Vehicle Body

Vehicle body with two axles in longitudinal motion

Libraries:
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. You can also optionally control the drag coefficient with a physical signal.

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

Inclined vehicle body diagram

Vehicle Model Variables

Symbol	Description
g	Gravitational acceleration parameter.
β	Road incline angle, parameter beta.
m	Mass of the vehicle, specified by either the Mass parameter or port M.
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.
V_x	Velocity of the vehicle. When V_x > 0, the vehicle moves forward. When V_x < 0, the vehicle moves backward.
V_w	Wind speed. When V_w > 0, the wind is headwind. When V_w < 0, the wind is tailwind.
n	Number of wheels per axle parameter. You can use an integer or a two-element vector where the elements represent the front and rear axles, respectively.
F_xf, F_xr	Longitudinal forces on each wheel at the front and rear ground contact points, respectively.
F_zf, F_zr	Normal load forces on each wheel at the front and rear ground contact points, respectively.
A	Effective frontal vehicle cross-sectional area.
C_d	Aerodynamic drag coefficient.
ρ	Mass density of air.
F_d	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.

$m {\dot{V}}_{x} = F_{x} - F_{d} - m g \cdot \sin β$

$\begin{matrix} Integer n & F_{x} = n (F_{x f} + F_{x r}) \\ Vector n & F_{x} = n (1) F_{x f} + n (2) F_{x r} \end{matrix}$

$F_{d} = \frac{1}{2} C_{d} ρ A {(V_{x} + V_{w})}^{2} \cdot sgn (V_{x} + V_{w})$

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

$\begin{matrix} Integer n & F_{z f} = \frac{- h (F_{d} + m g \sin β + m {\dot{V}}_{x}) + b \cdot m g \cos β}{n (a + b)} \\ Vector n & F_{z f} = \frac{- h (F_{d} + m g \sin β + m {\dot{V}}_{x}) + b \cdot m g \cos β}{n (1) a + n (2) b} \end{matrix}$

$\begin{matrix} Integer n & F_{z r} = \frac{+ h (F_{d} + m g \sin β + m {\dot{V}}_{x}) + a \cdot m g \cos β}{n (a + b)} \\ Vector n & F_{z r} = \frac{+ h (F_{d} + m g \sin β + m {\dot{V}}_{x}) + a \cdot m g \cos β}{n (1) a + n (2) b} \end{matrix}$

The wheel normal forces satisfy

$\begin{matrix} Integer n & F_{z f} + F_{z r} = m g \frac{\cos β}{n} \\ Vector n & n (1) F_{z f} + n (2) F_{z r} = m g \cos β \end{matrix}$

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:

$α = \frac{(f \cdot h) + (F_{z f} a) - (F_{z r} b)}{J}$

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.

Energy Accounting

You can track the power dissipated due to aerodynamic drag. The simlog workspace variable contains the power_dissipated variable.

Variables

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see Modify Nominal Values for a Block Variable.

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.

Examples

Vehicle with Dual Clutch Transmission

A vehicle with a five-speed dual-clutch transmission. 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

Ports

Input

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W — Headwind speed, m/s
physical signal

Physical signal input port for headwind speed, in m/s.

beta — Road incline angle, rad
physical signal

Physical signal input port for road incline angle, in rad.

CG — Center of gravity, m
physical signal

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

Dependencies

To enable this port, select Externally-defined additional mass.

M — Mass, kg
physical signal

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

Dependencies

To enable this port, select Externally-defined additional mass.

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

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

Dependencies

To enable this port, select Externally-defined additional mass and Pitch dynamics.

CD — Drag coefficient, unitless
physical signal

Physical signal input port for the drag coefficient.

Dependencies

To enable this port, select Variable drag coefficient.

Output

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V — Longitudinal velocity, m/s
physical signal

Physical signal output port for vehicle longitudinal velocity in the vehicle reference frame, in m/s.

NF — Front axle normal force, N
physical signal

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

NR — Rear axle normal force, N
physical signal

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

Conserving

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H — Horizontal motion
mechanical translational

Mechanical translational conserving port associated with the horizontal motion of the vehicle body. Connect tire traction motion to this port.

Parameters

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Main

Mass — Vehicle mass
`1200` `kg` (default) | positive scalar

Mass of the vehicle.

Number of wheels per axle — Wheel count
`2` (default) | positive integer or two-element vector

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) | positive scalar

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) | positive scalar

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) | positive scalar

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

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.

Externally-defined additional mass — Variable mass option
`off` (default) | `on`

Option to include mass as time- or event-variant that affects the mass and CG of the vehicle body. Use this option to represent vehicle occupants or unconfined bodies.

Negative normal force warning — Normal force warning
`off` (default) | `on`

Whether the block generates a warning when normal force calculations return negative values. 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) | positive 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.

Variable drag coefficient — Variable drag option
`off` (default) | `on`

Whether drag is variable. Control drag with port CD.

Drag coefficient — Aerodynamic drag coefficient
`0.4` (default) | nonnegative scalar

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) | nonnegative scalar

Density of the air that surrounds the vehicle.

Pitch

Pitch dynamics — Pitch dynamics option
`off` (default) | `on`

Whether to model pitch dynamics.

Pitch moment of inertia — Pitch moment of inertia
`4000` `kg/m^2` (default) | positive scalar

Moment of inertia for pitch calculations.

Dependencies

To enable this parameter, select Pitch dynamics.

Suspension model — Suspension parameterization method
`Linear` (default) | `By table lookup`

Parameterization method for suspension modeling. To specify the data using scalar values, select Linear. To specify the data using vector values, select By table lookup.

Dependencies

To enable this parameter, select Pitch dynamics.

Front suspension stiffness — Front suspension stiffness
`1e4` `N/m` (default) | positive scalar

Front suspension stiffness per axle.

Dependencies

To enable this parameter, select Pitch dynamics and set Suspension model to Linear.

Front suspension damping — Front suspension damping
`1e4N/(m/s)` (default) | positive scalar

Front suspension damping per axle.

Dependencies

To enable this parameter, select Pitch dynamics and set Suspension model to Linear.

Rear suspension stiffness — Rear suspension stiffness
`1e4` `N/m` (default) | positive scalar

Rear suspension stiffness per axle.

Dependencies

To enable this parameter, select Pitch dynamics and set Suspension model to Linear.

Rear suspension damping — Rear suspension damping
`1e4` `N/(m/s)` (default) | positive scalar

Rear suspension damping per axle.

Dependencies

To enable this parameter, select Pitch dynamics and set Suspension model to Linear.

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.