1、Y- Stock Distribution Optimization in I , 2000FTM8 1 a by: I C. Gosselin, Laval University and J. Masseth and Steve Noga, DANA Spicer Light Axle Group TECHNICAL PAPER Stock Distribution Optimization in Fixed Setting Hypoid Pinions Claude Gosselin, Laval University and J. Masseth and Steve Noga, DANA
2、 Spicer Light Axle Group The statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract In the spiral bevel and hypoid gear manufacturing industry, master pinions and gears are
3、 usually developed from initial machine settings obtained from computer software such as The Gleason Works TCA, which are then modified until a satisfactory bearing pattern is obtained. Once a satisfactory combination of master pinion and gear is obtained, the Stock distribution between the rough an
4、d finish cuts of fixed setting, face milled pinions is usually altered to the point where left-overs may appear at the finish cut. The operator must then modify the roughing machine settings until left-over disappears, which does not guarantee that Stock will be evenly distributed over the tooth fla
5、nk. Even Stock distribution allows increases in cutting speeds and feed, and produces gears with better tooth flanks while increasing tool life. This paper presents an algorithm to optimize the Stock distribution in two and three cut face milling processes for spiral bevel and hypoid gears. The algo
6、rithm minimizes the differences in terms of spiral angle, pressure angle, and tooth taper between the roughed and finished tooth flanks. An application example is presented to illustrate the usefulness of the met hod. Copyright O 2000 American Gear Manufacturers Association 1500 King Street, Suite 2
7、01 Alexandria, Virginia, 22314 October, 2000 ISBN: 1-55589-769-X STOCK DISTRIBUTION OPTIMIZATION IN FIXED SETTING HYPOID PINIONS Claude Gosselin (l), Jack Masseth (2), Steve Noga (3) () Department of Mechanical Engineering Laval University, Qubec, QC, Canada, G 1 K-7P4 Chief engineer, Gear Engineeri
8、ng DANA Spicer Light Axle Group Fort Wayne, IN 46801 (3) Gear development engineer, Gear Engineering DANA Spicer Light Axle Group Fort Wayne, IN 46801 1. Introduction Face milled hypoid pinions produced by the three cut, non completing, fixed setting system where roughing is done on one machine and
9、finishing for the concave-OB and convex43 tooth flanks is done on separate machines with different setups, are still in widespread use today. The undeveloped machine settings, for the finish and rough cuts, are normally obtained from a TCA program, such as those produced by Gleason and Klingelnberg
10、l. During the development process, where the finishing machine settings on the pinion are modified until the desired bearing pattern is obtained, the stock distribution between the roughing and finishing operations may be altered to the point where the finishing cut hardly touches the tooth flank in
11、 some areas of the tooth. The machine operators then either decrease the depth of the roughing operation, in which case an undesired lip or step may be left at the root of the tooth, or decrease the thickness of the finished tooth at the expense of increased backlash. This paper presents an algorith
12、m used to optimize the stock distribution between the roughing and finishing cuts for spiral bevel and hypoid members cut by the fixed setting method. The optimization is based on the Surface Match algorithm 2, where the differences between the roughing and finishing spiral angle, pressure angle and
13、 tooth taper are minimized in order to obtain rough and finished tooth flanks that are parallel. Application results of the optimization are shown. Better stock distribution usually results in : - both roughing and finishing tool life can be increased, at Finishing by more uniform and reduced stock
14、for the finishing cutters and, at Roughing, by being able to increase the point width of the roughing cutter; reduced development times; otherwise, both the rough and finish setups must be developed; improved productivity: it may be possible to increase the feeds and speeds with reduced chip loads;
15、improved tooth fatigue performance due to more uniform fillet radii in the roots. 2. Main Nomenclature - N S a, a3 Y v, r tooth flank normal unit vector relative speed vector position of a point on the blade edge cutter angular position work roll angle pressure angle error spiral angle error tooth t
16、aper error 3. Tooth Cutting Process The generating process of gear teeth are based on the basic concept of meshing elements between a cutter blade whose rotation represents the shape of one tooth of a theoretical generating gear, and the work itself. The fundamental equation of meshing is: - N.V, =O
17、 which states that the relative speed vector between contacting surfaces must be in a plane 1 tangent to the meshing surfaces at any contact point. When applied using the reference frames depicted in figure 1, eq. (1) yields a generated surface in the fixed reference frame X. The obtained surface eq
18、uation is a function of three variables il) Cutter tilt and swivel, respectively angles Tilt and Swivel in figure 1 ; iii) Work position, normally called Offset, Sliding base and Machine center to back, figure 1 ; iv) Decimal ratio, proportional to the ratio of roll between the work and the cradle.
