Loaded Tooth Contact Analysis of Cycloid Planetary Gear Drives
Transcription
Loaded Tooth Contact Analysis of Cycloid Planetary Gear Drives
The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS6.014 Loaded Tooth Contact Analysis of Cycloid Planetary Gear Drives Shyi-Jeng Tsai, Ching-Hao Huang, Hsian-Yu Yeh and Wei-Jhen Huang Department of Mechanical Engineering, National Central University, Jhong-Li, Taiwan e-mail: [email protected] Abstract: An approach of loaded tooth contact analysis is proposed for analyzing the load sharing and the load distribution of cycloid planetary gear drives with multiple tooth pairs contact. The mathematic equations of cycloid gear profile are derived from kinematic relation of the gear and the pin wheel. The analysis of tooth contact is based on instant center method. The contact stresses of the multiple engaged tooth pairs are calculated based on the influence coefficient method. An analytical, simple relation for calculation of the load sharing and the max. Hertzian contact stress with the assumption of constant meshing stiffness is also proposed for comparison. Finally, a study case was analyzed for load sharing and contact stress distribution of engaged tooth pairs by both methods. The characteristics of the contact stress of the cycloid planetary gear drives are clearly identified. Keywords: Cycloid planetary gear drives, Loaded tooth contact analysis, Influence coefficient method Nomenclature A matrix of influence cofficients F acting force H matrix of separation distance among engaged flanks P vector of contact pressure pitch circle radius of the pin wheel RC T transmitted torque radius of the pin rP e eccentricity of the cycloidal gear reducer h separation distance w deformation tooth number of the cycloidal disk zC tooth number of pin wheel zP reduction ratio iC approaching displacement transmission angle rotation angle of the crank shaft curvature generation angle for epitrochoid curve curvature radius pressure angle 1. Introduction The trends of gear transmission development such as precision, high power density and even also high reduction ratio can not be ignored. For example, the transmission drive of construction machinery should be designed as smaller as possible under requirement of high reduction ratio. On the other hand, the demand on precision transmission for automation machinery is also raised recently to reduce the labor cost and to increase the productivity. In general, the mentioned gear drives have the common features, such like high reduction ratio, small power density, oscillating transmission with high torque at low speed. The conventional application of planetary gear with involute gears can not fulfill the requirement apparently. In contrast to the involute gears, the differential type of planetary gear drive with cycloidal gears has the some significant advantages, i.e. possibility achieve small tooth number difference (at least one) to obtain high gear ratio; multiple tooth pair in contact for sharing the transmitted load to enhance the load capacity of the transmission and the high power density, so as suitable for application of small volume or weight; short teeth so as having good ability to absorb shock of oscillating transmission, and suitable for application of high torque at small speed. The cycloidal planetary gear drives, or usually known as the brand name “cyclo-drive”, haven been already applied in many branches of industry, for example the transmission of the pipe-jacking tunneling boring machine (TBM) shown in Fig. 1. Fig. 1 Pipe-jacking tunneling boring machine (TBM) and transmission The load analysis is an important issue for successful application of such the drives, but there are not many published research papers on the load, although more articles on geometrical and kinematic design can be found. Dong et al. [1] proposed a calculation approach for the acting forces on the rolling bearings of the cycloidal planetary gear drive. Blanche and Yang [2, 3] analysis the influence of the manufacturing errors on the transmitted load and transmission errors. Hidaka et al. [4] proposed an analytical method based on the assumption of contact mesh stiffness of tooth action to analyze the influence of manufacturing errors. The analytical approach is later also compared with FEM analysis [5]. Gorla et al. [6] develop an simplified approach to analyze the contact stress and also conducted an experiment to validate the theoretical analysis results. Another often applied method for analysis of contact stress of cycloidal gear drives is FEM, e.g. Blagojevic et al. [7] proposed a double stage of cycloidal