1、 STD-AGMA SbFTML-ENGL L79b Ob87575 0004775 375 = 96FIMl AComputer Based Approach Aimed at Reproducing Master Spiral Bevel and Hypoid Pinions and Gears by: Claude Gosselin, Laval University, Yoshio Shiono, Yutaka Seimitsu Kogyo, Ltd., Tetsuya Nonaka, Kyoto University, and Aizoh Kubo, Kyoto University
2、 TECHNICAL PAPER STD.AGMA SbFTML-ENGL 1b U b87575 000477b 221 m A Computer Based Approach Aimed at Reproducing Master Spirai Bevel and Hypoid Pinions and Gears Claude Gosselin “I, Yoshio Shiono I, Tetsuya Nonaka and Aizoh Kubo () Department of Mechanical Engineering Lavai University, Qubec, QC, Cana
3、 the surface differences are then used to evaluate by how much the vanous pinion or gear machine settings must be changed for the actual and target tooth surfaces to match Therefore, if a known master phion or gear is to be reproduced, corrective machine settings can be applied effectively oniy if t
4、he exact target tooth surface is known. The currently available corrective machine settings software packages, such as Gleasons GAGE, calculate correction data based either on the TCA-debed or actual target tooth surface. However, if the latter case is used, the theoretical machine settings are not
5、known, an unfavourable situation since the knowledge of the theoretical tooth surface is usually a desireable feature if oniy as a general process reference. This paper presents a general method to bypass such a limitation, when the basic cutting mdod and the target tooth flank form are known, but t
6、he corresponding theoreticai tooth surface is not exactly known. A computer based aigorithm is used to fmd the theoretical machine settings which will produce a simulated tooth surface matched to that of a measured surface within a given tolerance range, where the calculated theoretical surface is t
7、echnologically equivaient to the measured surface. The newly found theoretical surface is then used as a reference to calculate corrective machine settings by applying the same swfce matching algorithm, but in the opposite direction. The aigorithm can match, within a few pm, theoretical and measured
8、 tooth surfces up to Zd order differences for generated pinions and to 1 order differences for non-generated gear members, which is usuaiiy sufcient to obtain satisfactory tooth surfaces. A test case is presented in the paper: a target master pinion is initially measured on a CMM. The theoretical pi
9、nion is then matched to the measured target master pinion using 2“ order surface matching to define the reference tooth in operation, meshing occurs between the contacting teeth of the pinion and gear. The fundamenta equation of meshing is: which states that the relative speed vector between contact
10、ing surfaces must be in a plane tangent to the meshing surfaces at any contact point. z1 +- Gar /ph Figure 1 : General Simulation Reference Frames At the cutting level, let us consider a contact point common to I both the generating surface described by a rotating cutter blade edge and the generated
11、 surface attached to the work, and state at this point the expression for the reiative speed vector in a common general reference fiame Z (figures 1 and 2): , where is the relative speed vector between cutter and work, E is the speed vector of a point belonging to the cutter attached to a cradie rot
12、ating about its axis Wi, and rw is the speed vector of a corresponding point belonging to the work rotating about its axis X3. cimer Blade 1 k Di i I Figure 2: Simulation Reference Frames for Cutting Processes The work speed vector rw in reference he X, rigidiy connected to the work, is: 2 STD.AGMA
13、SbFTML-ENGL L7Sb b87575 OOOL1778 UT4 * . where Xw is the position of a common point between cutter and work, given in the work reference fiame X, and O, is the work rotation vector given by : (4) where Ra is the ratio of roil between the work and the cradle, such that: (5) as Ra=- LI where a3 is the
14、 work roil angle and LI is the cradle angle of rotation. Fm, the speed vector in general refmce fixme Z, is obtained by transforming the speed vector rm therefore, the expression of the cutter normal , in fixed reference fiame Z is sufficient to complete equation (i). Hence, eq. (1) yields the equat
15、ion of the generated surface in the general reference he Z. The obtained surface equation is a function of three variahies a+, a3 and S, respectively the cdtter angular position, the work roii angle and the position of a point along the cutter blade edge: (10) The position of any point P on the gene
16、rated tooth surface is therefore defined by a combination (ac, a3). The solution of equation (10) is a series of contact points between the nitter blade edge and the work desaibmg a line aiong the path of the cutter edge defined by its position angle uc. The bounded envelope, along the work roll ang
17、le a;, of a series of such lines in the work reference he X gives the generated pinion tooth shown in figure 3. Figure 3 also shows a non-generated gear tooth whose peculiarity is that no work to cradle rolling motion occurs during cutting. A Newton-aphson iterative method 1s used to wmeridy solve e
