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.

Vehicle free body diagram

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
rw [m] – wheel radius

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
Fs [N] – road slope (gradient) 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:
α [rad] – road slope angle
f [-] – rolling resistance coefficient

The road slope angle is calculated as:

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

The rolling resistance coefficient is calculated as:

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

The road slope (gradient) force is calculated as:

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

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.

Traction force - Xcos block diagram

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

NameValueDescription
GbxTq_NmGearbox torque [Nm]

Parameters

NameValueDescription
VehM_kg_C2255Vehicle mass [kg]
VehGrvygA_mps2_C9.81Gravitational acceleration [m/s2]
VehWhlFricCoeff_z_C1.0Driving wheels friction coefficient[-]
VehReWghtCoeff_z_C0.6Driving axle load coefficient [-]
LgtShaftEff_z_C0.994Longitudinal (propeller) shaft efficiency [-]
DftlGearRat_z_C2.769Final gear (differential) gear ratio [-]
DftlEff_z_C0.93Final drive (differential) efficiency [-]
VehWhlRollgRd_m_C0.31587Wheel (rolling) radius [m]

Outputs

NameValueDescription
VehTracF_NTraction 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.

Road resistance - Xcos block diagram

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

NameValueDescription
EnvRoadSlop_prcRoad slope (gradient) in [%]
BrkPedlPosn_prcBrake pedal position in [%]
VehV_kphVehicle speed [kph]

Parameters

NameValueDescription
VehM_kg_C2255Vehicle mass [kg]
VehGrvygA_mps2_C9.81Gravitational acceleration [m/s2]
VehDragCoeff_z_C0.29Aerodynamic drag coefficient [-]
VehFrntAr_m2_C2.138Vehicle frontal area [m2]
VehAirRho_kgpm3_C1.202Air density [kg/m3]
C01.3295·10-2Rolling resistance constant coefficient [-]
C1-2.8664·10-5Rolling resistance linear coefficient [s/m]
C21.8036·10-7Rolling resistance quadratic coefficient [s2/m2]

Outputs

NameValueDescription
VehTotDragF_NVehicle 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.

Speed calculation - Xcos block diagram

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

NameValueDescription
VehTracF_NTraction force (minimum between gearbox force and friction limit) [N]
VehTotDragF_NVehicle total drag force (road resistance) [N]

Parameters

NameValueDescription
VehM_kg_C2255Vehicle mass [kg]
VehRotCmpJCoeff_z_C1.25Rotational components inertia coefficient [-]
VehWhlRollgRd_m_C0.31587Wheel (rolling) radius [m]

Outputs

NameValueDescription
WhlN_rpmWheel speed [rpm]
VehV_kphVehicle speed [kph]

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