# Influence of Drivetrain Hybridization on Transmission Lifetime
## 2 Materials and Methods
### Lifetime Calculation  
In this paper, the bearing lifetime of the transmission is calculated on the basis of damage due to material fatigue. For this, the approach according to Figure 3 is applied. As shown in [28], the approach can also be adapted to other machine elements that fail due to fatigue, such as gears or shafts. By making suitable assumptions, wear-related failures such as those of friction clutches or seals can also be calculated [29], but these are omitted in this paper for brevity. The approach for considering component lifetime is based on existing standards and established procedures in the automotive industry. The core aspect of the approach is the comparison of occurring and bearable loads, which is necessary for the quantitative assessment of component lifetime [26]. Figure 3: Approach for determining the system lifetime [28]. Figure 2: Transmission layout and structure of its torsional rigid-body model. Here, a comparison is made computationally using a damage accumulation hypothesis. An alternative to simulation is testing, but this is usually associated with a high expenditure of time and money. Initially, driving cycles are defined to determine the bearing loads for the computational analysis. The cycles represent the assumed use of the transmission consisting of various boundary conditions such as driver, driving distance, special events and others. The influence of the vehicle is represented by a simulation model. The simulation model considers the interaction of the subsystems in a selected level of detail and outputs time-based records of the bearing loads (axial and radial forces) and their frequency of occurrence in the form of the rotational speed. The speed and load of the bearings depend on which gear is engaged, whether the vehicle accelerates or decelerates and whether gearshifts occur. In the case of measurement of the occurring loads, the bearing loads are typically calculated back from an easily measurable variable, such as the input torque of the transmission [26]. Bearings can be subjected to a combined load from radial \(\mathrm{F_{R}(t)}\) and axial forces \(\mathrm{F_{A}(t)}\). However, for the comparison of the occurring and bearable loads, a reduced quantity is required, i.e., the dynamically equivalent force \(\mathrm{P(t)}\). The calculation is carried out according to Equation (1). X and Y are the dynamic radial and axial load factor, respectively. They can either be obtained directly from the bearing manufacturer or calculated using analytical equations of ISO 281 [30]. \[\mathrm{P(t)}=\mathrm{X}\cdot\mathrm{F_{R}(t)}+\mathrm{Y}\cdot\mathrm{F_{A}(t)} \tag{1}\]  
The resulting time records \(\mathrm{P(t)}\) per bearing are then simplified to load spectra with the aid of the two-parametric time-at-level counting [31]. The occurring loads and speeds are divided into 64 classes according to the recommendation of [32]. The bearable loads of the bearings required for the accumulation of damage are determined using ISO 281 [30]. The possible number of rotations \(\mathrm{L_{i}}\) of a load class i with a failure probability of 10% is obtained in 10\({}^{6}\) bearing revolutions, according to Equation (2). \[\mathrm{L_{i}}=\mathrm{a_{ISO}}\cdot\left(\frac{\mathrm{C}}{\mathrm{P_{i}}} \right)^{\mathrm{P}} \tag{2}\]  
The life modification factor \(\mathrm{a_{ISO}}\) takes into account the type, quality and geometry of the bearing, as well as the fatigue limit of the raceway material [30]. C is the dynamic load rating of the bearing, and can be obtained from the bearing manufacturer. The exponent \(\mathrm{p}\) is 3 for ball bearings and \(\frac{10}{3}\) for roller bearings. \(\mathrm{P_{i}}\) is the dynamically equivalent bearing load of a load class. Information on the bearing data, taken from bearing manufacturer catalogues, can be found in Table 1.  
In the damage accumulation, occurring and bearable loads are compared and summed to yield damage D over all k load classes (Equation (3)).