# Influence of Drivetrain Hybridization on Transmission Lifetime

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Use Case

#### 2.2. Lifetime Calculation

^{6}bearing revolutions, according to Equation (2).

#### 2.3. System Model

## 3. Results

#### 3.1. Conventional Drivetrain

#### 3.2. Hybrid Drivetrain

#### 3.3. Sensitivity Analysis

## 4. Summary, Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BEV | Battery Electric Vehicle |

DMF | Dual Mass Flywheel |

EM | Electric Motor |

FD | Final Drive |

HCU | Hybrid Control Unit |

HEV | Hybrid Electric Vehicle |

ICE | Internal Combustion Engine |

ISO | International Organization for Standardization |

MB | Mechanical Brake |

PMSM | Permanent Magnet Synchronous Motor |

TCU | Transmission Control Unit |

WLTC | Worldwide Harmonized Light Vehicles Test Cycle |

## Nomenclature

${\mathsf{\alpha}}_{\mathrm{n}}$ | Normal pressure angle | (°) |

$\mathsf{\beta}$ | Helix angle | (°) |

$\mathsf{\varphi}$ | Angle | (1) |

$\dot{\mathsf{\varphi}}$ | Angular velocity | (s^{−1}) |

$\ddot{\mathsf{\varphi}}$ | Angular acceleration | (s^{−2}) |

${\mathsf{\varphi}}_{\mathrm{veh}}$ | Vehicle equivalent inertia angle | (1) |

${\dot{\mathsf{\varphi}}}_{\mathrm{veh}}$ | Vehicle equivalent inertia angular velocity | (s^{−1}) |

${\ddot{\mathsf{\varphi}}}_{\mathrm{veh}}$ | Vehicle equivalent inertia angular acceleration | (s^{−2}) |

${\mathsf{\varphi}}_{\mathrm{wheel},\mathrm{l}}$ | Left wheel angle | (1) |

${\dot{\mathsf{\varphi}}}_{\mathrm{wheel},\mathrm{l}}$ | Left wheel angular velocity | (s^{−1}) |

${\mathsf{\varphi}}_{\mathrm{wheel},\mathrm{r}}$ | Right wheel angle | (1) |

${\dot{\mathsf{\varphi}}}_{\mathrm{wheel},\mathrm{r}}$ | Right wheel angular velocity | (s^{−1}) |

${\mathrm{a}}_{\mathrm{ISO}}$ | Life modification factor | (1) |

$\mathrm{b}$ | Shape parameter | (1) |

${\mathrm{B}}_{10}$ | Bearing lifetime with a failure probability of 10% | (h) |

$\mathrm{c}$ | Stiffness | (Nm⸱rad^{−1}) |

$\mathrm{C}$ | Dynamic load rating | (N) |

$\mathrm{d}$ | Damping | (N⸱m⸱s⸱rad^{−1}) |

${\mathrm{d}}_{\mathrm{G}}$ | Gear diameter | (mm) |

${\mathrm{d}}_{\mathrm{m}}$ | Mean bearing diameter | (mm) |

$\mathrm{D}$ | Damage | (1) |

${\mathrm{f}}_{\mathrm{d}0}$ | Driving resistance parameter | (N) |

${\mathrm{f}}_{\mathrm{d}1}$ | Driving resistance parameter | (N⸱s⸱m^{−1}) |

${\mathrm{f}}_{\mathrm{d}2}$ | Driving resistance parameter | (N⸱s^{2}⸱m^{−2}) |

${\mathrm{F}}_{\mathrm{a}}$ | Axial gear force | (N) |

${\mathrm{F}}_{\mathrm{b}}$ | Bearing force | (N) |

${\mathrm{F}}_{\mathrm{A}}\left(\mathrm{t}\right)$ | Axial force | (N) |

${\mathrm{F}}_{\mathrm{B}}\left(\mathrm{t}\right)$ | Failure probability of all bearings over time | (1) |

${\mathrm{F}}_{\mathrm{drag}}$ | Driving resistance force | (N) |

${\mathrm{F}}_{\mathrm{i}}$ | Force i | (N) |

${\mathrm{F}}_{\mathrm{j}}\left(\mathrm{t}\right)$ | Failure probability of a single bearing j over time | (1) |

${\mathrm{F}}_{\mathrm{r}}$ | Radial gear force | (N) |

${\mathrm{F}}_{\mathrm{t}}$ | Tangential gear force | (N) |

${\mathrm{F}}_{\mathrm{R}}\left(\mathrm{t}\right)$ | Radial force | (N) |

$\mathrm{i}$ | Index | (1) |

${\mathrm{i}}_{\mathrm{G}}$ | Gear ratio | (1) |

$\mathrm{j}$ | Index | (1) |

$\mathrm{J}$ | Inertia | (kg⸱m^{2}) |

$\mathrm{k}$ | Number of load classes | (1) |

$\mathrm{l}$ | Length of a shaft section | (mm) |

${\mathrm{L}}_{\mathrm{i}}$ | Bearable number of rotations of a load class i | (10^{6}) |

