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

Abstract

The purpose of this research is to develop a new torque vectoring differential (TVD) for vehicle applications and investigate its effect on vehicle dynamic control. TVD is a technology that is able to distribute the engine torque to the left and right driving wheels at different ratios so that the yaw motion control can be realized. Attention has been paid to this technology in recent years because of its potential to improve the vehicle performance and driving safety. In this study, a new TVD design with a Ravigneaux gearset was developed. This new design is able to use only one pair of gearsets to generate two different speed ratios, and the weight and volume of the system can be reduced. To execute the research, current TVD designs were analyzed and their design principles were clarified. Next, a new TVD design with Ravigneaux gearset was proposed. Then the connecting manner and the gear ratio of the Ravigneaux gearset were discussed. The dynamic equation of the system was then derived and the operation of the system was simulated in a MATLAB program. Further simulation was performed with a vehicle dynamic model in SimulationX to demonstrate the effect of the new system. The results of this study show the potential of building a new TVD with a Ravigneaux gearset and can be helpful for further system development.

1. Introduction

Vehicle dynamic control is an essential task for improving driving performance and vehicle safety. Different control strategies such as direct yaw-moment control (DYC) have been proposed and well-discussed in the past [1,2]. Various systems, e.g., electronic stability program (ESP) [3], four-wheel steering (4WS) [4] have been developed to realize vehicle dynamic control in practical applications. Torque vectoring is a new technology that has been drawing the attention of the car industry in recent years. It is employed in automobile differentials and is able to distribute different driving torque values to different wheels so that traction distribution can be realized. Because of the operation of transferring the torque instead of diminishing the torque on the wheels, better system efficiency can be achieved. The torque vectoring technology becomes more competitive than the brake-based systems.

Different torque vectoring differential (TVD) patents have been revealed by car companies and suppliers, such as Ford, ZF, and Honda recently [5,6,7,8,9]. Most general designs consist of two pairs of gearsets in different speed ratios, and utilize clutches, brakes, or motors to control the direction of the transferred wheel torque [10,11,12,13,14]. However, the two pairs of gearsets or motors in current designs result in heavier system weight and larger space requirements. In this paper, a new TVD design with Ravigneaux gearset is proposed. This design is able to use only one pair of gearsets to generate two different speed ratios, and the weight and volume of the system can be reduced.

The research process of this study is described as follows. Firstly, the current TVD designs are analyzed and their design principle is understood. A new TVD design with Ravigneaux gearset is proposed. Then the connecting manner and gear ratio of the Ravigneaux gearset is discussed. The dynamic equation of the system is then derived and the operation of the system is simulated in a MATLAB program (The MathWorks, Inc., Natick, MA, USA). Further simulation is performed with vehicle dynamic model in SimulationX (ESI ITI GmbH, Dresden, Germany) to demonstrate the effect of the new system to vehicle dynamics. The results of this study are summarized in the conclusions.

2. Current Torque Vectoring Differential

Two current TVD designs are under investigation in this section to understand the design principles of the systems. The schematic diagrams of the superposition-clutch TVD (SPC-TVD) and the stationary-clutch TVD (STC-TVD) [15] are illustrated in Figure 1, and their system configurations transferred by means of a function power graph (FPG) [16] are shown in Figure 2. In Figure 1 and Figure 2, DG denotes the differential gearset; W₁ and W₂ are the left and right wheels; C₁ and C₂ are clutches; B₁ and B₂ are brakes; G, G₀, G₁ and G₂ are gear pairs; PG₁ and PG₂ are planetary gearsets; IN is the input of the engine power.

Figure 1. Schematic diagrams of current torque vectoring differential (TVD) designs: (a) superposition-clutch (SPC)-TVD and (b) stationary-clutch (STC)-TVD [15].

Figure 2. System configurations of current TVD designs: (a) SPC-TVD and (b) STC-TVD.

In Figure 2, the SPC-TVD consists of two gear pairs (G₁ and G₂), and two clutches (C₁ and C₂). When C₁ is engaged, the speed ratio of the left and right wheels is decided by the gear ratio of G₁. Similarly, when C₂ is engaged, the speed ratio of the two wheels is decided by the gear ratio of G₂. With the different gear ratios of G₁ and G₂, 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.

