# Characteristic Investigation and Torque Distribution Diagram-Based Computational Method for Continuously Variable Power-Split Transmission

Authors: Qingtao Li; Li Liang; Chao Li; Runtao Li; Chongyong Yan; Shengzhen Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The key parameter of power-split transmission is the planetary ratio. To obtain various planetary ratios in one transmission system, people use complex systems with multiple planetary gear trains, clutches, and brakes. Another solution is replacing the planetary gear train with a planetary traction drive. However, the participation of the planetary traction drive brings about creep and spin motion, which increase the difficulty of precise calculation. In this study, we investigate the relationships between torques and speeds. Then, we derive the torque and speed distribution formulas that consider the influence of spin torque and creep motion. This study is the first step to precisely calculate the emerging power-split transmission with the planetary traction drive. Based on a contact model and a proposed torque distribution diagram method, this study proposes a fast computational method for calculating the performance of the new continuously variable power-split transmission. The results show that the proposed torque distribution diagram-based computational method is feasible, and the planetary traction drive can be a competitive power-split device for power-split transmission.


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## Body

## 1. Introduction

Power-split transmission (PST) combines different transmission technologies [1]. It enjoys complementary advantages of different transmission technologies through optimization design and reasonable control strategy [2]. Thus, PST can be applied to severe working conditions that require high efficiency, continuously variable speed ratio, and automatic shifting. As the world’s environmental problem emerges, PST has attracted significant attention from researchers and manufacturers due to its lower fuel consumption and operation cost in hybrid vehicles [3, 4] and urban buses [5]. For off-highway vehicles, such as tractors, construction machinery, and tanks, PST is a competitive solution due to its high-torque capacity, the ability of power shifting, and continuous steering [6–8]. PST can also be applied in the wind turbine [9] to capture maximum energy from wind.To optimize PST configurations, several studies focused on the optimization of the planetary ratiok [10]. A sensitivity analysis study has been conducted by Zaremba et al. [11], indicating that the planetary ratio can affect operating points of the variators and efficiency. Then, people tend to add more planetary gear trains in PSTs. However, multiple planetary gear trains provide the transmission with multiple working modes [12]. The more planetary gear trains are mounted, the more planetary ratios are available. Ince and Guler [6, 13] designed and analyzed a proposed PST with three planetary gear trains offering different planetary ratios. They suggested that the designer can improve the total mechanical efficiency by choosing the appropriate planetary ratio. After comparing PSTs with multiple planetary gears, Zhuang et al. [14] concluded that triple planetary gear trains mounted on PST can improve the output torque and adaptability to SUVs, pickup trucks, or buses. However, this attempt increases the number of clutches and brakes, causing high complexity and high power consumption [15].A new solution to this problem is replacing the planetary gear train with a planetary traction drive (PTD). The new PTD is developed from the traction drive continuously variable transmission (CVT) by releasing the carrier component. Different from gear transmission, traction drive CVT is another branch of technology. The main working mechanism of traction drive CVT is the rolling traction (with limit creep motion) between the components whose shapes result in a nominal point or line contact [16–18]. There are many types of traction drive CVTs [19], such as full-toroidal CVT, half-toroidal CVT [20], NuVinci CVT [21], and Milner CVT [22]. McIndoe et al. [15, 23] investigated a VariGlide CVT that consists of PTD. They obtained its efficiency levels to be competitive with or better than current CVT technologies. Li et al. [18] proposed a half-toroidal PTD and found that the new PST has a higher output torque and a wider speed ratio range.Although the new PST with a continuously variable planetary ratio is considered an excellent candidate for future hybrid configurations [15, 18, 21, 23], people have not proposed a suitable method to precisely investigate the torques and speed distributions. This is because when PTD functions as a power-split device in PST, the torque distribution affects the speed distribution. It means that the speeds, torques, and power flows in the new PST system have a more complex relationship than normal ones. Additionally, all existing computational methods for traction drive CVT [20, 24, 25] do not consider the complex relationship.Therefore, to solve the problem, this study investigates the relationships between rotational speeds and torques in the new PST. Two relationship equations are considered, e.g., the Willis equation in the planetary gear train. We also propose a direct search method for the direction factors in the speed distribution method. Based on a contact model and the proposed torque distribution diagram method, we propose a fast computational method to calculate the performance of the new continuously variable PST.


## 2. The Characteristic of the Half-Toroidal PTD

The heart of PST is a planetary gear train [10, 26], which consists of three main components: sun gear, ring gear, and carrier carrying several planet gears. It has a well-known working mechanism given as follows:(1)ωs+kωr−1+kωc=0,(2)Ts:Tr:Tc=1:k:−1+k,where ωs, ωr, and ωc are the rotational speeds of the sun gear, ring gear, and carrier, respectively; Ts, Tr, and Tc are the torques acting on the sun gear, ring gear, and carrier, respectively; Tc always has a negative value because its direction is opposite to the direction of Ts and Tr; and k is the planetary ratio that is equal to the ratio of the radius of the ring gear to the radius of the sun gear in the planetary gear train.


### 2.1. Geometric Description and Parameters of the New Half-Toroidal PTD

Although two or three types of PTDs can be used as power-split devices, this study takes the half-toroidal PTD as a research object. The analysis process can be applied in other similar situations.As shown in Figure1, two discs form a cavity where trunnions with rollers are located. Trunnions are supported by a carrier mounted with two-axis pins whose center axis coincides with the center of the cavity. With the aid of the axis pins, trunnions with rollers can swing about the center of the cavity. Here, γ is the tilting angle, indicating the swing level of the roller; it is caused by the movement of the outer race. It has a positive value when the roller swings clockwise and a negative value otherwise. The roller contacts discs I and II at contact points I and II, respectively. The perpendicular from point I to the disc will cross the perpendicular from point II at the cavity center (swing center). The two perpendiculars form a cone angle (Figure 1). The half-cone angle is represented by θ. The distances from points I and II to the rotational axis are rI and rII, respectively. The radius of the cavity circle is r0. The discs can input or output power through gears and be supported by the middle shaft. The rollers are supported between the discs and rotated about their axes while revolving about the middle shaft. Rollers and discs I and II constitute a planetary topology where torques are transmitted through traction contacts at contact points. The three-dimensional (3D) geometry model of half-toroidal PTD is shown in Figure 2.Figure 1
Structural sketch of the half-toroidal PTD: (a) color drawing and (b) line drawing.
(a)(b)Figure 2
3D geometry model of half-toroidal PTD: (a) 3/4 section view and (b) the carrier with three rollers.
(a)(b)


