# Influence of Drivetrain Hybridization on Transmission Lifetime
## 2 Materials and Methods
### System Model  
To determine the operating conditions of the bearings in terms of force and rotational speed, a detailed forward oriented overall vehicle model is built in Matlab/Simulink using the Simscape library. The high-level structure of the model is shown in Figure 4. The submodels are divided into ICE, EM, mechanical drivetrain, vehicle, driver and hybrid control unit (HCU). On the input side, the model receives a driving cycle-dependent time-based desired velocity \(\mathrm{v_{des}}\). In the driver model, the desired velocity is compared to the actual velocity \(\mathrm{v_{act}}\) and the deviation is converted into an accelerator \(\mathrm{x_{thr}}\) or brake pedal position \(\mathrm{x_{brk}}\) by a PI-controller. The pedal positions are converted into desired torques of the ICE \(\mathrm{T_{des,ICE}}\), the EM \(\mathrm{T_{des,EM}}\) and the mechanical brake (MB) \(\mathrm{T_{des,MB}}\) by the HCU. In this paper, a simplified operating strategy is considered, since the influence of regenerative braking on the bearing lifetime of the transmission is investigated. The vehicle is driven purely by the ICE. The deceleration of the vehicle is implemented by splitting the braking torque between ICE, EM and the MB. For this, a required deceleration torque \(\mathrm{T_{dem,brk}}\) is calculated from the brake pedal position. This deceleration torque is applied with priority by the drag torque of the ICE. The desired ICE torque is calculated according to Equation (8) with the current transmission gear ratio \(\mathrm{i_{G}}\) (see Table 2). \[\mathrm{T_{des,ICE}}=\frac{\mathrm{T_{dem,brk}}}{\mathrm{i_{G}}} \tag{8}\]  
Figure 4: Modular structure of the system simulation model and exchanged quantities. If the maximum drag torque \(\mathrm{T_{max,\;ICE}}\) of the ICE is not sufficient for the deceleration of the vehicle, the EM is used as a generator (Equation (9)). \[\mathrm{T_{des,EM}}=\frac{\mathrm{T_{dem,brk}}}{\mathrm{i_{G}}}-\mathrm{T_{max, \;ICE}} \tag{9}\]  
For stronger braking which cannot be achieved by the drag torque of the ICE and the generator operation of the EM \(\mathrm{T_{max,EM,ICE}}\), the mechanical brake is applied (Equation (10)). \[\mathrm{T_{des,MB}}=\mathrm{T_{dem,brk}}-\mathrm{T_{max,EM,ICE}} \tag{10}\]  
In the ICE model, the desired torque \(\mathrm{T_{des,ICE}}\) is converted into an acting torque \(\mathrm{T_{act,ICE}}\) by a first-order delay with a time constant of \(\mathrm{t_{ICE}}=0.203\) s [33]. The maximum drive torque is limited in the model by a full load characteristic curve. For coast operation, an analytically derived thrust characteristic curve is used [26], which decreases in proportion to engine speed to \(-10\%\) of the nominal torque (Figure 5a). In the EM model, the desired torque \(\mathrm{T_{des,EM}}\) is converted into an acting torque \(\mathrm{T_{act,EM}}\) by a first-order delay with a time constant of \(\mathrm{t_{EM}}=0.04\) s [33]. The maximum deliverable motor and generator torques are limited by power hyperbolas and constant torque curves (Figure 5b). In the vehicle model, the effective driving resistance force \(\mathrm{F_{drag}}\) is calculated as a function of the vehicle velocity \(\dot{\mathrm{x}}_{\mathrm{veh}}\) according to Equation (11). \(\mathrm{f_{d0}}\) and \(\mathrm{f_{d1}}\) are driving resistance coefficients which cause constant and velocity proportional driving resistances. \(\mathrm{f_{d2}}\) causes a resistance force which is proportional to the square of the vehicle velocity. The values of the coefficients are listed in Table 10.  
\[\mathrm{F_{drag}}=\mathrm{f_{d0}}+\mathrm{f_{d1}}\cdot\dot{\mathrm{x}}_{ \mathrm{veh}}+\mathrm{f_{d2}}\cdot\dot{\mathrm{x}}_{\mathrm{veh}}^{2} \tag{11}\]  
The forces and torques described above act on a torsional vibration model of the mechanical drivetrain (Figure 6). Individual inertias are represented as cylinders connected by spring-damper combinations or clutches. External torques act through the ICE, EM, MB and driving resistance.