gear reducer. They developed an approach for load analysis for good load distribution and dynamic balance. The contact stress is however analyzed by FEM. Similar research can be also found in the works by Thube [8], Li [9] and Kim [10]. The aim of the paper is to propose an efficient method to calculate the loads acting in the cycloidal planetary gear drives. It will be expected to analyze the load sharing among the multiple contact tooth pairs, the distribution and variation of the contact stress, as well as the varied angular stiffness or the loaded transmission errors by using the proposed methods. Two different approaches for contact stress analysis of the multiple engaged tooth pairs are presented in the paper: one is a computerized approach based on the influence coefficient method; another is an analytical but simple approach with the assumption of constant meshing stiffness. The drive analyzed in the paper consists of two planetary stages, as the section view of the CAD model in Fig. 2 shows. The power is transmitted through the first planetary stage of involute gears and split into three crank shafts, on which the planet gears are fixed. The crank shafts drive the cycloidal disk of the second planetary stage in a eccentric motion to transmit the power further to the pin wheel or the carrier (see Fig. 3). In the case the carrier is fixed and the pin wheel plays the role of power output, here supply enough power to the primary and secondary cutter and grinder. Fig. 2 Section view of the cycloidal planetary gear for pipe-jacking TBM of various mesh stiffness of the contact tooth pairs. Besides the loaded contact analysis method based on the influence coefficients, an additional analytical method based on the assumption of the equal mesh stiffness are therefore proposed for simplified calculation. 2.1 Geometrical Relations 2.1.1 Epitrochoid profile The tooth profile of the cycloidal disk can be derived from the geometrical relation in Fig. 4 as the following vector equation: rC RC e i e e i( iC ) rP e i( ) , (1) where the pressure angle is the angle between the contact normal MiPi and the line of centers OPi, and is determined as z P e sin( zC ) . RC z P e cos( zC ) arctan (2) The initial position of the pin-wheel is based on the angle variable = 0. It can be also clearly identified that the tooth profile is also the equidistant curve of the the the epitrochoid curve with the distance rP (namely the pin radius). The vector rC in Eq. (1) can be also represented in the Cartesian coordinates as xC RC cos e cos(iC ) rP cos( ) (3) yC RC sin e sin(iC ) rP sin( ) (4) 2.1.2 Curvature The curvature relation of the contact tooth pair is important for the calculation of the Hertzian contact stress. While the curvature of the pin is constant, the curvature of the epitrochoid tooth profile can be calculated according to the definition xC ' ( ) yC " ( ) xC " ( ) yC ' ( ) . [ xC '2 ( ) yC '2 ( )]3 / 2 (5) However, it is somewhat complicated to differentiate Eq. (2) and (3) according to Eq. (4). Because of the property of the equidistant curve of the epitrochoid curve, the curvature radius of the cycloidal tooth profile is thus equal to the difference of the curvature radius of the epitrochoid curve and the pin radius. The equations of the epitrochoid curve are xCR RC cos e cos(iC ), yCR RC sin e sin(iC ). (6) And after differentiating Eq. (5), the curvature radius of the tooth profile can be obtained as Fig. 3 Structural scheme of the cycloidal planetary stage 2. Model of Loaded Tooth Contact Analysis In general the mesh stiffness of the drive is not constant. However, it is complicate to calculate the loaded deformation and the contact stresses under consideration RC [1 (iC e / RC )2 2 (iC e / RC ) cos((iC 1) )]3/ 2 1 iC (iC e / RC )2 (iC e / RC ) (1 iC ) cos[(iC 1) ] rP (7) When the curvature is equal to zero, or the curvature radius is infinite, the point on the tooth profile is the inflection point. The corresponding variable inf of the inflection point must be equal to inf 1 iC (iC e / RC ) 2 1 arccos . iC 1 (1 iC ) (iC e / RC ) (8) and finally OC I C (iC 1) e k e . (12) The location of the instantaneous point IC is changed with different rotation angle C. The locus of the points IC is a center with a radius k·e. It will be easy to determine the contact points Mi by using the line ICPi at each rotation angle C. 2.2.2 Tooth meshing To determine the positions of the contact points is essential for calculation of the contact stress of the loaded tooth pairs. The mesh analysis consists of two essential analysis works: the positions of contact points and the effective contact tooth pairs. (1) Positions of contact points. When the crank shafts rotate at an angle C, the center of the cycloidal disk translate along the circle around the center of the pin wheel with the angle C, and the pin wheel rotates at the angle C / iC., as the geometrical relation shown in Fig. 5. The position of contact point M1 can be determined from the relation in Fig. 5, the corresponding variable 1 for the point M1 on the tooth profile of the cycloidal disk is Fig. 4 Geometric and kinematic relation of the cycloidal planetary gear 1 P C / iC , and for the ith tooth pair i 1 ( j 1) P . 