### 2.2. Analysis of Forces and Torques

Are equations (1) and (2) suitable? We should doubt it at first. It is well known that the planetary gear train relies on the gear meshing, which can guarantee a fixed speed ratio that is not affected by the transmission torque. The scenario is different for PTD, where traction contact plays a key role (Figure 3). The linear velocities of driving and driven components cannot be equal and will be affected by the transmitting toque. In particular, the increase in the torque will decrease the linear velocity of driven components because the traction oil needs to suffer a more radical shear deformation to satisfy the higher torque requirement. Thus, equation (1) should be written as follows:(3)ωsT+kωrT−1+kωcT=0,where ω(T) is the angular velocity as a function of torque. Considering the difference between the planetary gear train and PTD, k represents the planetary ratio, which is equal to rII/rI, in equation (3). Due to the energy conservation, it can be inferred from equation (3) that the proportional relationship between torques should also be changed.Figure 3
Cross-section of traction contact.All forces and torques acting on the components of PTD are depicted in Figure4. We assume that the speeds of the PTD’s components remain unchanged. Then, the equilibrium equations for discs I and II are shown in [20]:(4)TIm=nFtracIrI+nMspinIsinθ+γ,and(5)TIIm+nMspinIIsinθ−γ=nFtracIIrII,where TI and TII are the torques acting on discs I and II, respectively; FtracI and FtracII are the traction forces acting on the roller; MspinI and MspinII are the corresponding spin moments; m is the number of cavities; and n is the number of rollers in one cavity. For simplicity, we only discuss a PTD with one cavity; thus, m = 1. The following derivation for more cavities can easily be inferred if we add the parameter m [20].(6)FtracI=TI/n−MspinIsinθ+γrI,and(7)FtracII=TII/n+MspinIIsinθ−γrII.Figure 4
Free body diagram of PTD.In steady-state conditions, the torque equilibriums of the roller at points I and II are given as follows:(8)2FtracIrroller−FCrroller−TB+MspinIcosθ+MspinIIcosθ=0−2FtracIIrroller+FCrroller−TB+MspinIcosθ+MspinIIcosθ=0,where rroller is the radius of the roller (see Figure 1); TB is the torque loss in bearing; and FC is the carrying force acting at the center of the roller provided by the carrier.(9)FC=TCn⋅2rI+rII.It is worth noting that the roller suffers some torques directly acting on it, such asMspinI, MspinII, and TB. It means that the quantities of the two tractions force will be affected. Therefore, to simplify the derivation, let:(10)Tresis=TB−MspinIcosθ−MspinIIcosθ.Here,Tresis is the resisting torque on the roller. Substituting equations (6), (7), (9), and (10) into equation (8), we obtain the following:(11)TC1+k=TI−nMspinIsinθ+γ−n2Tresisrroller/rI,and(12)TC1+k=TII+nMspinIIsinθ−γ+n/2Tresis/rroller/rIIk.Then, the proportional relationship between torques can be expressed as follows:(13)TI−nMspinIsinθ+γ−n2Tresisrroller/rI:TII+nMspinIIsinθ−γ+n2Tresisrroller/rII:TC=1:k:−1+k.It can be inferred from equation (13) that the resisting torque Tresis acting on the roller will affect the proportional relationship between torques. Additionally, even though we set Tresis = 0, which means that the operating condition of the roller is similar to that of the planet gear, TI, TII, and TC still cannot conform to a directly proportional relationship because of the existence of spin moments.


