COMPUTER-AIDED MANUFACTURING OF SPIRAL
BEVEL AND HYPOID GEARS WITH MINIMUM
SURFACE-DEVIATION
CHUNG-YUNN LIN and CHUNG-BIAU TSAY{
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan 30050, Republic of China
ZHANG-HUA FONG
Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi, Taiwan 62101, Republic of China
(Received 12 February 1996)
AbstractÐThe linear regression method to minimize the deviations of a real cut gear-tooth-surface is investigated in this paper. Based on the Gleason hypoid gear generator, a mathematical model of the tooth surface is proposed. Applying the proposed mathematical model, the sensitivities of tooth surface due to the variations of machine-tool setting are also investigated. The corrective machine-tool settings, calculated by using the sensitivity matrix and the linear regression method, are used to minimize the tooth-surface deviations. The minimization problem was solved by using the singular value decompo-sition (SVD) method. The result of this paper can improve the conventional development process and be also applied to dierent manufacturing machines and methods. Two examples are presented to demonstrate the proposed methodology. # 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION
Manufacturing spiral bevel and hypoid gears requires high precision and state-of-the-art ma-chinery because such gears have complex tooth-surface geometries. The fundamental geometry and characteristics of spiral bevel and hypoid gears have been investigated by many research-ers [1±10]. However, the inherent tooth surface deviations caused by machine-tool setting inac-curacies, machine constant errors, machine ¯exibility, and other factors are unavoidable in the manufacturing process. Therefore, it is desirable to build up a methodology that can minimize the deviations of a real cut gear-tooth-surface within the permissible tolerance.
In the recent years, several computer-aided inspection systems and closed-loop manufacturing systems that combine CNC coordinate measuring machines with theoretical gear-tooth-surface data, have been developed by Gleason Workers [11], M&M Precision Systems [12], Klingelnberg Soehne [13], and Lemanski [14]. Theoretical gear-tooth-surface data can be obtained from math-ematical models of bevel and hypoid gears. Krenzer [15] proposed computer-aided corrective machine-tool settings for manufacturing bevel and hypoid gear sets using ®rst-order and second-order sensitivity matrices. Litvin et al. [16, 17] and Zhang and Litvin [18] proposed a series of methodologies to minimize deviations in real cut gear-tooth surfaces and to analyze the meshing and contact of real cut gear-tooth surfaces. In Litvin's model the theoretical tooth-surfaces were derived based on basic machine-tool settings instead of actual machine-tool settings. Therefore, it was necessary to convert the basic machine-tool settings into the actual machine-tool settings and this could aect the precision of the results. Besides, characteristics of gear generators were more dicult to be controlled by using the basic machine-tool settings.
In this paper, the mathematical model of spiral bevel and hypoid gears cut by using Gleason No. 463 hypoid grinder is developed in terms of actual machine-tool settings and machine con-stants. Since the tooth-surface geometry of spiral bevel and hypoid gears is quite complex and
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sensitive to machine-tool settings, the sensitivity matrix between machine-tool settings and tooth-surface geometry is investigated and calculated. Using the sensitivity analysis technique, characteristics of gear generators can thus be obtained and controlled. In addition, the results of sensitivity analysis can be applied to decide what type of machine-tool setting or combi-nation should be chosen and modi®ed to minimize surface deviations of the real cut tooth surface. The theoretical tooth-surface data can be determined using the proposed gear-set mathematical model and then down-loaded to the CNC coordinate measuring machine. The real cut gear-tooth-surface data can be obtained using CNC coordinate measuring ma-chines to measure the coordinates of sampling points on tooth surfaces. The measured data can then be compared with the theoretical data to calculate gear-tooth surface deviations. Using the measured surface deviations and the sensitivity matrix, the corrective machine-tool settings that minimize the surface deviations within tolerances can be obtained by using the linear regression method. In this paper, the problem on how to solve the corrective machine-tool settings with minimum gear-surface-deviation is accomplished by applying the singular value decomposition (SVD) [19] method.
Fig. 2. Relationships among machine settings and basic cradle settings.
According to the proposed approaches, an improved procedure for the development of spiral bevel and hypoid gears is suggested. Two examples are presented to illustrate the sensitivity analysis and the proposed methodology.
