DESIGN, CONSTRUCTION AND TEST OF A TWO-STAGE PLANETARY TRACTION SPEED REDUCER

Transcription

1 DEIGN, CONTRUCTION AND TET OF A TWO-TAGE LANETARY TRACTION EED REDUCER Euaro Lobo Lustosa Cabral Escola olitécnica a Universiae e ão aulo Departamento e Engenharia Mecatrônica e istemas Mecânicos Av. Mello Moraes, 31 elcabral@usp.br Fabrício unahara Nagahashi Escola olitécnica a Universiae e ão aulo Departamento e Engenharia Mecatrônica e istemas Mecânicos Av. Mello Moraes, 31 fsnaga@uol.com.br Marcos Costa Hunol Escola olitécnica a Universiae e ão aulo Departamento e Engenharia Mecatrônica e istemas Mecânicos Av. Mello Moraes, 31 mahunol@usp.br Abstract. This work presents the esign an test of a two-stage planetary traction spee reucer. This is a precision reucer that presents low spee an torque fluctuations, low vibration level, an zero backlash. This reucer oes not have gears, transmission is mae by the friction between smooth surfaces, which enables the reucer to have zero backlash. A traction flui is use forming a film pa between the surfaces avoiing the metal-to-metal contact. Theoretical stuies concerning the principles of movement an torque transmission, the Hertz contact stresses, an the contact fatigue criteria are performe. The Hertz contact stresses combine with the contact fatigue criterion are use to preict the life of the reucer. The reucer s configuration, working principle, kinematics an force analysis are escribe. Finally, tests are performe to etermine the reucer precision. These tests are base on measurements of angular transmission error, velocity fluctuation, torsional stiffness an backlash. Results of the angular transmission error an backlash permitte to conclue that the reucer coul be classifie as a precise evice. Keywors: planetary rive, traction rive, contact stresses, position accuracy. 1. Introuction Many inustrial types of machines require high precision transmission systems. The angular spee reucers are among the most use evices for this en. The characteristics that etermine the precision of a spee reucer are: low torque an spee fluctuation; high torsional stiffness; zero backlash; an low vibration level. Toay the spee reucers most use in high precision machines are: worm an gear, harmonic, traction, an cycloial. Worm an gear reucers present low efficiency an high wear of the teeth that reuce its life. The precision cycloial reucers are very epensive because the gears profile is ifficult to manufacture with the require precision an surface roughness. The harmonic rivers present the highest torque to weight relation among all the others spee reucers, however they have low stiffness. All these reucers have a common rawback, they are mae by gears, which inevitably causes backlash an spee fluctuations limiting their use in high precision systems. A traction spee reucer oes not have teeth; the movement an torque transmission is performe by the friction between two surfaces separate by a film of a special lubricant flui. This kin of transmission eliminates backlash, thus improving the precision of the system. Also the absence of teeth ecreases the torque an spee fluctuations, as well as noise an vibration level. Therefore, the traction reucers present several avantages over the other types of reucers in almost all the aspects that etermine precision. A traction transmission is characterize by the presence of a flui film pa between the soli surfaces impeing the metal-to-metal contact. The metal-to-metal contact must be avoie to prevent surface amage. The torque an movement are transmitte from one surface to the other by the flui film shear resistance. The flui in the space between the two surfaces is submitte to high compression forces. The traction fluis have the property to increase their viscosity several times when they are submitte to high pressures. The traction coefficient efine as the relation between the tangential an the normal forces between two surfaces separate by a flui film is the most important esign parameter for a traction spee reucer. Its value together with the require maimum torque etermines the pre-loa forces (normal forces), the reucer life an the size of the reucer's parts. Accoring to Kraus (199) the traction coefficient provie by a given flui epens on several parameters involve in the transmission, such as: contact pressure, temperature, surface velocities, an contact geometry. The most use traction flui for transmissions is the antotrac-50, which provies the highest traction coefficient among all the traction fluis available accoring to a comprehensive stuy mae by Kraus (199) an Loewenthal an Rohn (1985).

