Design Principles and Traction Performance of a Novel Zero-spin Rolling Conical Traction Continuously Variable Transmission

 Abstract: In the existing traction continuously variable transmission (CVT), half toroidal CVT (HT-CVT) is considered to have a better traction performance. However, the HT-CVT has the spin losses due to its structural limitations, which significantly influences the traction efficiency. In this paper, the kinematic qualities and contact models of a novel zero-spin rolling conical CVT (RC-CVT) are studied, and then the rollers and conical disks are compactly designed through the proposed design principles. Subsequently, the transmission efficiency is investigated by using a detailed numerical model and compared with HT-CVT. Based on these computational models and parameters, the practical spin ratio, spin momentum and traction efficiency of RC-CVT are calculated and compared with HT-CVT. The results show that the practical spin ratio and spin momentum of RC-CVT are much smaller than that of HT-CVT, and the efficiency on fixed transmission ratio is consequently higher than that of HT-CVT.


Introduction
Traction continuously variable transmission(CVT) is an ideal mechanical transmission device to reduce vehicle emissions, improve the efficiency of the vehicle, and achieve environmental protection [1,2] , because the CVT can provide an infinite number of transmission ratios to allow the engine to operate closer to its optimal efficiency line [3][4][5] . The main principle of power transmitting of traction CVT is shown in Figure 1. The oil film is created in the contact ellipse between drive component and driven component, and the oil film can become highly viscous (known as elastohydrodynamic state) under the high normal force F to ensure the transmitting of a tangential force, which is also called traction force F . Generally, the traction force F can be calculated approximatively by the equation F = F , where the traction coefficient is determined by the rheological characteristics of given oil [6,7] .

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For a traction CVT, there are three main power losses existing in the elastohydrodynamic contact, namely, creep, sideslip and spin. Spin loss is a rotational slip element of the power transmitting contacts, which is determined by a geometrical relation of the input and output rotating components [6,8] . A study [6] has described this relationship of spin momentum and geometrical relation, as shown in Table 1. It is obvious that the rotational axes of two rotating components and the tangential line of the power transmitting contacts are parallel or intersecting at one point, spin is eliminated. Therefore, this special condition is always called zero-spin condition. A large number of studies [9][10][11][12] have shown that spin losses affect the traction performance of t-CVT significantly. Therefore, decreasing the spin losses can improve the reliability and fatigue life [13] , increase the power density and torque capacity [7,14] , and greatly improve the traction efficiency of traction CVT [15][16][17] .
Therefore, how to eliminate spin losses in traction CVT has become a hot research topic in this field. Typically, a kind of Shaft Drive CVT(SD-CVT) was proposed by Narita et al. [18] , where the spin was greatly decreased and the efficiency of SD-CVT was improved. Then, a novel double roller Full-Toroidal CVT(FT-CVT) was designed by Novellis et al. [17] , and this type of CVT can also significantly decrease the spin losses. However, the most representative research was the zero-spin design methodology developed by Li et al. [15,16] . The core idea of this methodology was that the original working surface of drive or driven components were redesigned to satisfy the zero-spin condition mentioned above. According to this method, the Logarithmic CVT(L-CVT) was firstly obtained by redesigning the original working surface of toroidal disk in HT-CVT (Li et al., 2015a;Li et al., 2015b). Sequentially, the integral CVT and Lambert CVT were obtained through the zero-spin design of FU-type CVT and tapered-roller and ring-disk CVT(TRCVT) (Li et al., 2017). Recently, a kind of conical CVT was also acquired by redesign the roller disk type of CVT according to the zero-spin methodology, then the Swing Conical CVT (SRC-CVT) and Rolling Conical CVT (RC-CVT) were proposed, and the kinematics and efficiency of SRC-CVT was studied [19] .
In this paper, the study of design principles and traction performance of RC-CVT is conducted. As for RC-CVT, the conical disks are used as intermediate transmission elements, the rollers are used as input and output elements. To ensure zero-spin transmission characteristic, the limiting mechanism is used to restrict the vertex of conical disks on the fixed-point. At first, the specific structure and working principle of RC-CVT are described in detail. Afterwards, the kinematic qualities, including creep and spin, and contact models of RC-CVT are developed, then the efficiency computational model is carried out. Based on these computational models and parameters, the traction efficiency of RC-CVT are calculated. The results show that the RC-CVT can eliminate the spin loss theoretically, and the efficiency on fixed transmission ratio is consequently higher than that of HT-CVT.