MATHEMATICAL MODEL OF THE GLEASON SPIRAL BEVEL AND HYPOID GEARS In this section, the proposed mathematical model for spiral bevel and hypoid gears is devel-oped based on the mechanism of Gleason No. 463 hypoid grinder. The Gleason No. 463 hypoid grinder is an automatic wet-type, high speed machine for grinding spiral bevel and hypoid gears. The grinding of bevel gears makes the extreme precision and heavy load capacity attainable that is required in modern gear production. The generating train of Gleason No 463 hypoid grinder is designed to perform the Modi®ed Roll Motion by means of a special cam reciprocator mechanism. The mechanism of Gleason No. 463 hypoid grinder can be divided into four major parts: (a) cup-shaped grinding wheel, (b) special Modi®ed Roll generating train, (c) feeding and driving mechanisms, and (d) work head assembly. The detailed description of the mechanism investigated by Lin, et al. [20] and the Gleason Works [21] is omitted here. The axial cross-section of the cup-shaped grinding wheel is straight edges in the a-a cross section as shown in Fig. 1, and it can be expressed in coor-dinate system S1(x1, y1, z1) as follows:
R1 uj; bj x1 y1 z1 8 < : 9 = ; rm2 W2 ujsincj ÿ sinbj rm2 W2 ujsincj ÿ cosbj ÿujcoscj 8 > < > : 9 > = > ;; 1
where j = i and o, and parameters ui, bi, uo and bo are the head cutter surface coordinates
of inside and outside blades, respectively. Subscript ``i'' indicates the inside blade, and ``o'' represents the outside blade; the ``2'' sign should be regarded as `` + '' sign for the out-side blade (j = o), and `` ÿ '' sign for the inout-side blade (j = i).
Coordinate systems of the Gleason No. 463 hypoid grinder mechanism are shown in Figs 2 and 3. The grinding wheel spindle is parallel to the cradle axis in the Modi®ed Roll Method. According to the geometry shown in Fig. 2, the tooth surface of the imaginary generating gear can be represented as follows:
Q 360ÿ f cf2e; 2 S 2e sin fe 2 ; 3 and R2 x2 y2 z2 8 < : 9 = ; S cos Q x1 ÿS sin Q y1 z1 8 < : 9 = ;; 4
where feis the cradle angle and is one of the machine settings for the Gleason hypoid grinder;
fe is the eccentric angle and is also a machine setting; e is the machine eccentric constant,
and e = 8 inches for the Gleason No. 463 hypoid grinder; S, is the basic radial distance setting; Q, is the basic cradle angle setting; and x1, y1 and z1 are the surface coordinates of the
cup-shaped grinding wheel represented in Equation (1).
As shown in Fig. 3 [20], coordinate systems S2(x2, y2, z2), S3(x3, y3, z3), S4(x4, y4, z4), S5(x5,
y5, z5), and S6(x6, y6, z6) are rigidly attached to the cradle, machine frame, sliding base, work
head, and workpiece, respectively. Therefore, the locus of the imaginary generating gear rep-resented in the workpiece coordinate system S6can be obtained by applying the following
coor-dinate transformation matrix equation:
where M65 1 0 0 ÿDx 0 cos xw ÿsin xw 0 0 sin xw cos xw 0 0 0 0 1 2 6 6 6 4 3 7 7 7 5; M53
cos gm 0 sin gm ÿEssin gm
0 1 0 ÿEv
ÿsin gm 0 cos gm Escos gm
0 0 0 1 2 6 6 6 4 3 7 7 7 5; M32 cos xc sin xc 0 0 ÿsin xc cos xc 0 0 0 0 1 0 0 0 0 1 2 6 6 6 4 3 7 7 7 5;
xc, is the cradle rotational angle; xw, is the work spindle rotational angle; gm, is the machine
root angle; Dx, is the increment of machine center to back; Es, is the sliding base setting; and
Ev, is the vertical oset setting.
The ratio between the work spindle rotation angle xw and cradle angle xc is called the roll
ratio, and is de®ned as Za=xw/xc. For the Gleason No. 463 hypoid grinder, Zais not a constant
and was derived by Lin, et al. [20].
Because the two mating surfaces are in tangency, the relative velocity of the mating surfaces lies on the common tangent plane during the generation process, and the following equation must be satis®ed [22]:
N V12 0; 6
where N is the normal vector of the generating tool surface, and V12 is the relative velocity
between the generating tool surface and the generated gear blank surface. Equation (6) is the so-called equation of meshing in the theory of gearing. The equation of meshing for spiral bevel and hypoid gears was developed by Fong and Tsay [7], and is given as follows:
AA BB sin xc CC cos xc 0; 7 where AA sin gmÿ1 Za y2nx2ÿ x2ny2 nz2Evcos gm;
BB ÿEvny2sin gmÿ nx2 z2ÿ Es ÿ x2nz2cos gm;
and CC ÿEvnx2sin gm ny2 z2ÿ Es2pL xc ÿ y2nc2 cos gm:
The tooth surface of the generated spiral bevel gear is determined by considering Equations (5) and (7) simultaneously. The unit normal vector njof the generated spiral bevel and hypoid gear
tooth surfaces can be obtained by applying the following equation: njj NNj
jj; j i; o; 8
Nj@R@uj j
@Rj
@b; j i; o;
and Rj represents the position vector of the generated spiral bevel and hypoid gear tooth
sur-faces, and Njindicates its surface normal.
