A Novel Finish Hobbing Methodology for Longitudinal Crowning of a Helical Gear with Twist-Free Tooth Flanks by Using Dual-Lead Hob Cutters

Transcription

1 A Novel Finish Hobbing Methodology for Longitudinal Crowning of a Helical Gear with Twist-Free Tooth Flanks by Using Dual-Lead Hob Cutters Van-The Tran Graduate Student vanct4.hut@gmail.com Ruei-Hung Hsu 1 Assistant Professor rhhsu@fcuoa.fcu.edu.tw Chung-Biau Tsay Professor cbtsay@mail.nctu.edu.tw Department of Mechanical and Computer-Aided Engineering, Feng Chia University 100 Wenhwa Rd., Seatwen, Taichung 40724, Taiwan Abstract In the gear finish hobbing process, to obtain a twist-free tooth flank of helical gears, a novel hobbing method for longitudinal crowning is proposed by applying a new hob s diagonal feed motion with a dual-lead hob cutter. Wherein the hob s diagonal feed motion is set as a second order function of hob s traverse movement and tooth profile of hob cutter is modified in a dual-lead form with pressure angle changed in it s longitudinal direction. We have verified our proposed method by using two computer simulation examples to compare topographies of the crowned work gear surfaces hobbed by the standard and dual-lead rack cutters. The results reveal the superiority of the proposed novel finish hobbing method. Keywords: Dual-lead hob cutter; gear finish hobbing; diagonal feed; twist-free tooth flank; longitudinal crowning; 1. Introduction In the gear finishing process, shaving is still the most efficiency and economy process especially in mass production. Besides, the achievable profile accuracy of the shaving process is also very high. However, more recently, finishing hobbing also has made great strides with improved accuracy of modern hobbing machines in combination with high-quality hob cutters (e.g., quality AAA or better). Therefore, the quality gap 1 Corresponding author. 1

2 between finish hobbing and shaving continues to narrow. Typically, the achievable quality of work gear surface can obtain in the level of DIN 6 7 after finish hobbing. However, a drawback of finish hobbing versus shaving remains natural twist of the tooth flanks if an involute helical gear is hobbed in longitudinal crowning. This twist is a defect that can result in more gear noise and less load carrying capacity of the mating gear pair. Therefore, this study proposes a methodology for finish hobbing of the twist-free tooth flank of helical gears in longitudinal tooth crowning by setting hob s diagonal feed motion as a second order function of the hob s traverse movement and using a dual-lead hob cutter with pressure angle changed in it s longitudinal direction. Recent technologies on CNC machines have much development that can extend potential gear hobbing process and gear finish hobbing process. In 2004, Abler et al. [1] presented some extensive available discussions on the gear hobbing process. And a mathematical model of curvilinear-tooth gear as a function of hob design and gear s generation motion parameters on basis of the CNC hobbing machine is proposed by Tseng and Tsay [2]. Two years later, Witte [3-4] has proposed a twist free hobbing solution by analyzing the width of convexity and amount of bias of the tooth flank as well as a finishing hob to cut external crowned helical gears without profile bias. Subsequently, Dimitriou et al. [5] studied the effective and factual simulation of gear hobbing based on virtual kinematics of solid models representing the cutting tool and the work gear. After that, Dimitriou and Antoniadis [6] constructed a CAD-based simulation of the hobbing process for the manufacturing of spur and helical gears. Also in this time, a novel face-hobbing method to generate spur gears with longitudinal crowning by two head cutters form an imaginary generating rack with cycloidal lengthwise tooth traces has proposed by Wang and Fong [7]. In 2009, a mathematical model with a two-parameter equation of meshing for simulation of gear hobbing process is proposed by Chen et al. [8]. And Lange [9] presented some important discussions on the tooth flank s bias errors of helical gears caused in conventional cutting methods and offered techniques used to limit and eliminate bias errors. In this year, the influences of axle deflection and bias error of tooth flanks on contact patterns and stress distributions of a hypoid gear set are investigated by Xu et al. [10]. One year later, a method for varying the hob profile angle along its lead and shifting the hob diagonally over the whole tooth face width during gear hobbing to reduce the twist of work gear tooth flanks is patented by Winkel [11]. Recently, Simon [12] proposed a method for determination of the optimal polynomial function for representation of machine tool setting variations for the pinion tooth finishing. Besides, Hsu and Fong [13] patented the design of a variable tooth thickness (VTT) hob in it s axis and the method of hobbing with a diagonal feed and without varying the center distance. Previously, a method for longitudinal crowning tooth flank of involute helical gear 2