Geometric description
As shown in Figure 2, this novel RC-CVT is composed of the input roller, the power transmitting conical disks and the output roller. The loading cam is a mechanism that utilizes balls constrained in wedges to create axial load, similar to most HT-CVT [9] . To ensure zero-spin transmission characteristic, the limiting mechanism is used to restrict the vertexes of conical disks on the fixed-point O. For this RC-CVT, the working principle is that the conical disks are driven to rotate around point O, as shown with the hollow arrow, then the input roller and output roller can be consequently driven to adaptively move along the direction of solid arrow by the loading cam mechanism, thus the transmission ratio is changed.

Figure 2 Geometrics of RC-CVT
To simplify the computation model, the geometric radii of output roller and input roller are set the same as 0 , which also means the cavity radius of RC-CVT.
Similarly, the output roller as: The rolling radii of conical disk at the input and output contacts are: From the figure 2, the following relationships can also be acquired: Define structural parameter as the ratio of the top part height of conical disk and the cavity radius 0 . Hence, can be calculated by: The theoretical transmission ratio is an important geometric parameter, which is defined as the ratio of input velocity and output velocity. For RC-CVT, it can be expressed as follows:

Creep and spin
The practical transmission ratio is different from the theoretical transmission ratio because of creep, which is the velocity difference on the contacting surfaces in the primary direction of traction. Generally, the creep is inevitable for producing the tangential force for traction, which can be represented by the creep coefficient. For input contact and output contacts, the creep coefficients are respectively defined as: Hence, the | 2 | | 1 | can be derived from the equation (11), it follows, Similarly, the Therefore, the practical speed ratio can be defined and represented as: Further, the spin velocities of input point and output contact are described as follows: At the same time, the corresponding spin ratios are defined as: σ 21 = 21 | 1 | and σ 23 = 23 | 3 | . Therefore, Substituting the (13) and (18), From the above equations (20) and (21), it is obvious that the spin ratios are zero when creep coefficients are not considered. So, the RC-CVT can eliminate the spin losses theoretically. But the practical spin ratio is not zero because of non-zero creep during the power transmitting.

Contact ellipse
According to the Hertz contact theory, when there is a normal loading force between a roller and a conical disk, the contact point should be an ellipse. The direction and dimension of the ellipse are related to the normal force of the contact point, the material properties and structural dimensions of contacted components.
For RC-CVT, the most critical problems for contact model is to determine the principal curvature radii of the direction of traction and side slip.

Figure 3
The reference frames and principal curvature radii for the rollers and conical disks The contact between the roller and conical disks, and the reference frames are shown in figure 3. The equivalent curvature radii of curvature along the x direction (traction direction) and y direction (side slip direction) are: where the subscript a refers to the roller and b to conical disk.
Then the equivalent radius of curvature is defined as For the input roller, and can be represented as, For the conical disk, the r and r are determined as: Then, the two equivalent curvature radii at the input contact along the x and y directions can be determined respectively: Similarly, the two principal curvature radii of output roller and conical disk at output contact can be determined as follows: Therefore, the equivalent curvature radii at the output contact along the x and y directions can be determined respectively: On this basis, the equivalent radii of curvature at input and output contact are then obtained, It is useful to define the dimensionless form of these equivalent radii of curvature as ρ = ρ 0 ⁄ , which has also described detailly in Carbone's article (2004).
According to the Hertz contact theory, the contact stress in the contact area can be expressed as follows: The maximum Hertz contact stress p is: where a is the long axis of the contact ellipse, and b is the short axis, and F is the normal force at contact point.
In the researches of elliptical contact at isothermal EHL state by Hamrock and Dowson [20,21] the iterative algorithm proposed by Harris [22] was used to determine the parameter a and b with higher precision, as follows: where ρ is the equivalent curvature radius, it can be determined by the ρ and ρ , or, ρ and ρ . While ′ is the equivalent elastic modulus, it can be determined by the following equation: According to the method by Carbone [23] , we define a characteristic length, The dimensionless semi-axis of contact ellipse can be described as,

Design principles
To compare with HT-CVT, the 0 and are given as the same as the research by Carbone [23] . Then the conical disks and rollers can be compactly designed through the following steps.