A similar operation principle is also observed from the STC-TVD design, in which two planetary gearsets PG₁ and PG₂ are involved, and two brakes B₁ and B₂ are used to select the direction of the torque vectoring effect. When B₁ is engaged, the speed ratio of the left and right wheels is decided by the gear ratio of PG₁. On the other hand, when B₂ is engaged, the speed ratio of the two wheels is decided by the gear ratio of PG₂.

According to the two current TVD designs discussed above, a TVD can be developed when two gear ratios between the left and right wheels are achievable, and the two ratios can be selected by controlling the engagement of the clutches or brakes. This is the design principle of the TVD understood by the authors and will be used for further TVD development in the later part of this study.

3. Design of New Torque Vectoring Differential

It is found that the STC-TVD uses two planetary gearsets in order to acquire the two different speed ratios between the two wheels. However, this may cause the system to require more space and have a heavier weight. Since the Ravigneaux gearset can reduce one ring gear compared to the two planetary gearsets design, there is a potential to reduce the space and weight of the STC-TVD design. Therefore, the authors of this study propose a new TVD design which involves a Ravigneaux gearset, and the new system is named Ravigneaux TVD (Rav-TVD) in this research.

Here, the feasible configuration of a Rav-TVD is discussed. The Ravigneaux gearset is a two degrees-of-freedom (DoF) mechanism which has four links, i.e., ring gear, carrier, large sun gear, and small sun gear, to connect to other devices. The twelve possible configurations of a Rav-TVD are listed in Table 1. In order to achieve the torque vectoring effect, the two brakes B₁ and B₂ have different effects on the speed relation between the input IN and the wheel W₁. This means when the B₁ engagement makes the input IN rotate faster than the wheel W₁, then the B₂ engagement should make IN rotate slower than W₁. 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.

Table 1. Possible arrangements of a Ravigneaux TVD (Rav-TVD).

No.	Connected Units				Effect of the Two Brakes to the Speed Ratio between IN-W1 Shafts
	IN	W1	B1	B2
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	●

∆: different effect, ●: same effect, *: selected configuration.

The configuration of the Rav-TVD is shown in Figure 3a and its schematic diagram is shown in Figure 3b. The Ravigneaux gearset is denoted as a square in the sketch of system configuration, and the four corners of the square are marked with the corresponding gears. The r is the ring gear, c is the carrier, ss is the small sun gear, and sl is the large sun gear. The Rav-TVD connects the ring gear to the input and differential carrier. The large sun gear is connected to wheel W₁ 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.

Figure 3. (a) System configuration and (b) Schematic diagram of the Rav-TVD.

The speed relation between each gear in the Ravigneaux gearset can be visualized in the lever diagram as shown in Figure 4, and can be calculated via Equations (1) and (2) [17,18], in which ω_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).

Figure 4. Lever diagram of a Ravigneaux gearset.

$${\omega }_{ss}-i_{ss}{\omega }_r=(1-i_{ss}){\omega }_c,$$

(1)

$${\omega }_{sl}+i_{sl}{\omega }_r=(1+i_{sl}){\omega }_c,$$

(2)

$$i_{ss}=\frac{Z_r}{Z_{ss}},$$

(3)

$$i_{sl}=\frac{Z_r}{Z_{sl}},$$

(4)

$$i_{ss}>i_{sl}>1,$$

(5)

To operate the Rav-TVD, when the brake B₁ 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 W₁) will be faster than the ring gear (Input IN), and hence the wheel W₁ will rotate faster than the wheel W₂. On the other hand, when the brake B₂ is engaged, a braking force will be applied to the carrier, and the speed of the large sun gear (wheel W₁) will be slower than the ring gear (Input IN). Thus, the wheel W₁ will rotate slower than the wheel W₂. 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

The dynamic equation of the Rav-TVD is a combination of the dynamic equations of a Ravigneaux gearset and a conventional differential. The dynamic equation of a Ravigneaux gearset is shown in Equation (6), where I is the inertia, R the radius, $\dot{\omega }$ the rotational acceleration, T the applied torque of each gear in the Ravigneaux gearset, and F₁ and F₂ 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).