2 . The two-stage planetary traction spee reucer The two-stage planetary traction spee reucer present in this paper consists on an improvement over other esigns evelope by Cisneros (1996) an Hunol (1999). Cisneros (1996) evelope a compose two-stage planetary traction reucer. The transmission ration of this reucer was 1:81, an it ha the uniqueness to be very compact, but its pre-loa system was base on geometric interference between the reucer elements, thus making the assembly process ifficult. Hunol (1999) evelope a single stage planetary traction reucer with a transmission ratio of 1:9, which ha essentially the same esign characteristics of the two-stage reucer presente in this paper. The two-stage planetary traction spee reucer is shown schematically in Fig. (1). The two stages are compose by a "sun", three "planets", a eternal ring, an a "arm". The reucer input shaft is the sun of the first stage. The operation of both stages is similar. The sun transmits torque an movement to the planets. The eternal ring is fie. The planets are in contact with the sun an the eternal ring, so as the sun rotates the planets rotates an their centers escribe a circular trajectory aroun the sun. The planets aes are connecte to the arm. The arm of the first stage is fie with the sun of the secon stage, an the arm of the secon stage is the output shaft. eternal ring arm planets output shaft sun of the º stage (fie to the arm of the 1 o stage) arm planets sun of the 1º stage (input shaft) Figure 1. chematic of the two-stage planetary traction spee reucer. The imensions of the sun, planets, an eternal ring are aopte to be the same for both stages. Therefore, in the kinematics analysis that follows this aoption is implicit..1. Kinematics analysis The relations among the iameters of the sun, planets an eternal rings are etermine by the esire transmission ratio through a kinematics analysis. Figure () shows a frontal view of one stage of the planetary reucer. eternal ring sun p s O B A v A planet Figure. Frontal view of one stage of the planetary traction reucer. ince point B is an instantaneous center of rotation its linear velocity is zero, thus, from simple kinematics consierations the velocity of point A (v A ) is given by: v A = =, (1)

3 where is the angular spee of the sun, which is the input shaft of the stage, is the iameter of the sun, p is the angular velocity of the planets, an p is the iameter of the planets. From Eq. (1) the angular velocity of the planets is, =. () Again, from simple kinematics consierations the velocity of point O (v O ), the center of the planets, is given by: v O = =. (3) 4 As the center of the planets are linke to the arm, the angular velocity of the arm ( b ) is given by: vo b = =, (4) r ( + ) b p where r b is the arm raius. As the output shaft for each stage is the arm, the transmission ratio for one stage (i) is given by: i = b = ( + ). (5) The transmission ratio efine for the spee reucer is 1:81, with each stage responsible for an equal transmission ratio of 1:9. Thus, the relation between the iameters of the planets an the sun obtaine from Eq. (5) is the following: ( + ) = 3.5. (6) 9 p = Aopting for the sun a iameter equals to 10mm results into planets with iameters of 35mm an an eternal ring with iameter of 80mm. 3. The two-stage planetary traction reucer esign ince the imensions of the sun, planets an eternal ring are efine from kinematics an imensional consierations, the esign of the traction planetary reucer is limite to etermine its life given a require output torque. This section presents the equations use to calculate the forces an stresses applie in the reucer elements, an the life calculation. As the imensions of the elements of both stages are the same an the torque applie in the secon stage is greater in the first stage, only the secon stage may be consiere in the stress an life analysis of the reucer Force analysis In a traction spee reucer the maimum output torque is function of the normal forces between the elements in contact. Therefore, the relation between the maimum torque transmitte an the require normal force between the planets an the sun is essential for the esign process. ince there are three planets in contact with the sun, the input torque in the secon stage of the reucer (T e ) is given by: 3 =, (7) Te Ft where F t is the tangential force in the planets of the secon stage. The tangential force in the planets is relate to the normal force (F n ) by the traction coefficient (µ) accoring to: F = µ F. (8) t n ubstituting Eq. (8) into (7) results into an epression for the input torque in the secon stage as function of the normal force in the planets, i.e., 3 = µ. (9) Te Fn

