound

• Lower bound

• Contact stiffness

• Contact damping

For more information, see the Pitch Parameter Dependencies table.

Front upper bound — Hardstop front suspension upper bound

.25 (default)

Upper bound of the hard stop for the front suspension.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Hard stop to On

For more information, see the Pitch Parameter Dependencies table.

Front lower bound — Vehicle front lower bound

-.25 (default)

Lower bound of the hard stop for the front suspension.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Hard stop to On

For more information, see the Pitch Parameter Dependencies table.

Front contact stiffness — Vehicle front contact stiffness

1e6 (default)

Contact stiffness of the hard stop for the front suspension.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Hard stop to On

For more information, see the Pitch Parameter Dependencies table.

Front contact damping — Vehicle front contact damping

150 (default)

Contact damping of the hard stop for the front suspension.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Hard stop to On

For more information, see the Pitch Parameter Dependencies table.

Rear upper bound — Vehicle rear upper bound

.25 (default)

Upper bound of the hard stop for the rear suspension.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Hard stop to On

For more information, see the Pitch Parameter Dependencies table.

Rear lower bound — Vehicle rear lower bound

-.25 (default)

Lower bound of the hard stop for the rear suspension.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Hard stop to On

For more information, see the Pitch Parameter Dependencies table.

Rear contact stiffness — Vehicle rear contact stiffness

1e6 (default)

Contact stiffness of the hard stop for the rear suspension.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Hard stop to On

For more information, see the Pitch Parameter Dependencies table.

Rear contact damping — Vehicle rear contact damping

150 (default)

Contact damping of the hard stop for the rear suspension.

Dependencies

This parameter is enabled when you set both of these options:

• Pitch dynamics to On

• Hard stop to On

For more information, see the Pitch Parameter Dependencies table.

Initial pitch — Pitch at the beginning of the simulation

0 rad (default)

Pitch at the beginning of the simulation.

Dependencies

This parameter is enabled when you set Pitch dynamics to On.

For more information, see the Pitch Parameter Dependencies table.

Initial pitch rate — Pitch rate at the beginning of the simulation

0 rad/s (default)

Pitch rate at the beginning of the simulation.

Dependencies

This parameter is enabled when you set Pitch dynamics to On.

For more information, see the Pitch Parameter Dependencies table.

Model Examples

Vehicle with Dual Clutch Transmission

A vehicle with a five-speed automatic dual-clutch transmission. The transmission controller converts the pedal deflection into a demanded torque. This demanded torque is then passed to the engine management. The pedal deflection and the vehicle speed are also used by the transmission controller to determine when the gear shifts should occur. Gear shifts are implemented via the two clutches, one clutch pressure being ramped up as the other clutch pressure is ramped down. Gear pre-selection via dog clutches ensures that the correct gear is fully selected before the on-going clutch is enabled.

Open Model

Vehicle with Four-Speed Transmission

A complete vehicle with Simscape™ Driveline™ components, including the engine, drivetrain, four-speed transmission, tires, and longitudinal vehicle dynamics. The transmission controller is implemented as a state machine in Stateflow®, selecting the gear based on throttle and vehicle speed.

Open Model

Vehicle with Four-Wheel Drive

A four-wheel drive vehicle starting from rest and ascending a 15 degree incline. Initially the vehicle rolls backwards until the engine develops sufficient torque to counter the slope. The tire compliance dynamics can be seen as the vehicle starts to accelerate. The model variant chosen for all of the tires can be set to the Simple, Friction Parameterized, or Magic Formula tire model using the hyperlinks in the model.

Open Model

Vehicle with Manual Transmission

A vehicle that has a four-speed manual transmission. The key elements of the transmission are four synchronizers. By engaging or disengaging these synchronizers and associated dog clutches, the transmission provides four ratios 3.581, 2.022, 1.384, and 1, respectively. The synchronizers are modeled using the Cone Clutch and Dog Clutch blocks.

Open Model

Extended Capabilities

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

Version History

Introduced in R2011a