Computer-aided manufacturing of spiral bevel and hypoid gears by applying optimization techniques

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Computer-aided manufacturing of spiral bevel and hypoid

gears by applying optimization techniques

Chung-Yunn Lin

a

_{, Chung-Biau Tsay}

a,*

_{, Zhang-Hua Fong}

b

a_{Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, ROC} b_{Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi 62101, Taiwan, ROC}

Received 7 June 1999

Abstract

A mathematical model of an ideal spiral bevel and hypoid gear-tooth surfaces based on the Gleason hypoid gear generator mechanism is proposed. Using the proposed mathematical model, the tooth surface sensitivity matrix to the variations in machine±tool settings is investigated. Surface deviations of a real cut pinion and gear with respect to the theoretical tooth surfaces are also investigated. An optimization procedure for ®nding corrective machine±tool settings is then proposed to minimize surface deviations of real cut pinion and gear-tooth surfaces. The results reveal that surface deviations of real cut gear-tooth surfaces with respect to the ideal ones can be reduced to only a few microns. Therefore, the proposed method for obtaining corrective machine±tool settings can improve the conventional development process and can also be applied to different manufacturing machines and methods for spiral bevel and hypoid gear generation. An example is presented to demonstrate the application of the proposed optimization model. # 2001 Elsevier Science B.V. All rights reserved.

Keywords: Spiral bevel; Hypoid gear; Machine±tool settings; Optimization; Surface deviation

1. Introduction

Manufacturing spiral bevel and hypoid gears requires state-of-the-art machinery and techniques because such gears have complex tooth-surface geometries. Many analy-tical efforts such as tooth contact analysis (TCA), loaded tooth contact analysis (LTCA), stress analysis, undercut checking, kinematic optimization, among others, are suc-cessfully applied to the design of spiral bevel and hypoid gears to obtain optimal tooth surfaces with permissible contact pattern position, length, bias, smoothness of motion, and adjustability of assembly. Therefore, it is signi®cant to develop a methodology that minimizes, within acceptable tolerances, the surface deviation of real cut spiral bevel and hypoid gear-tooth surfaces with respect to theoretical ones. Recent technology development on CNC machinery makes it possible to manufacture and inspect spiral bevel and hypoid gears using full quantitative and qualitative controls. Several computer-aided inspection systems and closed-loop manufacturing systems that combine CNC coor-dinate measuring machines with theoretical gear-tooth-sur-face data, have been developed by Gleason Works [1],

M&M Precision Systems [2], Soehne [3], and Lemanski [4] in the past few years. Theoretical gear-tooth-surface data can be obtained from mathematical models of bevel and hypoid gears. Krenzer [5] proposed computer-aided correc-tive machine settings for manufacturing bevel and hypoid gear sets using ®rst-order and second-order sensitivity matrices. Litvin et al. [6,7] and Zhang and Litvin [8] 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. However, all these studies investigated minimization of surface devia-tions by means of the so-called linear regression method. Since the tooth-surface geometry of spiral bevel and hypoid gears is quite complex and sensitive to machine±tool set-tings, gears with different characteristics should be gener-ated using different manufacturing machines and methods. Therefore, it is desirable to build up a methodology that has the following characteristics: (a) gear-set mathematical models represented in terms of machine±tool settings and machine constants; (b) calculated corrective machine±tool settings in terms of actual machine±tool settings; (c) better numerical ef®ciency, reliability, and robustness than the linear regression method.

In this paper, a methodology for simulating manufacture of theoretically correct tooth surfaces of Gleason spiral

*_{Corresponding author. Tel./fax: 886-3-5728450.}

E-mail address: cbtsay@cc.nctu.edu.tw (C.-B. Tsay).

