Proc. Natl. Sci. Counc. ROC(A) Vol. 23, No. 6, 1999. pp. 716-727
New Automatic Hybrid Transmissions for Motorcycles
KUEN-BAO SHEU*, SHEN-TARNG CHIOU**, WEN-MING HWANG**, TING-SHAN WANG**, AND HONG-SEN YAN**,†
*Department of Vehicle Engineering National Huwei Institute of Technology Yuenlin, Taiwan, R.O.C. **Department of Mechanical Engineering National Cheng Kung University Tainan, Taiwan, R.O.C.
(Received August 19, 1998; Accepted May 19, 1999)
ABSTRACT
This paper presents a systematic approach to designing new automatic transmission systems for motorcycles, including conceptual design, kinematic design, efficiency analysis, engine and transmission matching, and prototype developing and testing. The basic concept is to combine a stepped design and a stepless one into a hybrid transmission system. Four hybrid transmission systems, consisting of a two degrees of freedom planetary gear train and a rubber V-belt drive unit, are synthesized. Two operating regimes over the full speed ratio range are obtained by means of engaged clutches. Kinematic design of the hybrid transmission systems is employed to obtain the major dimensions. Based on the power flow and efficiency analysis, formulas for the mechanical efficiency of the transmission systems are derived. The transmission theory of the V-belt drive unit is studied. The design procedure for matching the power source and transmission is presented. A testing rig was designed and constructed. A prototype of this new hybrid transmission system was developed to measure the transmission efficiency and test the operat- ing characteristics. The results show that the proposed design is theoretically correct and practically feasible.
Key Words: motorcycle, hybrid transmissions, conceptual design, kinematic design, mechanical efficiency
I. Introduction
The transmission devices used in motorcycles can be divided into two categories: (1) stepped transmis- sion devices that work by alternating the gear drives, and (2) continuously variable transmissions (CVT) that transmit power by using a rubber V-shaped belt. CVTs have the advantages over gear drives of being simple Fig. 1. An input-coupled differential transmission. in construction, smooth in operation, easy to drive, low in cost, etc. However, the overall efficiency of CVTs is lower than that of gear drives. When a rubber V- CVU changes the overall speed ratio and the division belt CVT with a mechanical-type feedback control of power between the differential gear and CVU paths. system is used in a motorcycle, the transmission ef- Such a system can raise the overall transmission ef- ficiency is poor, especially during start-up and low ficiency while sacrificing some of the speed ratio range speed travel. of the CVU for higher overall efficiency. To obtain certain characteristics that are superior The concepts behind and applications of differ- to those of existing CVTs while retaining the variable ential transmissions date back to the early days of motor speed ratio property, the differential transmission has vehicles. In the Thomas transmission (Thomas, 1914- been used for many decades. It consists of a continu- 1915), used in locomotives and trucks, power from an ously variable unit (CVU) and a differential gear, as engine was split into two paths by a differential gear. shown in Fig. 1. Altering the speed ratio across the Subsequent development focused on automotive
†To whom all correspondence should be addressed.