2.2 Kinematic relations 2.2.1 Instantaneous centre The kinematic relation in Fig. 4 is so regarded that the cycloidal disk is stationary and the pin wheel moves relatively in two types of motions: Translation motion: the center of the pin wheel OP translates along the circle with the center OC, which is also the centre of cycloidal disk. This motion is equivalent to the revolution movement of the cycloidal disk around the center OP of the pin wheel. This motion is transmitted from the rotation of the crank shafts. The transmitted angle is denoted as C. Rotation motion: the pin wheel rotates around its center OP with an angle P which is equal to C/iC. Under this motion, the common normal on the contact point M1 or Mi of the pin and cycloidal tooth is in the direction of CP1M1 or CPiMi. Because the common normal the instantaneous center locates also on the normal, the instantaneous center IC must also lie on the line of the centers OPOC. In other words, the instantaneous center IC is the intersection point of the lines of OPOC and CP1M1. At the moment of mesh, as shown in Fig. 4, the translation velocity of the cycloidal disk is equal to vC e C , (9) and the direction is perpendicular to the line of the centers OPOC. With the definition of the instantaneous center, the relation of the velocities must be valid, or O P I C P e C , (10) O P I C e iC , (11) (13) (14) The coordinates and the related curvature radius of the contact point can be obtained by substituting the angle i into Eq. (3), (4) and (7), respectively. The angle i between the normal of contact tooth pair i and the line of the centers OPOC can be derived from the geometrical relation as i ( i i ) C . (15) (2) Approaching distance. Because the pressure angle of the contact tooth pair is varied during mesh, the approaching distances of the contact tooth pairs will be also different from each other. With assumption that the crank shafts rotate at an additional angle due to the loaded deformation of all the contact tooth pairs, a translational displacement e with an inclined angle (/2+C) will be generated correspondingly. The angle between the vector E of the translational displacement and the normal vector N can be obtained, i π C i i i . 2 2 (16) The effective approaching distance eqi for contact tooth pair i can be then determined as eqi e cos i e sin i qi e (17) (3) Effective contact tooth pairs. Because of multiple contact tooth pairs, it is also important to distinguish the tooth pairs in contact at each mesh position for further calculation of contact stresses. It can be classified into the following criteria according to the motion direction of the crank shaft: Counterclockwise rotation direction: the effective approaching distance eqi > 0 or the transmission angle i < 0 or the pressure angle 0. Clockwise rotation direction: the effective approaching distance eqi < 0 or the transmission angle i 0 or the pressure angle 0. 2.4 Influence coefficient method 2.4.1 Basic relation The contact of any engaged tooth pair can be regarded as the contact of two elastic bodies, as the simplified model shown in Fig. 6 (a). The tooth profiles are deformed due to the normal force P. The tooth will be then approached to each other along the normal direction with a distance 1 and 2, respectively. The point Q1,2 on the tooth profile 1 or 2 will be also deformed in a value of w1 and w2 under the contact pressure. In order to distinguish whether the two points Q1 and Q2 on the loaded teeth are coincided or not, the following equations must be valid, Fig. 5 Relation of loading and displacement 2.3 Method of equal mesh stiffness The loaded deformation wPi of contact tooth pair i can be expressed as the multiplication of the acting normal load FPi and the compliance fPi, wPi FPi f Pi . (18) With the assumption of equal mesh stiffness, all the compliance fPi will be equal to a constant, namely fPi = fP. Hence the following equation is valid for contact tooth pair i: (19) For n contact tooth pairs, the load equilibrium equation can be obtained with a given