Model longitudinal dynamics and motion of two-axle, four-wheel vehicle
Vehicle Components
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 F_{xf}, F_{xr} 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 V_{x} and the front and rear vertical load forces F_{zf}, F_{zr} 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.
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. |
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.
Mass m of the vehicle in kilograms (kg).
The default is 1200
.
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 b, in meters (m), from
the vehicle's center of gravity to the vehicle's rear wheel axle.
The default is 1.6
.
Height h, in meters (m), of the vehicle's
center of gravity from the ground. The default is 0.5
.
Effective cross-sectional area A, in meters
squared (m^{2}), presented by the vehicle
in longitudinal motion, for the purpose of computing the aerodynamic
drag force on the vehicle. The default is 3
.
The dimensionless aerodynamic drag coefficient C_{d},
for the purpose of computing the aerodynamic drag force on the vehicle.
The default is 0.4
.
The initial value of the vehicle's horizontal velocity, in meters/second
(m/s). The default is 0
.
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
Symbol | Meaning and Unit |
---|---|
g = -9.81 m/s^{2} | Gravitational acceleration (m/s^{2}) |
β | Incline angle (rad) |
m | Vehicle mass (kg) |
A | Effective frontal vehicle cross-sectional area (m^{2}) |
h | Height of vehicle CG above the ground (m) |
a, b | Distance of front and rear axles, respectively, from the vertical projection point of vehicle CG onto the axle-ground plane (m) |
V_{x} | Longitudinal vehicle velocity (m/s) |
F_{xf}, F_{xr} | Longitudinal forces on the vehicle at the front and rear wheel ground contact points, respectively (N) |
F_{zf}, F_{zr} | Vertical load forces on the vehicle at the front and rear ground contact points, respectively (N) |
C_{d} | Aerodynamic drag coefficient (N·s^{2}/kg·m) |
ρ = 1.2 kg/m^{3} | Mass density of air (kg/m^{3}) |
|F_{d}| = ½C_{d}ρAV_{x} ^{2} | Aerodynamic drag force (N) |
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.
$$\begin{array}{l}m{\dot{V}}_{x}={F}_{x}+\text{}{F}_{d}-mg\cdot \mathrm{sin}\beta ,\\ {F}_{x}={F}_{xf}+{F}_{xr},\\ {F}_{d}=-\frac{1}{2}{C}_{d}\rho A{{\displaystyle {V}_{x}}}^{2}\cdot \mathrm{sgn}({V}_{x})\end{array}$$
Zero vertical acceleration and zero pitch torque require
$$\begin{array}{l}{F}_{zf}=\frac{+h({F}_{d}-mg\mathrm{sin}\beta -m{\dot{V}}_{x})+b\cdot mg\mathrm{cos}\beta}{a+b}\\ {F}_{zr}=\frac{-h({F}_{d}-mg\mathrm{sin}\beta -m{\dot{V}}_{x})+a\cdot mg\mathrm{cos}\beta}{a+b}\end{array}$$
Note that F_{zf} + F_{zr} = 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. |
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.
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.