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systems. In an input-coupled system, as shown in Fig. 1, one shaft of the differential gear is linked to the input side. In an output-coupled system, one shaft of the differential gear is linked to the power output side. Theoretically six connection arrangements for the input- coupled and output-coupled system are possible, pro- vided that all three members of a simple planetary gear train can change positions. Figure 2 shows an example of a differential transmission with an input-coupled configuration. Fig. 2. An example of an input-couple differential transmission. To enhance the limited speed of motorcycles and provide excellent start-up acceleration, the concepts behind a hybrid transmission system can be used to applications. MacMillan and Davies (1965) modeled overcome the limitations placed on the speed ratio span six configurations of differential transmissions by using of the mechanical-type CVU used in differential block diagrams. White (1970, 1976) developed the transmissions. Rearrangement of the differential trans- multi-stage differential transmission and presented two mission system shown in Fig. 1 using clutches and families of differential transmissions that utilized two brakes results in the system shown in Fig. 3, where Ca coaxial differential gears. Mucino et al. (1995) pro- (Cv), B, and G are the clutches, brake, and chain device, posed a family of transmission which use two com- respectively. Alternative hybrid transmissions can be pound planetary gear trains as the differential gear and formed by employing different clutches and/or brakes a CVT unit as the CVU without using clutches and/ as described below. or chain drives. Other investigators, such as Beachley and Frank (1980), Stubbs (1981), Shockton (1984), and 1. Fixed Gear Ratio and Differential Transmis- Mucino et al. (1994), succeeded in designing differ- sion System ential transmission systems. Moreover, the configu- ration of many commercial differential transmissions A hybrid transmission system operates in two can be found in various US patents (Abbott, 1981; different regimes by engaging different clutch Gizard, 1985; Hirosawa, 1986; Itoh and Okada, 1986; sequences. Figure 4 shows an example of a fixed gear Macey and Vahabzadeh, 1987; Stockton, 1989; ratio and differential transmission system. In start-up Sakakibara and Hattori, 1989; Tervola, 1993; Brambilla, and low speed situations, power is transmitted via the 1994). Analyses of differential transmissions have high efficiency gear device by engaging clutch Ca and been numerous, including power flow analysis (Mac- applying brake B while clutch Cv is disengaged to Millan, 1961; White, 1967, 1977) and mechanical operate a fixed gear ratio regime. As vehicle speed efficiency analysis (Yu and Beachley, 1985; Hsieh and increases to and passes a medium speed, both clutches Yan, 1990; Hedman, 1993; Yan and Hsieh, 1994). Ca and Cv are engaged and brake B is released. The However, studies on differential transmissions for input power is transmitted to the chain device assembly motorcycles are not available. while the power flow is transferred from the variable The purpose of this work is to present a design ratio drive assembly to the planetary gear set to act as procedure for developing a new class of automatic a differential transmission regime. motorcycle transmissions, including a conceptual design, kinematic design, efficiency analysis, engine 2. Differential Transmission and CVT System and transmission matching, and prototype development and testing. Figure 5 shows an example of a differential
II. Conceptual Design
To obtain the benefits of low cost, small size, and manufacturing simplicity, the mechanical-type CVT is adopted here as the CVU of the differential transmission, and the simple planetary gear train (PGT) is adopted as the differential gear. The differential transmission systems can be grouped into input-coupled systems and output-coupled Fig. 3. A concept behind the hybrid transmission system.
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speed running. Therefore, the concepts behind the hybrid trans- mission system provide a new direction for studying the design motorcycle transmissions. The idea is to combine the stepped transmission function together with the stepless transmission function to obtain a hybrid system. In this way, the transmitting efficiency of the entire device can be significantly increased. A detailed discussion of these design procedures was given by Sheu et al. (1996). III. Kinematic Design
For the input-coupled configuration of the differ- ential transmission, the kinematic relationship among the speed ratio of the transmission (r), the relative Fig. 4. A fixed gear ratio and differential transmission system. speed ratio of the differential gear (R), the speed ratio of the chain device (K), and the speed ratio of the CVU (V) is (Sheu and Yan, 1996)
r ± KR V = . (1) 1±R
When the CVU of the differential transmission reaches the peak speed ratio, V=Vmax (V=Vmin), the transmission system achieves r=rmax (r=rmin). From Eq. (1), the speed range of the CVU, Vt, can be written as
V max r max ± RK V t = = . (2) V min r min ± RK
Let (Din)max ((Din)min) and (Dout)max ((Dout)min) be the maximum (minimum) pulley diameters of the input axis
Fig. 5. A differential transmission and CVT system.
transmission and CVT system. As clutch Ca is engaged and clutch Cd is disengaged, the transmission system first operates in a differential transmission mode. Then, as clutch Cd is activated while clutch Ca is disengaged, the transmission system operates in a CVT mode with the planetary gear train locked at a 1:1 ratio.