SENSITIVITY ANALYSIS
The constant sensitivity of spiral bevel and hypoid gears with respect to roll test machine set-tings has been studied by the engineers of Gleason Works [23], Fong and Tsay [9], and Litvin et al. [24]. In this paper, the sensitivity of spiral bevel and hypoid gear tooth-surfaces manufactured by the Gleason Modi®ed Roll method, due to machine-tool setting variations is studied by using the proposed mathematical model. Using the sensitivity analysis technique, the istics of machine-tool settings for gear generator can be obtained. Based on the repeat character-istic of the same gear generator in manufacturing process, the surface deviation of real cut gear-tooth surfaces can be minimized by choosing appropriate corrective machine-tool settings. The machine-tool settings of the Gleason No. 463 hypoid grinder used in sensitivity analysis include the machine root angle setting gm, cradle angle fc, eccentric angle fe, sliding base setting Es,
increment of machine center to back Dx, vertical oset Ev, mean grinding wheel diameter Dc,
cam setting d, and cam guide angle j. The ®rst variation on pinion and gear tooth surfaces due to the variations of machine-tool settings is de®ned as
dRi S@R@zi
jdzj 9
where dRi represents the variation on the tooth-surface position vector; parameter dzj indicates
the perturbation increment of machine-tool settings.
The perturbation increment of each machine-tool setting should be chosen according to the precision limitation of the Gleason spiral bevel and hypoid gears generator: 0.01 mm for linear displacements, and 1 min for angular displacements. In sensitivity analysis of spiral bevel and hypoid gear-tooth surfaces, m n discrete sampling points have been chosen to represent the tooth-surface geometric characteristics, as shown in Fig. 4. The values of m and n depend on tooth-surface geometry, sampling accuracy, machine precision as well as product requirements. The surface sampling points are numbered ascendant from the root to tip, and the heel to toe. The sensitivity coecient Sij is de®ned as the surface displacement variation along the normal
direction of tooth-surface point due to the perturbation of each machine-tool setting dzj.
Therefore, the system equation for the variation on the tooth-surface position vector at each sampling point can be rewritten as
fdRig Sijfdzjg;
and
Sij@R@zi
j i 1; . . . ; p; and j 1; . . . ; q; 10
where p = m n is the number of sampling points; q is the number of machine-tool settings. The sensitivity matrix [ Sij] of spiral bevel and hypoid gears can be used to calculate the
correc-tive machine-tool settings in the gear manufacture process. Example 1
A spiral bevel gear set cut on Gleason No. 463 hypoid grinder using the Modi®ed Roll method is chosen as an example to investigate the tooth-surface sensitivity due to machine-tool setting variations. The gear blank dimensions and grinding wheel speci®cations are shown in Table 1 while the machine-tool settings are listed in Table 2.
In this study, 45 surface sampling points have been chosen to represent the pinion and gear tooth-surface geometric characteristics, as shown in Fig. 4. The maximum displacement pertur-bations in each column of the sensitivity matrix { Sij} for the pinion is calculated by applying
Equation (10), and shown in Fig. 5. The abbreviation I.B. denotes the convex side of the pinion tooth-surface which is ground by the inside blade of the grinding wheel while O.B. denotes the concave side of the pinion tooth-surface. It is found that the maximum perturbation displace-ment is very sensitive to the variations of the machine root angle setting gm, eccentric angle fe,
sliding base setting Es, and cam setting d. Therefore, when installing the machine-tool settings,
these four machine settings should be carefully measured due to their high sensitivities to the real cut tooth-surface geometry.
The pinion surface perturbative diagrams of the machine root angle setting gm, eccentric angle
fe, sliding base setting Es, and cam setting d are investigated and shown in Figs. 6±9,
respect-ively. The rigid-body-rotation is extracted from the displacement variation by overlapping the sampling surface point No. 1 for each perturbation of machine-tool settings. The vertical displa-cements in Figs. 6±9 represent the surface perturbation of each sampling point measured along the surface normal direction. The scale is shown at the upper right corner of each diagram. The results of sensitivity analysis can be applied to decide what type of machine-tool setting or
com-Fig. 4. Sampling surface points on the spiral bevel gear convex side.