3 by varying the center distance is proposed by Litvin et al. [14]. Finally, the overall geometry, design, and manufacture of a conventional helical gear are detailed in textbook by Litvin and Fuentes [15]. A mathematical model for the tooth surface, hobbed with a new hob s diagonal feed motion and a new modified tooth profile of hob cutter, is proposed for finish hobbing process to obtain the twist-free tooth flank of helical gears in longitudinal crowning. Two numeral examples are presented to illustrate and verify the merits of the proposed gear hobbing method in longitudinal tooth crowning of helical gears with the twist-free tooth flank of work gears. 2. Mathematical model of standard and dual-lead hob cutter The profile of the generating surface of the standard hob cutter is the same as that of the standard helical gear. Assume that the helical gear is hobbed by a standard rack cutter. For generating a twist-free tooth flank of helical gear, the tooth profile of the standard rack cutter is modified in the form of dual-lead with pressure angle changed in it s longitudinal direction, as shown in Fig. 1. Normal sections of the standard and dual-lead rack cutters at positions A, B and C are shown in Fig. 2. The schematic generation mechanism of hob cutter is shown in Fig. 3, where the coordinate systems S7( x7, y7, z 7), S3( x3, y3, z 3), and S4( x4, y4, z 4) are rigidly connected to the rack cutter, hob cutter, and frame, respectively. The generated hob cutter rotates through an angle 1 about the z4 -axis, while the rack cutter translates a distance ro 1 1 without rotation. Accordingly, the position vector and unit normal vector of the rack cutter s right-hand side profile can be expressed in coordinate system S7( x7, y7, z 7) as follows: T T 7 [ x7, y7, z7,1] [ u1 cos on, u1 sin on bu11 v, v1,1], r (1) T T and 7 nx7 ny7 nz7 bv1 on on bu1 on n [,, ] [ sin,cos, cos ], (2) where u 1 and v 1 are the surface parameters, on is the pressure angle of the standard rack cutter, and b is duallead coefficient of rack cutter, as shown in Fig. 1. According to Fig. 3, by applying the homogenous coordinate transformation matrix equation, the locus and unit normal vectors of rack cutter surface represented in coordinate system S3( x3, y3, z 3) are attained by 3

4 Fig. 1. Surface parameters of the standard and dual-lead rack cutters Fig. 2. Normal sections of the standard and dual-lead rack cutters cos1 cos o1 sin1 sin o1 sin 1 ro 1(cos 1 1 sin 1) x7 sin1 cos o1 cos1 sin o1 cos 1 ro 1(sin1 1 cos 1) y 7 r3 M37 r 7, 0 sin o1 cos o1 0 z (3) and , (4) 4

5 where L 37 is the upper-left (3 3) submatrix of the (4 4) homogeneous coordinate transformation matrix M 37. Fig. 3. Coordinate systems for the schematic generation mechanism of hob cutter According to the theory of gearing, the equation of meshing between the right-hand side rack cutter surface and left-hand side hob cutter surface can be obtained by [ x3 ( u1, v1, 1), y3( u1, v1, 1), z3( u1, v1, 1)] f1( u1, v1, 1) n3 0 1 (5) Based on Eqs. (1) (5), the locus and unit normal vectors of the right-hand side rack cutter can be attained as r [ x ( u, v, ), y ( u, v, ), z ( u, v, ),1] T, (6)