To have a preliminary insight of the system, a simulation program was developed based on Equation (7) and written in the MATLAB program to demonstrate the operation of the Rav-TVD. In the simulation, the torque on the input shaft T_in and the two wheels T_w1 and T_w2 are set to be constant. The braking torque T_b1 or T_b2 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.

$$\left[\begin{array}{cccccc}{I}_c& 0& 0& 0& R_{sl}+R_r& R_{ss}-R_r\\ 0& I_r& 0& 0& -R_r& R_r\\ 0& 0& I_{sl}& 0& -R_{sl}& 0\\ 0& 0& 0& I_{ss}& 0& -R_{ss}\\ R_{sl}+R_r& -R_r& -R_{sl}& 0& 0& 0\\ R_{ss}-R_r& R_r& 0& -R_{ss}& 0& 0\end{array}\right]\left[\begin{array}{c}{\dot{\omega }}_c\\ {\dot{\omega }}_r\\ {\dot{\omega }}_{sl}\\ {\dot{\omega }}_{ss}\\ F_1\\ F_2\end{array}\right]=\left[\begin{array}{c}{T}_c\\ T_r\\ T_{sl}\\ T_{ss}\\ 0\\ 0\end{array}\right],$$

(6)

$$\left[\begin{array}{cccccccc}{I}_{in}& 0& 0& 0& 0& 2& R_r& -R_r\\ 0& I_{w1}& 0& 0& 0& -1& 0& -R_{sl}\\ 0& 0& I_{w2}& 0& 0& -1& 0& 0\\ 0& 0& 0& I_{b1}& 0& 0& -R_{ss}& 0\\ 0& 0& 0& 0& I_{b2}& 0& R_{ss}-R_r& R_{sl}+R_r\\ 2& -1& -1& 0& 0& 0& 0& 0\\ R_r& 0& 0& -R_{ss}& R_{ss}-R_r& 0& 0& 0\\ -R_r& -R_{sl}& 0& 0& R_{sl}+R_r& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\dot{\omega }}_{in}\\ {\dot{\omega }}_{w1}\\ {\dot{\omega }}_{w2}\\ {\dot{\omega }}_{b1}\\ {\dot{\omega }}_{b2}\\ F_1\\ F_2\\ F_3\end{array}\right]=\left[\begin{array}{c}{T}_{in}\\ T_{w1}\\ T_{w2}\\ T_{b1}\\ T_{b2}\\ 0\\ 0\\ 0\end{array}\right],$$

(7)

Table 2. Parameters of the MATLAB simulation.

Parameter	Value
Inertia of the input shaft Iin	0.18 (kg·m2)
Inertia of the wheels Iw1 and Iw2	2.70 (kg·m2)
Inertia of the brakes Ib1 and Ib2	0.01 (kg·m2)
Radii of the ring gear Rr	60 (mm)
Radii of the large sun gear Rsl	40 (mm)
Radii of the small sun gear Rss	30 (mm)
Torque on the input shaft Tin	20 (N·m)
Torque on the wheels Tw1 and Tw2	−10 (N·m)
Torque applied to the brakes Tb1 and Tb2	−5 (N·m)

The results of the MATLAB simulation are shown in Figure 5. Figure 5a is the case when B₁ is engaged at 1–2 s and B₂ is not activated. When B₁ is engaging, the wheel speed of W₁ becomes faster than W₂ and the speed difference constantly increases when B₁ is activated. Figure 5b shows the case when B₂ is engaged at 1–2 s when B₁ is not activated. In this case, an opposite situation happens for the two wheels’ speeds, and W₁ becomes slower than W₂. 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.

Figure 5. Operation of the Rav-TVD while (a) B₁ is engaged and (b) B₂ is engaged.

5. Simulation Results

In this section, to demonstrate the torque vectoring effect of TVDs and its influence while turning in vehicle applications, the SPC-TVD and the Rav-TVD systems were built and applied to a vehicle dynamic model that was compared with a solid axle (SA) and an open differential (OD) in the simulation software called SimulationX. SimulationX is software provides a modeling platform and libraries with customizable elements and tools which can be used for system development and analysis, and it is one of the most popular software in the field of multi-physics modeling and simulation. In automotive technology, for examples, transmission elements (e.g., clutches [19,20,21], continuously variable transmissions [22,23]), hydraulic systems [24,25,26,27], powertrain drivelines [28,29,30], and hybrid transmissions [31,32,33] etc. have been surveyed and modeled based on SimulationX. In addition to computer-aided engineering (CAE) analysis, computer aided design (CAD) with three-dimensional (3D) solid modeling for vehicles has also been established [34,35].

