A Direct Simulation Method for Continuous Variable Transmission with Component-wise Design Specifications

第十七屆全國機構機器設計學術研討會中華民國一零三年十一月十四日 國立勤益科技大學台灣 台中論文編號 :CSM048 A Direct Simulation Method for Continuous Variable Transmission with Component-wise Design Specifications Yu-An Lin 1,Kuei-Yuan Chan 1 Graduate Student, Department of Mechanical Engineering, National Cheng Kung University Associate Professor, Department of Mechanical Engineering, National Taiwan University Abstract Continuous variable transmissions (CVT), commonly implemented in scooters and motorcycles, have recently been applied to commercial vehicles for their ability to maintain high fuel economy compared with traditional transmissions. The performance of a CVT depends primarily on the design specifications of each components within the transmission. However, most existing methods in the literature use a backward simulation approach that considers the vehicle driving torque as one of the known inputs. This assumption requires the complete design information about a vehicle and therefore is impractical in the development stage of a new CVT. In this research, we develop a direct simulation method for the design of CVT components such that specific performance curve can be achieved. This direct simulation method takes the engine power and torque curves as known inputs to the CVT and systematically links the design specifications of each component within the information flow. The results is a system with 1 components and 69 design specifications. In this paper we demonstrate how this system can be used to match a target performance curve by the top level decision-makers. Although such a solution may not be unique, we show the strengths of the proposed system as a designers tool when making specific CVT design decisions. Keywords: continuous variable transmission, system integration, rubber V-belt, vehicle design 1 Introduction The application of continuous variable transmission (CVT) in automobiles can be traced back to the V-belt transmission developed by Benz and Daimler, the wellknown manufacturers in Germany in 1886 [1. However, the system-designing and parameter-adjusting work in the beginning was much difficult because it usually could only be executed by the rule of thumb, also called trial and error, therefore numerous re- Corresponding Author, chanky@ntu.edu.tw search groups investigated the transmission field due to the unfinished math model. The first theoretical model was developed in 1955 by Worley [ who proposed several semi-experimental formula of V-belts in CVT including the tension force of V-belts, the physical constraint of belt s length, and the axial force of both driving and driven parts. Since then, most researchers have focused on providing more comprehensive models for a component; for example, see [3,4. While most studies focus on the impact of one component on a CVT, Yu et al. developed a backward simulation method to link the driving and the driven parts within a CVT [5. The advancements in computation enabled the CVT simulation with more design details; for example, Zeng used a quadratic function to represent the contour of pulleys in [6; Takeuchi, Koide, and Sunayama modeled V-belts using Euler s equations and included the initial tension and the effective tension in their analysis [7; Shen integrated several CVT formula in [8. These research activities have enabled more component-specific parameters in the performance analysis of a CVT. Existing methods that analyze a CVT system base primarily on the backward simulation method given environment resistance information such as the drag resistance as a function of vehicle speed and the grade resistance [5. However, these environment resistances can only be known when critical components of a CVT and the driving cycle of the vehicle have all been determined. In the design stage of a CVT, the requirements of these information contradicts to the actual design practices in developing a new CVT system. Therefore in this research, we propose a direct simulation method for CVT systems with comprehensive component specifications. In Section, we provide detail component models in our integration. The proposed direct simulation method is introduced in Section 3. Results and comparisons are shown in Section 4, followed by the conclusions in Section 5. 1

