# Modeling and simulation of a vehicle with automatic transmission

### 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.

Image: Vehicle free body diagram

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

$F_{g}=F_{w} \tag{10}$
where:
Fg [N] – Gearbox (traction) force
Fw [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.

$F_{g}= \frac{T_{g} \cdot i_{0} \cdot \eta_{0} \cdot \eta_{l}}{r_{w}} \tag{11}$
where:
η0 [-] – final drive (differential) efficiency
ηl [-] – longitudinal (propeller) shaft efficiency

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

$F_{lim}= m_{v} \cdot g \cdot \mu_{w} \cdot c_{w} \tag{12}$
where:
mv [kg] – vehicle mass
g [m/s2] – gravitational acceleration
μw [-] – driving wheels friction coefficient
cw [-] – 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].

$F_{w} = F_{i} + F_{a} + F_{r} + F_{s} + F_{b} \tag{13}$
where:
Fi [N] – inertia force
Fa [N] – aerodynamic force
Fr [N] – rolling resistance force
Fb [N] – braking force

The vehicle inertia force is calculated as:

$F_{i} = m_{v} \cdot c_{i} \cdot \frac{dv_{v}}{dt} \tag{14}$
where:
vv [m/s] – vehicle speed
ci [-] – 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 ci. 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:

$F_{a} = \frac{1}{2} \cdot C_{d} \cdot A \cdot v_{v}^{2} \cdot \rho \tag{15}$
where:
Cd [-] – aerodynamic drag coefficient
A [m2] – vehicle frontal area
ρ [kg/m3] – air density

The rolling resistance force is calculated as:

$F_{r} = m_{v} \cdot g \cdot \cos(\alpha) \cdot f \tag{16}$
where:
f [-] – rolling resistance coefficient

The road slope angle is calculated as:

$\alpha = \arctan \left ( \frac{s}{100} \right ) \tag{17}$
where:

The rolling resistance coefficient is calculated as:

$f = C_{0} + C_{1} \cdot v_{v} + C_{2} \cdot v_{v}^{2} \tag{18}$
where:
C0 [-] – rolling resistance constant coefficient
C1 [s/m] – rolling resistance linear coefficient
C2 [s2/m2] – rolling resistance quadratic coefficient

$F_{s} = m_{v} \cdot g \cdot \sin(\alpha) \tag{19}$

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

$H(s) = \frac{K}{T \cdot s +1} = \frac{100}{0.1 \cdot s +1} \tag{20}$
where:
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.

$x_{v} = \int v_{v} \cdot dt \tag{21}$
where:
xv [m] – vehicle offset

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

$V_{v} = 3.6 \cdot v_{v} \tag{22}$
where:
Vv [kph] – vehicle speed

The wheel speed in [rpm] is calculated as:

$N_{w} = \frac{v_{v}}{r_{w}} \cdot \frac{30}{\pi} \tag{23}$
where:
Nw [rpm] – wheel speed

#### 5.1 Traction force

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

Image: Traction force – Xcos block diagram

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/s2] 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]

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

Image: Road resistance – Xcos block diagram

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/s2] VehDragCoeff_z_C 0.29 Aerodynamic drag coefficient [-] VehFrntAr_m2_C 2.138 Vehicle frontal area [m2] VehAirRho_kgpm3_C 1.202 Air density [kg/m3] 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 [s2/m2]

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.

Image: Speed calculation – Xcos block diagram

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]