# Design and Analysis of a New Torque Vectoring System with a Ravigneaux Gearset for Vehicle Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Current Torque Vectoring Differential

_{1}and W

_{2}are the left and right wheels; C

_{1}and C

_{2}are clutches; B

_{1}and B

_{2}are brakes; G, G

_{0}, G

_{1}and G

_{2}are gear pairs; PG

_{1}and PG

_{2}are planetary gearsets; IN is the input of the engine power.

_{1}and G

_{2}), and two clutches (C

_{1}and C

_{2}). When C

_{1}is engaged, the speed ratio of the left and right wheels is decided by the gear ratio of G

_{1}. Similarly, when C

_{2}is engaged, the speed ratio of the two wheels is decided by the gear ratio of G

_{2}. With the different gear ratios of G

_{1}and G

_{2}, tire slip ratio of the two tires can be controlled with different engagements of the clutches, so that different traction distributions can be achieved and the torque vectoring effect can be realized.

_{1}and PG

_{2}are involved, and two brakes B

_{1}and B

_{2}are used to select the direction of the torque vectoring effect. When B

_{1}is engaged, the speed ratio of the left and right wheels is decided by the gear ratio of PG

_{1}. On the other hand, when B

_{2}is engaged, the speed ratio of the two wheels is decided by the gear ratio of PG

_{2}.

## 3. Design of New Torque Vectoring Differential

_{1}and B

_{2}have different effects on the speed relation between the input IN and the wheel W

_{1}. This means when the B

_{1}engagement makes the input IN rotate faster than the wheel W

_{1}, then the B

_{2}engagement should make IN rotate slower than W

_{1}. According to Table 1, configurations 1, 2, 4, and 6 satisfy this requirement and can be developed as a TVD. Since the gear ratio of the Ravigneaux gearset in configuration 2 is the most practical in realistic applications, configuration 2 is considered to be the most feasible configuration for a Rav-TVD, and was further investigated in this study.

_{1}and one of the differential shafts. The small sun gear and the carrier of the Ravigneaux gearset are connected to brakes individually so that different torque vectoring effects can be controlled by the engagement of the brakes.

_{r}, ω

_{c}, ω

_{sl}, and ω

_{ss}are the rotating speed of the ring gear, carrier, large sun gear, and small sun gear, respectively. i

_{ss}is the gear ratio between the ring gear and the small sun gear, as presented in Equation (3), and i

_{sl}is the gear ratio between the ring gear and the large sun gear, as shown in Equation (4), where Z

_{r}, Z

_{ss}and Z

_{sl}are the number of teeth on the ring gear, small sun gear, and large sun gear respectively. The constraint of i

_{ss}and i

_{sl}is indicated in Equation (5).

_{1}is engaged, a braking force will be applied to the small sun gear and reduce its speed; according to the lever diagram, this operation results in a trend such that the speed of the large sun gear (wheel W

_{1}) will be faster than the ring gear (Input IN), and hence the wheel W

_{1}will rotate faster than the wheel W

_{2}. On the other hand, when the brake B

_{2}is engaged, a braking force will be applied to the carrier, and the speed of the large sun gear (wheel W

_{1}) will be slower than the ring gear (Input IN). Thus, the wheel W

_{1}will rotate slower than the wheel W

_{2}. With the opposite speed trends between the two wheels controlled by the two brakes, the torque vectoring effect can be realized by the Rav-TVD.

## 4. Modeling

_{1}and F

_{2}respectively are the internal forces between the ring gear and the planet gears, and between the planet gears and the large and small sun gears. To establish the Rav-TVD model, parameters in Equation (6) were replaced with the parameters of the Rav-TVD system according to the connection relation introduced in the previous section. The dynamic model of the Rav-TVD is shown in Equation (7).

_{in}and the two wheels T

_{w}

_{1}and T

_{w}

_{2}are set to be constant. The braking torque T

_{b}

_{1}or T

_{b}

_{2}are initially zero and will be applied to the system at 1–2 s. The speed of the input shaft, the two wheels and the two brake discs were recorded. The parameters of this simulation are arranged in Table 2.

_{1}is engaged at 1–2 s and B

_{2}is not activated. When B

_{1}is engaging, the wheel speed of W

_{1}becomes faster than W

_{2}and the speed difference constantly increases when B

_{1}is activated. Figure 5b shows the case when B

_{2}is engaged at 1–2 s when B

_{1}is not activated. In this case, an opposite situation happens for the two wheels’ speeds, and W

_{1}becomes slower than W

_{2}. According to these results, the proposed Rav-TVD can control the speed ratio between the two wheels symmetrically with different engagement of the brakes, and the design requirement of a TVD mentioned previously is fulfilled.