Figure 6 shows the interface of SimulationX and a vehicle dynamic model with a steering control for front wheels, and an OD for rear wheels. In the diagram view of our model, connection lines can be categorized into three types: (a) Signal Blocks (blue lines for control signals), (b) 1D Mechanics (black lines for power transmission), and (c) Multi-Body System Mechanics (yellow lines for 3D vehicle that can be visualized in the 3D view). Due to the convenience of the SimulationX-based model, only the transmission part needs to be altered. For example, a vehicle dynamic model for a SA can be established easily by directly removing the differential element. In this simulation, the vehicle dynamics of the same vehicle equipped with different transmissions (SA, OD, SPC-TVD, and Rav-TVD) are investigated, where the vehicle is assumed be to a rear-wheel-drive vehicle. The simulation scenario is that the vehicle is driven, starting from 0 km/h at −6 s (time = −6 s) at first and accelerated to a constant speed at 60 km/h before 0 s (time < 0 s). Then, a P controller (Proportional controller) is used to maintain the vehicle speed at 60 km/h during the entire process (time > 0 s). More detailed numerical data is shown in Table A1 of Appendix A.

Figure 6. The interface and vehicle dynamic model for the open differential (OD) in SimulationX.

Figure 7 shows the traction, normal force, tire speed of rear tires, and the vehicle trajectory for SA and OD cases while a 5-degree constant steering angle to the left is applied to the front wheels at 0.2 s (time = 0.2 s). Both SA and OD models show the normal forces are shifted from the inside (rear left, RL) tire to the outside (rear right, RR) tire due to the centrifugal force while cornering (Figure 7a). The traction of the inside tire is always larger than the outside tire; this causes the difficulty while turning for the SA model, whereas the differential function is achieved such that both rear tires always share the similar traction forces for the OD model (Figure 7b). As a result, the vehicle with an OD will slow down the inside tire and accelerate the outside tire, making a turn easier than the one with an SA (Figure 7c), and it has a smaller turning radius for better cornering ability (Figure 7d).

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.

Next, one of the current TVD designs (SPC-TVD) was selected to understand what the torque vectoring effect is and how this effect influences the vehicle while turning. A vehicle dynamic model of the SPC-TVD shown in Figure 8 was transferred from the system configuration of the SPC-TVD shown in Figure 2a, and the only difference compared to the OD model was the transmission as shown in the diagram view. Except for the open differential, two clutches and two pairs of gearsets were added in the SPC-TVD for torque distributions.

Figure 8. The vehicle dynamic model for the SPC-TVD in SimulationX.

In this simulation, the scenario was similar to the SA and OD models. However, in order to examine the torque vectoring effect, the steering angle was fixed to zero all the time, and three cases were simulated: (a) no actuation (NA) in the SPC-TVD will happen, (b) the clutch C₁ will start to be engaged at 0.2 s, and (c) the clutch C₂ 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 C₁ 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 C₂ 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 C₁ and C₂ 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, C₁, and C₂ engaged cases shown in Figure 9e were denoted as neutral-steering (NS), over-steering (OS), and under-steering (US).

Figure 9. Torque vectoring effect and influence of the vehicle turning of the SPC-TVD model, (a) no engagement; (b) C₁ is engaged; (c) C₂ is engaged; (d) without steering; (e) with a 5-degree constant steering angle.

In Figure 10, the Rav-TVD model was also transferred from the system configuration, as shown in Figure 3a, where the transmission was a differential plus one pair of the Ravigneaux gearset with two brakes (B₁ and B₂). 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 B₁ will start to be engaged at 0.2 s, and (c) the brake B₂ 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.

Figure 10. The vehicle dynamic model for the Rav-TVD in SimulationX.

Figure 11. Torque vectoring effect and influence of vehicle turning of the Rav-TVD model, (a) no engagement; (b) B₁ is engaged; (c) B₂ is engaged; (d) without steering; (e) with a 5-degree constant steering angle.

Figure 12. Comparison of cornering for the SA, OD, SPC-TVD, and Rav-TVD models.