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bevel and hypoid gears is proposed. The sensitivity of tooth surfaces to variations of machine±tool settings is investi-gated using the sensitivity analysis technique. Therefore, the characteristics of the gear generator can thus be obtained and controlled. Using the proposed gear-set mathematical model, theoretical tooth-surface data can be determined and then down-loaded to CNC coordinate measuring machines. Using CNC coordinate measuring machines to measure sampling points on tooth surfaces, real cut gear-tooth-surface data can be obtained. The measured data can then be compared with the theoretical data to calculate gear-tooth-surface deviations. Using the measured surface devia-tions and the sensitivity matrix, corrective machine settings that minimize the surface deviations to within tolerances can be obtained by means of a quadratic optimization algorithm. Using this optimization technique to calculate corrective machine±tool settings is shown to be more ef®cient and successful than using the linear regression method.

In the optimization procedure, the maximum gear-tooth-surface deviations is chosen as the objective function, instead of the least-squares sum used by the linear regression method [7], and perturbations of the machine±tool settings are chosen as the design variables to be updated automatically by sequential quadratic programming (SQP). The tooth-surface characteristics vary with cutting machines and methods, and surface characteristics such as tilted root angle, tooth thick-ness, backlash, etc., should be considered in the optimization development procedure. In practice, the corrective machine± tool settings are bounded to a permissible range to match the practical machine±tool relationship. In this study, a prototype optimization software program: multifunctional optimization system tool (MOST) [9], based on the SQP method, is used as an optimization tool because of its accuracy, reliability, and ef®ciency [10]. An improved procedure for development of spiral bevel and hypoid gears based on the proposed approach is suggested. An example is presented to demonstrate the optimization technique and its applications.

2. Mathematical model of the Gleason spiral bevel and hypoid gears

In practice, spiral bevel and hypoid gears can be cut using the Gleason hypoid gear generator. The Gleason hypoid gear generator mechanism can be divided into four major parts: face-mill cutter; cradle assembly; feed and drive mechan-isms; work-head assembly. Detailed description of the mechanism as presented by Fong and Tsay [11], Litvin et al. [12], and the Gleason Works [13±15] is omitted here. Cross-section a±a of the head cutter can be considered to be two straight lines, as shown in Fig. 1, and it can be expressed in coordinate system S1(x1, y1, z1) as follows:

x1 rm 1₂W ujsin jj sin b;

y1 rm 1₂W ujsin jj cos b;

z1 ujcos jj; (1)

where j i and o, and parameters ui, bi, uo, bo, are the

head-cutter surface coordinates of the inside and outside blades, respectively. Subscript ``i'' indicates the inside blade, and ``o'' represents the outside blade; the ``'' sign should be considered a ``'' sign for the outside blade j o, and a `` '' sign for the inside blade j i.

The Gleason spiral bevel and hypoid gears generating mechanism coordinate systems are shown in Fig. 2. The position vectors of the spiral bevel and hypoid gear-tooth surfaces

~

Riand their surface unit normals

~niwere developed by Fong and Tsay [16].

3. Sensitivity analysis

The sensitivity of spiral bevel and hypoid gear-tooth surfaces to machine±tool settings was studied by Fong and Tsay [17], Krenzer [5], and Huston et al. [18]. Using the sensitivity analysis technique, the machine±tool settings characteristics of each spiral bevel and hypoid gear gen-erator were obtained. Based on the repeatability of the same gear generator in the manufacturing process, the surface deviation of real cut gear-tooth surfaces can be minimized by choosing optimal corrective machine±tool settings.

Fig. 1. Relationship between the coordinate system S1and the face mill

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Detailed descriptions of the gear-tooth surface sensitivity to variations in machine±tool settings were discussed by Fong and Tsay [17], and are omitted here. The machine±tool settings chosen for sensitivity analysis include the cradle angle fc, eccentric angle fe, cutter spindle rotation angle ft,

swivel angle fs, sliding base setting Es, increment of

machine center to back Dx, vertical offset Ev, machine root

angle setting gm, ratio of Helical motion change gears Zh, and

ratio of roll change gears Zr. The ®rst variation in gear-tooth

surfacesduetovariationsinmachine±toolsettingsisde®nedas d ~ RiX @_@BR~i jdBj; (2) where d ~