Figure 4 The design flowchart of RC-CVT
In this paper, the following relationship is assumed to simplify the design process: Step 1: The given parameters are 0 = 40 , = /3, and the transmission ratio range is 0.5~2. According to the equation (10), the range of can be obtained, which is -0.19 rad -0.19 rad.
Step 2: According to the Hertz contact theory described in section 4, the relationship of i and p at the situation of different 1 can be acquired, as shown in figure 5. It is found that d=2 is a better choice to make the maximum Hertz contact stress p at a relatively small level, which is less than 3Gpa, similar to HT-CVT.
Then, the roller can be designed, as shown in the following figure 6, where the theoretical working curve of roller is 1 and 2 . So, we can determine one segment from this curve to design the final roller. For example, 1 1 2 2 forms a roller, and the . represents the rotation axis.  Step 3: When the conical disk is tilted to extreme position, that means the tilting angle of conical disk has reached its maximum γ , the no separation relationships of rollers and conical disk should be satisfies.
Based on the relationship in geometric description, the following can be acquired, Hence, ≤ 27.0976 mm, ≥ 94.1271mm can be acquired. To design a compact conical disk, the should take the maximum value and takes the minimum. Considering a safety margin, the and can be determined. Then, the and ℎ can be obtained based on equation (8).
Step 4: For RC-CVT, the speed changing motion of conical disks is rotation around point O, which can be realized by lots of mechanisms actually. In this paper, a simple shifting mechanism [24] is used, where the shifting gears can be driven by a couple of synchronous motors.
Finally, the RC-CVT can be designed. However, this design principle just provides a simple and general method, which may be further refined when conduct a specific design of variable RC-CVTs.

Traction efficiency
In terms of efficiency calculation, Carbone [23] has given a more accurate computational method, which has been widely used especially in the efficiency calculation of toroidal CVT [15-17, 25, 26] . In this method, the thickness of the oil film is explained with Hamrock and Dowson formulas [27] , the oil temperature is assumed at the flash point and pressure distribution in the contact area obeys the Hertz contact theory. The rheological behavior of traction fluid is described using the Bair and Winer Non-Newtonian model [28] , and the influence of the pressure on the fluid viscosity is computed with Roelands model [28] , which is more suitable for the situation of high pressure in the contact area. This computational method is also developed in this paper to calculate the efficiency of RC-CVT.
The traction efficiency of RC-CVT is defined as follows: where 1 − = (1 − )(1 − ) , and are the input and output creep coefficients. Additionally, and are the input and output traction coefficients, and are the spin momentum coefficients, these four parameters can be calculated by the numerical integration in the region of contact area, which are all described in detail by Carbone [23] .
According to the aforementioned calculation models, the traction performance is analysed, where the geometrical quantities are reported in Table 2. To compare with HT-CVT, the geometrical quantities and fluid properties are set as similar as that of HT-CVT by Carbone [23] . As developed in section 5, the maximum Hertz contact stress p has been optimized to less than 3Gpa by compactly designing the roller. Then the Hertz contact stress of RC-CVT is similar to HT-CVT, which is also the equality condition to conduct the comparation.
For the designed RC-CVT, the parameter reflects the structural compactness. In fact, the also influences the maximal tilting angle of conical disk γ , which further determines the transmission ratio range. According to the compactly designed roller and conical disk, the structural parameter can be determined as 0.3175. Subsequently, the practical spin ratio of RC-CVT is calculated and compared with HT-CVT, the results are shown in Figure 8. It can be seen that the practical spin ratio of RC-CVT is not zero when creep coefficients are considered, but the value is far less than that of HT-CVT, which is close to zero. Therefore, the RC-CVT is theoretically eliminate the spin losses.
(a) (b) Figure 8 The practical spin ratio comparation between HT-CVT and RC-CVT: (a) The practical spin ratio at input contact; (b) The practical spin ratio at output contact From the equations (61), it is obvious that the spin momentum coefficients is mainly determined by the spin moment . The input torque coefficient is the dimensionless form of , which represents the input torque. The relationship between the input spin momentum coefficients and the input torque coefficient of the HT-CVT and RC-CVT are shown in Figure 9. From this figure, it is easy to find that the spin moment of the RC-CVT is greatly reduced compared with the HT-CVT. But the fact is that the practical are not zero, because of the larger , the greater (see in figure 15), which could cause the additional spin losses. Additionally, the situation of output contact is similar to this of input contact.