19、zi Di T Machine plane t O z2 D3 i/ y Machine center to back I Figure 1 : Reference frames for the simulation of gear manufacturing Pinion Tooth Figure 2: Mesing pinion and gear teeth 2 4. The Stock Distribution Graph and its Interpretation The Stock distribution is the amount of material that is to
20、be removed at the finishing cut. Ideally, this should be constant over the tooth flank in order to provide adequate performance of the cutting tool. If the tooth taper is Duplex, nearly constant sotck distribution may be achieved; otherwise, stock distribution may be biased. The Gleason and Klingeln
21、berg TCA Softwares initially provide a good stock distribution by properly selecting the roughing machine settings. In practice, because the roughing and finishing cutter diameters and pressure angles may be different, and because the machine settings may have been modified during the development cy
22、cle, Stock is not uniformly distributed. Stock distribution may be shown by two superimposed surfaces, figure 3, respectively representing the topography of the tooth flank after roughing (solid lines) and finishing (dotted lines). O mm 03853 o4554 cut n n n nn n “ “ n Profilewise direction O Length
23、wise direction Figure 3: Stock distribution graph For clarity, the tooth flanks are unwrapped; thus the horizontal lines are in the tooth lengtwise direction, the slanted lines are in the tooth profilewise direction, and the vertical lines represent the difference between the roughed and finished to
24、oth flank surfaces. The differences between the roughed and finished tooth flanks of figure 3 are called Surface Error Plots and show the error in the direction of the local tooth flank normal vector. The actual differences between the tooth thicknesses of the rough and - finished tooth can readily
25、be appreciated. Each data point gives the local difference between the rough and finished surfaces; global 1 order trends can be observed in the lengthwise and profilewise directions (figure 4): the lengthwise trend depicts an error in spiral angle, which is the average tilt of roughed data lines re
26、lative to the corresponding finished data lines; a crowning error is shown as a curve between the finished and roughed data lines; the profilewise trend depicts an error in pressure angle, which is the average tilt of the finished data lines relative to the corresponding roughed data lines; a profil
27、e curvature error is shown as a curve between the finished and roughed data lines; taper error is seen as a difference in spiral angle between the IB and OB tooth flanks. While 2“d order errors may be appreciated in the Stock distribution graph, they are neglected in the optimization because of the
28、limited freedom normally available at roughing. Pressure angle mor L Spirai angle error Figure 4: Common Stock distribution errors 5. Error Surface Sensitivity to Machine Setting Changes The Surface Match algorithm 2, used in this paper to optimize Stock distribution, relies on the global response o
29、f the Error surface to changes in machine settings. This section shows how the Error surface may respond to such changes, and global behaviors are established. 3 In order to demonstrate the sensitivity of the Error surface to changes in machine settings, the basic Stock distribution of a hypoid pini
30、on is used, figure 5, which shows negligible pressure, spiral and taper errors. 784 843 Figure 5: Basic Stock distribution Figures 6 to 10 use the same basic Stock distribution, except that the roughing machine settings are changed to reflect tooth flank topography modifications. The following machi
31、ne settings are modified separately: Machine root angle, Spiral angle, cutter Tilt, work Offset and Machine center to back. For each machine setting change, Stock distribution is recalculated and the behavior changes are identified. For any given change in machine setting, the sliding base is modifi
32、ed such as to keep tooth depth constant at mid-facewidth. The actual values in machine setting changes are not reported since the sensitivity of the Error surface depends on the actual geometry. Figure 1 may be consulted to link the changes in machine settings to the simulation model. Figures 6 to 1
33、0 show what are called 1 order changes 5, 61, e.g. with minimal curvature or surface twist effects. In figure 6, the Machine root angle is changed; the resulting error surface is a combination of spiral angle error, tooth taper error (spiral angle error difference between the IB and OB tooth flanks)
34、, and pressure angle error, all of which can be appreciated through the changes in the corner values of the Stock distribution graph. ,- Pressure angle error Figure 6: Change in Machine root angle Modifying the spiral angle, figure 7, induces spiral angle error; thus the spiral angle will be the cho