torque T, n iC e ( FPi sin i ) T . (20) i 1 Together with Eq. (18) and (19), the shared load on each individual contact tooth pair can be calculated as FPi n w1 w2 h 1 2 , (23) out of contact: w1 w2 h 1 2 , (24) where h is the initial separation distance between two engaged tooth flanks, and can be calculated from the geometrical relations of the mathematical equations of the flanks. Besides the relations of loaded deformation and displacement in Eq. (22) and (23), another boundary condition for loaded contact analysis is also essential. The contact pressure p' on arbitrary position (x’, y’) of the contact region A must be positive and the sum of the contact pressure p' on each area must be also equal to the normal force P, i.e. A p'(x', y')dx'dy' P , p'(x',y') 0 (25) The deformation on arbitrary position k (see Fig. 6(b)) due to all the acting loads on the contact region can be defined with aid of influence coefficients, namely r wk w(1) k w(2) k f H,kj p j , (26) j 1 FPi f P e 0 . sin i T sin i in contact: . iC e sin 2 i (21) where fH,kj is influence coefficient for the Hertzian contact deformation on unit k due to the load acting on unit j. Hence the set of the deformation-displacement equations and the load equilibrium equation can be expressed in a form of matrix equation as [11], A1 0 0 s1I1 n 1 0 0 A2 0 0 An s2I1 n 2 - q1I n 1 P1 H1 H - q2I n 1 P2 2 , (27) - q p I n 1 Pp Hp T / e 0 e 1 2 P s p I1 n P P i 1 The max. contact stress of each contact tooth pair can be also calculated according to the equation, Hi ,max FPi E 1 1 ( ). 2 2 b (1 ) C P (22) Fig. 6 (a) Relation of deformation and load (b) meshing on the common tangent plane where the approaching distance of each contact tooth pair is equal to e·qi. The transformation coefficient qi can be calculated according to Eq. (17). 2.4.2 Separation distance between engaged teeth The separation hj distance between the cycloidal flank and the circular pin in the normal direction consists of two parts: one is the distance hMj of point Mj on the cycloidal flank to the common plane, and the other is hCj of point Cj on the pin to the same plane, Fig. 7., h j hCj hMj . (28) The distance hCj can be determined under a given distance lMj on the tangential plane as hCj rp rp2 2 lM j , (29) The distance lMj and the separation hMj can be determined by using the following vector relations rMij t Mi lMj , (30) rMij n Mi hMj , (31) where the vector rMij is equal to the difference of the position vector rC(j) of point Mj and rC(i) of Mi: rMij rC ( j ) rC ( i ) . (32) On the other hand, the tangential vector tMi and the normal vector nMi can be derived from the geometrical relations in Fig. 7, namely, t Mi e i , (33) n Mi e with i ( ) 2 , 2 (34) . (35) The unknown variable j for point Mj is solved from Eq. (30) with a given value of the distance lMj. The separation distance hMj is thus determined with the solved j. Fig. 7 Separation distance among the engaged flanks 3. Numerical Example 3.1 Gearing data The gearing data of the drive for the analysis example is listed in Table 1. The difference of tooth number is 1 in this case. Table 1 Gearing data for analysis example Items / symbols value Pitch circle radius of pin wheel RC Radius of the pin rP remarks 162.5 mm 8 mm Eccentricity e 3.5 mm Tooth number of the cycloidal disk zC 39 Tooth number of the pin wheel zP 40 Reduction ratio iC (Carrier fixed) 40 Thickness of the cycloidal disk t 31.5 mm Input torque T 4000 Nm zP/(zPzC) 3.2 Variation of load sharing and contact stress The shared loads among the engaged tooth pairs are calculated at the rotation angle of the crank shaft C = 0. The load sharing factor K is defined as the portion of the load acting on the tooth pair i to the average acting load. The average acting load is determined from the total normal load Fi evenly shared by the half of tooth number zP. Thus the load sharing factor can be expressed as K i z P Fi 2 Fi (35) The result is illustrated in Fig. 8. It can be identified that the results calculated both by the proposed approaches are similar in trends but the max. load sharing factors are quite different. The contact tooth pair #1 is the tooth pair which engages each other at the beginning of contact. From the result by the method of influence coefficient (IC), contact tooth pair #3 bears the max. load, while #4 by the method of equal mesh stiffness (EMS). The results reveal also that the load is mainly shared to the region of the concave flanks. The variation of the load sharing and the contact stress with the rotation of the input crank shaft are illustrated in Fig. 9. The analysis results for two proposed methods are