Models of CVT Components Nowadays the rubber V-belt CVT,which is applied to most of the scooter motor, can be divided into three parts, driving, driven and V-belt shown in figure 1. The driving part is connected with the input source, engine, to drive the system s behavior. The V-belt is the transportational medium for the purpose of tansporting the power from engine to output shaft. Last the driven part is not only a torque sensor but also a buffer device for adjusting the rotation radius of V-belt. In the following we will discuss the facility and formula of every important component in the three parts. Figure : Driving part components and the important forces cause by pulleys. Driven Part The driven part shown in figure 3 equally has two sheaves but controlled the movable sheave s axial deformation by a torque cam and spring attached at the output shaft. When the engine starts to accelerate, the V-belt s rotation radius at driving part may increase, due to the constant length of V-belt, simultaneously the driven one will decrease and cause the movable sheave expanded. But there are still some factors to resist the radius changing, one is resistance force F cam by the torsion T cam and the other is the spring force F s (including the preload force). After classifying the resistance forces, we obtain the axial force F dr [6: Figure 1: The rubber V-belt CVT model.1 Driving Part Driving part is mainly composed of the input shaft, two sheaves(one is movable in axial direction, and the other is fixed) and the pulleys, which control the movable sheave s axial deformation, shown in figure. When the shaft accepts the input power from the engine, the rotation speed will make the pulleys generate the centrifugal force F c, this force changes to a horizon force F d via the geometric shape of sheaves to push the movable sheave and squeezes the belt indirectly. The pushing force F d can be written as following: F dr = F cam + F s = T cam(sin β µ sc cos β) r cam (cos β + µ sc sin β) + [ F s0 + k d (r + dr r dr) tan α () Note that r cam is the radius of torque cam, β is the tilt angle of the cam slot, F s0 is the preload force of the torque spring, k d is the spring coefficient, r + dr is the maximum V-belt rotation radius in the driven part, r dr is the V-current belt rotation radius, α is the sheave angle, and µ sc is the friction coefficient between cam and slot. nmωin F d = r c cos β l +µ a sin β l (1) sin βr µa cos βr sin β l µ a cos β l + cos β r µ a sin β r Where n is the quantity of pulleys, m is the pulley s weight, ω in is the rotation speed of input shaft, β l is the tilt angle of movable sheave, β r is the tilt angle of pulley cover, r c is the rotation radius of pulleys which is related with the rotation radius of V-belt in the driving part. Figure 3: Driven part components and the important forces cause by torque cam and spring

.3 V-belt V-belt is the transportational medium of tension caused by the tautness between driving and driven parts. In other words, the centrifugal arrangement, pulleys, are used to locate position of V-belt in driving sheave, and the tension will be maintained by spring on driven sheave. When the radius attached to the sheaves change, the reduction ratio transforms automatically. Assume that the total length of belt L would keep constant, unlike the rubber band, we can derive the restrict function of V-belt s length. r d (π θ ) + r θ dr + C sin(π θ ) = L (3) Where r d, θ 1 and r dr, θ are the rotation radius and angle of the driving and driven part respectively, C is the distance between two axle. F d.b = F t1θ 1 F dr.b = F t1 F t µ sin φ ( 1 µ tan α ) µ + tan α ( cos α µ cos φ sin α.4 Reduction Ratio of a CVT ) (5) (6) The reduction ratio is define as the ratio of the belt s radius attached at the driving and driven sheaves. With the transformation of reduction ratio N and gearbox ratio N out, finally we can get the output angular velocity ω out of the wheel shaft. ω out = ω in NN out = ω in r d r dr N out (7) 3 The Proposed Direct Simulation Method The simulating processes for CVT performance are shown in figure 5, the difference between backward and direct simulating is mainly about the calculating order for the components mentioned in Subsection 3.3 and 3.4. We start from the engine and sequentially go through the driving part, V-belt, the driven part and finally the output shaft, like what we consider for the direct simulation. Figure 4: V-belt components and the cross-section view of the belt with axial forces When CVT begins to work, the power will be transported from driving to driven part by the belt, however the belt loss problem occurs in the same time. The reason is the two sides of tensions(the taut tension F t1 and slack tension F t ) are different and make friction angle φ 0 shown in figure 4. The correlation between the two different tensions can be written as the following [8, Where µ is the friction coefficient between V-belt and sheaves. ( ) F t1 θ µ sin φ = exp F t µ cos φ cos α + sin α (4) By the previous discussion, we can find the sheave always give the axial force no matter in driving or driven part. Similarly, the belt also generates the axial forces in order to resist the pressing from the sheaves, note that these force are much related with the tension forces F t1 and F t. The resistance forces(f d.b in the driving part and F dr.b in the driven part) shown in figure 4 can be found according to the reference [ and [9. Figure 5: The direct simulating method for modeling CVT performance 3.1 Preparation First we have to define the CVT components geometry and size such as the pulley s weight and the limit of the belt rotation radius, then build the data base of Belt length model, which includes every reduction ratio 3