6 T and 3 x y z n [ n ( u, v, ), n ( u, v, ), n ( u, v, )], (7) where x ( r u cos ) cos ( r (sin bv ) u cos v sin )sin, (8) 3 o1 1 on 1 o1 1 on 1 1 o1 1 o1 1 y ( r u cos )sin ( r ( sin bv ) u cos v sin ) cos, (9) 3 o1 1 on 1 o1 1 on 1 1 o1 1 o1 1 z v cos (sin bv ) u sin, (10) 3 1 o1 on 1 1 o1 n (sin bv ) cos cos ( cos bu sin )sin, (11) x3 on 1 1 on o1 1 o1 1 ny cos cos ( cos o bu sin ) (sin bv ) sin, (12) 3 on o1 on 1 1 n cos ( bu cos sin ). (13) and z3 on 1 o1 o1 The tooth profile and its unit normal of the dual-lead hob cutter are defined by considering Eqs. (5), (6) and (7), simultaneously. The locus tooth profile and its unit normal of the standard hob cutter are determined by setting coefficient, b, equal zero. 3. Mathematical model of the crowned involute helical gear In gear finish hobbing process, the longitudinal crowning of an involute helical gear can be performed by applying the standard hob cutter with a new hob s diagonal feed motion. However, the using standard hob cutter will induce natural twist of the tooth flank on hobbed gears. Therefore, the tooth profile of hob cutter is modified in a form of dual-lead with the pressure angle changed in it s longitudinal direction, as shown in Fig. 1. Coordinate systems for the finish hobbing of work gear with longitudinal crowning teeth are shown in Fig. 4. Wherein coordinate systems S1 ( x1, y1, z 1), S2( x2, y2, z 2) and Sa( xa, ya, za) are rigidly connected to the hob cutter, work gear and frame of hobbing machine, respectively. While Sb ( xb, yb, z b), Sc ( xc, yc, z c), Sd ( xd, yd, z d ), Se( xe, ye, z e) and S f ( x f, y f, z f ) are auxiliary coordinate systems for simplification of coordinate transformation. For the proposed new finish hobbing process, there are just two hob s movements: traverse movement along the axis of the work gear za () t, and diagonal feed along the axis of hob z () t. To crown a tooth flank in longitudinal direction, the new hob s diagonal feed zs () t is set as a second-order function of the hob s traverse feed za () t. The relationship between the hob s diagonal feed z () t and traverse movement za () t is proposed as follows: s s 6

7 2 s 1 a 2 a z ( t) c z ( t) c z ( t). (14) According to Fig. 4 and Eq. (14), the hob cutter is shifted diagonally in the process of gear hobbing if c1 0 or c2 0. Fig. 4. Coordinate systems for the finish hobbing of work gear with longitudinal crowning teeth The position vector r 1 and unit normal n 1 of the hob cutter equal the position vector r 3 and unit normal n3 of the rack cutter, respectively, as defined in Eqs. (6) and (7). By applying the homogeneous coordinate transformation matrix equation, the locus of hob cutter can be represented in coordinate system S 2 as follows: r ( u, v,,, z ( t)) M (, z ( t)) r ( u, v, ) a 21 1 a cos γsin1 sin2 cos1 cos2 cos2 sin1 cos γcos1 sin2 sin γsin2 Eocos2 z s(t)sinγsin2 x1 cos γcos2 sin1 cos1 sin2 sin1 sin2 cos γcos1 cos2 cos2 sin γ z s(t)cos2 sin γ Eosin2 1. y, sin sin1 sin cos1 cos zs( t)cos za( t) z (15) 7

8 After some mathematical operations, the locus of hob cutter can be simplified as: r ( u, v,,, z ( t)) [ x ( u, v,,, z ( t)), y ( u, v,,, z ( t)), z ( u, v,,, z ( t)),1], (16) a a a a where x cos ( E x cos y sin ) sin [( z ( t) z )sin cos ( y cos x sin )], (17) 2 2 o s y cos [( z ( t) z )sin cos ( y cos x sin )] sin ( E x cos y sin ), (18) 2 2 s o z z ( t) ( z ( t) z )cos ( y cos x sin )sin. (19) and 2 a s T 1 The rotating angle of work gear 2( 1, z ( t)) is set as a linear function of both the hob s rotation angle a and the amount of hob s traverse feed z () t. This rotational relationship between the hob and work gear is defined as a N (, z ( t)) ( z ( t)), (20) a 1 20 N2 a where symbols N 1 and N 2 indicate the number of teeth of the hob cutter and gear, respectively. The parameter 20 ( za ( t )) is the proposed additional rotation angle of the work gear at respective different traverse position of the hob cutter z () t. It assures that the tooth of hob cutter passes continuously through the work gear s tooth a space at middle facewidth in the finish hobing process. It is noted that the new hob s diagonal feed zs () t is set as a second-order function of the hob s traverse feed z () t, as expressed in Eq. (14). However, in Eq. (14), the a second order coefficient c2 is usually much smaller than the first order coefficient c 1. Therefore, the exact solution for the additional rotation angle of work gear 20 ( z ( t )) is a linear function of a traverse movement along the axis of hob cutter za () t. a Since the finish hobbing process has two independent motion parameters, 1 and za () t, there are two equations of meshing between the hob and work gear that can be obtained and expressed as follows: 8