represented: those from the method of equal mesh stiffness (EMS) is shown in blue, and the method of influence coefficients (IC) in red. The range of the rotation angle C of the crank shaft in the diagram corresponds to the profile variable , where the relation = C /iC is valid. With other words, the range of the angle C is equal to 18040/39 = 184.615 in this case, while the range of angle = 180 for one flank side. The max. load sharing factor calculated from the method of influence coefficients occurs at the angle of 14.3077° with the value of 2.3179, while that from the method of equal mesh stiffness occurs at 29.5384° with 2.005. It can be also identified that the shared load varies continuously. The two calculated results for contact stress from both the methods are similar to each other. But the max. values of the contact stress are different, although they 2.5 Load Sharing EMS 2 1,200 Contact Stress IC Contact Stress EMS 1.5 900 1 600 0.5 300 0 -200 max. Contact Stress [MPa] 1,500 Load Sharing IC Load Sharing Factor occur at the similar angle about -55.385°. The max. contact stress calculated by the method of EMS is 1320.87 N/mm2, by the method of IC is 1206.739 N/mm2. In order to explore the contact stress on the flanks, the variation of the contact stress and the curvature with the rotation angle of the crank shaft are compared, as the diagram in Fig. 10 shows. The max. curvature 0.455 mm-1 occurs at the same angle where the max. contact stress occur. This result reveals that the main factor affecting the contact stress is the curvature, not the shared load, although the mesh stiffness affects the shared load. The reason can be also identified from the calculated result by the method of IC that the max. shared load occurs at the rotation angle 14.3077°, where the curvature is negative and the result contact stress is reduced. 0 -150 -100 -50 0 Rotation Angle of the Crank Shaft [deg] Fig. 9 Variation of the load sharing and the contact stress of the engaged tooth pair during meshing 2.5 Method of EMS 2 1200 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Number of Cycloid Tooth Fig. 8 Load sharing of the engaged teeth 1000 0.5 Contact Stress -- IC Contact Stress -- EMS Curvature 0.4 0.3 Curvature [1/mm] 1400 Contact Stress [MPa] Load Sharing Factor Method of IC 800 0.2 600 0.1 400 0 200 -0.1 0 -200 -180 -160 -140 -120 -100 -80 -0.2 -60 -40 -20 0 Rotation Angle of the Crank Shaft [deg] Fig. 10 Relation of the contact stress and the curvature 3.3 Characteristics of Contact Stress 3.3.1 Distribution on the tooth flanks In order to explore the contact pattern, and the distribution of the contact stress on flanks, the contact stresses calculated by the method of IC are illustrated in 3D representation. The distributed contact stress for the contact point near the inflection point is shown in Fig. 11, for the contact point with the max. contact stress (max. curvature) in Fig. 12. The inflection point on the flank occurs at the rotation angle -30.4373°. From the results, it is can be clearly identified that the contact pattern is similar to a saddle form. The concentrated stresses occur on the edges of the face-width of the cycloidal disk. The area of the contact pattern of the tooth pair with the max. contact stress is smaller than that on the inflection point. Fig. 11 Contact stress distribution: tooth acting on the inflection point Fig. 12 Contact stress distribution: tooth acting on the point with max. contact stress -55.4° -36° -29.5° -18.4° -11.5° -0.46° Contact Stress (MPa) 1200 1000 800 600 400 0 -0.4 -0.2 0 0.2 0.4 1000 800 600 400 200 0 -0.04 -0.02 0 0.02 0.04 0.06 Minor Axis of Contact Pattern [mm] Fig. 14 Contact stress in the profile direction: from the contact point with max. contact stress to the contact end 3.3.3 Loaded transmission errors The angular stiffness of the cycloidal gear drive can be also calculated by the proposed method of influence coefficient when the deformations of all the essential components are considered. In the paper, only the compliance of the contact tooth pairs due to Hertzian contact is involved in the calculation. The rotation range of the crank shaft for each contact cycle of the engaged tooth pair is equal to 360/39 = 9.23077. The angular compliance on the input side is calculated from Eq. (27). With expressed for output side, i.e. the angular compliance equal to /iC, the loaded transmission error can be obtained, as the diagram shown in Fig. 15. The quasi sinusoid can be identified, where the transmission errors varies in the range of 11.247 arcsec and 11.345 arcsec. The value of the error amplitude is equal to 0.098 