Ri represents the variation in the tooth surface

position vector and parameter dBjindicates the perturbation

increment of the 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 displacement; 1 min for angular displacement; 0.0001 for the ratio of change gears. 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. 3. 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 top, and the heel to toe. The sensitivity coef®cient Sij is de®ned as the displacement

variation along the normal direction of each tooth-surface point due to the perturbation of each machine±tool setting dBj. Therefore, Eq. (2) can be rewritten as

fd ~

Rig SijfdBjg;

Sij@_@B~Ri

j i 1; . . . ; p and j 1; . . . ; q; (3)

where p m n is the number of sampling points, q the number of machine±tool settings. The sensitivity matrix [Sij]

can be applied to calculate corrective machine settings in the manufacturing development process for spiral bevel and hypoid gears.

4. Corrective machine±tool settings for gear set manufacturing

Deviations between the theoretical gear-tooth surfaces and real cut gear-tooth surfaces may exist for a number of reasons, such as machine±tool setting inaccuracies, machine constant error, machine ¯exibility, among others. Whatever the reason, the corrective machine±tool settings are required

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to minimize tooth surface deviations to within permissible levels. Conventionally, the rolling test development was used to obtain the corrective machine±tool settings and compensate for tooth surface deviations [13±15]. However, it is a time-consuming and inef®cient process for manufac-turing development of spiral bevel and hypoid gears.

In this section, a quadratic optimization procedure is applied to reduce shop time during the development stage.

Using the proposed gear-set mathematical model, theore-tical tooth surfaces can be represented by the meshed sampling points, as shown in Fig. 3. The coordinates of the sampling points

~

Ri are down-loaded to the CNC

coordinate measuring machine, which then measures the corresponding points on the real cut gear-tooth surfaces and records the coordinates

~ R

i on a data diskette. The

measurement data are then compared with the theoretical

Fig. 4. Spiral bevel and hypoid gear development ¯ow chart using the linear regression method. Fig. 3. Surface sampling points on the spiral bevel gear convex side.

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data, and any deviations in the normal direction deter-mined according to

DRi R_~i R_~i _~ni; (4)

where subscript i designates the number of sampling points, and

~ Riand

~ni represent the theoretical position vector and unit normal vector of the sampling points on the tooth surface, respectively.

Based on the calculated surface deviations DRi and the

sensitivity matrix [Sij] of the generator, the governing

equa-tion used to minimize deviaequa-tions in real cut gear-tooth surfaces can be written as follows:

DR1 DR2 DRp 8 > > > > < > > > > : 9 > > > > = > > > > ; @R1 @B1 @R1 @Bq @R2 @B1 @R2 @Bq @Rp @B1 @Rp @Bq 8 > > > > > > > > > < > > > > > > > > > : 9 > > > > > > > > > = > > > > > > > > > ; DB1 DB2 DBq 8 > > > > < > > > > : 9 > > > > = > > > > ; ; (5) or fDRig SijfDBjg; (6)

where {DRi} represents the surface deviation of sampling

points, [Sij] the sensitivity matrix of partial derivatives

@ ~

Ri=@Bj, and {DBj} the corrective machine±tool settings.

The system in Eq. (5) is over-determined, since p @ q. Generally, an over-determined system of equations will not have a solution, so the linear regression method can be adopted to solve it. A schematic ¯ow chart for spiral bevel and hypoid gear development using the linear regression method is shown in Fig. 4. However, from a practical point of view the linear regression method is not adequate for solving this problem because the corrective machine±tool settings obtained by the linear regression method cannot be constrained to a permissible range that matches the physical machine±tool relationship. In some situations, the obtained solutions may be meaningless because the corrective machine±tool settings are yielded out-of-range. In addition, Eq. (6) becomes linearly dependent when the cradle angle

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and sliding base setting perturbations exist simultaneously during Helical motion. This results in a numerical diver-gence instability problem.