9 x2 ( u1, v1, 1, 1, za ( t)) d1 f2( u1, v1, 1, 1, za( t)) n2 ( y2( 1, v1, 1, 1, za( t)) ) 0, dt u 1 z 2( u1, v1, 1, 1, za ( t)) T (21) and x2 ( u1, v1, 1, 1, za ( t)) dza () t f3( u1, v1, 1, 1, za( t)) n2 ( y2( 1, v1, 1, 1, za( t)) ) 0, dt za () t u z 2( u1, v1, 1, 1, za ( t)) T (22) where nx 1( u1, v1, 1) n2( u1, v1, 1, 1, za ( t)) L21( 1, za ( t)) ny1( u1, v1, 1), nz1( u1, v1, 1) (23) and the transformation matrix L 21( 1, za( t)) is the submatrix of M 21, as defined in Eq. (15), by deleting the last column and row. The tooth surface of the crowned helical gear can be determined by solving Eqs. (5), (15), (16), (21), (22), and (23), simultaneously. The surface locus depends on the hob s diagonal feed motion function zs () t, as expressed in Eq. (14). This function can be used for longitudinal crowing of an involute helical gear with twistfree tooth flanks in gear finish hobbing process. 4. Numeral Examples 4.1. Example 1 To compare the tooth flank twist level of the crowned and uncrowned work gears, an index of evenness of lengthwise crowning along the longitude of work gear is proposed. This index is called the crowning evenness ratio, R ce, and it is defined by the smallest crowning amount divided by the largest crowning amount at different position on each longitude of work gear surfaces. Therefore, it is preferable if the crowning evenness ratio approaches one. 9

10 Table 1. Basic data of the gear and hob cutter Work gear data Number of teeth ( N 2 ) 50 Normal module ( m ) 3 pn Normal circular-tooth thickness ( s pn2 ) mm Normal pressure angle ( ) 20 o pn Helix angle ( p2 ) 21 o R.H. Face width ( F ) 14 mm Hob data w2 Number of teeth ( N 1 ) 1 Lead angle ( p1 ) 1 o R.H. Normal circular-tooth thickness ( s pn1 ) mm Finish hobbing process data Dual-lead coefficient b Hob diagonal shift first order coefficient c Hob diagonal shift second order coefficient c The purpose of this example is to validate the proposed longitudinal crowning method. The basic data for work gear, hob, dual-lead coefficient and coefficients of new hob s diagonal feed function are given in Table 1. The normal deviations and tooth surface topographies of the crowned helical gear, generated by applying the standard and dual-lead hob cutters, are calculated and shown in Tables 2-3 and Figs. 5-6, respectively. The simulated results reveal that the tooth flank of work gears can be crowned in longitudinal direction by setting new hob s diagonal feed motion zs () t as a second-order function of the hob s traverse feed za () t, as expressed in Eq. (14). However, the maximum tooth flank twist for the hobbed work gear tooth flank generated by using the standard hob cutter, m, is much higher than that of the hobbed work gear tooth flank generated by using the dual-lead hob cutter, 0.62 m. Besides, the crowning evenness ratio for the tooth flank surface generated by using the standard hob cutter, as shown in Table 2 and Fig. 5 ( R ce 0.13), is much smaller than that of the tooth flank surface generated by using the dual-lead hob cutter, as shown in Table 3 and Fig. 6 ( ce R 0.98). It is verified that the twist of the tooth flank almost equals zero by applying the proposed duallead hob cutter. 10