3. Fixed Gear Ratio and CVT System
The feature of this design is that the transmission system operates in two modes, a fixed gear ratio mode and a continuously variable speed ratio mode. Figure 6 shows an example. In start-up and low/medium speed situations, the transmission first operates in the 1st gear range of the fixed gear ratio and then operates in the 2nd gear range. Finally, the V-belt device with a stepless transmission function is employed for high Fig. 6. A fixed gear ratio and CVT system.
− 718 − Hybrid Transmissions for Motorcycles and the output axis of the CVU, respectively; we have ± T (ω ± ω ) ηc r r c ω −ω sr = ω ω for Ts( s c)>0, (7) × T s( s ± c) V max (D in)max (D out)max V t = = . (3) V (D ) × (D ) ηc ηc ω −ω min in min out min sr = rs =1 for Ts( s c)=0, (8) According to the performance requirements of ω ω ± T s( s ± c) motorcycles and the space constraints placed on the ηc = for T (ω −ω )<0. (9) rs T (ω ± ω ) s s c transmission, the maximum speed ratio of the trans- r r c mission system (r ) and the speed ratio of the dif- max Here, ηc ( ηc ) is the efficiency of the power flowing ferential gear (R) can be chosen. If the pulley diameters sr rs from the sun gear to the ring gear (the ring gear to the of the CVU are selected, the minimum speed ratio of sun gear) with the planet carrier fixed. the transmission system (r ) and the speed ratio of min When the simple differential gear is used for the chain device (K) can be obtained from Eqs. (2) and input-coupled configuration differential transmission (3). systems (Fig. 3), the sun gear (planet carrier, ring gear) of the differential gear is adjacent to the input (output, IV. Mechanical Efficiency CVU). The subscript s, c, and r are rewritten as a , b, and v, respectively. Hence, the relative power of It is well known that there are three types of the the differential gear can be expressed as T (ω −ω ) power flow in the differential transmission: no s s c = T ( ω ± ω ). Substituting r=ω /ω and K= ω /ω recirculation, positive recirculation, and negative a a b b a a a into this expression gives recirculation. In analyzing the mechanical efficiency of differential transmissions, these three types should ω ω r T s( s ± c)= T a ω a (1 ± ). (10) be identified first. Let T a (Tb, Tv), Pa (Pb, Pv), and K ω (ω , ω ) be the torque, the power, and the angular a b v Based on Eq. (10), the directions of the relative velocity adjacent to the input axis (output axis, CVU) power of the differential gear, over the full transmis- of the differential gear, respectively, as shown in Fig. sion speed ratio range, can be identified as T (ω −ω )> 3. With no energy loss, the relation among the external s s c 0 for r
From Eq. (6), these three types of power flow in Using Eq. (4), the torque equilibrium requirement, and an input-coupled differential transmission can be iden- Eqs. (5) and (12), we have tified by using the ratio of the transmission input power to the power carried by the CVU of 0
1, and Pcv/Pin<0, respectively. = (13) T ηc The concept of relative power for computing the b sr(R ±1)±R mechanical efficiency of the differential gear is used and here to derive the efficiency formulas of the differential ω ω ω c transmission. Let Ts(Tr, Tc) and s( r, c) be the torque T η (1 ± R) v = sr . (14) and angular velocity adjacent to the sun gear (ring gear, ηc T b sr(R ±1)±R planet carrier) of the simple differential gear, respectively. Then, the mechanical efficiency of the As seen in Fig. 3, if there is no power loss in the differential gear can be expressed as joints of the links and the power flow along a single
− 719 − K.B. Sheu et al. link has an algebraic sign, then the relationships of the power flow for the no recirculation system are
Pin=−Pa−Pcv,
ηvPcv=−Pv, (15)
η − kPa= Pa ,
Pout=Pb.
