### 5. Plant model: Vehicle

The study vehicle is considered to have Rear Wheel Drive (RWD). The vehicle model is relatively simple, defined as a mass in translation, which receives the gearbox torque (force) as traction force and the sum of road loads as resistive force.

The longitudinal dynamics of the vehicle is based on the following equation:

\[F_{g}=F_{w} \tag{10}\]F

_{g}[N] – Gearbox (traction) force

F

_{w}[N] – Wheel (resistive) force

The **gearbox force** is considered after the final (differential) gear, being the force which goes in the wheel hubs and propels the vehicle.

η

_{0}[-] – final drive (differential) efficiency

η

_{l}[-] – longitudinal (propeller) shaft efficiency

r

_{w}[m] – wheel radius

The **traction force** is limited to the friction force, which can be applied to the wheels.

m

_{v}[kg] – vehicle mass

g [m/s

^{2}] – gravitational acceleration

μ

_{w}[-] – driving wheels friction coefficient

c

_{w}[-] – driving axle load coefficient

The load coefficient shows how much weight of the vehicle is on the rear (driving) wheels. If the value is 0.65, means that 65% of the total vehicle weight is on the rear wheels.

The **wheel (resistive) force** is the sum of the inertial force, aerodynamic drag, rolling friction force, road slope (gradient) resistance and braking force [2].

F

_{i}[N] – inertia force

Fa [N] – aerodynamic force

F

_{r}[N] – rolling resistance force

F

_{s}[N] – road slope (gradient) force

F

_{b}[N] – braking force

The vehicle **inertia force** is calculated as:

v

_{v}[m/s] – vehicle speed

c

_{i}[-] – rotational components inertia coefficient

The inertia of the components which are rotating, for example wheels, longitudinal shaft, gears, etc., are taking into account through the coefficient *c _{i}*. If this coefficient is 1.1, means that the vehicle mass is artificially increased with 10% in order to account for the inertia of the rotational components.

The **aerodynamic drag force** is calculated as:

C

_{d}[-] – aerodynamic drag coefficient

A [m

^{2}] – vehicle frontal area

ρ [kg/m

^{3}] – air density

The **rolling resistance force** is calculated as:

α [rad] – road slope angle

f [-] – rolling resistance coefficient

The road slope angle is calculated as:

\[\alpha = \arctan \left ( \frac{s}{100} \right ) \tag{17}\]s [%] – road slope (gradient)

The rolling resistance coefficient is calculated as:

\[f = C_{0} + C_{1} \cdot v_{v} + C_{2} \cdot v_{v}^{2} \tag{18}\]C

_{0}[-] – rolling resistance constant coefficient

C

_{1}[s/m] – rolling resistance linear coefficient

C

_{2}[s

^{2}/m

^{2}] – rolling resistance quadratic coefficient

The **road slope (gradient) force** is calculated as:

The **braking force** is modelled as a first order transfer function:

K [-] – gain

T [s] – time constant

The input into the transfer function is the brake pedal position in [%]. This means that, for the brake pedal fully pressed 100 %, the braking force will be 10000 N. The time constant will filter at a small degree the pedal position coming from the Driver model.

The vehicle speed is calculated by integrating the vehicle acceleration, the result being in m/s. Further, if we integrate the speed, we can determine the offset of the vehicle, which is the distance travelled by the vehicle from the starting point.

\[x_{v} = \int v_{v} \cdot dt \tag{21}\]x

_{v}[m] – vehicle offset

We can also calculate the vehicle speed in [kph], as:

\[V_{v} = 3.6 \cdot v_{v} \tag{22}\]V

_{v}[kph] – vehicle speed

The wheel speed in [rpm] is calculated as:

\[N_{w} = \frac{v_{v}}{r_{w}} \cdot \frac{30}{\pi} \tag{23}\]N

_{w}[rpm] – wheel speed

#### 5.1 Traction force

Equations (11) and (12) are used for the Xcos model of the Traction force.

The traction force applied at the wheel is the minimum between the raw traction for and the friction limit force.

**Inputs**

Name | Value | Description |

GbxTq_Nm | – | Gearbox torque [Nm] |

**Parameters**

Name | Value | Description |

VehM_kg_C | 2255 | Vehicle mass [kg] |

VehGrvygA_mps2_C | 9.81 | Gravitational acceleration [m/s^{2}] |

VehWhlFricCoeff_z_C | 1.0 | Driving wheels friction coefficient[-] |

VehReWghtCoeff_z_C | 0.6 | Driving axle load coefficient [-] |

LgtShaftEff_z_C | 0.994 | Longitudinal (propeller) shaft efficiency [-] |

DftlGearRat_z_C | 2.769 | Final gear (differential) gear ratio [-] |

DftlEff_z_C | 0.93 | Final drive (differential) efficiency [-] |

VehWhlRollgRd_m_C | 0.31587 | Wheel (rolling) radius [m] |

**Outputs**

Name | Value | Description |

VehTracF_N | – | Traction force (minimum between gearbox force and friction limit) [N] |

#### 5.2 Road resistances

Equations (13), (14), (15), (16), (17), (18), (19) and (20) are used for the Xcos model of the Road resistances.

The road resistances can also be called as drag forces, because the oppose the movement of the vehicle.

**Inputs**

Name | Value | Description |

EnvRoadSlop_prc | – | Road slope (gradient) in [%] |

BrkPedlPosn_prc | – | Brake pedal position in [%] |

VehV_kph | – | Vehicle speed [kph] |

**Parameters**

Name | Value | Description |

VehM_kg_C | 2255 | Vehicle mass [kg] |

VehGrvygA_mps2_C | 9.81 | Gravitational acceleration [m/s^{2}] |

VehDragCoeff_z_C | 0.29 | Aerodynamic drag coefficient [-] |

VehFrntAr_m2_C | 2.138 | Vehicle frontal area [m^{2}] |

VehAirRho_kgpm3_C | 1.202 | Air density [kg/m^{3}] |

C0 | 1.3295·10^{-2} | Rolling resistance constant coefficient [-] |

C1 | -2.8664·10^{-5} | Rolling resistance linear coefficient [s/m] |

C2 | 1.8036·10^{-7} | Rolling resistance quadratic coefficient [s^{2}/m^{2}] |

**Outputs**

Name | Value | Description |

VehTotDragF_N | – | Vehicle total drag force (road resistance) [N] |

#### 5.3 Speed calculation

Equations (10), (21), (22) and (23) are used for the Xcos model of the Speed calculation.

The road resistances can also be called as drag forces, because the oppose the movement of the vehicle.

**Inputs**

Name | Value | Description |

VehTracF_N | – | Traction force (minimum between gearbox force and friction limit) [N] |

VehTotDragF_N | – | Vehicle total drag force (road resistance) [N] |

**Parameters**

Name | Value | Description |

VehM_kg_C | 2255 | Vehicle mass [kg] |

VehRotCmpJCoeff_z_C | 1.25 | Rotational components inertia coefficient [-] |

VehWhlRollgRd_m_C | 0.31587 | Wheel (rolling) radius [m] |

**Outputs**

Name | Value | Description |

WhlN_rpm | – | Wheel speed [rpm] |

VehV_kph | – | Vehicle speed [kph] |