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.

**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**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 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 compute 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 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 = 1.18 kg/m^{3} |

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.

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

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. As such, the equations assume that the wheels never lose contact. This constraint can result in negative normal forces.

Port | Description |
---|---|

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 |

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

Was this topic helpful?