4 Assuming that there is no energy loss in the reucer, the input torque in the secon stage can be relate to the output torque (T s ) through the transmission ratio of the secon stage, i, i.e., T s = T e i. Thus, the normal force in the contact between the planets an the sun of the secon stage as function of the output torque is given by: F n = Ts 3µ i. (10) 3.. tress istribution an contact fatigue ince normal forces press the elements of the reucer to guarantee their contact the Hertz contact stress theory forms the basis to calculate these elements. However, the Hertz contact stress theory is applie only for static conitions an pure rolling (without sliing) between two surfaces. When rooling is accompanie by sliing the stress fiel is istorce by the tangential force (the traction force in the case of the traction reucer). mith an Lui (1953) evelope an anlysis incluing the effects of the tangential force over the stress istribution in the contact region. They conclue that the tangential force has a significant effect on the stress fiel an the stresses can be separate into normal an tangential components. The theory presente in this section is taken from Norton (1998) an from mith an Lui (1953). Nagahashi (00) performe a stuy where he verifie that the principal stresses appear at the surface of the contact region. Therefore, the equations presente here are for the stresses at the surfaces in contact an are specialize for the case of cylinrical boies. For two cylinrical boies presse by a normal force, the contact region assumes a rectangular format as shown in Fig. (3). Figure 3. ressure istribution an contact region forme between two cylinrical boies in contact (Norton, 1998). The half of the with of the contact region (a) is given by the following epression: a = F m1 + m πl B n, (11) where F n is the normal force, L is the length of the contact region, B is a geometric constant, an m 1 an m are functions of the material of the cyliners. For two cyliners in contact the parameter B is given by: B = + R1 R, (1) where R 1 an R are the raius of the two cyliners. The parameters m 1 an m are epresse by: 1 ν 1 m1 = ; E 1 m 1 ν = ; (13) E where ν 1 an ν are the oisson coefficients, an E 1 an E are the Young Moulus of the materiais of cyliners 1 an respectivelly. The maimum contact pressure (p ma ) is relate with the normal force an the with of the contact region accoring to the following epression:

5 Fn pma =. (14) πal For points at the surface the stresses ue to the normal force are calculate by the following epressions: pma 1, for a; σ = = zn σ n a (15) 0, for > a. The stresses ue to the tangential force for points at the surface are the following: σ t zt f = f f f = 0, ma ma ma ma a + a, a a a 1, a 1, 1, for for for a; for > a; a; -a; for a; (16) where f ma is the maimum tangential pressure efine as f ma = µ p ma. Aing the stresses ue to the normal an tangential forces results into the total stresses in the, y an z irections as follows: σ = σ + σ ; z n zn t σ = σ + σ ; (17) y zt ( σ σ ) σ = ν + ; = +. z zn zt z Note that the normal stress σ zt an the shear stresses zn, y an yz are always zero at the surface for any point in the region of contact. The principal stresses σ 1, σ an σ 3, which are use to calculate the life of the surfaces in contact, are given by the eigen-values of the stress tensor matri, efine as: [ T ] σ = y z σ y y yz σ z yz z. (18) The maimum shear stress, 13, is given by: 13 1 σ 3 = σ, (19) where σ 1 an σ are largest an smallest principal stresses respectively. Below the surface the compression stresses ue to the normal force ecrease. However, the shear stress increases bellow the surface reaching a maimum at a eepth equals to 0.5a. The peak to peak value of the shear stress ue to the normal force is 0.5p ma for any traction coefficient. The main failure mechanisms for surfaces in contact are wear an pitting. ome authors consier that the peak to peak value of the shear stress is responsible for most of the failure that happens below the surface. However, there is a big controversy about which is the most important failure mechanism an also about the location of the failure.