## 5. Simulation Results

_{1}will start to be engaged at 0.2 s, and (c) the clutch C

_{2}will start to be engaged at 0.2 s, where the speed of the vehicle is still maintained at 60 km/h over the whole process, and the simulation results including the traction of tires and the trajectory of the vehicle are recorded and illustrated in Figure 9. As shown in the simulated results, there was no turning maneuver observed when no clutch was engaged (Figure 9a). In the case of C

_{1}engaged (Figure 9b), the traction of the inside (rear left) tire was transferred to the outside (rear right) tire, causing a torque moment to the vehicle body, and the vehicle began to turn left. Conversely, in the case of C

_{2}engaged (Figure 9c), a similar phenomenon occurred but in the opposite direction, and the vehicle began to turn right. According to this simulation, the torque vectoring effect of the SPC-TVD was demonstrated, and the effect of the C

_{1}and C

_{2}engagement was shown to be opposite to the vehicle behavior (Figure 9d). Therefore, this effect could also be observed for the vehicle during cornering, while a 5-degree constant steering angle to the left was applied to the front wheels, where the curves of NA, C

_{1}, and C

_{2}engaged cases shown in Figure 9e were denoted as neutral-steering (NS), over-steering (OS), and under-steering (US).

_{1}and B

_{2}). The scenario was the same as in the SPC-TVD simulation, under the same vehicle speed (60 km/h) without steering for the whole process, and the three cases were: (a) no brake in the Rav-TVD will be engaged, (b) the brake B

_{1}will start to be engaged at 0.2 s, and (c) the brake B

_{2}will start to be engaged at 0.2 s. Figure 11 shows the simulated results for the Rav-TVD model that also had the ability to adjust the torque distributions during cornering. Therefore, the vehicle during a NS condition could be shifted to a US or OS condition by generating a torque vectoring effect to conquer or facilitate the steering maneuver. Finally, four vehicle dynamic models (SA, OD, SPC-TVD, and Rav-TVD) were put together for comparison of cornering ability, as shown in Figure 12. All of the vehicles could be observed clearly from the top view of the 3D illustration, and the trajectories showed that the SPC-TVD with the smallest turning radius had the best performance. Although the cornering ability of Rav-TVD was not as good as the SPC-TVD, however, it was still better than the SA and OD.

## 6. Analysis of the Numerical Simulation

_{sl}and i

_{ss}were changed with a variation (±10%) from the benchmark value 1.5 and 2.0. With the same scenario in previous section (for the case of steering angle = 0 degree), the simulated results are shown in Figure 13. Also, for the Max. press-on force of the brakes B

_{1}and B

_{2}, the values were changed with a variation (±10%) from the benchmark value 7200 N, and the simulated results are shown in Figure 14. The results at t = 5 s among these parameters are collected and listed in Table 3, and i

_{ss}had the largest increment and decrement on the y-direction displacement of the vehicle trajectory. As a result, adjusting the gear ratio of ring gear and small sun gear was the most efficient way to change the torque vectoring effect.

_{sl}and i

_{ss}) directly affected the characteristics of the torque vectoring effect. For example, after selecting the size of the ring gear, a gearset combined by a larger large sun gear (smaller i

_{sl}) and a smaller small sun gear (larger i

_{ss}) had more torque vectoring power. From the viewpoint of press-on force of the brakes, the force is proportional to the torque applied on brake. Therefore, the larger press-on force, the larger the torque that is transmitted to the other side to make a turn. From the viewpoint of the vehicle trajectory, the radius of curvature is changed from various vehicle speeds and steering angles; the higher the speed or the smaller the steering angle, the lager the radius of curvature of the vehicle trajectory. Moreover, the vehicle dynamics of this model showed vehicle stability, agility, and safety at speeds under 75 km/h, and at steering angles below 5 degrees.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