Table 1. Gear blank and grinding wheel used in the example
Items Pinion Gear
Blank data
Number of teeth 27 28
Face width 25.00 mm 25.00 mm
Pitch angle 228 4' 228 56'
Outside diameter 128.73 mm 132.91 mm
Pitch apex to crown 148.38 mm 147.45 mm
Grinding wheel speci®cations
Mean diameter of grinding wheel Ð 250.00 mm
Point diameter of grinding wheel (I.B.) 250.95 mm Ð
Point diameter of grinding wheel (O.B.) 251.46 mm Ð
Inside blade angle 218 0' 218 0'
Outside blade angle 198 0' 198 0'
Point width Ð 2.41 mm
bination should be chosen and modi®ed to minimize surface deviations of the real cut gear-tooth surface in the spiral bevel gear development process.
CORRECTIVE MACHINE-TOOL SETTINGS FOR GEAR SET MANUFACTURING
Deviations between the theoretical gear-tooth surfaces and the real cut gear-tooth surfaces may exist for a number of reasons, such as the machine-tool setting inaccuracies, machine con-stant errors, machine ¯exibility, and so on. Whatever the reason, the corrective machine-tool settings are required to minimize tooth-surface deviations within a permissible level. Conventionally, the rolling test development was used to obtain the corrective machine-tool set-tings and compensate the tooth-surface deviations [25]. However, it is a time-consuming and inecient process for the manufacture development of spiral bevel and hypoid gears.
In this section, a linear regression method that calculates the corrective machine-tool settings to minimize the surface deviations is applied to reduce the shop time during the development process. Using the proposed gear-set mathematical model, theoretical tooth surfaces can be rep-resented by the meshed sampling points, as shown in Fig. 4. The coordinates of the sampling points Ri are down-loaded to the CNC coordinate measuring machine, then measuring the
cor-responding sampling points on the real cut gear-tooth surface and recording the measured coor-dinates R*
i on a data diskette. The measurement data are then compared with the theoretical
data, and surface deviations for each sampling point measured along its normal direction can be Table 2. Machine-tool settings used in the examples
Machine-tool Settings Pinion I.B. Pinion O.B. Gear
Machine root angle 208 26' 208 26' 218 15'
Machine center to back ÿ9.01 mm ÿ8.46 mm ÿ2.40 mm
Sliding base 3.15 mm 2.96 mm 0.87 mm
Vertical oset ÿ11.64 mm ÿ11.28 mm 2.96 mm
Cam setting 101.60 mm 101.60 mm 76.20 mm
Eccentric angle 328 49' 328 38' 358 44'
Cradle angle 678 3' 718 45' 3248 59'
Cam guide angle 58 10' 08 0' ÿ08 40'
Feed cam setting 68 0' 08 0' 08 0'
Generating cam no. #23 #23 #29
Index interval 10 10 9
Index gears 48/80 50/81 48/80 50/81 42/70 45/84
I.B.: Convex side of pinion tooth-surface ground by inside blade of grinding wheel O.B.: Concave side of pinion tooth-surface ground by outside blade of grinding wheel
determined according to the following equation:
DRi Riÿ Ri ni; 11
where subscript i designates the number of sampling points; Ri and ni represent the theoretical
position vector and unit normal vector of sampling points on the pinion and gear tooth-surface, respectively.
Based on the calculated surface deviations DRi and the sensitivity matrix [ Sij] of the
genera-tor, the governing equation used to minimize surface deviations in the real cut pinion and gear tooth-surface can be written as follows:
DR1 DR2 . . . . . . DRp 8 > > > > < > > > > : 9 > > > > = > > > > ; @R1 @z1 . . . @R1 @zq @R2 @z1 . . . @R2 @zq . . . . . . . . @Rp @z1 . . . @Rp @zq 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 Dz1 Dz2 . . . . . . Dzq 8 > > > > < > > > > : 9 > > > > = > > > > ; ; 12 or fDRig SijfDzjg i 1; . . . ; p; and j 1; . . . ; q; 13
Fig. 6. Surface perturbations on the pinion convex and concave sides due to the variation of machine root angle gm.
where {DRi} represents the surface deviation of sampling points, [ Sij] is the sensitivity matrix of
partial derivatives@Ri
@z1; and {Dzj} represents the corrective machine-tool settings.