11 Table 2. Normal deviations of crowned work gear surfaces generated by the standard hob cutter Top Land (Unit: m ) E D C B A Gear Root Note: The maximum tooth flank twist = m and longitudinal crowning evenness ratio = 0.13 Table 3. Normal deviations of crowned work gear surfaces generated by the dual-lead hob cutter Top Land (Unit: m ) E D C B A Gear Root Note: The maximum tooth flank twist = 0.62 m and longitudinal crowning evenness ratio = 0.98 Fig. 5. Topography of crowned work gear surfaces hobbed by the standard hob cutter Fig. 6. Topography of crowned work gear surfaces hobbed by the dual-lead hob cutter 11

12 4.2. Example 2 In this example, the influences of coefficient b of the tooth profile dual-lead hob cutter, coefficients c1 and c 2 of the new hob s diagonal feed function on the twist of the hobbed tooth flank are investigated. The basic data for work gear and hob are also the same as those given in Table 1. The effects of each coefficient on the twist of the crowned gear tooth flank are investigated with the same abatement amount of 10% of the coefficients. The sensitivities of coefficients b are shown in Table 4 and Fig. 7. It reveals that the twist of tooth flank is quite sensitive to the change of coefficient b. The longitudinal crowning evenness ratio is reduced from 0.98 (Table 3) to 0.86 (Table 4). The evenness ratio for this case has reduced about 12.24%. Therefore, the duallead coefficient b should be chosen agreeably with the data of the crowned work gear to obtain a twist-free tooth flank of crowned work gears. Table 4. Normal deviations of crowned work gear surfaces with an abatement amount 10% of coefficient b Top Land (Unit: m ) E D C B A Gear Root Note: The maximum tooth flank twist = 4.73 m and longitudinal crowning evenness ratio = 0.86 Fig. 7. Topography of crowned work gear surfaces hobbed with an abatement amount 10% of coefficient b 12

13 The sensitivities of coefficients c1 and c 2 are shown in Tables 5 6 and Figs. 8 9, respectively. It is found that the twist of tooth flank is also quite sensitive to the change of coefficients c1 and c 2, and the coefficient c2 has a slightly sensitive than that of the coefficient c 1. The longitudinal crowning evenness ratios are reduced from 0.98 (Table 3) to 0.85 (Table 5) and 0.82 (Table 6), respectively. The evenness ratios have reduced 13.27% and 16.33%, respectively. It reveals that the coefficients c1 and c2 of tooth flank. So these coefficients are usually determined by applying optimal programs. are much effect on the twist Table 5. Normal deviations of crowned work gear surfaces with an abatement amount 10% of coefficient c 1 Top Land (Unit: m ) E D C B A Gear Root Note: The maximum tooth flank twist = 4.95 m and longitudinal crowning evenness ratio = 0.85 Table 6. Normal deviations of crowned work gear surfaces with an abatement amount 10% of coefficient c 2 Top Land (Unit: m ) E D C B A Gear Root Note: The maximum tooth flank twist = 5.38 m and longitudinal crowning evenness ratio =

14 Fig. 8. Topography of crowned work gear surfaces hobbed with an abatement amount 10% of coefficient c 1 Fig. 9. Topography of crowned work gear surfaces hobbed with an abatement amount 10% of coefficient c 2 5. Conclusions In this paper, mathematical models for the dual-lead hob cutter and tooth profile of crowned work gear have constructed for the finish hobbing process. According to the simulated results of two numeral examples, it leads to the following conclusions: 1. The tooth flank of involute helical gear can be crowned longitudinally by applying a new hob s diagonal feed motion and using the standard hob cutter. However, the using standard hob cutter will induce natural twist of the tooth flank on the hobbed gear. Therefore, tooth profile of hob cutter is modified in dual-lead form with the pressure angle changed in it s longitudinal direction. 2. The twist-free tooth flanks can be obtained by setting hob s diagonal feed motion as a second-order function of the hob s traverse feed and using the dual-lead hob cutter in the finish hobbing process. 3. The sensitivities of all coefficients to the twist of the hobbed work gear tooth flanks are investigated. It is found that the twists of tooth flanks are much sensitive to the change of all coefficients. 14