18、quation (IO) 1,2. “ Pinion Tooth Fimre3: Contach ne Pinion and Gear Teeth The above presented simulation includes adjustments aid movements found in most existing generators 61 Fonnate/Helixform machines. These adjustments can bi- classified in four broad categories: i) Cutter helical advance motion
19、, such as that found in noix generating Helixfom machines, where the cutter is advanced while cutting one tooth space. ii) Cutter tilt and swivei, respectively angles T and K in figure 2 Cutter tilt is used when the available cutter blades do nob match the desired pressure angle at the mean point M
20、It is also used when generating hypoid pinions meshing with nor. generated gear members, as the cutter blades must duplicati: the shape and orientation of the mating non-generated toot:$ of the cutter tilt axis, Da in figure 2, such that the desre. pressure angle is obtained at the mean point M, and
21、 that t-. tooth rootline is parallel to its design value. iii) Work position reiative to reference fi-ame Z. Such w: k positional changes are normally called offset, Sliding YXQF 3 * and Machine Center to Back, respectively Ofket, SiBase and Mctb in figure 2. offset and Mcb are normally used to chau
22、ge the shape of the cut surface, while SiBase is used to maintain tooth depth when cutter tilt z, swivel or Mctb are changed. iv) Decimal ratio, or DRatio, proportional to the ratio of roll between the work and the cradle. 4. TOOTH SURFACE MEASUREMENT AND ERROR SURFACE INTERPRETATION This section wi
23、ll briefly introduce the main aspects considered in the measurement and interpretation of tooth flank error surfaces. Tooth surface measurement is performed by a Coordinate Measurement Mache, or CMM, using a high precision probing head which is displaced in Werent directions and detects when contact
24、 is made with an obstacle, such as a tooth flank (figure 4). The probe is a small sphere of known radius, and the CMM knows at each instant the exact position of the center of the probe 4,5,6. At measutement time, the CMM probe is programmed to aim for a theoreical position, and then to search for a
25、 contact point with the tooth surface; when this contact point is found, the coordinate of the probe center is memorized until ali the measurement points have been picked up. The comparison between the measured and simuiated surfaces is then based on the theoretical and measured coordinates. To obta
26、in the error surface, e.g. the difference between the theoretical and measured coordinates, the tooth surface nod for each measurement data point is calculated and the measured coordinate is compensated for the probe radius using the foiiowing relation: where X- is the compensami tooth flank coordin
27、ate, X- is the measured coordinate, R is the probe radius, and N is the tooth flank nod vector, what is caiied dynamic compensatior since the tooth flank nod vector N is recalculated each time a compensated point is needed. For example, when machine settings are modified, the tooth fiank nod at a gi
28、ven radiai- axial position dong the tooth flank is changed such that the resuits of the compensation will be merent. , Figure 5 shows typical tooth flank measurement results. The results are represented as error surfaces, in dotted lines, compared to the theoretical surfaces in solid lines, for both
29、 the Convex and Concave tooth fanlcs simuitaneously. The vertical distance between the theoretical and measured surfaces corresponds to the error in the direction of the local tooth Bank nod vector, with the error at a preset point equal to zero, nody the dataset middle point when the numbers of bot
30、h prolewise and lengthwise measured data points are odd. 0.0520 O 4 -0.0210 Fime 4: Tooth Flank Measurement v Fksdmgleh a) Spirai and Dressure ande errors b) Tooth lengthwise crownine and bias errors 4 STD-AGMA SbFTMZ-ENGL 199b Ob87575 0004800 582 = Piuion ConvadB c Measurement Grid Figure 5: Tooth
31、error surface intmretation The calculated errors can be caused by heat treatment distorsion, cutter blade, machine setup, machine age and maintenance condition, or internai machine errors. While each data point gives the local error between the measured and theoretical surfaces, more global 1 order
32、trends are shown in figure 5 a), both in the lengthwise and proiewise directions: the lengthwise trend depicts an error in spirai angle, which is the average tiit of lengthwise measurement data lines relative to the corresponding theoretical lengthwise data lines; an error in cutter point diameter i
33、s shown by a me between the lengthwise measurement data hes relative to the corresponding theoretical lengthwise data lines; the proiewise trend depicts an error in pressure angle, which is the average tilt of profiewise measurement data lines relative to the corresponding theoretical profilewise da
