Development of self-adaptively loading for planetary roller traction-drive transmission

Transcription

1 Available online Journal of Chemical and Pharmaceutical Research, 013, 5(9): Research Article ISSN : CODEN(USA) : JCPRC5 Develoment of self-adatively loading for lanetary roller traction-drive transmission Desheng Chen and Xuewen Du College of Vocational and Technical Education, Zhejiang University of Technology, angzhou, China ABSTRACT Traction drives offer certain advantages over gear drives. In articular, they generally run more quietly than geared units and rovide an oortunity for higher seed and higher efficiency by eliminating sliding motion between contacting elements. Generating adequate normal load at the frictional contact is essential for traction drives. The best way to generate the normal force is the slef-adatively loading to revent the rollers under a continuous and high-stress load. In this aer, a novel design of lanetary traction-drive device was develoed. It is a combination of a lanetary cylindrical roller configuration and a logarithmic siral rofile connection. In this device, the sun roller is hollow, and connected to the inut shaft by a logarithmic siral rofile. So as the inut shaft rotates relative to the sun roller according a certain direction, a wedging action takes lace between the inut shaft and the sun roller. This makes the sun roller generate a certain amount of elastic deformation, so an areciable amount of contact load is generated between the contacting elements. The greater the inut torque is, the stronger the strength of the wedging action, and the greater the contact load, and vice versa. So this device is self-actuating and self-adative. This design offers an alternative way to solve the roblem of self-adatively loading of the lanetary cylindrical roller traction drive. Keywords: Traction-drive, lanetary roller mechanism, self-adatively loading, rofile connection, logarithmic siral INTRODUCTION Traction drive ossesses desired erformance attributes that ertain to its unique oerating rinciles. These attributes include high mechanical efficiency, little or no backlash, low noise and vibration and high-seed caability. The ower density of a traction drive can be very high when oerated at elevated seeds. These erformance features, in conjunction with its inherent manufacturing simlicity, make traction drives suitable candidates for a host of alications. Traction drives can be broadly classified into variable-seed ratio drives and fixed-seed ratio drives. Generating adequate normal load at the frictional contact is essential for traction drives. Planetary configuration has been the referred arrangement for fixed-seed ratio drives because the internal forces are well balanced. A common ractice is to use taered raceways along the axial direction. By moving these raceways axially, a radial dislacement and thus normal force are generated. Examles of such designs were disclosed in [1, ]. Since the enveloes of the taered raceways in most design configurations do not converge to a common oint, this results in a so-called sin motion between the contacting surfaces. The sin motion not only offsets the high-efficiency otherwise offered by friction drives, but also causes comonent wear and high break away torque. Recently, a zero-sin design of lanetary traction drive was roosed by Ai and Rybkoski [3]. This design emloyed the on-aex concet similar to that of a taered roller bearing and offered sin free rolling motion at the frictional contacts. Two rows of lanetary rollers were used in a back-to-back arrangement to balance the internal axial force 498

2 on the lanetary rollers. This design featured a torque-actuated roller-ram loading mechanism, offering imroved torque caability. It looks as if the roblem of sin motion has been solved, but when the shae of the taered lanetary rollers were revised in shae in order to imrove the contact stress distribution, the sin motion would reoccur. Cylindrical annular rollers could eliminate the sin motion. owever, the torque-actuated roller ram mechanism is not alicable to this case. Instead, the normal load was generated by using oversized rollers before. The magnitude of the load was controlled by recisely re-set internal interference. This imoses severe constraints on manufacturing and assembling. Moreover, when a traction drive of such design is oerated at artial torque, contact comonents are unnecessarily overloaded. This laces the rolling comonents under a continuous, high-stress load and reduces the durability of the traction drive. So in a word, if taered rollers were used in the lanetary roller mechanism, the sin motion would occur likely; while if cylindrical rollers were used, the sin motion would not haen, but it would be required to solve the roblem of self-adative loading for the lanetary roller mechanism. This roblem has been studied by Ai et al [4] who develoed an eccentric cylindrical lanetary design and by Flugrad and Qamhiyah [5] who used a different rincile involving airs of intermediate rollers to roduce the normal force on torque demand. Also, this roblem is the subject of this aer, in which a new way to solve this roblem will be resented. (a) Initial state (b) Working state Fig. 1 A rofile connection with three logarithmic sirals Fig. A lanetary cylindrical roller traction-drive device 1. Inut shaft.. Sun roller. 3. Planetary rollers. 4. Planet carrier. 5. Outer ring. OPERATING PRINCIPLES With the develoment of numerical control machining technology, logarithmic sirals have been used for rofile keyless connections [6], as seen in Figure 1. Insired by this, a new tye of lanetary cylindrical roller traction-drive device was develoed, as shown in Figure. It is comosed of a sun roller, three lanetary rollers, an outer ring, an inut shaft, and a lanet carrier (outut shaft). The sun roller is hollow, and is attached to the inut shaft through a logarithmic siral rofile. When the inut shaft rotates relative to the sun roller in a certain direction, a wedging action takes lace between the inut shaft and the hollow sun roller. This makes the sun roller roduce a certain amount of radial dislacement. Thus, an areciable quantity of normal load is generated between the sun roller and the lanetary rollers as well as between the lanetary rollers and the outer ring. The greater the inut torque, the stronger the strength of the wedging action, and then the greater the contact load, and vice versa. So this 499