Figure 9
The spin moment as a function of input traction coefficient Moreover, the efficiency of the RC-CVT and HT-CVT as a function of the input torque coefficient are shown in Figure 10. It is apparent that the efficiency of the RC-CVT is higher than that of the HT-CVT, especially when the input torque coefficient is at the smaller stage. This is indeed the advantage of eliminating spin losses. It is worth pointed out that the efficiency rapidly drops as approaches the limited value, because the creep coefficient fast increase at this situation (see in figure  15).

Figure 10
The efficiency of the CVTs as a function of the input traction coefficient The figure 11 gives the efficiency as a function of the overall creep coefficient . Compare with HT-CVT, the efficiency of RC-CVT is much higher than that of HT-CVT when is smaller than 0.03, the reason is that the traction losses are dominated by spin at this situation, but the spin of RC-CVT is eliminated theoretically. However, as further increases, the creep losses become the dominator, which is considered much greater than spin, the traction efficiency therefore linearly drops to the similar level of these two CVTs. It should be pointed out that the same input torque coefficient would cause the different creep for this two CVTs. In fact, the creep coefficient of HT-CVT is more sensitive to , which means a small input torque of HT-CVT may cause a relatively larger creep compared with RC-CVT. In particular, as shown in figure 11, when increases at a smaller stage, the of HT-CVT increases obviously faster than RC-CVT. Only when the increases to greater than 0.085, the of RC-CVT increases rapidly, and the HT-CVT has a similar feature.

Figure 11
The efficiency of the CVTs as a function of the creep coefficient Figure 12 The relationship between and of HT-CVT and RC-CVT Totally, the RC-CVT has a higher traction efficiency on fixed transmission ratio than that of HT-CVT, especially when or is relatively small. Therefore, the RC-CVT is considered to have the potential to be a new generation of traction CVT with higher efficiency.

Conclusions
In this paper, the design principles and traction performance are studied. The following conclusions can be drawn: (1) The kinematics and contact model of RC-CVT are studied, a design flowchart of RC-CVT is given. By the proposed design principles, the roller and conical disk, as well as the shifting mechanism are designed. Especially, the edge radius of roller significantly influences the maximum Hertz contact stress, and the recommended of designed RC-CVT is = 2 0 .
(2) The RC-CVT just theoretically eliminates the spin losses. When the creep coefficients are considered, the practical spin ratio of RC-CVT is not zero. However, the spin moment is far less than that of HT-CVT, reduced by more than 90% at the input contact.
(3) The RC-CVT has been proved to have relatively higher traction efficiency on fixed transmission ratio than that of HT-CVT, especially when the and are at a small stage.
Although the traction efficiency performance of RC-CVT is studied, it is just the efficiency calculation on fixed transmission ratio without considering the shifting loss. Therefore, the overall efficiency including the shifting loss of the dynamic speed changing process and the experimental verification are the next important research work. In addition, the influence of different parameters to traction efficiency has not been discussed completely, which also deserves to be studied in the future.

Availability of data and materials
The datasets supporting the conclusions of this article are included within the article. Figure 1 The main principle of power transmitting of t-CVT The reference frames and principal curvature radii for the rollers and conical disks The design owchart of RC-CVT Figure 5 The design owchart of RC-CVT Figure 6 The designed curve of roller for RC-CVT The compact design of roller for RC-CVT The practical spin ratio comparation between HT-CVT and RC-CVT: (a) The practical spin ratio at input contact; (b) The practical spin ratio at output contact The spin moment χ_in as a function of input traction coe cient t_in Figure 10 The e ciency of the CVTs as a function of the input traction coe cient t_in Figure 11 The e ciency of the CVTs as a function of the creep coe cient C_r Figure 12 The relationship between t_in and C_r of HT-CVT and RC-CVT