### 2.3. Analysis of Speeds

The speed relationship can be derived by analyzing the speed distribution on the roller (Figure5). Here, vrollerI, vrollerII, and vrollerC are the velocities of the roller at contact points I and II, and the center of the roller. CrI and CrII are creep coefficients at contact points I and II; vDiscI and vDiscII are the velocities of discs I and II at their contact points, respectively.(14)vDiscI=ωIrIvDiscII=ωIIrII,where ωI and ωII are the angular speeds of discs I and II, respectively. The velocity at the center of the roller is slightly complex because the location and attitude of the roller vary with the variation of planetary ratio. The motion of the roller can be divided into revolution and rotation motions. The angular speed of the revolution motion is equal to the rotational angular speed of the carrier ωC. Thus,(15)vrollerC=rI+rII2ωc.Figure 5
One of the working conditions of the roller: the power flows into the roller and out of the two discs.The creep coefficient indicates the sliding rate at contact points and should be equal to the proportion of the sliding velocity to the total traction velocity. For the scenario in Figure5, the roller acts as an input component, and discs I and II act as output components. Thus, we have the following:(16)vDiscI=vrollerI1−CrIvDiscII=vrollerII1−CrII.The roller should be regarded as rigid to obtain the following:(17)vrollerI+vrollerII=2vrollerC.Substituting equations (14), (15), and (16) into equation (17), we obtain the following:(18)ωIrI1−CrI+ωIIrII1−CrII=rI+rIIωC,(19)ωI1−CrI+kωII1−CrII=1+kωC,where k is equal to rII/rI.However, equation (19) cannot consider all working conditions of PTD because the directions of power flows always change. Therefore, we introduce two-direction factors, DfI and DfII, of power flows at contact points I and II, respectively. For instance, if the power is transmitted from a disc to the roller, the direction factor is equal to 1. In contrast, if power is transmitted from the roller to a disc, the direction factor is equal to −1. Then, equation (19) can be written in general form as follows:(20)ωI1−CrIDfI+kωII1−CrIIDfII−1+kωC=0,where CrI and CrII are affected by working conditions.One PTD has six possible connections toward the two transmission paths: mechanical and variable ratio [18]. Each connection has three types of power flows: types, I, II, and III [27]. Type I indicates that the power has a recirculation from the variable ratio path; type II indicates that the recirculation is from the mechanical path; and type III means there is no power recirculation.The most visualized method for determining the values ofDfI and DfII is drawing power flows on the structural diagram. As shown in Figure 6, a transmission system using PTD as its power-split device works with the three different power flows. In the type I power flow, the power is inputted from the input shaft and outputted from the output shaft while generating a power recirculation through the variable ratio path. Summarizing the power flows across the two paths, we can conclude that the power goes from disc II to the roller and then flows out toward disc I. Consequently, DfI = −1 and DfII = 1. For type II power flow, DfI = 1 and DfII = −1. The type III power flow is the simplest in that the roller becomes the only input component, meaning that DfI = −1 and DfII = −1. If we rearrange the mechanical structure or connect one of the PTD components to the output shaft, we obtain a similar analysis process, which will not be listed in this study.Figure 6
Structural diagram of PST system working with different power flows: (a) type I power flow; (b) type II power flow; and (c) type III power flow.
(a)(b)(c)Although the above method can be visualized, it is complex sometimes, especially when the spatial arrangement is not easy to be depicted. Since the possible connections of the planetary components to the two paths have already been enumerated, we propose a direct search method in matrix form to quickly determine the values ofDfI and DfII for all basic arrangements.The first step is to establish a 3 × 3 matrix with elements EXY. Here, X is the number of rows, and each row represents a power flow type; Y is the number of columns, and each column represents a planetary component (see the top left corner of Figure 7(a)). Every element can only 1 or −1: 1 means the power is flowing in and −1 means the power is flowing out.Figure 7
Procedure for determining direction factors in PSTs: (a) input split transmission and (b) output split transmission.
(a)(b)The second step is to address the connections of the planetary components. Here, MEC, VAR, IN, and OUT represent that the component is connecting to the mechanical path, variable ratio path, input shaft, and output shaft, respectively. As shown in Figure7(b), the disc I, disc II, and roller are connected to the mechanical path, variable ratio path, and the input shaft, respectively. The connection in Figure 7(a) is similar to that in Figure 6.The third step is to determine the rows of type III and the column of the roller. The power flow of type III is a pure power-split mode without power recirculation; thus, all power from the input shaft is split into two power flows toward the two paths. To be specific, the row of type III should be [−1, −1, 1]. Then, the column of the roller should be [1, 1, 1]T, and the engine braking situation is not considered in this study.The fourth step is to determine the rows of types I and II. As stated above, we can determine the rest of the elements’ values. Type I means the power recirculate from the variable ratio path, leading toE11 = E31 and E12 = −E32. The scenario of type II is opposite to that of type II, leading to E21 = −E31 and E22 = E32.The final step is to pick out the values forDfI and DfII shown at the bottom left corner of Figure 7(a). Although the procedure for this method is a long description, the steps are simple if the rules are well understood. Additionally, the method is suitable for all possible arrangements for the continuously variable PST. This method is easy to use, especially for a complicated spatial connection.If PTD is located at the output side, meaning that one of the three planetary components should be connected to the output shaft, the process is similar. For instance, when PTD is located at the output side, the third step is slightly different from when PTD is located at the input side (see the top right corner of Figure7(b)). The procedure for an output split transmission is illustrated in Figure 7(b).Using the proposed method, we enumerate values ofDfI and DfII in twelve possible PST with one PTD working with three power flow types. The results are presented in Table 1.Table 1
Value table for direction factors in different situations.
Disc IDisc IICarrierType IType IIType IIIDfIDfIIDfIDfIIDfIDfIIMECVARIN−111−1−1−1VARMECIN1−1−11−1−1MECINVAR−1111−11VARINMEC11−11−11INMECVAR1−1111−1INVARMEC111−11−1MECVAROUT1−1−1111VARMECOUT−111−111MECOUTVAR1−1−1−11−1VAROUTMEC−1−11−11−1OUTMECVAR−11−1−1−11OUTVARMEC−1−1−11−11


## 2.1. Geometric Description and Parameters of the New Half-Toroidal PTD

Although two or three types of PTDs can be used as power-split devices, this study takes the half-toroidal PTD as a research object. The analysis process can be applied in other similar situations.As shown in Figure1, two discs form a cavity where trunnions with rollers are located. Trunnions are supported by a carrier mounted with two-axis pins whose center axis coincides with the center of the cavity. With the aid of the axis pins, trunnions with rollers can swing about the center of the cavity. Here, γ is the tilting angle, indicating the swing level of the roller; it is caused by the movement of the outer race. It has a positive value when the roller swings clockwise and a negative value otherwise. The roller contacts discs I and II at contact points I and II, respectively. The perpendicular from point I to the disc will cross the perpendicular from point II at the cavity center (swing center). The two perpendiculars form a cone angle (Figure 1). The half-cone angle is represented by θ. The distances from points I and II to the rotational axis are rI and rII, respectively. The radius of the cavity circle is r0. The discs can input or output power through gears and be supported by the middle shaft. The rollers are supported between the discs and rotated about their axes while revolving about the middle shaft. Rollers and discs I and II constitute a planetary topology where torques are transmitted through traction contacts at contact points. The three-dimensional (3D) geometry model of half-toroidal PTD is shown in Figure 2.Figure 1
Structural sketch of the half-toroidal PTD: (a) color drawing and (b) line drawing.
(a)(b)Figure 2
3D geometry model of half-toroidal PTD: (a) 3/4 section view and (b) the carrier with three rollers.
(a)(b)