DYC | direct yaw-moment control |

ESP | electronic stability program |

4WS | four-wheel steering |

TVD | torque vectoring differentials |

DG | differential gearset |

SPC-TVD | superposition clutch TVD |

STC-TVD | stationary clutch TVD |

Rav-TVD | Ravigneaux TVD |

FPG | function power graph |

DoF | degree-of-freedom |

SA | solid axle |

OD | open differential |

P controller | Proportional controller |

NA | no actuation |

CAE | computer-aided engineering |

CAD | computer aided design |

3D | three-dimensional |

RL | rear left |

RR | rear right |

NS | neutral-steering |

OS | over-steering |

US | under-steering |

Subscripts | |

W_{1}, W_{2} | left and right wheels |

C_{1}, C_{2} | clutches |

B_{1}, B_{2} | brakes |

G, G_{0}, G_{1}, G_{2} | gear pairs |

PG_{1}, PG_{2} | planetary gearsets |

IN | input of the engine power |

r | ring gear |

c | carrier |

ss | small sun gear |

sl | large sun gear |

## Appendix A

Parameters | Value |
---|---|

1. Dimensions of the original Rav-TVD | |

Radii of ring gear R_{r} | 60 mm |

Radii of large sun gear R_{sl} | 40 mm |

Radii of small sun gear R_{ss} | 30 mm |

2. Dimensions of the original vehicle | |

Mass | 1200 kg |

Wheel radius (unloaded) | 310 mm |

Tire width | 210 mm |

Wheel base | 2.6 m |

Truck width | 1.8 m |

Inertia of the wheel | 0.3381 kg·m^{2} |

Inertia of the powertrain | 1 kg·m^{2} |

Rolling resistance coefficient of the tire | 0.01 |

Aerodynamic drag coefficient C_{d} | 0.3 |

Air density ρ | 1.204 kg/m³ |

Frontal area of the vehicle A | 1.5 m² |

3. Setting of the original system parameters | |

Gear ratio of ring gear and large sun gear i_{sl} (R_{r}/R_{sl}) | 1.5 |

Gear ratio of ring gear and small sun gear i_{ss} (R_{r}/R_{ss}) | 2.0 |

Max. press-on force of the brakes B_{1} & B_{2} (Corresponding braking torque) | 7200 N (230 N·m) |