Here, ηv and ηk denote the efficiency of the CVU and the chain device, respectively. Solving Eq. (15), the overall mechanical efficiency of the transmission sys- Fig. 7. Force balance of the V-belt device. tem becomes
± P out ± P b along with the speed ratio of the CVU are four key η = = . (16) 0 P P P variables that can be used to control the drive. The in a + v η η η force balance between both Fvr and Fvn determines the ka v kv actual speed ratio of the CVT in a running drive. It For the speed ratio range as r
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operating condition. The housing contact angle is related to the location of the centrifugal roller as shown in Fig. 9(a). The axial displacement of the movable flange (Sr) can be expressed as
Sr=x0−x1+Smsinδ
ρ 2 2 =(± r mo) ±(r mo + y off)
ρ 2 δ 2 δ ± ( ± r mo) ±(r mo + y off+ S mcos ) + S msin , (20)
where
Sm displacement of the roller along locus of the roller center,
ρ radius of curvature of the roller housing,
rmo radius of the centrifugal roller,
yoff distance between the center point of the roller housing and the shaft surface indicated in Fig. 9(b).
Fig. 8. A mechanical-type CVT.
Rc normal force exerted by the roller housing,
µb coefficient of friction between the roller and roller contact plate,
µc coefficient of friction between the roller and roller housing,
m total mass of the centrifugal roller,
ym distance between the center of the roller and the shaft centerline,
δ angle between the roller back contact plate and a line perpendicular to the shaft centerline,
γ angle between the axial direction and the tan- gent at the point of contact between the centrifu- gal roller and the roller housing,
ω input angular velocity.
From Eq. (19), if the housing contact angle (γ) is found, the axial force can be obtained for a given Fig. 9. Control parameters of the CVU driver pulley.
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torque-sensing mechanism and the shaft centerline,
Da diameter of the helical cam,
Fp compression preload of the torsion-compres- sion spring,
Kn spring rate of the torsion-compression spring,
Fig. 10. Control parameters of the CVU driven pulley. µa coefficient of friction on the helical cam.
When a driver and driven pulley are combined in According to the relationship between the pulley a drive, the axial force applied to the belt by the driven axial travel (Sr) and belt pitch diameter (Dr), the axial pulley is transmitted to the driver pulley. When the displacement of the movable flange can be expressed axial force (Fvr) of driver pulley exerted by the belt as and the axial force (Fcl) of the driver pulley supplied α by the centrifugal roller is balanced, the drive is op- S =( D ± D ) tan , (21) r r ro 2 erated under a steady state conditions. The equations that relate the belt tension to the where Dro denotes the minimum pitch diameter of the axial force of pulley are (Worley, 1955) driver pulley and α is the groove angle of the pulley. θ µ α From Eqs. (20) and (21), the travel of the roller along F 1 r 1± tan( /2) F vr = [ ], (24) the locus of the roller center point (Sm) can be obtained 2 µ + tan(α/2) as a running condition that is operating at a given pitch cos(α/2) ± µsinφ sin(α/2) diameter Dr. Then, the angle between the axial direc- F =(F ± F )[ ], (25) tion and the tangent at the point of contact between the vn 1 2 2µcosφ centrifugal roller and the roller housing is µθ φ F 1 ncos δ = exp [µ φ α α ], (26) r mo + y + S mcos F 2 sin cos( /2) + sin( /2) cosγ = off . (22) ρ ± r mo where Referring to Fig. 10, the driven pulley has a θ movable flange that can slide axially along the shaft. r belt wrap angle on the driver pulley, The vehicle load on the driven shaft is converted to θ an axial force on the belt in the groove by the helical n belt wrap angle on the driven pulley, cam. Based on the force equilibrium exerted on the movable flange by the torsion-compression spring (Fs), F1 belt tension of the tight side, the belt force (Fa), and the torque (Ft), due to the vehicle load, the axial force of the driven pulley (Fvn) operating F2 belt tension of the slack side, under an impending opening condition with a pitch φ diameter (Dn) and the belt tension difference (F1−F2) friction angle, can be expressed as µ coefficient of friction between the belt and pulley, Fvn=Fa+Fs
F ± F cosβ ± µ sinβ α groove angle of the pulley. D n × 1 2 × a = β µ β D a 2 sin ± acos In conclusion, the procedure for analyzing the +Fp+Kn(Dno−Dn)tan(α/2), (23) speed ratio of a mechanical-type CVT used in a dif- ferential transmission for a motorcycle can be summa- where rized as follows: (1) Based on the external load of the motorcycle and β angle between the helical cam surface of the the torque ratio of the differential gear, from Eq.