## 2.2. Analysis of Forces and Torques

Are equations (1) and (2) suitable? We should doubt it at first. It is well known that the planetary gear train relies on the gear meshing, which can guarantee a fixed speed ratio that is not affected by the transmission torque. The scenario is different for PTD, where traction contact plays a key role (Figure 3). The linear velocities of driving and driven components cannot be equal and will be affected by the transmitting toque. In particular, the increase in the torque will decrease the linear velocity of driven components because the traction oil needs to suffer a more radical shear deformation to satisfy the higher torque requirement. Thus, equation (1) should be written as follows:(3)ωsT+kωrT−1+kωcT=0,where ω(T) is the angular velocity as a function of torque. Considering the difference between the planetary gear train and PTD, k represents the planetary ratio, which is equal to rII/rI, in equation (3). Due to the energy conservation, it can be inferred from equation (3) that the proportional relationship between torques should also be changed.Figure 3
Cross-section of traction contact.All forces and torques acting on the components of PTD are depicted in Figure4. We assume that the speeds of the PTD’s components remain unchanged. Then, the equilibrium equations for discs I and II are shown in [20]:(4)TIm=nFtracIrI+nMspinIsinθ+γ,and(5)TIIm+nMspinIIsinθ−γ=nFtracIIrII,where TI and TII are the torques acting on discs I and II, respectively; FtracI and FtracII are the traction forces acting on the roller; MspinI and MspinII are the corresponding spin moments; m is the number of cavities; and n is the number of rollers in one cavity. For simplicity, we only discuss a PTD with one cavity; thus, m = 1. The following derivation for more cavities can easily be inferred if we add the parameter m [20].(6)FtracI=TI/n−MspinIsinθ+γrI,and(7)FtracII=TII/n+MspinIIsinθ−γrII.Figure 4
Free body diagram of PTD.In steady-state conditions, the torque equilibriums of the roller at points I and II are given as follows:(8)2FtracIrroller−FCrroller−TB+MspinIcosθ+MspinIIcosθ=0−2FtracIIrroller+FCrroller−TB+MspinIcosθ+MspinIIcosθ=0,where rroller is the radius of the roller (see Figure 1); TB is the torque loss in bearing; and FC is the carrying force acting at the center of the roller provided by the carrier.(9)FC=TCn⋅2rI+rII.It is worth noting that the roller suffers some torques directly acting on it, such asMspinI, MspinII, and TB. It means that the quantities of the two tractions force will be affected. Therefore, to simplify the derivation, let:(10)Tresis=TB−MspinIcosθ−MspinIIcosθ.Here,Tresis is the resisting torque on the roller. Substituting equations (6), (7), (9), and (10) into equation (8), we obtain the following:(11)TC1+k=TI−nMspinIsinθ+γ−n2Tresisrroller/rI,and(12)TC1+k=TII+nMspinIIsinθ−γ+n/2Tresis/rroller/rIIk.Then, the proportional relationship between torques can be expressed as follows:(13)TI−nMspinIsinθ+γ−n2Tresisrroller/rI:TII+nMspinIIsinθ−γ+n2Tresisrroller/rII:TC=1:k:−1+k.It can be inferred from equation (13) that the resisting torque Tresis acting on the roller will affect the proportional relationship between torques. Additionally, even though we set Tresis = 0, which means that the operating condition of the roller is similar to that of the planet gear, TI, TII, and TC still cannot conform to a directly proportional relationship because of the existence of spin moments.