35、sen parameter to control spiral angle errors. 7 Spirai angle error Figure 7: Change in Spiral angle Changing cutter Tilt, figure 8, produces a combination of spiral and pressure angle errors. A change in work Offset, figure 9, produces a combination of spiral angle, tooth taper, lengthwise crowning
36、and pressure angle errors. A slight profile curvature error is also visible. Likewise, a change in work Machine center to back, figure 10, results in a combination of spiral angle, tooth taper, lengthwise crowning, and pressure angle errors. 4 793 524 Convex-IB Figure 8: Change in Cutter Tilt Pressu
37、re angle error Toe O7275 7 o 4747 t A- mm O3323 Figure 9: Change in work Offset I Pressure angle error Toe - O 8386 7 n 977 Figure 1 O: Change in work Machine center to back From the above, the following conclusions are drawn: 0 spiral angle errors are effectively controlled by a change in Spiral an
38、gle at the mean point, with negligible side effects; 0 pressure angle errors may be controlled by changes in cutter Tilt, Machine root angle, work Offset and Machine center to back; 0 tooth taper errors may be controlled by changes in Machine root angle, work Offset and Machine center to back. Spira
39、l angle Pressure angle looth taper Bias Lcngthwise crownuig I Proecrownuie I Machine rmt I Figure 11 : Mapping of machine settings cross-influences Figure 11 maps the cross-influences of the machine settings on the error surface. Relative cross-influence sensitivity is not shown since it varies with
40、 the actual tooth geometry. While the effects of changes in some machine settings may appear predictable when used one at a time, combined changes include such Error surface side effects that the results cannot be predicted directly. In the above, it is assumed that tooth thickness between rough and
41、 finish is adequate and, therefore, that cutter point width is not changed. In order to quantify the differences between the rough and finish states of a tooth, the following values are defined: 5 pressure angle error: = col=l 1 j spiral angle error: j YJ = row=1 1 tooth taper error: c=YB-YOB where:
42、 -TI IL, Y-T, IL? 5-T, IL, where and Y are the averaged pressure and spiral angle errors, 6 is the taper error, TI, T2 and T3 are desired deviations between the rough and finish cuts, L, L2 L3 are the tolerance ranges within which the objective functions can be considered satisfied. In practice, dev
43、iations Ti are normally null. A Newton-Raphson numerical solution is used to solve the above objective functions, equation (7), where the partial derivatives of the objective functions are calculated in relation to machine setting changes to produce the Jacobian matrix. (3 b) (3 cl The Jacobian matr
44、ix, the sought machine setting changes APs and the objective functions a, Y and Ct form the following systems solved using Gaussian elimination: is the index of row data, along the tooth flank; is the index of column data, across the tooth flank; is the error value at point ij of the grid; is the di
45、stance between data points along the tooth flank; is the distance between data points across the tooth flank. a ao a a, a, ap, ay av av a, ap, a, ar ac ar - - Equations 3 a) to c) are used to quantify the differences between the rough and finish tooth surfaces. Whenever the rough surface is changed,
46、 the Error surface is altered and the above defined quantities are recalculated accordingly. 7. Application -a -Y -5 (7) The algorithm presented above is used to optimize 6. Solution the roughing machine settings of a hypoid pinion. The basic roughing machine settings of the pinion, obtained from a
47、production summary, are given in Table 1. These were established initially when the tooth geometry was defined using a TCA computer The objective is to find a combination of machine settings that minimizes the differences in Spiral angle, Pressure angle and Tooth taper between the rough and finish c
48、uts of a tooth. angle and Spiral angle errors, to changes in Thus, the following objective functions must be satisfied: machine settings while maintaining tooth depth. Figure 12 shows the calculated Stock distribution graph for the above pinion, with left-over material at the toe end of the Convex-I
49、6 tooth flank and at the 6 tip of the OB-Concave tooth flank, which can clearly tooth taper errors between the roughed and finished be seen after the actual cut in figure 13. In order to tooth flanks. bypass this problem, the machine operator may change the roughing machine settings, or increase The final roughing machine settings, Table 2, show finishing cutting depth, thereby reducing tooth changes in every setting except Ratio of roll and thickness and introducing an undesired lip or step at Cutter tilt. the root of the tooth. 81 8 81 4 Fi