For the above-mentioned reasons, a development proce-dure using the proposed optimization proceproce-dure was adopted to overcome the weakness of the linear regression method. The ¯ow chart for spiral bevel and hypoid gear development procedures using the optimization technique is proposed and is shown in Fig. 5. In this study, corrective machine±tool settings were regarded as problem constraints to be con-strained to reasonable and desirable ranges. All machine± tool setting perturbations were chosen as design variables and updated automatically. Unlike the least-squares-error method used in linear regression analysis, the maximum deviation of real cut gear-tooth surfaces was chosen as the objective function. In the proposed development procedure, the sensitivity matrix is also updated automatically to obtain the most accurate corrective machine±tool settings. There-fore, the problem of how to obtain the corrective machine± tool settings is transformed into an optimization subproblem expressed as follows: min DBk j f DBk_j max i jS k ij fDBkj g fDRki gj

i 1; . . . ; p; j 1; . . . ; q and k iteration counter; (7)

that is subject to the constraints: (a) the tooth thickness is constant; (b) the root angle is kept in the range to obtain an admissible gear clearance; (c) DBjl DBj DBju; where DBj

j 1; . . . ; q are the design variables including the pertur-bation increments of cradle angle fc, eccentric angle fe,

cutter spindle rotation angle ft, swivel angle fs, sliding base

setting Es, increment of machine center to back Dx, vertical

offset Ev, machine root angle setting gm, ratio of Helical

motion change gears Zh, and ratio of roll change gears Zr. All

these perturbations are treated as design variables and can be represented as follows:

DBjfDfc; Dfe; Dft; Dfs; DEs; DDx; DEv; Dgm; DZh; DZrg:

(8) Constraint (a) indicates that the gear and pinion tooth thicknesses are held constant, and therefore the backlash is also kept within the design range. Constraint (b) means that the root angle of the gear generated by the corrective machine±tool settings is limited to the desired range to maintain clearance and to avoid interference between mat-ing gears. Regardmat-ing constraint (c), the corrective machine± tool settings DBi must be bounded to within a reasonable

range between DBiland DBiuto match the practical machine±

tool relationship. The values of DBiland DBiuare speci®ed by

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the characteristics of the Gleason spiral bevel and hypoid gears generators.

The above optimization subproblem was solved by using MOST [9], a prototype optimization software pro-gram, in which the SQP method is adopted as an optimizer because of its accuracy, reliability, and ef®ciency [10]. The SQP algorithm is a generalized gradient-descent optimiza-tion method, and subsequently converges to a local rather than a global optimum. A conceptual ¯ow chart of the SQP algorithm is shown in Fig. 6, which re¯ects the characteristics of the direct iterative optimization

method. The steps of the algorithm are brie¯y summarized as follows:

Step 1. Initialization: set k 0 and estimate x(k)_{. Select}

two small numbers for allowable maximum constant violation (e1) and convergence parameter (e2),

respec-tively. Let approximate Hessian matrix H(0) _{be an}

identity matrix.

Step 2. Search direction: linearize the objective and constraint functions about the current x(k)_{and a quadratic}

step size constraint is imposed for the linearized subproblem. This problem can be defined as a quadratic programming (QP) subproblem. Thus, solution of the QP problem yields a direction vector Dx(k)_.

Table 1

Gear blank dimensions, cutter speci®cations, and machine±tool settings (Gleason No. 106 hypoid generator settings)

Items Pinion Gear Blank dimensions

Number of teeth 22 22 Face width 24.000 mm 24.000 mm Pitch angle 45800 ₄₅₈₀0

Outside diameter 93.311 mm 93.311 mm Pitch apex to crown 41.344 mm 41.344 mm Cutter specifications

Mean cutter diameter 125.000 mm 125.000 mm Inside blade angle 268450 ₂₆₈₄₅0

Outside blade angle 138150 ₁₃₈₁₅0

Point width 1.900 mm 1.900 mm Tip fillet 0.500 mm 0.500 mm Initial machine±tool settings

Eccentric angle 40850 ₄₀₈₃₇0

Cutter spindle rotation angle 138350 ₁₈₈₁₁0

Swivel angle 1898420 ₂₆₅₈₅₅0

Cradle angle 1498230 ₃₅₅₈₁₂0

Machine root angle 358240 ₄₁₈₉0

Machine center to back MDa_{, 0.829 mm} _MDa_{, 0.404 mm}

Blank offset 0.884 mm 0.000 mm Sliding base 1.455 mm 5.081 mm Nc/50 ratio gears 43/6357/64 48/6656/66 Helical motion gears 67/4573/46 ±

Helical motion position No. 2 ±

a_{Mounting distance.}

Fig. 7. Gear surface perturbations due to machine±tool setting variations.