## 2.3. Analysis of Speeds

The speed relationship can be derived by analyzing the speed distribution on the roller (Figure5). Here, vrollerI, vrollerII, and vrollerC are the velocities of the roller at contact points I and II, and the center of the roller. CrI and CrII are creep coefficients at contact points I and II; vDiscI and vDiscII are the velocities of discs I and II at their contact points, respectively.(14)vDiscI=ωIrIvDiscII=ωIIrII,where ωI and ωII are the angular speeds of discs I and II, respectively. The velocity at the center of the roller is slightly complex because the location and attitude of the roller vary with the variation of planetary ratio. The motion of the roller can be divided into revolution and rotation motions. The angular speed of the revolution motion is equal to the rotational angular speed of the carrier ωC. Thus,(15)vrollerC=rI+rII2ωc.Figure 5
One of the working conditions of the roller: the power flows into the roller and out of the two discs.The creep coefficient indicates the sliding rate at contact points and should be equal to the proportion of the sliding velocity to the total traction velocity. For the scenario in Figure5, the roller acts as an input component, and discs I and II act as output components. Thus, we have the following:(16)vDiscI=vrollerI1−CrIvDiscII=vrollerII1−CrII.The roller should be regarded as rigid to obtain the following:(17)vrollerI+vrollerII=2vrollerC.Substituting equations (14), (15), and (16) into equation (17), we obtain the following:(18)ωIrI1−CrI+ωIIrII1−CrII=rI+rIIωC,(19)ωI1−CrI+kωII1−CrII=1+kωC,where k is equal to rII/rI.However, equation (19) cannot consider all working conditions of PTD because the directions of power flows always change. Therefore, we introduce two-direction factors, DfI and DfII, of power flows at contact points I and II, respectively. For instance, if the power is transmitted from a disc to the roller, the direction factor is equal to 1. In contrast, if power is transmitted from the roller to a disc, the direction factor is equal to −1. Then, equation (19) can be written in general form as follows:(20)ωI1−CrIDfI+kωII1−CrIIDfII−1+kωC=0,where CrI and CrII are affected by working conditions.One PTD has six possible connections toward the two transmission paths: mechanical and variable ratio [18]. Each connection has three types of power flows: types, I, II, and III [27]. Type I indicates that the power has a recirculation from the variable ratio path; type II indicates that the recirculation is from the mechanical path; and type III means there is no power recirculation.The most visualized method for determining the values ofDfI and DfII is drawing power flows on the structural diagram. As shown in Figure 6, a transmission system using PTD as its power-split device works with the three different power flows. In the type I power flow, the power is inputted from the input shaft and outputted from the output shaft while generating a power recirculation through the variable ratio path. Summarizing the power flows across the two paths, we can conclude that the power goes from disc II to the roller and then flows out toward disc I. Consequently, DfI = −1 and DfII = 1. For type II power flow, DfI = 1 and DfII = −1. The type III power flow is the simplest in that the roller becomes the only input component, meaning that DfI = −1 and DfII = −1. If we rearrange the mechanical structure or connect one of the PTD components to the output shaft, we obtain a similar analysis process, which will not be listed in this study.Figure 6
Structural diagram of PST system working with different power flows: (a) type I power flow; (b) type II power flow; and (c) type III power flow.
(a)(b)(c)Although the above method can be visualized, it is complex sometimes, especially when the spatial arrangement is not easy to be depicted. Since the possible connections of the planetary components to the two paths have already been enumerated, we propose a direct search method in matrix form to quickly determine the values ofDfI and DfII for all basic arrangements.The first step is to establish a 3 × 3 matrix with elements EXY. Here, X is the number of rows, and each row represents a power flow type; Y is the number of columns, and each column represents a planetary component (see the top left corner of Figure 7(a)). Every element can only 1 or −1: 1 means the power is flowing in and −1 means the power is flowing out.Figure 7
Procedure for determining direction factors in PSTs: (a) input split transmission and (b) output split transmission.
(a)(b)The second step is to address the connections of the planetary components. Here, MEC, VAR, IN, and OUT represent that the component is connecting to the mechanical path, variable ratio path, input shaft, and output shaft, respectively. As shown in Figure7(b), the disc I, disc II, and roller are connected to the mechanical path, variable ratio path, and the input shaft, respectively. The connection in Figure 7(a) is similar to that in Figure 6.The third step is to determine the rows of type III and the column of the roller. The power flow of type III is a pure power-split mode without power recirculation; thus, all power from the input shaft is split into two power flows toward the two paths. To be specific, the row of type III should be [−1, −1, 1]. Then, the column of the roller should be [1, 1, 1]T, and the engine braking situation is not considered in this study.The fourth step is to determine the rows of types I and II. As stated above, we can determine the rest of the elements’ values. Type I means the power recirculate from the variable ratio path, leading toE11 = E31 and E12 = −E32. The scenario of type II is opposite to that of type II, leading to E21 = −E31 and E22 = E32.The final step is to pick out the values forDfI and DfII shown at the bottom left corner of Figure 7(a). Although the procedure for this method is a long description, the steps are simple if the rules are well understood. Additionally, the method is suitable for all possible arrangements for the continuously variable PST. This method is easy to use, especially for a complicated spatial connection.If PTD is located at the output side, meaning that one of the three planetary components should be connected to the output shaft, the process is similar. For instance, when PTD is located at the output side, the third step is slightly different from when PTD is located at the input side (see the top right corner of Figure7(b)). The procedure for an output split transmission is illustrated in Figure 7(b).Using the proposed method, we enumerate values ofDfI and DfII in twelve possible PST with one PTD working with three power flow types. The results are presented in Table 1.Table 1
Value table for direction factors in different situations.
Disc IDisc IICarrierType IType IIType IIIDfIDfIIDfIDfIIDfIDfIIMECVARIN−111−1−1−1VARMECIN1−1−11−1−1MECINVAR−1111−11VARINMEC11−11−11INMECVAR1−1111−1INVARMEC111−11−1MECVAROUT1−1−1111VARMECOUT−111−111MECOUTVAR1−1−1−11−1VAROUTMEC−1−11−11−1OUTMECVAR−11−1−1−11OUTVARMEC−1−1−11−11


## 3. Torque Distribution Diagram and a Computational Method for PTD

As stated in Section2, the torque and speed relationships of PTD are different from that of the planetary gear train, which is shown in (13) and (20). The existing analytic methods for PSTs always rely on the Willis equation, indicating that there is no suitable computational method for the PTD deployed in PST. Moreover, directly replacing the Willis equation in the existing analytic methods may not be the best solution. This is because the situation in the traction contact is very complicated. However, the elastohydrodynamic lubrication calculation method should be employed. The rotational speed and torque have an interaction effect through creep motion.This section presents a fast computational method for PTD based on the elastohydrodynamic lubrication theory that considers the interaction effect. The computational method consists of a torque distribution diagram and a direct computational procedure.