$\mathrm{m}$ | Vehicle mass | (kg) |

${\mathrm{M}}_{\mathrm{i}}$ | Moment i | (N⸱m) |

${\mathrm{n}}_{\mathrm{i}}$ | Number of bearing revolutions in a load class i | (1) |

$\mathrm{p}$ | Lifetime exponent | (1) |

$\mathrm{P}\left(\mathrm{t}\right)$ | Dynamically equivalent bearing load | (N) |

${\mathrm{P}}_{\mathrm{i}}$ | Dynamically equivalent bearing load of a load class i | (N) |

${\mathrm{r}}_{\mathrm{tire}}$ | Tire radius | (m) |

$\mathrm{t}$ | time | (s) |

${\mathrm{t}}_{0}$ | Failure free time | (h) |

${\mathrm{t}}_{\mathrm{EM}}$ | EM time constant | (s) |

${\mathrm{t}}_{\mathrm{ICE}}$ | ICE time constant | (s) |

${\mathrm{T}}_{\mathrm{cycle}}$ | Cycle duration | (s) |

${\mathrm{T}}_{\mathrm{act},\mathrm{EM}}$ | Acting EM torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{act},\mathrm{ICE}}$ | Acting ICE torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{act},\mathrm{MB}}$ | Acting MB torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{dem},\mathrm{brk}}$ | Demanded deceleration torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{des},\mathrm{EM}}$ | Desired EM torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{des},\mathrm{ICE}}$ | Desired ICE torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{des},\mathrm{MB}}$ | Desired MB torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{drag}}$ | Equivalent driving resistance torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{G}}$ | Gear torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{i}}$ | Torque i | (N⸱m) |

${\mathrm{T}}_{\mathrm{max},\mathrm{EM},\mathrm{ICE}}$ | Sum of maximum EM generator and ICE drag torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{max},\mathrm{ICE}}$ | Maximum ICE drag torque | (N⸱m) |

${\mathrm{T}}_{\mathrm{j}}$ | Characteristic lifetime of a single bearing j | (h) |

${\mathrm{v}}_{\mathrm{act}}$ | Actual velocity | (km⸱h^{−1}) |

${\mathrm{v}}_{\mathrm{des}}$ | Desired velocity | (km⸱h^{−1}) |

${\mathrm{x}}_{\mathrm{brk}}$ | Brake pedal position | (1) |

${\mathrm{x}}_{\mathrm{thr}}$ | Throttle pedal position | (1) |

${\dot{\mathrm{x}}}_{\mathrm{veh}}$ | Vehicle velocity | (km⸱h^{−1}) |

$\mathrm{X}$ | dynamic radial load factor | (1) |

$\mathrm{Y}$ | dynamic axial load factor | (1) |

## Appendix

**Table A1.**Bearing parameters. Position according to Figure 2. Dynamic load rating C and mean diameter ${\mathrm{d}}_{\mathrm{m}}$.

Position | Type | C (N) | ${\mathrm{d}}_{\mathrm{m}}\left(\mathrm{mm}\right)$ |
---|---|---|---|