3 configuration is self-actuating. ere, as cylindrical rollers are used in the lanetary traction-drive device, such a sin motion as would resent in the case where taered rollers were used will not occur. GEOMETRIC DESIGN FOR PROFILE CONNECTION In mathematics, a logarithmic siral refers to the track of a moving oint its moving direction kees a fixed angle λ with its olar radius, as seen in Figure 3. The olar coordinate equation of logarithmic sirals can be exressed m as ae, where a and m are constants, is referred to as the olar angle, and is referred to as the olar radius. In a logarithmic siral, a decides the size of the logarithmic siral, while m decides the shae. When a logarithmic siral is used in a rofile connection, the value of m decides the magnitude of the wedge angle, which in turn decides the frictional self-locking status of the curved wedge. ere, the rofile connection is used for self-adative loading, so it is not ermissible for it to be self-locking. That is to say, the following relationshi must be satisfied: tan tan g(90 ) 1/ tan m f where f is the coefficient of friction between the shaft and the hub. Fig. 3 Geometry of logarithmic siral If a logarithmic siral rofile connection is used to transmit torque, its cross-section contour only requires being caable of centring, and normally consists of three segments of logarithmic siral. ere, the logarithmic siral rofile connection is to be used for the lanetary cylindrical roller traction drive. The aim of doing so is to obtain the radial elastic deformation of the sun roller and then to obtain the normal force needed for traction drive through the rofile connection between the inut shaft and the hollow sun roller. Under the condition that the inut torque is constant, the greater the amount of radial elastic deformation of the hollow sun roller, the greater the normal forces between the rollers of the traction-drive device, and then the higher the caacity of the transmission. In order to obtain a greater elastic deformation under the condition that the minimum wall thickness of the hollow sun roller satisfies the tensile strength, a thin average wall thickness is desired. If only three segments of logarithmic siral were emloyed in the cross-section outline of the rofile connection, the length of each segment of logarithmic siral would be longer, and the average wall thickness of each segment of wall would be thicker, and then the elastic deformation of the hollow sun roller would be smaller. Increasing the number of segments of logarithmic siral would effectively reduce the average wall thickness. Considering that the wedging stroke is not long, six segments of logarithmic siral could be selected. Figure 4 shows the cross-section contour of the rofile connection. It consists of six segments of logarithmic siral, and in each segment of logarithmic siral, the value of is defined as from 1 0to 11π/36, and a straight line or arc is adoted to connect the two adjacent segments in order to reduce the stress concentration. 500

4 m ae 11 π (0 ) 36 Fig. 4 Interface contour of rofile connection FORCE ANALYSIS In order for the lanetary traction-drive device to work roerly, relative sliding is not ermissible between the rolling elements. This can be referred to as the frictional self-locking condition for this device, and can be further exlained as follow: When a torque is alied to the inut shaft, a wedging action takes lace between the inut shaft and the hollow sun roller. The wedging action makes the hollow sun roller roduce a certain amount of radial elastic deformation. This makes the hollow sun roller aly a contact load to the lanet rollers and outer ring. This contact load is needed for the lanetary traction drive. To revent sliding between the rolling elements, the frictional forces generated by the contact load should overcome the working torque alied to the lanetary traction-drive device. Contact Stress of Profile Connection Provided there is an ideal condition to the fit between the inut shaft and the hollow sun roller, and a torque is alied to the inut shaft, as shown in Figure 5, then an areciable amount of contact stress is thus generated at the six sections of working surface of the rofile connection. Taking any oint P from the AB section, so in the infinitesimal ds that contains the oint P, the summing of the moments about oint o will roduce the following exression: dt ( sin lds f cos lds) (sin f cos ) lds (1) where is contact stress between two fitted surfaces, is olar radius of the oint P, l is fit length between two fitted surfaces, is included angle between and, and f is coefficient of friction between two fitted surfaces. Fig. 5 Calculation model of contact stress of rofile shaft From the characteristics of the logarithmic siral, we have m π /, ae and ds d Substituting these relationshis into Equation (1) leads to dt (sin f cos ) ld (sin f cos ) ld 501