Vehicle speed | 60 km/h |

Steering angle (starts at t = 0.2 s) | 5 degree |

## References

- Shibahata, Y.; Shimada, K.; Tomari, T. Improvement of vehicle maneuverability by direct yaw moment control. Veh. Syst. Dyn.
**1993**, 22, 465–481. [Google Scholar] [CrossRef] - Geng, C.; Mostefai, L.; Denaï, M.; Hori, Y. Direct yaw-moment control of an in-wheel-motored electric vehicle based on body slip angle fuzzy observer. IEEE Trans. Ind. Electron.
**2009**, 56, 1411–1419. [Google Scholar] [CrossRef] [Green Version] - Yim, S.; Park, Y.; Yi, K. Design of active suspension and electronic stability program for rollover prevention. Int. J. Automot. Technol.
**2010**, 11, 147–153. [Google Scholar] [CrossRef] - Furukawa, Y.; Yuhara, N.; Sano, S.; Takeda, H.; Matsushita, Y. A review of four-wheel steering studies from the viewpoint of vehicle dynamics and control. Veh. Syst. Dyn.
**1989**, 18, 151–186. [Google Scholar] [CrossRef] - Shibahata, Y. Torque Distributing Mechanism in Differential. U.S. Patent No. 5387161, 7 February 1995. [Google Scholar]
- Richardson, J.A. Speed Reduction Gearset and Torque Split Differential Mechanism. U.S. Patent No. 5,643,129, 1 January 1997. [Google Scholar]
- Gumpoltsberger, G.; Baasch, D. Transmission for Distributing a Drive Torque. U.S. Patent No. 7,056,252, 6 June 2006. [Google Scholar]
- Gradu, M. Differential with Torque Vectoring Capabilities. U.S. Patent No. 7,238,140, 3 July 2007. [Google Scholar]
- Platt, W. Continuously Variable Torque Vectoring Axle Assembly. U.S. Patent No. 7,951,035, 31 May 2011. [Google Scholar]
- Wheals, J.C.; Baker, H.; Ramsey, K.; Turner, W. Torque vectoring AWD driveline: Design, simulation, capabilities and control (No. 2004-01-0863). SAE Tech. Pap.
**2004**. [Google Scholar] [CrossRef] - Kakalis, L.; Cheli, F.; Sabbioni, E. The Development of a Brake based Torque Vectoring System for a Sport Vehicle Performance Improvement. In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics, Intelligent Control Systems and Optimization, Milan, Italy, 2–5 July 2009; pp. 298–304. [Google Scholar]
- Lin, C.; Xu, Z. Wheel torque distribution of four-wheel-drive electric vehicles based on multi-objective optimization. Energies
**2015**, 8, 3815–3831. [Google Scholar] [CrossRef] - De Pinto, S.; Camocardi, P.; Sorniotti, A.; Gruber, P.; Perlo, P.; Viotto, F. Torque-Fill Control and Energy Management for a Four-Wheel-Drive Electric Vehicle Layout With Two-Speed Transmissions. IEEE Trans. Ind. Appl.
**2017**, 53, 447–458. [Google Scholar] [CrossRef] - De Novellis, L.; Sorniotti, A.; Gruber, P.; Shead, L.; Ivanov, V.; Hoepping, K. Torque vectoring for electric vehicles with individually controlled motors: State-of-the-art and future developments. In Proceedings of the 26th Electric Vehicle Symposium, Los Angeles, CA, USA, 6–9 May 2012. [Google Scholar]
- Deur, J.; Ivanović, V.; Hancock, M.; Assadian, F. Modeling and analysis of active differential dynamics. J. Dyn. Syst. Meas. Control
**2010**, 132, 061501. [Google Scholar] [CrossRef] - Chen, I.M.; Yang, T.H.; Liu, T. Function Power Graph A Novel Methodology for Powertrain and Hybrid System Conceptual Design and Analysis. In Proceedings of the 14th IFToMM World Congress, Taipei, Taiwan, 25–30 October 2015; pp. 544–552. [Google Scholar]
- Wang, C.; Zhao, Z.; Zhang, T.; Dai, X.; Yuan, X. Development of a compact compound power-split hybrid transmission based on altered Ravigneaux gear set. SAE Tech. Pap.
**2014**. [Google Scholar] [CrossRef] - Zhang, Y.; Ma, X.; Yin, C.; Yuan, S. Development and Simulation of a Type of Four-Shaft ECVT for a Hybrid Electric Vehicle. Energies
**2016**, 9, 141. [Google Scholar] [CrossRef] - Zhao, L.; Zhou, Y.; Zheng, L. Modeling and simulation of AMT clutch actuator based on simulationX. In Proceedings of the CiSE 2009. International Conference on Computational Intelligence and Software Engineering, Wuhan, China, 11–13 December 2009; pp. 1–5. [Google Scholar]
- Chen, L.; Xi, G.; Yin, C.L. Model referenced adaptive control to compensate slip-stick transition during clutch engagement. Int. J. Automot. Technol.
**2011**, 12, 913–920. [Google Scholar] [CrossRef] - Guo, W.; Wang, S.H.; Su, C.G.; Li, W.Y.; Xu, X.Y.; Cui, L.Y. Method for precise controlling of the at shift control system. Int. J. Automot. Technol.
**2014**, 15, 683–698. [Google Scholar] [CrossRef] - Farkas, Z.; Jóri, I.J.; Kerényi, G. The Application and Modelling Possibilities of CVT in Tractor. In Proceedings of the 5th International Conference Multidisciplinary, Baia Mare, Romania, 23–24 May 2003; pp. 145–150. [Google Scholar]
- Ji, J.; Jang, M.J.; Kwon, O.E.; Chai, M.J.; Kim, H.S. Power transmission dynamics in micro and macro slip regions for a metal v-belt continuously variable transmission under external vibrations. Int. J. Automot. Technol.
**2014**, 15, 1119–1128. [Google Scholar] [CrossRef] - Kim, D.M.; Kim, S.C.; Noh, D.K.; Jang, J.S. Jerk phenomenon of the hydrostatic transmission through the experiment and analysis. Int. J. Automot. Technol.
**2015**, 16, 783–790. [Google Scholar] [CrossRef] - Tang, P.; Wang, S.; Liu, Y.; Xu, X. Analysis of the oil pressure rule during the shift process of automatic transmission. In Proceedings of the 2010 Seventh International Conference on Fuzzy Systems and Knowledge Discovery (FSKD), Yantai, China, 10–12 August 2010; Volume 1, pp. 109–113. [Google Scholar]
- Pengxiang, T.; Shuhan, W.; Xiangyang, X.; Wenyong, L.; Lin, S.; Guoru, Z. Notice of Retraction Design of system pressure valve of 8-speed automatic transmission. In Proceedings of the 2010 International Conference on Computer Application and System Modeling (ICCASM), Taiyuan, China, 22–24 October 2010; Volume 4. [Google Scholar]
- Wang, S.H.; Xu, X.Y.; Liu, Y.F.; Dai, Z.K.; Tenberge, P.; Qu, W. Design and dynamic simulation of hydraulic system of a new automatic transmission. J. Cent. South Univ. Technol.
**2009**, 16, 697–701. [Google Scholar] [CrossRef] - Wei, G.; Xiangyang, X.; Yongxin, C.; Yang, Y. Simulation of powertrain and dynamics of automobile based on SimulationX. In Proceedings of the 2011 6th IEEE Conference on Industrial Electronics and Applications (ICIEA), Beijing, China, 21–23 June 2011; pp. 2326–2330. [Google Scholar]
- Dai, Z.; Liu, Y.; Xu, X.; Wang, S. The Application of Multi-domain Physical System Simulation Method in the Study of Automatic Transmissions. In Proceedings of the WCSE’09. WRI World Congress on Software Engineering, Xiamen, China, 19–21 May 2009; Volume 2, pp. 504–508. [Google Scholar]
- Belmon, L.; Yan, J.; Abel, A. Modelling and Simulation of DCT Gearshifting for Real-Time and High-Fidelity Analysis. In Proceedings of the FISITA 2012 World Automotive Congress, Beijing, China, 27–30 November 2013; pp. 399–411. [Google Scholar]
- Li, W.; Abel, A.; Todtermuschke, K.; Zhang, T. Hybrid vehicle power transmission modeling and simulation with simulationX. In Proceedings of the ICMA 2007, International Conference on Mechatronics and Automation, Harbin, China, 5–8 August 2007; pp. 1710–1717. [Google Scholar]
- Ma, X.; Zhang, Y.; Yin, C. Kinematic Study and Mode Analysis of a New 2-Mode Hybrid Transmission. In Proceedings of the FISITA 2012 World Automotive Congress, Beijing, China, 27–30 November 2013; pp. 309–318. [Google Scholar]
- Abel, A.; Adir, A.; Blochwitz, T.; Greenberg, L.; Salman, T. Development and verification of complex hybrid systems using synthesizable monitors. In Proceedings of the Haifa Verification Conference, Haifa, Israel, 5–7 November 2013; pp. 182–198. [Google Scholar]
- Bös, M. Subsystem- and full-vehicle-simulation of mobile machines using Simulation X. In Proceedings of the 15th ITI Symposium, Dresden, Germany, 14–15 November 2012. [Google Scholar]
- Tüschen, T. SIMULATORS–‘auto. mobile-driving simulator’–suspensions design of a wheel-based driving simulator. In Proceedings of the 7th International Munich Chassis Symposium 2016, Munich, Germany, 14–15 June 2016; Springer Fachmedien Wiesbaden: Wiesbaden, Germany, 2017; pp. 411–434. [Google Scholar]