### 3.1. Torque Distribution Diagram

The torque distribution diagram can be regarded as a database of the possible torque distribution situations. As shown in equation (13), TI, TII, and TC cannot conform to a directly proportional relationship because of the existence of spin moments and Tresis. It means that directly deriving torques acting on the planetary components is difficult.The torque distributions are directly influenced byMspinI, MspinII, γ, k, and Tresis. Here, γ is determined by k, and the other factors represent torques that are also determined by the creep motion according to the elastohydrodynamic lubrication theory. Thus, the three independent variables can be determined, i.e., k, CrI, and CrII. The procedure for calculating a torque distribution diagram is shown in a flow chart in Figure 8, which consists of a triple loop. The inner loop is a double loop for CrI and CrII. It determines proper values for torques and spin moments using a traction contact model.(21)TIFNmna˜X|Ia˜Y|I=∫01∫02πrIτ˜X|IR+sinθ+γΛa˜X|Iτ˜Y|Icosf−a˜Y|Iτ˜X|IsinfR2dfdR−TIIFNmna˜X|IIa˜Y|II=∫01∫02πrIIτ˜X|IIR+sinθ−γΛa˜X|IIτ˜Y|IIcosf−a˜Y|IIτ˜X|IIsinfR2dfdRMspinIFNa˜X|Ia˜Y|I=∫01∫02πΛa˜X|Iτ˜Y|Icosf−a˜Y|Iτ˜X|IsinfR2dfdRMspinIIFNa˜X|IIa˜Y|II=∫01∫02πΛa˜X|IIτ˜Y|IIcosf−a˜Y|IIτ˜X|IIsinfR2dfdR.Figure 8
Flow chart for the torque distribution diagram.The traction contact model is based on the elastohydrodynamic lubrication theory; several researchers have proposed and verified some traction contact models for different traction CVTs [20, 21, 28]. This contact model is fully flooded isothermal. It uses the Hertz contact model to describe the pressure distribution in the contact area. Additionally, it uses the Roelands model to describe the pressure influence on the limiting shear stress and fluid viscosity. Furthermore, it uses the Bair and Winer non-Newtonian model to describe the rheological behavior of the fluid. Finally, it uses the Hamrock and Dowson formulas to describe the film thickness. By employing performance parameters (shown in Table 2) and creep coefficients (CrI and CrII) to determine the contact situation and shear deformation at contact points, respectively, we can derive the values of torques and spin moments as follows.Table 2
Performance parameters of PST.
Performance parametersAssumed valuesCavity radius40 mmRadius of curvature of the roller32 mmHalf-cone angleπ/3Value range of the tilting angle[−π/3, π/3]Speed ratio at the gear pair besides disc IiI = −1Speed ratio at the gear pair besides disc IIiII = 1Speed ratio at the gear pair besides output shaftiOUT = −1Input speed−2000 RPMDistance from the cavity to the rotational axis25 mmThe number of cavitiesm = 1The number of rollers per cavityn = 3Absolute viscosity at the atmospheric pressure3.25 × 10−3 Pa sViscosity-pressure index0.85 Pa−1Pressure-viscosity coefficient1.71 × 10−8 Pa−1Limiting shear stress at the atmospheric pressure0.02 × 109 PaLimiting shear stress constant0.085Pole pressure constant of the Roelands viscosity model1.96 × 108 PaPole viscosity of the Roelands viscosity model6.31 × 10−5 Pa sHere, the subscriptsX and Y indicate the relevant coordinate axis; I and II indicate the contact points I and II, respectively; and a˜ and τ˜ indicate the dimensionless length of semi-axis and dimensionless shear stress acting on the roller, respectively. For example, τ˜X|I is the X-axis component of shear stress acting on the roller at the contact point I. FN is the normal force acting at contact points, and Λ is the contact length parameter. All parameters in equation (21) have detailed descriptions in the reference [20].Equation (13) is a judgment condition in the procedure for determining proper torque distributions. The outer loop is for k, because different k means different postures of the components, which causes a variation of the contact situation. The procedure is applied to PST in Section 4 to show how to derive the torque distribution diagram.


### 3.2. A Diagram-Based Computational Method

As shown in equation (21) and Figure 8, the computational workload of the torque distribution can be very much if small step sizes and high-precision integral calculations are required. Thus, it makes no sense to repeat the inner loop in Figure 8. Therefore, we propose a diagram-based computational method to achieve a fast calculation as soon as possible.However, there is another problem of special speed distribution shown in (20). The speed relationship is influenced by creep motion and power flow direction. Thus, the first step of the computational method is to determine the parameters of (20).There are some rules we should introduce in PST. The torque distribution diagram can find other two-component torques from one component torque that can be directly calculated fromTin or Tout. For example, the output torque can directly decide one of TI, TII, and TC if the planetary mechanism is connected to the output shaft. Similarly, if ωin and ωout are known, two of the component speeds can be calculated. Furthermore, the component speeds can be calculated in the same form if ωin and ωout are given in speed ratio form [7] or normalized form [29].As shown in Figure9, after inputting the initial parameters, ωin, ωout, k, Tin, and Tout, the procedure is divided into two paths.Figure 9
Flow chart for the diagram-based computational method.The left one is to calculateTI, TII, TC, CrI, and CrII by interpolating in the torque distribution diagram using one of the component torques decided by Tin or Tout. The calculated TI, TII, and TC can also determine the power recirculation situation.The right one is to calculate the power recirculation situation. First, it usesωin and ωout to determine two of ωI, ωII, and ωC. Then, the power recirculation situation can be calculated by comparing the power passing through the mechanical path with the total power. Additionally, the power recirculation situation can offer the direction factors DfI and DfII using the method in Section 2.3.After inputting the calculated parameters into (20), we can calculate the last speed parameter in (20). The last part of the procedure is normal because there are many calculation methods for PST if all speeds and torques of the planetary mechanism are determined.Although the calculated amount of the torque distribution diagram is large, the diagram-based computational method only needs to conduct an interpolation calculation on the diagram. The method is linear, meaning a small amount of calculation.