5 So, the total moment exerted by the three ieces of working surface of the rofile connection is as follows: T 6 11π 36 0 dt 6 11π 36 0 a l(sin cos )e m d 11mπ 18 3a l(sin f cos )(e m 1) Thus, the contact stress is mt () 11m π 18 3al(sin f cos )(e 1) Contact Load between Rollers P In this design of a lanetary roller traction drive, as the contact load for the traction drive is roduced through the elastic deformation of the hollow sun roller, so its value can be calculated by analysing the elastic deformation of the hollow sun roller. In order to simlify the theoretical calculation, the hollow sun roller its wall thickness is actually not uniform, can be seen a uniform thickness sleeve, as seen in Figure 6. In this simlified model, the inner wall of the sleeve is subjected to a uniformly distributed load and constrained by the inut shaft; the outer wall is constrained by three equally distributed lanetary rollers, so it is subjected to three equally distributed contact loads (reactions) P. Obviously, this sleeve is an internal-force statically indeterminate system. According to the rincile of deformation suerosition, two deformation quantities, u and, can be calculated resectively. ere, u is the radial deformation quantity of the sleeve roduced by the internal ressure, and is the contact deformation quantity between the rollers caused by the three contact loads P. Then, according to the deformation comatibility condition of the sleeve at oint C, that is, the contact load P could be calculated. u Fig. 6 Free-body diagram of sleeve Calculation of Radial Deformation Quantityu : The calculating model of the radial deformation quantity of this thin-walled sleeve is shown in Figure 7. The calculating formula of the radial deformation of a thin-walled sleeve is [7] u r1 Et (3) where r 1 is the radius of the sleeve, t is the wall thickness of the sleeve. 50

6 Fig. 7 Calculation model of radial deformation of sleeve Calculation of Contact Deformation Quantity : The calculation model of the contact deformation quantity of this lanetary roller traction-drive device is shown in Figure 8. There are two laces at which the contact deformation occurs: one is between the sun roller and the lanetary rollers; the other is between the lanetary rollers and the outer ring. The quantities of contact deformation at the two laces are named and i e resectively. In engineering, Palmgren s formula is usually used to calculate the line-contact elastic deformation [8], so we get P i e (4) 0.8 L where L is the contact length of two contacting bodies. Fig. 8 Calculation model of contact deformation of traction-drive device Deformation comatibility equation: Provided that only the contact elastic deformation is considered while the whole elastic deformation is neglected for the lanetary roller and the outer ring, then the deformation comatibility equation can be exressed as u (5) i e Self-locking Condition for Traction-drive Device If is referred to as the coefficient of friction between the rollers of the traction-drive device, the condition to revent sliding between the rollers is as below: 3 Pr 1 T (6) Trying to solve the equations (), (3), (4), (5) and (6) simultaneously, and noting that, theoretically the frictional self-locking condition for the lanetary roller traction-drive device can be obtained, but it is difficult to 503

7 write out the analytic solution. STRENGT DESIGN In order to simlify calculations, the following strength conditions are based on an assumtion of simle stress. Comressive Strength Condition The comressive strength condition for the rofile connection air is [ ] (7) where is the comressive stress of the rofile connection air, [ ]is the allowable comressive stress. Tensile Strength Condition The hollow sun roller is taken as a thin-walled sleeve, and squeezed by the rofile shaft, so it is subjected to a circumferential tensile stress. The circumferential tensile stress may make the hollow sun roller roduce the lastic deformation and even ruture. Therefore, the tensile strength condition for the hollow sun roller is r1 [ ] (8) t where is the circumferential tensile stress of the hollow sun roller, [ ] is the allowable tensile stress. Contact strength Condition According to the ertz theory, the contact strength condition for two cylindrical bodies is P 1 1 ( ) L 1 [ π E1 E ] (9) in which, is maximum contact stresses for two bodies, [ ] is allowable contact stresses for two bodies, 1 and are radius of curvature for two bodies resectively, 1 and are Poisson s ratios for two bodies resectively, E1 and E are modulus of elasticity for two bodies resectively. DESIGN EXAMPLE If the device as shown in Figure is to be used for a seed reducer, and taking the sun roller as the inut member, the lanet carrier as the outut member and the outer ring as the stationary member, then the seed ratio of this device would be i 1 r3/ r1, where r1 and r 3 are the radius of the sun roller and the outer ring. Suosing i 4 is required for a certain alication, then a radius of 1mm for the sun roller and 63mm for the outer ring will satisfy the seed requirement. Once the radius of the sun roller is known, the thickness of the sleeve at the thinnest wall t and the constant of logarithmic siral a could be determined. The thinnest wall t can be calculated through the use of Eq. (8), and the constant of the logarithmic siral a can be calculated through the following formulas: a r ( t ) 1 11mπ m in which, ( e m1 36 a e ) a (e 1). After the geometric arameters of the lanetary roller traction-drive device are determined, solving equations (), (3), (4), (5) and (6) simultaneously, and noting that, the required minimum friction coefficient to ensure self-locking for the lanetary roller traction drive can be obtained. The inut shaft is made of oil-quenched and temered (OQ&T) 5140 steel (American Iron and Steel Institute (AISI)), and the hollow sun roller is made of OQ&T 5100 steel. From the American Society of Metal(ASM) Metals Reference Book [9], the yield strength of OQ&T 5140 steel is =550, and the yield strength of OQ&T S 504