**Figure 1.**Schematic diagrams of current torque vectoring differential (TVD) designs: (

**a**) superposition-clutch (SPC)-TVD and (

**b**) stationary-clutch (STC)-TVD [15].

**Figure 7.**The influence to vehicle turning of the solid axle (SA) and OD models, (

**a**) tire traction; (

**b**) tire normal force; (

**c**) tire speed; (

**d**) vehicle trajectory.

**Figure 9.**Torque vectoring effect and influence of the vehicle turning of the SPC-TVD model, (

**a**) no engagement; (

**b**) C

_{1}is engaged; (

**c**) C

_{2}is engaged; (

**d**) without steering; (

**e**) with a 5-degree constant steering angle.

**Figure 11.**Torque vectoring effect and influence of vehicle turning of the Rav-TVD model, (

**a**) no engagement; (

**b**) B

_{1}is engaged; (

**c**) B

_{2}is engaged; (

**d**) without steering; (

**e**) with a 5-degree constant steering angle.

**Figure 13.**Sensitivity analysis of the Rav-TVD model for different gear ratios: (

**a**) i

_{sl}and (

**b**) i

_{ss}.

**Figure 14.**Sensitivity analysis of the Rav-TVD model for different Max. press-on forces: (

**a**) brake B

_{1}and (

**b**) brake B

_{2}.

**Figure 15.**Vehicle dynamics of the Rav-TVD model for different vehicle speeds while (

**a**) B

_{1}is engaged and (

**b**) B

_{2}is engaged.