## 3.1. Torque Distribution Diagram

The torque distribution diagram can be regarded as a database of the possible torque distribution situations. As shown in equation (13), TI, TII, and TC cannot conform to a directly proportional relationship because of the existence of spin moments and Tresis. It means that directly deriving torques acting on the planetary components is difficult.The torque distributions are directly influenced byMspinI, MspinII, γ, k, and Tresis. Here, γ is determined by k, and the other factors represent torques that are also determined by the creep motion according to the elastohydrodynamic lubrication theory. Thus, the three independent variables can be determined, i.e., k, CrI, and CrII. The procedure for calculating a torque distribution diagram is shown in a flow chart in Figure 8, which consists of a triple loop. The inner loop is a double loop for CrI and CrII. It determines proper values for torques and spin moments using a traction contact model.(21)TIFNmna˜X|Ia˜Y|I=∫01∫02πrIτ˜X|IR+sinθ+γΛa˜X|Iτ˜Y|Icosf−a˜Y|Iτ˜X|IsinfR2dfdR−TIIFNmna˜X|IIa˜Y|II=∫01∫02πrIIτ˜X|IIR+sinθ−γΛa˜X|IIτ˜Y|IIcosf−a˜Y|IIτ˜X|IIsinfR2dfdRMspinIFNa˜X|Ia˜Y|I=∫01∫02πΛa˜X|Iτ˜Y|Icosf−a˜Y|Iτ˜X|IsinfR2dfdRMspinIIFNa˜X|IIa˜Y|II=∫01∫02πΛa˜X|IIτ˜Y|IIcosf−a˜Y|IIτ˜X|IIsinfR2dfdR.Figure 8
Flow chart for the torque distribution diagram.The traction contact model is based on the elastohydrodynamic lubrication theory; several researchers have proposed and verified some traction contact models for different traction CVTs [20, 21, 28]. This contact model is fully flooded isothermal. It uses the Hertz contact model to describe the pressure distribution in the contact area. Additionally, it uses the Roelands model to describe the pressure influence on the limiting shear stress and fluid viscosity. Furthermore, it uses the Bair and Winer non-Newtonian model to describe the rheological behavior of the fluid. Finally, it uses the Hamrock and Dowson formulas to describe the film thickness. By employing performance parameters (shown in Table 2) and creep coefficients (CrI and CrII) to determine the contact situation and shear deformation at contact points, respectively, we can derive the values of torques and spin moments as follows.Table 2
Performance parameters of PST.
Performance parametersAssumed valuesCavity radius40 mmRadius of curvature of the roller32 mmHalf-cone angleπ/3Value range of the tilting angle[−π/3, π/3]Speed ratio at the gear pair besides disc IiI = −1Speed ratio at the gear pair besides disc IIiII = 1Speed ratio at the gear pair besides output shaftiOUT = −1Input speed−2000 RPMDistance from the cavity to the rotational axis25 mmThe number of cavitiesm = 1The number of rollers per cavityn = 3Absolute viscosity at the atmospheric pressure3.25 × 10−3 Pa sViscosity-pressure index0.85 Pa−1Pressure-viscosity coefficient1.71 × 10−8 Pa−1Limiting shear stress at the atmospheric pressure0.02 × 109 PaLimiting shear stress constant0.085Pole pressure constant of the Roelands viscosity model1.96 × 108 PaPole viscosity of the Roelands viscosity model6.31 × 10−5 Pa sHere, the subscriptsX and Y indicate the relevant coordinate axis; I and II indicate the contact points I and II, respectively; and a˜ and τ˜ indicate the dimensionless length of semi-axis and dimensionless shear stress acting on the roller, respectively. For example, τ˜X|I is the X-axis component of shear stress acting on the roller at the contact point I. FN is the normal force acting at contact points, and Λ is the contact length parameter. All parameters in equation (21) have detailed descriptions in the reference [20].Equation (13) is a judgment condition in the procedure for determining proper torque distributions. The outer loop is for k, because different k means different postures of the components, which causes a variation of the contact situation. The procedure is applied to PST in Section 4 to show how to derive the torque distribution diagram.


## 3.2. A Diagram-Based Computational Method

As shown in equation (21) and Figure 8, the computational workload of the torque distribution can be very much if small step sizes and high-precision integral calculations are required. Thus, it makes no sense to repeat the inner loop in Figure 8. Therefore, we propose a diagram-based computational method to achieve a fast calculation as soon as possible.However, there is another problem of special speed distribution shown in (20). The speed relationship is influenced by creep motion and power flow direction. Thus, the first step of the computational method is to determine the parameters of (20).There are some rules we should introduce in PST. The torque distribution diagram can find other two-component torques from one component torque that can be directly calculated fromTin or Tout. For example, the output torque can directly decide one of TI, TII, and TC if the planetary mechanism is connected to the output shaft. Similarly, if ωin and ωout are known, two of the component speeds can be calculated. Furthermore, the component speeds can be calculated in the same form if ωin and ωout are given in speed ratio form [7] or normalized form [29].As shown in Figure9, after inputting the initial parameters, ωin, ωout, k, Tin, and Tout, the procedure is divided into two paths.Figure 9
Flow chart for the diagram-based computational method.The left one is to calculateTI, TII, TC, CrI, and CrII by interpolating in the torque distribution diagram using one of the component torques decided by Tin or Tout. The calculated TI, TII, and TC can also determine the power recirculation situation.The right one is to calculate the power recirculation situation. First, it usesωin and ωout to determine two of ωI, ωII, and ωC. Then, the power recirculation situation can be calculated by comparing the power passing through the mechanical path with the total power. Additionally, the power recirculation situation can offer the direction factors DfI and DfII using the method in Section 2.3.After inputting the calculated parameters into (20), we can calculate the last speed parameter in (20). The last part of the procedure is normal because there are many calculation methods for PST if all speeds and torques of the planetary mechanism are determined.Although the calculated amount of the torque distribution diagram is large, the diagram-based computational method only needs to conduct an interpolation calculation on the diagram. The method is linear, meaning a small amount of calculation.