Fig. 8. Pinion surface perturbations due to machine±tool setting variations.

Table 2

Machine±tool setting changes

Items Pinion Gear

Using linear regression method

Eccentric angle 0840 ₀₈₂0

Cutter spindle rotation angle 08320 ₀₈₃₉0

Swivel angle 618510 ₁₀₈₃₇0

Cradle angle 111078150 _5873378320

Machine root angle 08150 ₀₈₄₁0

Machine center to back 0.37 mm 0.04 mm Blank offset 0.43 mm 0.11 mm Sliding Base 96.25 mm 0.79 mm Helical motion gears ratio 0.13 0.20 Nc/50 ratio gears 0.01 0.01 Using SQP method

Eccentric angle 08 80 _0.00

Cutter spindle rotation angle 08 70 _{08 7}0

Swivel angle 0.00 0.00 Cradle angle 0.00 0.00 Machine root angle 0810 _{08 1}0

Machine center to back 0.00 0.06 mm Blank offset 0.03 mm 0.05 mm Sliding base 0.03 mm 0.15 mm Helical motion gears ratio 0.00 0.00 Nc/50 ratio gears 0.00 0.00

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Step 3. Convergence criteria: if the maximum constraint violation and convergence parameter are less than the given accuracy e1and e2, then stop the iteration and exit.

Step 4. Line search: a step size (ak) along the direction is

estimated based on a suitable line search method using descent functions.

Step 5. Update H: the BFGS (Broyden±Fletch±Goldfard± Shanno) updated formula [20] is selected to guarantee a positive definite updated Hessian H(k)_.

Step 6. Design updated: the new design is updated with step size (ak) and search direction Dx(k) as iterative

formula Xk1_xk_a_k_Dxk_.

5. Examples

In this section, a spiral bevel gear set generated by the Gleason No. 106 hypoid generator using the Duplex±Helical method is used as an example to demonstrate the proposed optimization procedure. This example investigates the sen-sitivity of surface characteristics to machine±tool setting variations, and illustrates the optimization procedure that calculates corrective machine±tool settings for minimizing deviations of real cut gear-tooth surfaces. The gear blank dimensions, cutter speci®cations, and machine±tool settings are listed in Table 1.

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The maximum displacement perturbations in each col-umn of the sensitivity matrix {Sij} for the gear and pinion

tooth surfaces are calculated according to Eq. (3), and shown in Figs. 7 and 8. The abbreviation I.B. denotes the convex side of the tooth surface, which is cut by the inside blade of the face-mill cutter, while O.B. denotes the concave side of the tooth surface. It was found that the maximum perturba-tion displacement is very sensitive to variaperturba-tions in the eccentric angle fe, sliding base setting Es, vertical offset

Ev, and machine root angle gm. Therefore, when making

these machine±tool settings, care should be taken in mea-suring because they greatly affect real cut tooth-surface geometry. For this example, a spiral bevel gear set generated by the Duplex±Helical method was used. It is usually dif®cult to improve the surface characteristics of gears generated by the Duplex method because, as shown in Figs. 7 and 8, most of the parameters have con¯icting effects on the displacement perturbations from side to side. Thus, improving one side on the tooth may result in a correspond-ing degradation of the other side. However, it is more

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convenient and ef®cient to use the proposed optimization technique to minimize surface deviations than to use the linear regression method.

