# Vehicle Body

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

Tires & Vehicles

## Description

This block models a vehicle with two axles in longitudinal motion. The axles can have different wheel counts. For example, two wheels on the front axle and one wheel on the rear axle. The vehicle wheels are assumed identical in size. Vehicle properties and effects that you specify include mass, geometry, and drag.

Port H represents the vehicle body. Physical signal input ports W and beta provide the means to specify the headwind speed and road incline angle. Physical signal output ports V, NF, and NR provide the measurements of the vehicle longitudinal velocity, front-axle normal force, and rear-axle normal force. The signal units are the Simscape™ defaults—`m/s` for velocities, `rad` for angles, and `N` for forces.

### Vehicle Body 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

SymbolDescription and Unit
gGravitational acceleration
βIncline angle
mMass of the vehicle
hHeight of vehicle CG above the ground
a, bDistance of front and rear axles, respectively, from the normal projection point of vehicle CG onto the common axle plane
VxVelocity of the vehicle. When Vx > 0, the vehicle moves forward. When Vx < 0, the vehicle moves backward.
VwWind speed. When Vw > 0, the wind is headwind. When Vw < 0, the wind is tailwind.
nNumber of wheels on each axle
Fxf, FxrLongitudinal forces on each wheel at the front and rear ground contact points, respectively
Fzf, FzrNormal load forces on each wheel at the front and rear ground contact points, respectively
AEffective frontal vehicle cross-sectional area
CdAerodynamic drag coefficient
ρMass density of air
FdAerodynamic drag force

#### Vehicle Dynamics and Motion

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.

`$\begin{array}{l}m{\stackrel{˙}{V}}_{x}={F}_{x}-\text{​}{F}_{d}-mg\cdot \mathrm{sin}\beta ,\\ {F}_{x}=n\left({F}_{xf}+{F}_{xr}\right),\\ {F}_{d}=\frac{1}{2}{C}_{d}\rho A{\left(}^{{V}_{x}}\cdot \mathrm{sgn}\left({V}_{x}+{V}_{w}\right)\end{array}$`

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

`$\begin{array}{l}{F}_{zf}=\frac{-h\left({F}_{d}+mg\mathrm{sin}\beta +m{\stackrel{˙}{V}}_{x}\right)+b\cdot mg\mathrm{cos}\beta }{n\left(a+b\right)},\\ {F}_{zr}=\frac{+h\left({F}_{d}+mg\mathrm{sin}\beta +m{\stackrel{˙}{V}}_{x}\right)+a\cdot mg\mathrm{cos}\beta }{n\left(a+b\right)}\end{array}$`

The wheel normal forces satisfy Fzf + Fzr = mg·cosβ/n.

### 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 (Simscape).

Unlike block parameters, variables do not have conditional visibility. The Variables tab lists all the existing block variables. If a variable is not used in the set of equations corresponding to the selected block configuration, the values specified for this variable are ignored.

### 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

`H`

Translational conserving port associated with the horizontal motion of the vehicle body.

`W`

Physical signal input port for specifying the headwind speed.

`beta`

Physical signal input port for specifying the road incline angle.

`V`

Physical signal output port for measuring the vehicle longitudinal velocity.

`NF`

Physical signal output port for measuring the normal force on the front axle.

`NR`

Physical signal output port for measuring the normal force on the rear axle.

## Parameters

Mass

Mass m of the vehicle. The default is `1200` `kg`.

Number of wheels per axle

Wheel counts on the front and rear axles, specified as a scalar number or a two-element array. 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.

The default is `2`, corresponding to two wheels on each axle.

Horizontal distance from CG to front axle

Horizontal distance a from the center of gravity to the front wheel axle of the vehicle. The default is `1.4` `m`.

Horizontal distance from CG to rear axle

Horizontal distance b from the center of gravity to the rear wheel axle of the vehicle. The default is `1.6` `m`.

CG height above ground

Distance, h, between the center of gravity of the vehicle and the ground. The default is `0.5` `m`.

Frontal area

Effective cross-sectional area A presented by the vehicle in longitudinal motion, to compute the aerodynamic drag force on the vehicle. The default is `3` `m^2`.

Drag coefficient

The dimensionless aerodynamic drag coefficient Cd, for the purpose of computing the aerodynamic drag force on the vehicle. The default is `0.4`.

Gravitational acceleration

Acceleration due to gravitational force acting at the center of gravity of the vehicle. The default is `9.81` `m^2`.

Air Density

Density of the air that surrounds the vehicle. The default is `1.18` `kg/m^3`.