6 Nevertheless there eists a consensus that the largest principal stress is the one that shall be compare to the fatigue limit of the material to etermine the life of surfaces in contact. The following epression etermines the life of surfaces in contact: ( ζ λlog ) = 10, (0) N 10 k where N is the number of cycles until failure, ζ an λ are the resistance factors of the material, an k is the eperimental loa factor given by: ( m m ) k = π +, (1) 1 σ princ_má where σ princ_má is the maimum principal negative (compression) stress. In Eq. (1) the unit use for stress is psi. Analysing Eq. (15) an (16), associate with Eq. (11), (1) an (14), it may be observe that the largest stresses will appearin the contact between the sun an the planets ue to the smallest raius of theses elements. In the planetary traction reucer, for each turn of the output shaft, a given point of the sun of the secon stage is submitte to 3 loas ue to the presence of 3 planets. Thus, the sun is the most solicite element of the reucer Calculation of the reucer life For calculation of the reucer life the following ata are aopte: The iameters of the sun an planets are 10mm an 35mm respectively; The material of the sun is 860 cemente steel with surface harness 60 HRC; The material of the planets is AII 5100 steel with surface harness 197 HRC; The transmission ratio per stage is 9; The oisson coefficient an Young Moulus of the sun an planets materials are 0.3 an 07Ga respectively; Accoring to the stuy mae by Kraus (199) the estimate traction coefficient provie by the antotrac-50 for the operating conitions presente in the reucer is 0.06; Accoring to Morrison (1968), consiering rolling with 9% of sliing, for the material use for the sun the fatigue factors are: k = 1700psi, λ = 7.39, an ζ = The reucer life is function of the maimum output torque. Figure (4) shows the principal stresses, the Von Misses stress (σ VM ) an the maimum shear stress in the sun as function of the istance from the center of the contact region, an Tab. (1) presents a summary of the calculation results for three ifferent output torques. The normal force applie to the sun an planets to transmit a esire output torque is calculate from Eq. (10). The maimum contact pressure is calculate from Eq. (11), (1), (13) an (14). The principal stresses are calculate as function of the istance from the center of the contact using Eq. (15), (16), (17), (18) an (19). Giving the maimum principal stress the reucer life is calculating using Eq. (0) an (1) σ 1 σ σ 3 σ VM stress/pma /a Figure 4. Distribution of principal stresses, Von Mises stress an maimum shear stress in the contac region. Fig. (4) shows that the stress istribution is not symmetric aroun the center of contact; this is cause by the presence of the tangential force. Also, observe that the maimum principal stress is greater than the maimum contact pressure; again this is cause by the tangential stress component that is ae to the normal stress component to form the total stress.

7 Output Torque (Nm) Table 1. ummary of the calculation of the reucer life. Normal force (N) Maimum contact pressure (N/m ) Maimum principal stress (N/m ) Estimate life (number of cycles of the output shaft) Observe that the results for the reucer life presente in Tab. (1) are calculate consiering that there is metal-tometal contact between the sun an the planets. ince the metal-to-metal contact is avoie by the presence of the antotrac-50 film pa between the elements, the actual life of the reucer will be greater than this calculate value. This was one because no ata was foun for the fatigue factors for the case of contact using antotrac re-loa system To apply the require normal force to obtain the esire output torque, a pre-loa system as shown in Fig. (5) is use. This system is compose by two rings with conical formats an several screws. When the conical surfaces are approimate by the screws a pressure is applie on the eternal ring an on the boy of the reucer. The pressure on the eternal ring is transmitte to the planets an the sun, resulting in the require normal force. screws planets screws conical rings reucer boy sun arm arm planet ais conical ring eternal ring Figure 5. Frontal an lateral views of the pre-loa system. The arms have cuts an holes aroun the positions where the planets aes are fie. These cuts an holes allow small raial movements of the planets aes. The raial movements permit applying the pre-loa force without causing stresses ue to the eformation of the planets aes, an also make simpler the assembly process. Figure (6) shows a frontal view of the arm showing the cuts an holes. arm lanets aes cuts an holes Figure 6. Frontal view of the arms showing the cuts an holes aroun the planets aes.