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(14), the tension difference of the belt operating Table 1. Specifications of the CVU of the Prototype under a pitch diameter Dn is (F1−F2)=Tv×Dn/2. β ° (2) From Eq. (23), the axial force (Fvn) of the driven Torque ramp angle ( ) 50.76 pulley supplied by the vehicle load can be solved. Preload of torsion-compression spring (Fp) 39.25 kg (3) Solving Eqs. (23), (25), and (26), the tight side Spring rate of torsion-compression spring (Kn) 219.33 kg/m Radius of curvature of roller housing (ρ) 0.0252 m tension (F1) can be obtained. Location of center of curvature of movable flange (4) From Eq. (24), the axial force of the driver pulley housing (yd) 0.019 m (Fvr) due to the external load converted by the Mass of centrifugal roller (m) 0.0909 kg belt can be obtained. Angle of roller back contacted plane (δ) 27.7° (5) Based on Eq. (19), the axial force of the driver Location of roller at zero rpm (ymo) 0.0251 m pulley (Fcl) due to the input angular speed (ω) can be obtained. (6) When the axial force (Fcl) of the driver pulley due to the input angular speed and the axial force (Fvr) of the driver pulley due to the external load are balanced, i.e., Fcl=Fvr, the input angular speed under static state conditions can be identified. The computer program for analyzing and design- ing the mechanical-type CVT and the CVU of a hybrid transmission can be developed on the basis of the design procedure described above.
VI. Prototype Developing and Test- ing Fig. 11. Analytical operating characteristics of the hybrid transmis- In this investigation, four new design concepts of sion system. hybrid motorcycle transmissions are proposed. A pro- totype of the fixed gear ratio and differential transmis- sion system is developed. An existing CVT that has (8) Parameters of the control unit of the CVU, the following characteristics is used: (9) Clutches and brakes. (1) The limited speed is 90 km/hr. The minimum speed ratio of the transmission (2) The start-up acceleration capability is greater system affects start-up acceleration. Based on the than 0.3 g, where g denotes the gravity acceler- mechanical efficiency of the differential transmission, ation. from Eq. (6), if RK tends to be a maximum, then Pcv/ (3) The minimum speed ratio of the CVT is 0.38, Pin tends to be a minimum and the mechanical effi- the maximum speed of the CVT is 1.14, the final ciency of the differential transmission tends to be a reduction ratio is 0.128, and the rear wheel maximum. Therefore, considering the start-up accel- diameter is 0.42 m. eration capability of and the space constraint placed (4) The engine speed for the variable ratio range of on the existing CVT, the speed ratio of the fixed gear the CVT is 3500-6500 rpm. ratio (r1), the speed ratio of the differential gear (R), The fixed gear ratio and differential transmission and the speed ratio of the final drive assembly (rf) of system shown in Fig. 4 operates in two modes: the fixed the prototype are: R=r1=0.384 and rf=1/7.8=0.1282. gear ratio regime and the differential transmission An existing simple differential gear used in a 3- regime. The design items include: speed motorcycle is adopted in this work. The teeth (1) The speed ratio of the fixed gear ratio regime of the simple differential gear are: ring gear, Zr=85; (r1), sun gear, Zs=53; and planet pinion, Zp=15. The teeth (2)The minimum speed ratio of the differential of the final drive assembly are: Zf1=15, Zf2=42, Zf3=14, transmission regime (rmin), and Zf4=39. The maximum speed ratio of the differ- (3)The maximum speed ratio of the differential ential regime is rmax=1.11. transmission regime (rmax), After the ratio range of the transmission system (4) The gear teeth of the simple differential gear, is selected, the pulley diameter and the center distance (5) The gear teeth of the final drive assembly, must be checked against the available space envelope (6) The gear teeth of the chain device, of the motorcycle. An existing pulley in the scooter (7) The Pulley diameters of the CVU, that has a center distance of 230 mm is adopted here:
