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.

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

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

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

\[F_{g}= \frac{T_{g} \cdot i_{0} \cdot \eta_{0} \cdot \eta_{l}}{r_{w}} \tag{11}\]

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

\[F_{lim}= m_{v} \cdot g \cdot \mu_{w} \cdot c_{w} \tag{12}\]

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].

\[F_{w} = F_{i} + F_{a} + F_{r} + F_{s} + F_{b} \tag{13}\]

The vehicle inertia force is calculated as:

\[F_{i} = m_{v} \cdot c_{i} \cdot \frac{dv_{v}}{dt} \tag{14}\]

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:

\[F_{a} = \frac{1}{2} \cdot C_{d} \cdot A \cdot v_{v}^{2} \cdot \rho \tag{15}\]

The rolling resistance force is calculated as:

\[F_{r} = m_{v} \cdot g \cdot \cos(\alpha) \cdot f \tag{16}\]

The road slope angle is calculated as:

\[\alpha = \arctan \left ( \frac{s}{100} \right ) \tag{17}\]

The rolling resistance coefficient is calculated as:

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

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:

\[H(s) = \frac{K}{T \cdot s +1} = \frac{100}{0.1 \cdot s +1} \tag{20}\]

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.

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

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

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

The wheel speed in [rpm] is calculated as:

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

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/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]

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/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.

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]