The system Equation (12) is overdetermined since p>>q. An overdetermined system equations will generally not have a solution. For that reason, we seek a solution Dz*
j that possesses the
clo-sest possible solution in the least square sense. In other words, we solve Equation (12) and ®nd Dz
j which minimizes jj SijDzj ÿ DRi jj2: 14
The problem was solved by the singular value decomposition (SVD) method because of the SVD dealing with system equations that are either singular or else numerically very close to singular. The SVD method will not only diagnose the problem, it will also solve the problem and give a useful numerical solution. The details on how to derive and solve equations are dis-cussed by Press et al. [19] and are omitted here.
Example 2
The spiral bevel gear set cut by Gleason No. 463 hypoid grinder using the Modi®ed Roll method is used as an example to demonstrate the proposed development procedure. The initial
Fig. 7. Surface perturbations on the pinion convex and concave sides due to the variation of eccentric angle fe.
design values for this example are the same as those listed in Tables 1 and 2. The gear was ®n-ished using the duplex grind (i.e. Spread blade) process while the pinion was ground side-by-side using dierent machine-tool settings (i.e. Fixed settings).
Again, 45 surface sampling points have been chosen to represent the pinion and gear tooth-surface geometric characteristics. The sample gear was cut using the primary machine-tool set-tings shown in Table 2 and the coordinates of the tooth-surface sampling points on the real cut gear-tooth surfaces were measured using CNC coordinate measuring machines. The measured data were then compared with the theoretical data obtained from the proposed gear-tooth math-ematical model. For considerations of precision and minimization of runout errors, four actual teeth were measured and the average measurement values were taken as the actual surface data. Surface deviations at each sampling point on the real cut gear surface are shown in Fig. 10. The maximum deviation on the real cut gear-tooth surface was 0.0045 mm, and occurred at sampling point A1 on the convex side; the tooth thickness deviation at the basic reference point E3 was 0.1578 mm. Using the proposed development procedure and the developed computer simulation programs, modi®cations or changes in machine-tool settings were calculated, and are listed in Table 3. Based on these corrective machine-tool settings, a spiral bevel gear was cut using the same Gleason No. 463 hypoid grinder. The surface deviations at sampling points on the real cut gear-tooth surface are shown in Fig. 11. The tooth thickness deviation at the basic reference point E3 was reduced to 0.0231 mm, and the maximum surface deviation of about 0.005 mm was occurred at sampling point I5 on the convex side. These surface deviations are within the permissible tolerances and no further gear development is required.
Fig. 8. Surface perturbations on the pinion convex and concave sides due to the variation of sliding base setting Es.
On the other hand, the pinion was cut according to the primary machine-tool settings shown in Table 2 and pinion-surface coordinates of the sampling points were also measured. Since the convex and concave pinion sides were ground side-by-side with dierent machine-tool settings. The surface deviations at sampling points on the convex and concave sides of the real cut pinion-surfaces are shown in Figs. 12 and 13, respectively. The maximum surface deviations were 0.033 mm on the convex side and 0.081 mm on the concave side, respectively. Using the proposed development procedure and the calculated machine-tool setting changes shown in Table 3, a pinion was cut and its surface sampling point deviations are shown in Fig. 14 and Fig. 15. The maximum surface deviations were substantially reduced to 0.013 mm on the convex
Fig. 9. Surface perturbations on the pinion convex and concave sides due to the variation of cam set-ting d.
Table 3. Machine-tool setting changes
Items Pinion I.B. Pinion O.B. Gear
Machine root angle 08 26' ÿ08 23' ÿ08 15'
Machine center to back ÿ0.42 mm 0.16 mm Ð
Sliding base Ð Ð 0.75 mm
Vertical oset ÿ0.59 mm ÿ0.78 mm Ð
Cam setting 0.24 mm 0.24 mm ÿ0.27 mm
Eccentric angle Ð Ð 08 9'
side and 0.023 mm on the concave side, respectively. Therefore, the proposed method for obtain-ing corrective machine-tool settobtain-ings to minimize surface deviations of the real cut pinion and gear has proved to be very useful. This indicates that the sensitivity analysis and linear re-gression method were successfully applied in the proposed methodology.
CONCLUSION
By applying the developed mathematical model of Gleason spiral bevel gear generator, the sensitivity of tooth surfaces due to manufacture errors has been investigated. Based on the results of sensitivity analysis and using a CNC coordinate measurement machine, the linear
gression method for corrective machine-tool settings calculation that minimize deviations on the real cut gear and pinion tooth-surface within the permissible tolerances has been developed. The results of this paper can be applied to improve the quality and quantity controls of the manu-facture of spiral bevel and hypoid gears.
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