## 4. Results and Discussion

Following the proposed steps, this section conducts a computational analysis on a continuously variable PST (Figure10).Figure 10
Structural diagram of a continuously variable PST.In the continuously variable PST, the input shaft is connected to the carrier; the two discs output their power through two gears; discs I and II are connected to the variator and output shaft, respectively. The variator is a hydraulic pump-motor system whose speed ratio ranges from −1 to 1. The speed ratioi represents the ratio of output speed to input speed and subscripts I, II, OUT, and v, respectively. Table 2 presents the performance parameters of the PST.The input speed is −2000 RPM because the input shaft is connected to the carrier (Figure10). Additionally, the torque acting on the carrier also has a negative value because its direction is opposite to the direction of Ts and Tr. Then, the input power, which is the product of input speed and torque, has a positive value.As shown in Figure11, we investigate −Tc instead of Tc for simplicity. This is because the value of Tc always has an opposite sign compared with TI and TII. When k = 0.7, TI is higher than TII, and the opposite happens when k = 1.5 because the position of the roller is opposite.Figure 11
Torques as functions of creep coefficient: (a) whenk = 0.7 and (b) when k = 1.5.
(a)(b)CrII increases with CrI. When PTD transfers a higher torque, it suffers a higher creep motion. It means that torques are monotonic functions with respect to the creep coefficients. If we can determine one of the parameters of CrI, CrII, TI, TII, and Tc, the other four parameters can easily be obtained. Therefore, the proposed torque predicted that the computational method is feasible.After repeating the procedure for drawing Figure11 several times with different values of k, we derive the torque distribution diagram as depicted in Figure 12. Figure 12 introduces CrI + CrII as an independent variable; the other one is k. As shown in Figure 11, CrI increases linearly with CrII; CrI + CrII can represent the overall creep motion and torque level. Figures 12(a) and 12(b) are asymmetrical because k varies from 0.5 to 2, whose range is asymmetrical. The top of diagrams is decided by our options. Additionally, larger values of CrI + CrII are forbidden regions in the calculation because of its low-speed efficiency in the contact traction point. For the geometrical arrangement, Tc enjoys a relatively higher torque value. This characteristic should be carefully considered and used when designing the continuously variable PST because the torque limit of PSD is lower than the planetary gear train in the same size. Only the diagram of CrI is carried out because that of CrII can be easily calculated.Figure 12
Torque diagram of the continuously variable PST: (a)TI diagram; (b) TII diagram; (c) Tc diagram; and (d) CrI diagram.
(a)(b)(c)(d)The torque distribution diagrams can easily be used to calculate the transmission situations of PTD. For example, if the input or output torque of continuously variable PST is known, eitherTI, TII, or Tc is known, which can easily determine a working point in the torque distribution diagram. Employing the position of the working point, TI, TII, TcCrI, and CrII can be directly calculated using the interpolation method without repeating the procedure in Figure 8.Figure13 shows the proportional relationship between TII and Tc. The results show that the ratio of TII to Tc cannot be constant in the PTD transmission. As shown in Figure 13(a), TII/Tc should be −0.412 in a planetary gear train and cannot be constant in the PTD transmission. A similar conclusion can be obtained using Figure 13(b).Figure 13
Proportional relationship betweenTII and Tc: (a) when k = 0.7 and (b) when k = 2.0.
(a)(b)Figure14 shows that if we replace the planetary gear train of the PST system with the proposed PTD, the relationships between iPST and iv are nearly the same, especially when the input torque and creep coefficient are sufficiently small. However, if the input torque increases to an acceptable level, the difference between the speed ratios of variator can increase to about 0.07. Thus, the influence of torque level on the speed ratio is limited; however, the difference increases as the torque level increases.Figure 14
Relationship betweeniPST and iv: (a) the full speed ratio range and (b) a larger version at the rectangle area.
(a)(b)The efficiency of continuously variable PST can be calculated after determining the power flow situation, mechanical path loss, and variable path loss. The mechanical path loss consists of the gear and traction contact losses. In the previous study, the traction contact loss cannot be calculated precisely, and the creep motion is neglected [18]. In contrast, traction contact loss and creep motion can easily be obtained using the results of this study. We take traction point II as an example since the traction contact working conditions at points I and II are similar. Here, ηTorque|II indicates the torque efficiency at point II, which is equals to the ratio of TII to ideal torque divided into disc II.(22)ηTorque|II=TIIkTc/−1+k=−TIITII+nMspinIIsinθ−γ+n/2Tresis/rroller/rII.The speed efficiency can easily be calculated using the creep coefficient; thus, the traction contact efficiency at point II can be expressed as follows:(23)ηTraction|II=ηTorque|II1−CrII=−TII1−CrIITII+nMspinIIsinθ−γ+n/2Tresis/rroller/rII.Similarly, the traction contact efficiency at point I can be expressed as follows:(24)ηTractionI=ηTorque|I1−CrI=−TI1−CrITI−nMspinIsinθ+γ−n/2Tresis/rroller/rI.Here,ηTraction indicates the overall traction efficiency, and |I and |II indicate points I and II, respectively.These factors in the efficiency calculation formulas can be obtained from the torque distribution diagrams. Figure15 shows the calculation results.Figure 15
Traction contact efficiency of PTD: (a) the torque efficiency and (b) the overall efficiency.
(a)(b)We can see that the torque efficiency can reach about 0.96 to 0.98 when creep motion is relatively high. This is an essential characteristic caused by the fact that when creep motion is relatively high, the traction oil will appear with its limit shear strength, and the spin moment reduces [19]. It means that if the load torque is acceptable, the PTD will have relatively high efficiency.The traction efficiency is affected byk because k determines the spin moment. When load torque is low and creep motion is small, the efficiencies of traction contact points are low. This is because the spin moment plays an essential role in this working condition. Although some zero-spin traction drives CVTs, we still recommend PTD as a power-split device in PST to gain variable planetary ratio and relatively higher overall efficiency.


## 5. Conclusions

By conducting a mechanism kinematic analysis, this study investigated the relationships between torques and speeds in PTD. We obtained particular torques and speed distribution rules in PTD transmission, enhancing further investigation of this device possible. Based on a widely used contact model and the proposed distribution rules, we conducted a set of analytical methods for the continuously variable PST system. The analytical method was applied to a continuously variable PST system.(i)
The torque distribution equation of PTD is directly influenced by the spin moments and the resisting torques acting on the rollers ((13)). Torques are monotonic functions with respect to the creep coefficients.(ii)
Direction factors,DfI and DfII, were introduced to summarize three different speed distribution equations into a one-speed distribution equation. We also proposed a direct search method for the direction factors in matrix form.(iii)
A torque distribution diagram and its procedure were performed as calculation tools for computing the transmission working conditions in PST.(iv)
The PTD has a relatively high efficiency of about 0.94 to 0.97 when it transfers normal torque. When load torque is low, the efficiencies of traction contact point are low due to spin moment, which dramatically decreases the torque efficiency.The main purpose of this study is to analyze the relationships between torques and speeds of the continuously variable PST and propose a set of fast computational methods. However, continuously variable PST, a new transmission mode, may have more attractive characteristics and needs to be analyzed more systematically and comprehensively, which is also our future research direction and calls for more researchers to participate.



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*Source: 6136766-2022-02-28.xml*