**Figure 16.**Vehicle dynamics of the Rav-TVD model for different steering angle: (

**a**) 3 degrees and (

**b**) 1 degree.

No. | Connected Units | Effect of the Two Brakes to the Speed Ratio between IN-W_{1} Shafts | |||
---|---|---|---|---|---|

IN | W_{1} | B_{1} | B_{2} | ||

1 | r | c | sl | ss | ∆ |

2 * | r | sl | ss | c | ∆ |

3 | r | ss | c | sl | ● |

4 | c | r | sl | ss | ∆ |

5 | c | sl | r | ss | ● |

6 | c | ss | r | sl | ∆ |

7 | sl | r | c | ss | ● |

8 | sl | c | r | ss | ● |

9 | sl | ss | r | c | ● |

10 | ss | r | c | sl | ● |

11 | ss | c | r | sl | ● |

12 | ss | sl | r | c | ● |

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

Inertia of the input shaft I_{in} | 0.18 (kg·m^{2}) |

Inertia of the wheels I_{w}_{1} and I_{w}_{2} | 2.70 (kg·m^{2}) |

Inertia of the brakes I_{b}_{1} and I_{b}_{2} | 0.01 (kg·m^{2}) |

Radii of the ring gear R_{r} | 60 (mm) |

Radii of the large sun gear R_{sl} | 40 (mm) |

Radii of the small sun gear R_{ss} | 30 (mm) |

Torque on the input shaft T_{in} | 20 (N·m) |

Torque on the wheels T_{w}_{1} and T_{w}_{2} | −10 (N·m) |

Torque applied to the brakes T_{b}_{1} and T_{b}_{2} | −5 (N·m) |

Parameter | Variation | Value | Result (y-Axis at t = 5 s) | Difference |
---|---|---|---|---|

Gear ratio of ring gear and large sun gear i_{sl} | +10% | 1.65 | 1.910 m | −5.45% |

0 | 1.5 | 2.020 m | 0% | |

−10% | 1.35 | 2.144 m | +6.14% | |

Gear ratio of ring gear and small sun gear i_{ss} | +10% | 2.2 | −1.642 m | +19.35% |

0 | 2 | −2.036 m | 0% | |

−10% | 1.8 | −2.432 m | −19.45% | |

Max. press-on force of the brake B_{1} | +10% | 7920 N | 2.213 m | +9.55% |

0 | 7200 N | 2.020 m | 0% | |

−10% | 6480 N | 1.826 m | −9.60% | |

Max. press-on force of the brake B_{2} | +10% | 7920 N | −2.233 m | +9.68% |

0 | 7200 N | −2.036 m | 0% | |

−10% | 6480 N | −1.840 m | −9.63% |

Parameter | Value | The Radius of Curvature of the Vehicle Trajectory | ||
---|---|---|---|---|

No Brakes Engaged | Brake B_{1} Is Engaged | Brake B_{2} Is Engaged | ||

Vehicle speed | 30 km/h | - | 1273 m | 1268 m |

60 km/h | - | 1450 m | 1436 m | |

75 km/h | - | 1589 m | 1570 m | |

90 km/h | - | × | × | |

Steering angle | 1 degree | 167.1 m | 149.7 m | 188.9 m |

3 degree | 56.4 m | 53.7 m | 59.0 m | |

5 degree | 37.8 m | 36.7 m | 38.6 m | |

7 degree | × | × | × |

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## Share and Cite

**MDPI and ACS Style**

Chen, Y.-F.; Chen, I.-M.; Chang, J.; Liu, T.
Design and Analysis of a New Torque Vectoring System with a Ravigneaux Gearset for Vehicle Applications. *Energies* **2017**, *10*, 2157.
https://doi.org/10.3390/en10122157

**AMA Style**

Chen Y-F, Chen I-M, Chang J, Liu T.
Design and Analysis of a New Torque Vectoring System with a Ravigneaux Gearset for Vehicle Applications. *Energies*. 2017; 10(12):2157.
https://doi.org/10.3390/en10122157

**Chicago/Turabian Style**

Chen, Yu-Fan, I-Ming Chen, Joshua Chang, and Tyng Liu.
2017. "Design and Analysis of a New Torque Vectoring System with a Ravigneaux Gearset for Vehicle Applications" *Energies* 10, no. 12: 2157.
https://doi.org/10.3390/en10122157