1, 2 | tapered roller bearing | 64,000 | 65 |

3 | axial needle roller bearing | 30,000 | 65 |

4 | needle roller bearing | 26,500 | 30 |

5 | needle roller bearing | 27,500 | 36 |

6, 15, 17 | ball bearing | 32,000 | 49.5 |

7 | cylindrical roller bearing | 33,500 | 53.3 |

8 | needle roller bearing | 33,500 | 45.5 |

9 | needle roller bearing | 33,500 | 45.5 |

10 | needle roller bearing | 33,500 | 45.5 |

11 | needle roller bearing | 33,500 | 45.5 |

12 | needle roller bearing | 33,500 | 45.5 |

13 | needle roller bearing | 33,500 | 45.5 |

14 | needle roller bearing | 33,500 | 37 |

16 | cylindrical roller bearing | 59,000 | 53.9 |

18 | cylindrical roller bearing | 33,500 | 56 |

19, 20 | cylindrical roller bearing | 28,500 | 29.5 |

Gear | Value |
---|---|

1 | 14.07 |

2 | 7.66 |

3 | 4.68 |

4 | 3.26 |

5 | 2.52 |

6 | 2.04 |

Parameter | Value and Unit |
---|---|

${\mathrm{f}}_{\mathrm{d}0}$ | 160 N |

${\mathrm{f}}_{\mathrm{d}1}$ | 4 N⸱s⸱m^{−1} |

${\mathrm{f}}_{\mathrm{d}2}$ | 0.37 N⸱s^{2}⸱m^{−2} |

Parameter | Value and Unit |
---|---|

${\mathrm{r}}_{\mathrm{tire}}$ | 0.302 m |

${\mathrm{J}}_{\mathrm{Veh}}$ | 72.96 kg⸱m^{2} |

${\mathrm{J}}_{\mathrm{ICE}}$ | 0.3 kg⸱m^{2} |

${\mathrm{J}}_{\mathrm{DMF}}$ | 0.3 kg⸱m^{2} |

${\mathrm{J}}_{\mathrm{EE}}$ | 0.2 kg⸱m^{2} |

${\mathrm{J}}_{\mathrm{wheel}}$ | 0.5 kg⸱m^{2} |

${\mathrm{c}}_{\mathrm{DMF}}$ | 580 N⸱m⸱rad^{−1} |

${\mathrm{d}}_{\mathrm{DMF}}$ | 50 N⸱m⸱s⸱rad^{−1} |

${\mathrm{c}}_{\mathrm{shaft},\mathrm{l}}$ | 1.1E4 N⸱m⸱rad^{−1} |

${\mathrm{c}}_{\mathrm{shaft},\mathrm{r}}$ | 9.5E3 N⸱m⸱rad^{−1} |

${\mathrm{d}}_{\mathrm{shaft},\mathrm{l}}$ | 6 N⸱m⸱s⸱rad^{−1} |

${\mathrm{d}}_{\mathrm{shaft},\mathrm{r}}$ | 6 N⸱m⸱s⸱rad^{−1} |

${\mathrm{c}}_{\mathrm{tire}}$ | 2.3E4 N⸱m⸱rad^{−1} |

${\mathrm{d}}_{\mathrm{tire}}$ | 1.15 N⸱m⸱s⸱rad^{−1} |

Parameter | Value and Unit |
---|---|

Duration ${\mathrm{T}}_{\mathrm{cycle}}$ | 1800 s |

Distance | 23,266 m |

Average velocity | 46.5 km⸱h^{−1} |

Maximum velocity | 131 km⸱h^{−1} |

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**Figure 1.**Arrangements of the EM (electric machine) in basic parallel hybrid topologies (P0-P4) and exemplary arrangement through combination (P0P2).

**Figure 3.**Approach for determining the system lifetime [28].

**Figure 5.**(

**a**) Full-load characteristic curve (blue) and thrust characteristic curve (red) of the ICE model; (

**b**) traction (blue) and generator (red) torque limits of the EM model.

**Figure 6.**Structure of the torsional vibration model used for the hybrid drivetrain. For the conventional drivetrain the torque ${\mathrm{T}}_{\mathrm{act},\mathrm{EM}}$ is omitted and the inertia at the transmission input is added to the inertia of the clutch drum.

**Figure 7.**Force diagram of the first intermediate shaft. All forces are rotated into the section plane.

**Figure 8.**Load spectrum of the transmission input torque for the conventional drivetrain in the WLTC.

**Figure 9.**Failure probabilities of the individual bearings and the bearings as a group for the conventional drivetrain. The position numbers of the bearings are shown framed (see Figure 2). The dotted line indicates the ${\mathrm{B}}_{10}$-lifetime with 195,000 km.

**Figure 10.**Load spectrum of the transmission input torque for the hybrid drivetrain without limitation of the regeneration torque in the WLTC. Scaling of the color bar identical to Figure 8.

**Figure 11.**Failure probabilities of the individual bearings and the bearings as a group for the hybrid drivetrain. The position numbers of the bearings are shown framed (see Figure 2). The dotted line indicates the ${\mathrm{B}}_{10}$-lifetime of the bearing group with 123,000 km.

**Figure 12.**Load spectrum of the transmission input torque for the hybrid drivetrain with a regenerative torque limit of the electric machine of -100 Nm in the WLTC. Scaling of the color bar identical to Figure 8.

**Figure 13.**Results of the sensitivity analysis. Bearing lifetime over maximum regenerative torque in the WLTC.

Parameter | Value |
---|---|

Vehicle class | C-segment |

Vehicle mass ${\mathrm{m}}_{\mathrm{veh}}$ | 1600 kg |

ICE Power | 125 kW |

EM Power | 83 kW |

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## Share and Cite

**MDPI and ACS Style**

Habermehl, C.; Jacobs, G.; Neumann, S.; Weißenfels, K.
Influence of Drivetrain Hybridization on Transmission Lifetime. *Appl. Sci.* **2020**, *10*, 7086.
https://doi.org/10.3390/app10207086

**AMA Style**

Habermehl C, Jacobs G, Neumann S, Weißenfels K.
Influence of Drivetrain Hybridization on Transmission Lifetime. *Applied Sciences*. 2020; 10(20):7086.
https://doi.org/10.3390/app10207086

**Chicago/Turabian Style**

Habermehl, Christian, Georg Jacobs, Stephan Neumann, and Kevin Weißenfels.
2020. "Influence of Drivetrain Hybridization on Transmission Lifetime" *Applied Sciences* 10, no. 20: 7086.
https://doi.org/10.3390/app10207086