6. Analysis of the Numerical Simulation

In order to verify the soundness of the simulation model and understand the vehicle performance, numerical simulation with different design parameters were demonstrated. The Rav-TVD model with the original design parameters (Table A1) was selected as a benchmark, and then the simulation results with different design parameter values (gear ratios of the Ravigneaux gearset, Max. press-on force of the brakes, vehicle speed, and steering angle) were adjusted to compare. First, for the gear ratios of the Ravigneaux gearset, the values of i_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 B₁ and B₂, 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.

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 B₁ and (b) brake B₂.

Table 3. Sensitivity analysis for the Rav-TVD model.

Parameter	Variation	Value	Result (y-Axis at t = 5 s)	Difference
Gear ratio of ring gear and large sun gear isl	+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 iss	+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 B1	+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 B2	+10%	7920 N	−2.233 m	+9.68%
	0	7200 N	−2.036 m	0%
	−10%	6480 N	−1.840 m	−9.63%

In Figure 15, the original vehicle speed (60 km/h) of the Rav-TVD model was changed to 30 km/h and 75 km/h. The simulation time was also extended to 475 s to make sure that the vehicle trajectory was a circular shape after one of the brakes was engaged at t = 0.2 s. This means the Rav-TVD provided the same torque vectoring effect on vehicle to make a turn. The simulated results showed that the radius of curvature of the vehicle trajectory differed from vehicle speed; the higher vehicle speed, the larger the radius of curvature. On the other hand, the parameter of constant steering angle also varied, and the original steering angle (5 degree) is already shown in Figure 11e. Under the same vehicle speed (60 km/h), different steering angles such as 1 degree and 3 degrees were also simulated, and shown in Figure 16a,b. Finally, Table 4 shows the calculated radius of curvatures from the vehicle trajectories for different conditions. In addition, a higher vehicle speed (90 km/h) and a larger steering angle (7 degree) have also been simulated; however, the vehicles flipped and turned over during torque vectoring due to the larger centrifugal force while cornering. Nevertheless, the vehicle dynamics model still had good stability, agility, and safety for most situations.

Figure 15. Vehicle dynamics of the Rav-TVD model for different vehicle speeds while (a) B₁ is engaged and (b) B₂ is engaged.

Figure 16. Vehicle dynamics of the Rav-TVD model for different steering angle: (a) 3 degrees and (b) 1 degree.

Table 4. Vehicle dynamics of the Rav-TVD model for different conditions.

Parameter	Value	The Radius of Curvature of the Vehicle Trajectory
		No Brakes Engaged	Brake B1 Is Engaged	Brake B2 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	×	×	×

-: vehicle trajectory is flat (no curvature), ×: vehicle flips or turns over (simulation failed).

In this section, four design parameters were selected for a sensitivity analysis and for vehicle dynamics. From the viewpoint of the mechanical dimensions of the Ravigneaux gearset, the gear ratios (i_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

The development of a new vehicle component design and torque vectoring differential is demonstrated in this paper. First, the design principles of current TVDs are clarified, and their system configurations transferred from schematic diagrams are illustrated. Second, the new TVD design, which consists of a Ravigneaux gearset and two brakes, is introduced, and the feasible configuration of the Rav-TVD is discussed. Next, the interface of SimulationX software is introduced, and examples of the solid axle and open differential have been demonstrated. Due to the convenience of the system configuration, SimulationX-based models for the same vehicle equipped with different transmissions (SA, OD, SPC-TVD, and Rav-TVD) are established intuitively and directly. Simulations have been performed to show the operation of the system and the torque vectoring effects of the SPC-TVD and Rav-TVD have been demonstrated. According to the simulation results, both the SPC-TVD and Rav-TVD are able to obtain different traction distributions when different clutches or brakes are engaged, and can be used for DYC for vehicle applications. Furthermore, the cornering ability of four transmissions are simulated and compared, and numerical simulations for the Rav-TVD model have been analyzed.

In summary, our mechanical design, the Rav-TVD, has the potential for torque vectoring ability. The vehicle dynamics can also be improved with the system to fulfil driving requirements such as stability and agility, and better driving safety and performance are considered to be realizable with the new Rav-TVD design for vehicle applications.