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analytical results for the mechanical efficiency of the prototype hybrid transmission and CVT under the condition of a constant driving load, and Fig. 13 shows the results for the operating line on an engine operation map. A test rig was designed and constructed to mea- sure the mechanical efficiency of the prototype, as illustrated in Fig. 14. This system is powered by a 3- phase a.c. induction motor. The maximum power output and speed of the motor are 10 hp and 9000 rpm, respectively. The vehicle loads, such as aerodynamic drag, rolling friction, and the mass of the motorcycle, were simulated by a magnetic brake. The input and output power were measured by speed and torque sensors that were attached to the input and output shafts. A Fig. 12. Analytical results for the mechanical efficiency of the hybrid personal computer was used to control the motor speed, transmission system and CVT. to apply the simulated vehicle load, and to collect the test data. Figure 15 shows one of the results for the oper- ating characteristics of the hybrid transmission system, where the solid line represents the analytical results and the small circles the experimental results. Under the
Fig. 13. Analytical operating lines of the hybrid transmission system and CVT. Fig. 14. The transmission testing rig.
(Dr)min=48 mm, (Dr)max=96.7 mm, (Dn)min=82.3 mm, and (Dn)max=126.7 mm. From Eqs. (2) and (3), if the pulley diameter of the CVU is selected, the minimum speed ratio of the differential regime is rmin=0.62, and the speed ratio of the chain device is K=1.0. The teeth of the drive sprocket (Zcr) and those of the driven sprocket (Zcn) of the chain device are 22. As seen in Fig. 4, clutches Ca and Cv are the idle clutch and the transferred clutch of the transmission system, respectively. The existing centrifugal clutch in the scooter is used as the transferred clutch, and an overrunning clutch is used as the brake B. The idle clutch, Ca, is ignored. Table 1 shows the specifications of the CVU of the prototype. Figure 11 shows the analytical results for the operating characteristics, Fig. 12 shows the Fig. 15. Operating characteristics of the hybrid transmission system.
− 724 − Hybrid Transmissions for Motorcycles
The kinematic design of the hybrid transmission sys- tems has been employed to obtain the relationship between the major dimensions. Equations of mechani- cal efficiency of the differential transmission have been derived. A test rig has been constructed to measure the transmission efficiency and to test the operating characteristics. A prototype of the hybrid transmission for motorcycles has been built and tested. The results show that the operating characteristics of the hybrid transmission meet the design goal. However, the trans- mission efficiency is not better than that of the existing CVT since the parts of the prototype were roughly fabricated. The overall results show that the proposed designs are theoretically correct and practically feasible. A patent was granted (Yan and Sheu, 1998), and several others have been filed. Fig. 16. Mechanical efficiency of the hybrid transmission system and CVT. Acknowledgment condition of a constant driving load, the analytical and The authors are grateful to the National Science Council of experimental results for the operating characteristics the Republic of China for its support of this research through Grants of the prototype are generally in good agreement. NSC 85-2221-E-006-033, NSC 86-2212-E-006-052, and NSC 87- 2218-E-006-020 to National Cheng Kung University. Figure 16 compares the experimental results for the mechanical efficiency of the prototype and an existing CVT. As shown in Fig. 16, the mechanical Nomenclature efficiency of the hybrid transmission system was not better than that of the CVT. Comparing Fig. 12 and Ca, Cv Clutches adjacent to the input shaft and CVU of the transmission Fig. 16, the analytical and experimental results are B Brake adjacent to the axis of the differential gear quite different especially for the low speed situation. Din, Dout Diameters of the pulley adjacent to the input and Referring to Fig. 4, as the hybrid transmission system output