8 5. Tests an results The following characteristics of the spee reucer were calculate through eperimental tests to verify its performance: Angular transmission error; Angular velocity fluctuation; Backlash; Torsional stiffness. The eperimental apparatus use to test the reucer is escribe in etails in Nagahashi (00), this apparatus is similar to the one use by Mitsubishi (19 ) to test its high precision spee reucers. The spee reucer is riven by a DC electric motor through a flat belt. The flat belt is use to minimize vibrations that may be cause by the motor to be transfer to the reucer. To measure the angular positions of the input an the output shafts optical encoers are use with resolutions of 36 an 18 secons of arc respectively. The eperimental results presente in this section are obtaine with a pre-loa force require for an maimum output torque of 30Nm. Figure (7) presents a typical result for the angular velocities for the input an output shafts for clockwise rotation. All the tests are preforme with the motor at constant angular velocity. ince optical encoers are use to measure the angular position, the angular velocity is calculate as follows: θ ( t + T ) θ ( t) =, () T where is the angular velocity of the input or the output shaft, θ is the angular position rea from the encoer, an T is the sampling perio, i.e., the time interval between two consecutive reaings of the encoer s counter. Angular velocity (ra/s) Input velocity Time (secons) Angular velocity (ra/s) Output velocity Time (secons) Figure 7. Eperimental ata for the angular velocity for the input an output shafts for clockwise rotation. Figure (8) presents the angular transmission error an the angular velocity fluctuation calculate using the ata shown in Fig. (7). Error (minutes of arc) Angular transmission error Angular position (turns) Angular velocity fluctuation (%) Angular velocity fluctuation Angular position (turns) Figure 8. Calculate values for the angular transmission error an the angular velocity fluctuation for one turn of the output shaft for clockwise rotation. The angular transmission error (θ) is calculate from eperimental ata accoring to the following equation: θ = θ θ 1 / i T, (3)

9 where θ 1 an θ are the angular position of the input an output shafts respectively, an i T is the total transmission ratio of the spee reucer. The angular velocity fluctuation is calculate using eperimental ata as follows: δ 1 / i = = T, (4) where δ is the rate of the angular velocity fluctuation, is the angular velocity fluctuation, an 1 an are respectively the angular velocity of the input an output shafts. The results of angular velocity an angular velocity fluctuation show small oscillations typical of quantization errors. The encoer use to measure the angular position of the output shaft oes not have the aequate resolution. An encoer with resolution of at least 5 secons of arc woul be necessary to eliminate these quantization errors. Thus, from the measurements of angular velocity fluctuation it is not possible to conclue if these fluctuations are cause by the reucer operation. Table () presents a summary of the results obtaine from several tests with the spee reucer. Observe that the error between the measure transmission ratio (1:80.6) an the nominal value (1:81) is very small, about 0.5%. Table. Results for the angular transmission error an the angular velocity fluctuation. Measure transmission ratio 80,6 Angular transmission error (minutes of arc) 1,5 Angular velocity fluctuation (ra/s) less than 0,09 Rate of angular velocity fluctuation (%) less than 5,57% To etermine the backlash an the torsional stiffness the input shaft is blocke an torque is applie at the output shaft. The torque is applie in the following sequence: (1) starting from zero the torque is increase up to 50% of the maimum torque in one irection; () then, the torque is graually ecrease until returning to zero; (3) the torque is now applie in the other irection following the same proceure, i.e., it is increase from zero to 50% of the maimum torque an then graually ecrease to zero. For each value of torque the angular position of the output shaft an the torque itself are measure. Figure (9) presents a typical result obtaine in this test where it is possible to observe that the reucer presents no backlash. A backlash woul be represente by a iscontinuity (large horizontal variation) in the angular position when the applie torque changes its irection, i.e., when the torque passes through zero. The variation of the angular position is ifferent uring the loa an the unloa stages of the test, causing a small hysteresis. This hysteresis is cause by a small amount of sliing between the rolling surfaces. The torsional stiffness of the traction planetary reucer is given by the tangent of the curve of the angular position versus torque, resulting in a value about,5 Nm/min. 6 Output shaft angular position (minuts) Output torque (Nm) Figure 9. Typical result for the test of torsional stiffness. For comparison purposes Tab. (3) presents a classification of spee reucers accoring to their precision, as given by Mitsubishi (19 ). Comparing the information given by Tab (3) with the results obtaine for the traction planetary reucer it is possible to conclue the following: The results of angular transmission error classifies the reucer as belonging to the precision class; The results of backlash allows to classify the reucer as belonging to the high precision class; Due to the quantization errors the results of angular velocity fluctuation cannot be use to classify the reucer accoring to the precision.

