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

Tires & Vehicles

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.

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

This figure and table define the vehicle motion model variables.

**Vehicle Dynamics and Motion**

**Vehicle Model Variables**

Symbol | Description and Unit |
---|---|

g | Gravitational acceleration |

β | Incline angle |

m | Mass of the vehicle |

h | Height of vehicle CG above the ground |

, ab | 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 on each axle |

, 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 |

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.

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

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

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

The wheel normal forces satisfy * F_{zf} + F_{zr} = mg·cosβ/n*.

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.

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.

`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.

**Mass**Mass

of the vehicle. The default is*m*`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

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

`m`

.**Horizontal distance from CG to rear axle**Horizontal distance

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

`m`

.**CG height above ground**Distance,

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

`m`

.**Frontal area**Effective cross-sectional area

presented by the vehicle in longitudinal motion, to compute the aerodynamic drag force on the vehicle. The default is*A*`3`

`m^2`

.**Drag coefficient**The dimensionless aerodynamic drag coefficient

*C*_{d}, 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`

.

Tire (Friction Parameterized) | Tire (Magic Formula) | Tire (Simple) | Tire-Road Interaction (Magic Formula)

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