axes of the CVU was operated in the fixed gear ratio regime, the power F1 Belt tension of the tight side was transmitted via the high efficiency components of F2 Belt tension of the slack side Fcl Axial force of the driver pulley supplied by the cen- the chain device, the differential gear, and the final trifugal roller drive assembly. For reasons of low cost and simplicity, Fvr Axial force acting on the driver shaft the components of the prototype of the hybrid trans- Fvn Axial force acting on the driven shaft mission system were employed by modifying the com- Fp Preload of the torsion-compression spring ponents used in existing motorcycles. Apparently, the G Chain device assembly K Speed ratio of the chain device components used in the prototype of the hybrid trans- Kn Spring rate of the torsion-compression spring mission system are rough products. We believe this m Mass of the centrifugal roller is the main reason why no high efficiency advantage P a (Pb, P v) power adjacent to the input axis (output axis, CVU) for the hybrid transmission system is apparent. of the differential gear Pcv Power carried by the CVU r Speed ratio of the transmission VII. Conclusions rf Speed ratio of the final drive assembly rmax, rmin Maximum and minimum speed ratios of the transmis- A systematic approach to designing new auto- sion matic transmission systems for motorcycles, including rmo Radius of the centrifugal roller a conceptual design, kinematic design, efficiency R Relative speed ratio of the planetary gear set Sm Displacement of the roller along locus of the roller analysis, engine and transmission matching, and pro- center totype developing and testing has been proposed in this T a ( Tb, T v) Torque adjacent to the input axis (output axis, CVU) study. Four design concepts of the hybrid transmission of the differential gear systems have been synthesized for differential trans- Tvr Torque acting on the driver shaft mission systems with different clutch sequences. Tvn Torque acting on the driven shaft V Speed ratio of the CVU Kinematic analysis has been carried out to obtain the Vmax, Vmin Maximum and minimum speed ratios of the CVU range of the relative speed ratio of the differential gear. Vt Speed range of the CVU
− 725 − K.B. Sheu et al.
yd Distance from the center of curvature of the movable MacMillan, R. H. and P. B. Davies (1965) Analytical study of flange housing and the shaft centerline systems for bifurcated power transmission. Journal of Mechani- ymo Distance between the center of the roller and the shaft cal Engineering Science, 7(1), 40-47. centerline when the driver pulley is not rotating Mucino, V. H., J. E. Smith, B. Cowan, and M. Kmicikiewicz (1994) Z Number of the gear teeth Parametric Modeling and Analysis of a Planetary Gear-CVT α Groove angle of the pulley Mechanism. SAE 940519, SAE, Detroit, MI, U.S.A. β Angle between the helical cam surface of the torque- Mucino, V. H., J. E. Smith, and P. K. Sharma (1995) A Double sensing mechanism and the shaft centerline Planetary Gear Train-CVT Transmission with Multiple γ Angle between the axial direction and tangent at the Applications. SAE 950094, SAE, Detroit, MI, U.S.A. contact point of the centrifugal roller and the roller Oliver, L. R. and D. D. Henderson (1972) Torque Sensing Variable housing Speed V-belt Drive. SAE 720708, SAE, Milwaukee, WI, δ Angle between the roller back contact plane and a line U.S.A. perpendicular to the shaft centerline Oliver, L. R., K. G. Hornung, J. E. Swenson, and H. N. Shapiro, θn Belt wrap angle on the driven pulley (1973) Design Equations for a Speed and Torque Controlled µb Coefficient of friction between the belt and pulley Variable Ratio V-belt Transmission. SAE 730003, SAE, Detroit, µb Coefficient of friction on the helical cam MI, U.S.A. µb Coefficient of friction between the roller and roller Sakakibara, S. and M. Hattori (1989) Continuously variable contact plate transmission. U.S. Patent 4,864,889. µc Coefficient of friction between the roller and roller Sheu, K. B. and H. S. Yan (1996) Kinematic design of hybrid housing transmissions for motorcycles. 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