10 Table 3. Classification of spee reucers accoring to the precision (Mitsubishi, 19 ). arameter High recision recision Normal Rate of angular spee fluctuation (δ - %) δ < 0,% 0, % < δ < 3,0 % δ > 3,0 % Angular transmission error (θ - secons of arc) θ < < θ < 100 θ >100 Backlash (Cθ - minutes of arc) Cθ < < Cθ < 10 Cθ >10 6. Conclusions In this work a traction planetary spee reucer has been esigne an test. The reucer presents zero backlash an an angular transmission error small enough to be classifie as a precision reucer. Although it was not possible to measure with the require resolution the angular velocity fluctuation is observe to be very small. The measure transmission ratio is slightly ifferent than the nominal value. This ratio was measure with no loa applie to the output shaft an the ifference, which is about 0.5%, is probably cause by the presence of sliing uring the operation. The stiffness of the reucer is relatively low, about 50% lower than the stiffness of a precision gear reucer with the same maimum output torque an about the same weight. The reucer stiffness in highly affecte by the normal force applie to generate the traction force. Giving a esire life the normal force is limite by the iameter of the sun, thus, the stiffness of the traction reucer may be enhance by increasing the sun iameter. It is important to observe that the reucer life is calculate consiering that there is metal-to-metal contact between the sun an the planets. ince the metal-to-metal contact is avoie by the presence of the antotrac-50 the actual life of the reucer is greater than the calculate value. A more realistic value for the reucer life was not possible to calculate ue to the lack of ata concerning contact fatigue factors in the presence of antotrac-50. Also, note that a sliing of 9% was assume to calculate the reucer life an this value is greater than it was observe. Future work is unerway to measure the angular velocity fluctuation without quantization problems an the maimum output torque. The measurement of the maimum output torque is important because all the calculations are base on this torque. Finally, to conclue this work it is still necessary to measure eperimentally the real life of the reucer uner operating conitions to valiate the calculations performe. 7. References Cisneros, C. A. F., 1996, Design of a Compose lanetary Traction pee Reucer, Master Dissertation, Department of Mechanical Engineering, Escola olitécnica a U, ão aulo, Brazil (in ortuguese). Hunol, M. C., 1999, Rotative neumatic ositioning ystem, Doctoral Thesis, Department of Mechanical Engineering, Escola olitécnica a U, ão aulo, Brazil (in ortuguese). Kraus, C. E., 1985, Rolling Traction Analysis an Design,. Ecelermatic Inc., Teas, UA. Loewenthal,. H. an Rohn, D.A., 1985, Elastic Moel of the Traction Behavior of Two Traction Lubricants, ALE Transactions, v.7,, pp Morrison, R. A., 1968, Loa/Life Curves for Gear an Cam Materials, Machine Design, Vol. 40, pp Mitsubishi, 19, Rotation Accuracy of Mitsubishi lanetary Traction Drive Unit, Tokyo, Japan. Nagahashi, F.., 00, Design, Construction an Test os a Two-tage lanetary Traction pee Reucer, Master Dissertation, Department of Mechanical Engineering, Escola olitécnica a U, ão aulo, Brazil (in ortuguese). Norton, R. L., 1998, Machine Design: An Integrate Approach, rentice-hall, New Jersey. mith, J. O. an Lui, C.K., 1953, tresses Due to Tangential an Normal Loas on an Elastic oli with Application to ome Contact tress roblems, Journal of Applie Mechanics. Transactions of The AME, 75: pp Nagahashi, F.., Design, Construction an Tests of a Tw-tage lanetary Traction pee Reucer, Master Dissertation, Department of Mecatronic Engineering, Escola olitécnica a U, ão aulo, Brazil (in ortuguese).