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

Tires & Vehicles

The Vehicle Body block models a two-axle vehicle, with an equal number of equally sized wheels on each axle, moving forward or backward along its longitudinal axis.

The model includes the following vehicle properties.

Mass

Number of wheels on each axle

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

For model details, see Vehicle Body Model.

You specify the headwind speed *V*_{W} (in
meters/second) and the road inclination angle *β* (in
radians) through physical signal inputs at ports W and beta, respectively.

The block reports the longitudinal vehicle velocity *V*_{x} and
the front and rear normal forces (load on wheels) *F*_{zf}, *F*_{zr} as
physical signal outputs at ports V, NF, and NR, respectively.

The horizontal motion of the vehicle is represented by the translational conserving port H.

**Mass**Mass

*m*of the vehicle. The default is`1200`

.From the drop-down list, choose units. The default is kilograms (

`kg`

).**Number of wheels per axle**Number

*n*of equally-sized wheels on each axle, forward and rear. The default is`2`

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

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

.From the drop-down list, choose units. The default is meters (

`m`

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

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

.From the drop-down list, choose units. The default is meters (

`m`

).**CG height above ground**Height

*h*of the vehicle's center of gravity from the ground. The default is`0.5`

.From the drop-down list, choose units. The default is meters (

`m`

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

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

.From the drop-down list, choose units. The default is meters-squared (

`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`

.**Initial velocity**The initial value

*V*_{x}(0) of the vehicle's horizontal velocity. The default is`0`

.From the drop-down list, choose units. The default is meters/second (

`m/s`

).

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**

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

g | Gravitational acceleration = 9.81 m/s^{2} |

β | Incline angle |

m | Vehicle mass |

h | Height of vehicle 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} | Longitudinal vehicle velocity |

V_{W} | Headwind speed |

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 the 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 = 1.2 kg/m^{3} |

F_{d} | Aerodynamic drag force |

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
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.

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

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

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

The wheel normal forces satisfy *F*_{zf} + *F*_{zr} = *mg*·cos*β*/*n*.

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.

These SimDriveline™ example models contain working examples of vehicle bodies:

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