arcsec. This small value, of course, is not the realistic result of such the drive, but it can be thus verified that the proposed LTCA method can be used for analysis of angular stiffness. -11.2 200 -0.6 -55.4° -75.7° -97° -144° -174° -184° 1200 -0.06 0.6 Minor Axis of Contact Pattern [mm] Fig. 13 Contact stress in the profile direction: from the contact begin to the contact point with max. contact stress LTE of the Pin Wheel [arcsec] 1400 1400 Conatct Stress [MPa] 3.3.2 Variation along the epitrochoid profile It can be also found from Fig. 11 that the contact stress of the tooth pair on the inflection point is asymmetrically distributed in the profile direction. Therefore the variation of such the contact stress distribution at different rotation angle of the crank shaft is shown in Fig. 13 for the region of stress increasing (Region I) and Fig. 14 for the region of stress decreasing (Region II). At contact begin, because the contact type of the tooth pair is concave-convex, the contact pattern is enlarged and the contact stress is reduced correspondingly. On the other hand, the contact near the inflection point is not only concave–convex, but also convex–convex on the contact area. As consequence, the contact stresses distributed asymmetrically in the profile direction, for example the distribution curves at = -18.4°, -29.5°, -36° in Fig. 13. The asymmetrical distribution of the contact stress will disappear and will be symmetrical away from the inflection point. The length of minor axis of the contact pattern in the region I is increased from contact begin until to the inflection point due to the small shared load and convexconcave contact and then decreased until to the contact point with the max. stress due to the large shared load and curvature. The length of minor axis in region II, on the other hand, are at first increased and then decreased until to the contact end. The reason is can be explained from the relation of the curvature and the shared load. After the point with max. contact stress, both of them are reduced. At first the contact pattern is enlarged a little because the reduced gradient of the curvature is larger than that of the shared load. But after the contact point at the angle about 100°, the gradient of the curvature becomes smaller, while the gradient of the reduced shared load is enlarged. Consequently the length of minor axis is reduced until to zero, see Fig. 14. -11.225 -11.25 -11.275 -11.3 -11.325 -11.35 -11.375 -11.4 -45 -40 -35 -30 -25 -20 -15 -10 -5 Rotation Angle of the Crank Shaft [deg] Fig. 15 Loaded transmission errors of the output shaft 0 4. Conclusion The loaded tooth contact of the cycloidal planetary gear is analyzed by using the method of influence coefficients and the method of equal mesh stiffness. The analysis results enable us to draw the following conclusions: The contact stress of the pin and the epitrochoid flank is influenced mainly by the curvature of the epitrochoid flank. The mesh stiffness, on the other hand, affects the load sharing among the contact tooth pairs. A small shared load and a larger contact pattern can be found in the contact region of the concave profile of the cycloidal disk, and thus smaller contact stress occurs. The max. contact stress occurs at the point with max. curvature. The concentrated stresses occur on the edges of the face-width of the cycloidal disk. The corresponding contact pattern is a saddle form. The analytical equations of constant mesh stiffness are suitable only for calculation of the max, contact stress. The difference comes from the deviation of mesh stiffness in the concave-convex contact region. The proposed LTCA method based on influence coefficient can efficiently calculate the contact pattern, the contact stress distribution, load sharing and loaded transmission error. The paper presents also a possibility for LTCA of the compound planetary gear drive with cycloidal and involute gears, when the deformations of the other relevant components, such as the rolling bearings, the crank shafts, the involute planet gears and the sun gear, are further integrated in the proposed model. Acknowledgment The authors would like to thank Transmission Machinery Co., Ltd. for their financial support. References [1] Dong, X.; Deng, J.; Chen, J.: “Force Analysis of RV Transmission Mechanism”, Journal of Shanghai Jiao Tong University, Vol.30, No.5 1996, pp. 65-70, 84. [2] Blanche, J. G., D.; Yang, C. H.: “Cycloid Drives With Machining Tolerances”, Journal of Mechanisms, Transmissions, and Automation in Design, No. 3, Vol. 111, 1989, pp. 337-344. 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