N corresponding to the driving and driven part s belt rotation radius r d and r dr. Assume we capture n points from the radius data, the radius r d and r dr can be presented below, note that all the data points should be in the initial limitation(it means r dmin r d r dmax and r drmin r dr r drmax ). With these data points, we can get the rotation angles θ 1 and θ by Eq.(3), the restrict function of V-belt s length. The result can be seen in table 1. In addition, we can consider both rotation radius r d and r dr to be the input parameter. r d = [ r d1 r d... r dn r dr = [ r dr1 r dr... r drn N = [ r d r r dr... dn r drn r d1 r dr1 Table 1: The data of every rotation radius corresponding to the rotation angle Input Rotation radius r d or r dr Output r d1 (r dr1 ) r d (r dr )... r dn (r drn ) Driving angle θ 1 θ 11 θ 1... θ 1n Driven angle θ θ 1 θ... θ n 3. The N ω out Data Base After the preparation, the next step is to build the relationship between every possible reduction ratio N and output angular velocity ω out. According to the simulating process, assume we already have the data of input torque T in and angular velocity ω in sourced by engine, the data of every output angular velocity ω out corresponding to the reduction ratio N would be obtained by Eq.(7). T in = [ T 1 T T 3... T n ω in = [ ω 1 ω ω 3... ω n Table : The data of every output angular velocity corresponding to the reduction ratio Input Reduction ratio N = r d /r dr Output N 1 N... N n Angular velocity ω out ω out1 ω out... ω outn 3.3 V-belt Tension Model The intent for the tension model is to calculate the taut tension F t1 and slack tension F t, then to get the optimal friction angle φ in every different rotation radius. Because it s difficult to determined the belt s tension directly without considering other components in CVT, we start from calculating the axial pushing force F d acted by pulleys, note that the force F d can be regarded as f(ω in, r c ) shown in Eq.(1). Besides, the pulley s rotation radius r c is also related with the belt rotation radius of driving part r d, thus we can consider F d = f(ω in, r d ) and build the data base of (r c, F d ). When ω in = ω i, T in = T i (i [1,, 3,... n), we can obtain the data of r c and F d shown in table 3. With the information of pulley pushing force F d, the next work is to determined the resistant force F d.b acted by the belt. In general case, when (F d F d.b ) = 0, it means the belt s rotation radius between two parts would keep constant in this moment owing to the axial force balance. Therefore, finding the balance condition by calculate the belt resistant force F d.b would be an important process. According to Eq.(4), (5) and (6), we can find Ft1 F t = f(θ dr, φ) and (F t1 F t ) = f(t in, θ dr ). Consequently, under the condition of ω in = ω i, T in = T i (i [1,, 3,... n), the input factors for calculating the resistant force F d.b would be the rotation angle of driven part θ dr and the friction angle φ, note that we divides the domain of φ into k point in order to carry on the optimization work in the next step. The relationship between the input factors and determined factors can be seen in table 4 and 5. By series of calculating, we already have the data base of resistant force F d.b and pulley pushing force F d, the final work of this section is to optimize the objective function, then to get the optimal friction angle φ opt like the information in table 6. Table 3: The data of every pulley s rotation radius and axial pushing force corresponding to the belt rotation radius of driving part Input Belt rotation radius r d Output r d1 r d... r dn Pulley radius r c r c1 r c... r cn Pushing force F d F d1 F d... F dn Table 4: The input factors relation for determining the taut tension force Input1 Rotation angle θ dr Input θ dr1 θ dr... θ drn φ 1 F t1,11 F t1,1... F t1,1n Friction φ F t1,1 F t1,... F t1,n angle φ..... φ k F t1,k1 F t1,k... F t1,kn 4

Table 5: The input factors relation for determining the driving part belt resistant force Input 1 Rotation angle θ dr Input θ dr1 θ dr... θ drn φ 1 F d.b,11 F d.b,1... F d.b,1n Friction φ F d.b,1 F d.b,... F d.b,n angle φ..... φ k F d.b,k1 F d.b,k... F d.b,kn Table 6: The optimization work for friction angle Input Input 1 Friction angle φ Rotation angle θ dr θ dr1 θ dr... θ drn φ 1 F d.b,11 F d1 F d.b,1 F d... F d.b,1n F dn φ F d.b,1 F d1 F d.b, F d... F d.b,n F dn..... φ k F d.b,k1 F d1 F d.b,k F d... F d.b,kn F dn Objective function (F d.b F d ) = 0 Optimization result φ opt1 φ opt... φ optn coverage the changing region occupies, the better CVT facility performs. For the purpose of verifying the feasibility of the direct method, we also consult the relative CVT coefficients and the real curve example. The direct simulating and experiment data can be observed in figure 7 (owing to the non-disclosure agreement with KWANG YANG Industrial, the scale is hided), this result shows that the biggest difference between these two curve occurs at the speed changing region, and the reason causes this phenomenon we will mention in the next section. 