8 5100 steel is S =1700Ma. If the safety factor of the rofile connection is selected as S 1. 5, then the allowable comressive stress between the inut shaft and the hollow sun roller is s / S =550/1.5=440Ma, and the allowable tensile stress of the hollow sun roller is s / S =1700/1.5=1360Ma. As stated in [5], the allowable contact stress of the rollers and raceways is [ ] =4000~4500Ma. aving determined the geometric arameters of the traction-drive device, substituting the inut torquet into Equation (), is obtained; substituting into Equation (3), u is obtained; substituting i e u into Equation (4), P is obtained; substituting P andt into Inequality (6), the frictional locking condition, that is, the minimum value of could be obtained; substituting substituting P into the left term of Inequality (9), 1 and Table. Table 1 Calculations at into the left term of Inequality (8), is obtained; is obtained. The calculations have been summarized in Table t.3mm T N m a mm Table Calculations at t 1.8mm T N m a mm The results of calculation for in Table 1 and Table show: (1) Under a certain wall thickness condition, as the inut torque increases, the value of the minimum friction coefficient needed to satisfy the self-locking condition for the traction drive will decline slightly. This exlains that the self-locking condition for the traction drive is mainly relevant to the geometric arameters and the elastic modulus of the material, and has little relevance to the inut torque. () Under a certain inut torque condition, as the wall thickness reduces, the value of the minimum friction coefficient needed to satisfy the self-locking condition for the traction drive will decline obviously. This is because, as the wall thickness reduces, the radial elastic deformation of the sun roller increases. Thus, the contact loads between the rollers increases, so the value of the minimum friction coefficient needed declines. But the reduction in the wall thickness is limited by the tensile strength of the hollow sun roller. Based on the receding analysis, it is shown that the calculation results are logical. This shows that the foregoing theoretical calculation has certain reliability. The results of calculation for, and have also been listed in Tables 1 and, so as to comare them with their allowable stress, and then to judge the allowable maximum value of the inut torque. CONCLUSION Based on the logarithmic siral rofile connection, a novel cylindrical roller traction drive device was develoed. As cylindrical rollers are used in this design, such a sin motion as would resent in the lanetary taered roller traction drives will not occur, and as a logarithmic siral rofile connection is utilized in this device, the device is self-actuating and self-adative, so a new way of self-adative loading for lanetary cylindrical roller traction drives has been obtained. In this way, the device has higher caacity to transmit ower comared to the eccentric cylindrical lanetary device [4] due to its multi-roller drive and has fewer transmission elements comared to the 505

9 device roosed by Flugrad and Qamhiyah [5]. This research not only exands the method of self-adative loading for cylindrical roller lanetary traction drives, but also extends the utilization of logarithmic siral rofile connections. Future research is lanned to exerimentally measure the efficiencies for the roosed design. The stress distribution between the rolling elements will be determined by finite element analysis. The stresses will then be used to redict the exected fatigue life of the system, and exerimental tests will be conducted to verify the life redication. Acknowledgment This research was suorted by Zhejiang Provincial Natural Science Foundation of China (ZJNSF No. Y ). REFERENCES [1] LO ewko. AIAA J. ydronaut, 1968,, [] R Zhou. U.S. atent, 1997, No. 5, 688,01. [3] X Ai; T J Rybkoski. U.S. atent, 000, No. 6, 095,940. [4] X Ai; M Wilmer; D Lawrentz. ASME J. Tribol., 005, 17, [5] DR Flugrad; AZ Qamhiyah. ASME J. Mech. Des., 005, 17, [6] YY Zheng. Journal of Machine Design, 003, 0(1), (In Chinese). [7] JR Barber. Intermediate Mechanics of Materials, McGraw-ill Comanies, Inc., U.K, 001. [8] A Palmgren. Ball and Roller Bearing Engineering, 3 rd Edition, Philadelhia, Burbank, [9] M Bauccio. ASM Metals Reference Book, 3st Edition, O, ASM,