# Longitudinal Vehicle Dynamics

Model longitudinal dynamics and motion of two-axle, four-wheel vehicle

## Library

Vehicle Components

## Description

The Longitudinal Vehicle Dynamics block models a two-axle vehicle, with four equally sized wheels, moving forward or backward along its longitudinal axis. You specify front and rear longitudinal forces Fxf, Fxr applied at the front and rear wheel contact points, as well as the incline angle β, as a set of Simulink® input signals. The block computes the vehicle velocity Vx and the front and rear vertical load forces Fzf, Fzr on the vehicle as a set of Simulink output signals. All signals have MKS units.

You must specify the vehicle mass and certain geometric and kinematic details:

• Position of the vehicle's center of gravity (CG) relative to the front and rear axles and to the ground

• Effective frontal cross-sectional area

• Aerodynamic drag coefficient

• Initial longitudinal velocity

See Vehicle Model following for details of the vehicle dynamics.

### Limitations

The Longitudinal Vehicle Dynamics block lets you model only longitudinal (horizontal) dynamics. Depending on the initial configuration, the block might implement inconsistent initial conditions for the vertical load forces, causing spurious transient dynamics just after the simulation starts.

 Caution   The Longitudinal Vehicle Dynamics block does not correctly simulate with sudden changes in the external (longitudinal and gravity) forces. It correctly models only slowly changing external conditions.

### Using Vehicle Component Blocks

Use the blocks of the Vehicle Components library as a starting point for vehicle modeling. To see how a Vehicle Component block models a driveline component, look under the block mask. The blocks of this library serve as suggestions for developing variant or entirely new models to simulate the same components. Break the block's library link before modifying it and creating your own version.

## Dialog Box and Parameters

Mass

Mass m of the vehicle in kilograms (kg). The default is `1200`.

Horizontal distance from CG to front axle

Horizontal distance a, in meters (m), from the vehicle's center of gravity to the vehicle's front wheel axle. The default is `1.4`.

Horizontal distance from CG to rear axle

Horizontal distance b, in meters (m), from the vehicle's center of gravity to the vehicle's rear wheel axle. The default is `1.6`.

CG height from ground

Height h, in meters (m), of the vehicle's center of gravity from the ground. The default is `0.5`.

Frontal area

Effective cross-sectional area A, in meters squared (m2), presented by the vehicle in longitudinal motion, for the purpose of computing the aerodynamic drag force on the vehicle. The default is `3`.

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`.

Initial longitudinal velocity

The initial value of the vehicle's horizontal velocity, in meters/second (m/s). The default is `0`.

## Vehicle Model

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

This figure and table define the vehicle motion model variables.

Vehicle Dynamics and Motion

Vehicle Model Variables and Constants

SymbolMeaning and Unit
g = -9.81 m/s2Gravitational acceleration (m/s2)
mVehicle mass (kg)
AEffective frontal vehicle cross-sectional area (m2)
hHeight of vehicle CG above the ground (m)
a, bDistance of front and rear axles, respectively, from the vertical projection point of vehicle CG onto the axle-ground plane (m)
VxLongitudinal vehicle velocity (m/s)
Fxf, FxrLongitudinal forces on the vehicle at the front and rear wheel ground contact points, respectively (N)
Fzf, FzrVertical load forces on the vehicle at the front and rear ground contact points, respectively (N)
CdAerodynamic drag coefficient (N·s2/kg·m)
ρ = 1.2 kg/m3Mass density of air (kg/m3)

|Fd| = ½CdρAVx 2

Aerodynamic drag force (N)

### Vehicle Dynamics and Motion

The vehicle motion is determined by 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 either pulls it 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.

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

Zero vertical acceleration and zero pitch torque require

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

Note that Fzf + Fzr = mg·cosβ.

 Caution   The Longitudinal Vehicle Dynamics block is implemented with a transfer function that imposes a small delay on the vertical force reaction to changes in the horizontal forces. The vertical and pitch equilibria hold only on average.

## Examples

The example model drive_4wd_dynamics combines two differentials with four tire-wheel assemblies to model the contact of tires with the road and the longitudinal vehicle motion.

The example model drive_vehicle models an entire one-wheel vehicle, including Tire and Longitudinal Vehicle Dynamics blocks.

## References

Centa, G., Motor Vehicle Dynamics: Modeling and Simulation, Singapore, World Scientific, 1997.

Pacejka, H. B., Tire and Vehicle Dynamics, Society of Automotive Engineers and Butterworth-Heinemann, Oxford, 2002.