The sample gear was cut using the primary machine±tool settings shown in Table 1 and the coordinates of the surface sampling points on the real cut gear-tooth surfaces were measured using the CNC coordinate measuring machine. The measured data were then compared with the theoretical

data obtained from the proposed gear-tooth mathematical model. For considerations of precision and minimization of run-out errors, four actual teeth were measured and the average measurement values were taken as actual surface data. Surface deviations at the sampling points on the real cut gear surface are shown in Fig. 9. The maximum devia-tion on the real cut gear-tooth surfaces was 0.029 mm, and occurred at sampling point A1 on the convex side; the tooth

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thickness deviation at the basic reference point E3 was 0.158 mm. On the other hand, the pinion was cut according to the primary machine±tool settings shown in Table 1 and surface coordinates of the sampling points were measured. Surface deviations at the sampling points on the real cut pinion surface are shown in Fig. 10. The maximum deviation on the real cut pinion-tooth surface was 0.048 mm, and occurred at reference point A1 on the concave side. The

tooth thickness deviation at the basic reference point E3 was 0.078 mm. Using the proposed development procedure and the developed computer simulation programs, modi®cations or changes in machine±tool settings were calculated by using linear regression method and SQP method, and are listed in Table 2, respectively. The computer simulation results are also shown in Figs. 11 and 12: in these ®gures the residual surface deviations are almost the same by using

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linear regression method and SQP method. However, since the linear regression method solves this problem dealing with unconstraint problem, the results of corrective machine±tool setting such as cradle angle and swivel angle setting shown in Table 2 were out-of-range and meaningless. Besides, the pinion was generated by Helical motion method. This also induced the linearly dependent problem of cradle angle and sliding base setting changes. In this study, the changes in machine±tool settings calculated by means of linear regression method using the singular value decomposition (SVD) [19] algorithm were typical during the course of trial-and-error. All of above-mentioned cases

illustrated that the linear regression method existed weak-ness to solve this problem. This is why the SQP method is adopted in this problem.

Using the proposed optimization technique and the devel-oped computer simulation programs, optimum changes in machine±tool settings were calculated, and are listed in Table 2. Based on these corrective machine±tool settings, a spiral bevel gear was cut using a Gleason No. 106 hypoid generator. The surface deviations at sampling points on the real cut gear-tooth surfaces are shown in Fig. 13. The tooth thickness deviation at the basic reference point E3 was reduced to 0.025 mm, and the maximum surface deviation

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of about 0.01 mm occurred at sampling point A1 on the convex side. These surface deviations are within tolerance and no further gear development is required. In addition, using the proposed optimum machine±tool setting changes shown in Table 2, a pinion was cut and its surface sampling point deviations are shown in Fig. 14. The tooth thickness deviation at the basic reference point E3 was reduced to 0.013 mm and the maximum surface deviation of 0.011 mm occurred at sampling point A5 on the convex side, which is acceptable. Therefore, the proposed method for obtaining corrective machine±tool settings to minimize surface devia-tions of real cut pinions and gears proved to be very useful.

This indicates that the sensitivity analysis and optimization techniques were successfully applied in the proposed meth-odology.

6. Conclusion

Using a proposed mathematical model of Gleason spiral bevel and hypoid gears and a CNC measuring machine, sensitivity analysis of generated pinion and gear-tooth sur-faces due to machine±tool settings have been investigated. Since the proposed mathematical model was derived

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directly in terms of the machine settings and machine constants, it is very easy to implement the mathematical model and to establish a closed-loop manufacturing system for the spiral bevel and hypoid gears.

Based on the sensitivity analysis and using a CNC coor-dinate measuring machine, an optimization procedure for corrective machine±tool setting calculation that minimizes surface deviations on real cut pinion and gear-tooth surfaces to within the permissible tolerances has been developed. The optimization method, which uses the SQP technique instead of the conventional linear regression method, has also been successfully applied to ®nd corrective machine±tool settings within reasonable setting constraints. The developed opti-mization procedure is applicable to improving quality and quantity controls during manufacture of spiral bevel and hypoid gear sets generated by the Gleason±Duplex method, the Helical±Duplex method, the ®xed setting method, and the modi®ed roll method.

Acknowledgements

The authors are grateful to the National Science Council of the ROC for their grant. Part of this work was performed under Contract No. NSC-84-2212-E-009-018. The authors are also thankful to Prof. C.H. Tseng for his courtesy and kind help in using the developed optimization software MOST version 1.1.

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