3.4 The Optimal Wheel Velocity V fac The last step of modeling CVT performance is to determine the optimal wheel velocity V fac. In this section, we will take advantage of the formula of driven part to obtain our target. According to Eq.() and (6), we can easily find the axial force F dr and the belt resistant force in driven part F dr.b. Note that the value of friction angle φ is what we obtained in V-belt tension model (φ opt ) when calculating. Likewise, we take the second optimization here, and the objective function is the force balance of driven part (F d.b F d ) = 0. By this process, we finally get the optimal belt rotation radius r dr.opt shown in table 7 in every input engine speed. Because of knowing the rotation radius r dr.opt, the reduction ration N opt and wheel velocity V fac can also be obtained. Figure 6: Example for CVT reduction ratio performance curve Table 7: The input factors relation for determining the pushing force and driven part belt resistant force Input Belt rotation radius r dr and friction angle φ opt Output r dr1, φopt1 r dr, φ opt... r drn, φ optn Pushing force F dr F dr1 F dr... F drn Resistant force F dr.b F dr.b1 F dr.b... F dr.bn Objective function (F dr.b F dr ) = 0 Optimization result Final target r dr.opt N opt and V fac 4 Results and Comparisons After illustrating the processes of direct simulation, we use MATLAB to compile our performance modeling program. Before observing the result, we define the performance curve into 3 regions, speed unchanged, speed changing and speed changed region, the example is shown in figure 6. In general, the more extensive Figure 7: CVT performance curves for simulation and experiment data 5

5 Conclusions and Future Work Our research proposes another thinking for CVT performance simulating, it makes designers be more flexible when facing the design problem with different known conditions. However, the research still has much work to achieve, take figure 7 for example, it is obvious the experiment and simulation data do not match completely especially in the range of speed changing region. The reason is that the driving part parameter, pulleys movement trajectory, does not set up like the actual one. On the other hand, the shape of torque cam s slot is also an interesting tissue to discuss, like figure 8. In general, the slot used in CVT would be linear, but it seams possible that modifying the slot shape to change cam s position in every moment to further control the corresponding axial pushing force. Therefore, the follow-up work we will concentrate on is studying how to improve the reduction ratio performance with effectively adjusting the parameters such as adding the real trajectory function to the model and make the experiment and simulating data be matched. [3 B. Gerbert. Some notes on v-belt drives. American Society of Mechanical Engineers (Paper), (80 -C/DET-91), 1980. [4 D. Sun. Performance analysis of a variable speedratio metal v-belt drive. pages 31 43, Chicago, IL, USA, 1989. [5 K.-H. Yu. Parameter Design and Numerical Simulation of the Power System of Electric Motorcycle. Master s thesis, National Taiwan University, Taipei, Taiwan, 1999. [6 W.-C. Zeng. A Study on the Application of Parabola on the Driving Dish Design for Continuously Variable Transmission. Master s thesis, National Chung Cheng University, Chiayi, Taiwan, 010. [7 M. Takeuchi, M. Koide, and Y. Sunayama. Prediction method of speed characteristics of v-belt cvt. 400 Commonwealth Drive, Warrendale, PA 15096-0001, United States, 011. [8 P.-H. Shen. Analysis on the Effect of Torque Cam and Pulley Balls toward the Performance of Electric Motorcycle. Master s thesis, National Taiwan University, Taipei, Taiwan, 011. [9 F.-R. Tsai. Computer-aided design of V-belt countinuously variable transmission. Master s thesis, National Tsing Hua University, Hsinchu, Taiwan, 1995. 摘要 Figure 8: Different slot shapes for the torque cam 6 Acknowledgements This work was supported by the Ministry of Science and Technology of Taiwan and Kwang Yang Motor Co., Ltd.. References [1 S. Ashley. Is cvt the car transmission of the future? Mechanical Engineering, 116(11):64 68, 1994. 無段變速器 (CVT) 相對於一般的傳統變速器, 由於擁有較高的機械效益與低油耗率, 故被廣泛的應用在速可達與摩托車領域 CVT 的性能會因為系統內元件的幾何或尺寸而有很大影響, 根據參考文獻, 目前關於分析無段變速器性能之方法, 大部分採用以行車扭矩作為輸入參數的反向動力模擬法 然而, 反向模擬卻需要相對完整的設計參數作為已知資料, 對於新型的 CVT 設計開發而言, 是相對受拘束且不方便的 在本研究中, 我們導入 CVT 系統的 1 種元件與 69 個設計參數, 並提出一個以引擎扭矩與轉速作為輸入參數, 配合元件參數設計來達成變速器性能調整的設計方法, 順向模擬法 本文中我們將演示如何模擬 CVT 系統, 並設計出符合決策者需求的性能曲線 雖然最後得出的方案並非唯一解, 但我們仍具體提供了一個工程師在進行 CVT 設計決策時, 能夠有效辨識系統性能優缺的工具 關鍵字 : CVT 無段變速器 轉速比 橡膠 V 型皮帶 系統整合 車輛設計 [ W. Worley. Designing adjustable-speed v-belt drives for farm implements. Milwaukee, WI, United states, 1955. 6