34、ta lines; a profiie curvature error is shown by a curve between the proiewise measurement data lines and the corresponding theoretical prolewise data iines; Second order errors take the more complex shape of a saddle which is a combination of lengthwise crowning error and twisting of the error surfa
35、ce that is cailed bias in this paper (figure 5b). Figure 5c shows a Spicai measurement grid on a pinion tooth flank. The density of the measurement grid wiil affect the resolution and the types of errors which can be detected. For example, a 5x9 grid, e.g. 5 profilewise by 9 lengthwise points, is no
36、nnally sufficient for 1 and 2“ order global surface errors, but may not reveal surface waviness or proie error. 5. ERROR SURFACE SENSITMTY TO MACHINE SETTING CHANGES Traditiody in the gearing industry, changes to tooth surface geometry have relied on proportional changes coefficients that may be fou
37、nd at the end of a gear set summary, fiom The Gleason Works for example, which are based on the differential tooth surface geometq at the mean point and on the blank geometry. Such changes are nody applied to the pinion finishing process, as it is usuaiiy cut in Fixed Setting mode and the convex and
38、 concave tooth flanks may be treated separately. Proportional changes may be used in the bearing pattern development process, where the pinion finishing machine settings are progressively modified until a satisfactory bearing pattern is obtained, by converting the V-H test values into quimient chang
39、es on the pinion tooth surhce 7. The surface matching method presented in this paper relies on the global response of the error surface to changes in machine settings. Therefore, this section wili show how an error suIace may respond to such changes, and global trends wiii be established. lu order t
40、o demonstrate the sensitivity of the error surhce to changes in machine settings, a theoretical measurement datatile is created, which contains a 5x9 grid of the phiion convex and concave tooth flank coordinates.When compared to the theoretical tooth flank without machine setting changes, the error
41、surface shows no error, as on the convex tooth fiank of figures 7 to 12. Figures 6 to 12 use the same basic measurement datale, except that the concave tooth flank data is changed to rdect tooth flank topography modifications due to changes in machine settings. The following machine settings are mod
42、ified separately to show how the tooth flank is affected: machine root angie, spiral angie, cutter tilt, cutter point diameter, work offset, machine center to back and decimal ratio. For each machine setting change, the error surhe is recatcuiated and the global trends are identied. Figure 2 should
43、be consulted hereafter to link the changes in machine settings to the simulation model. Figures 6 to 8 show what are caiied I* order changes 5, e.g with minimal curvature or surface twist effects. in figure 6, the machine root angle of the concave and convex tooth flanks is changed by 5; the resulti
44、ng error surface is a combition of spiral angle error on both tooth flanks, tooth taper error which is a difference in spiral angle error between the tooth fianks, pressure angle error and slight surface twist or bias. Fieure 6: 5 Machine root ande change 5 STD-ALMA SbFTML-ENGL 1SSb Therefore, the m
45、achine root agie could be used very thus the spiral angle will be the chosen parameter to control spiral angle mors. Since in ciassical generators the spiral angle is controlled by an eccentric mechanism, a change in spiral angle will result in changes in eccentric, cradie and swivel angies. In hypo
46、id pinions meshing with non-generated gear members, however, a change in spiral angle must be compensated by changes in machine root angle to maintain tooth rootline paraleiism, and sliding base for tooth depth. /-*- Fimire 7: 5 Suiral ande chanae A 5 cutter tilt change, figure 8, produces a combina
47、tion of spiral and pressure angle errors. In classical generators, cutter tilt is produced by the cutter spindle SI, a tilted plane which, when rotated, increases or decreases cutter tilt. Cutter spindle rotation must be compensated by cutter swivel to maintain the direction of the cutter tilt axis
48、relative to the tooth mean point, while sliding base is used to maintab tooth depth. Ob87575 000480L 419 = A, 0.010 i cutter point diameter change, figure 9, produces a lengthwise crowning error combined to a slight spiral angle error. While cutter diameter change produces both crowning and slight s
49、pirai angle errors, broadly speaking it can be thought of as a le order change since the change in curmure is proportiomal to the change in cutter diameter. If the cutter is tilted, as is usuaiiy the case for pinions, the change in cutter tilt requires a change in sliding base to maintain tooth depth. However, cutter diameter change is not usually a parameter of choice to control the tooth lengthwise crowning error as cutter diameter adjustment is a lengthy process. Therefore, another control parameter is needed. Figures 1