15 Acknowledgments The authors are grateful to the National Science Council of the R.O.C for the financial support. Part of this work was performed under Contract No. Nomenclature s jki circular tooth thickness, j = b, o, p; k = t, n; i = 1, 2 jki helix angle, j = b, o, p; k = t, n; i = 1, 2 crossed angle jki lead angle, j = b, o, p; k = t, n; i = 1, 2 m jki module, j = b, o, p; k = t, n; i = 1, 2 p on normal circular pitch measured at the operating pitch plane N i number of teeth, i = 1, 2 E o operating center distance jki pressure angle, j = b, o, p; k = t, n; i = 1, 2 r jki radius of the pitch circle, j = b, o, p; k = t, n; i = 1, 2 b tooth modified coefficient c i hob diagonal shift coefficient, i = 1, 2 Subscripts b n t p 1 hob base circle measured in normal section measured in transverse section pitch circle 2 work gear References [1] Abler, J., Felten, K., Kobialka, C., et al., 2004, Gear Cutting Technology: Practice Handbook, Liebherr-Verzahntechnik GmbH, Kempten. [2] Tseng, J.-T., Tsay, C.-B., 2004, Mathematical Model and Surface Deviation of Cylindrical Gears with Curvilinear Shaped Teeth Cut by a Hob Cutter, ASME 7th Biennial Conference on Engineering Systems Design and Analysis, Manchester, England, pp [3] Witte, D., 2006, 2006, Finish Hobbing Crowned Helical Gears without Twist, Gear Solutions 15

16 Magazine, pp [4] Witte, D., 2006, A Revolution in Hobbing Technologies, Gear Solutions Magazine, pp [5] Dimitriou, V., Vidakis, N., Antoniadis, A., 2007, Advanced Computer Aided Design Simulation of Gear Hobbing by Means of Three-Dimensional Kinematics Modeling, Manufacturing Science and Engineering, Vol. 129, pp [6] Dimitriou, V., Antoniadis, A., 2008, CAD-Based Simulation of the Hobbing Process for the Manufacturing of Spur and Helical Gears, Advanced Manufacturing Technology, Vol.41, pp [7] Wang, W.-S., Fong, Z.-H., 2008, A Dual Face-Hobbing Method for the Cycloidal Crowning of Spur Gears, Mechanism and Machine Theory 43, pp [8] Chen, K.-H., Chen, C.-J., Fong, Z.-H., 2009, Computer Simulation for the Cutting Process of a CNC Hobbing Machine, Chinese Society of Mechanism and Machine Theory Conference, Chia-Yi, Taiwan, pp [9] Lange, J., 2009 How Are You Dealing with the Bias Error in Your Helical Gears AGMA Fall Technical Meeting. [10] Xu, H., Chakraborty, J., Wang, J.C., 2009, Effects of Axle Deflection and Tooth Flank Modification on Hypoid Gear Stress Distribution and Contact Fatigue Life, AGMA Fall Technical Meeting, 05FTM09, Detroit, Michigan. [11] Winkel, O., 2010, New Developments in Gear Hobbing, Gear Technology, pp [12] Simon, V., 2011, Optimized Polynomial Function for Inducing Variation to Machine Tool Settings in Manufacturing Hypoid Gears, 13 th World Congress in Mechanism and Machine Science, Guanajuato, Mexico. [13] Hsu, R.-H., Fong, Z.-H., 2011, Novel Variable-tooth-thickness Hob for Longitudinal Crowning in the Gear-hobbing Process, Thirteenth World Congress in Mechanism and Machine Science, Gearing and transmissions, Guanajuato, Mexico. [14] F.L. Litvin, A. Fuentes, I.G. Peres, L. Carvenali, K. Kawasaki, R.F. Handschuh, Modified Involute Helical Gears: Computerized Design, Simulation of Meshing and Stress Analysis, Computer Methods in Applied Mechanics and Engineering 192, 2003, pp [15] Litvin, F.L., Fuentes, A., 2004, Gear Geometry and Applied Theory, the second edition, Cambridge University Press. 16