AGMA 13FTM12-2013 Practical Considerations for the Use of Double Flank Testing for the Manufacturing Control of Gearing.pdf

1、13FTM12 AGMA Technical Paper Practical Considerations for the Use of Double Flank Testing for the Manufacturing Control of Gearing By E. Reiter, Web Gear Services Ltd. and F. Eberle, Hi-Lex Automotive Center2 13FTM12 Practical Considerations for the Use of Double Flank Testing for the Manufacturing

2、Control of Gearing Ernie Reiter, Web Gear Services Ltd. and Fred Eberle, Hi-Lex Automotive Center 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 The gearing ind

3、ustry has developed many unique measuring techniques for the production control of their products. Each technique has inherent advantages and limitations which should be considered by designers and manufacturers when selecting their use. Double flank composite inspection, (DFCI) is one such techniqu

4、e that can functionally provide quality control results of test gears quickly and easily during manufacturing. The successful use of DFCI requires careful planning from product design, through master gear design and gage control methods in order to achieve the desired result in an application. This

5、document explains the practical considerations in the use of double flank testing for the manufacturing control of spur, helical, and crossed axis helical gearing including: a general description of double flank inspection equipment including an explanation of what can be measured. recommendations o

6、n practical master gear design. the calculation of tight mesh center distance and test radius limits. the resulting backlash that can be anticipated in gear meshes based on applying double flank tolerances in a design. initial and ongoing statistical techniques in double flank testing and how they c

7、an be practically used to improve gear quality. double flank gage measurement system analysis including case studies of gage repeatability and reproducibility (R mnnormal module of the system, mm; zwnumber of teeth on the test gear; NOTE: For external gears, use a positive value for zwand for intern

8、al gears, use a negative value. z3number of teeth on the master gear; whelix angle of the test gear, degrees or radians; 3helix angle of the master gear, degrees or radians; NOTE: For spur gears 0w3, degrees or radians NOTE: For right hand helical gears, worms, and worm gears use a positive value fo

9、r the helix angle. For left hand helical gears, worms, and worm gears use a negative value for the helix angle. nnormal pressure angle for the mesh, degrees or radians; snw maxmaximum normal circular tooth thickness of the test gear, mm; snw minminimum normal circular tooth thickness of the test gea

10、r, mm; sn3normal circular tooth thickness of the master gear, mm; FidTwtotal composite tolerance for the test gear, mm. The calculation procedure that follows is sufficiently general to account for gears with non-standard tooth thicknesses and heavily modified profiles. Step 1. Calculation of the st

11、andard center distance, a The standard center distance, a, of an external or internal test gear when meshed with an external master gear on a double flank tester is: 2cos cos3wn wwwzzm zaz(1)where a is the standard center distance between the test gear and the master gear, mm. NOTE: These equations

12、are sufficiently general to account for external or internal spur, helical, crossed axis helical and worm gears Parallel axis double flank tight mesh center distance limits The following additional steps are needed for the calculation of tight mesh center distance test limits for external and intern

13、al parallel axis spur and helical gear meshes. 7 13FTM12 Step 2. Calculation of the transverse pressure angle, tThe transverse pressure angle, t, for the mesh on the double flank tester is: tantancos-1 ntw(2) where tis the transverse pressure angle for the mesh in degrees or radians NOTE: For spur m

14、eshes tn Step 3. Calculation of the maximum tight mesh center distance limit, ad maxThe maximum tight mesh center distance, ad max, of the test gear with the master gear for a spur and parallel axis helical double flank mesh is: cos2cos inv inv2costidTwwd maxw-1 nn3nw maxwtwwaFzazms szza(3) where ad

15、 maxis the maximum tight mesh center distance of the test gear with the master gear, mm; inv is the involute function and inv = tan - with expressed, radians. inv-1x is the inverse involute function where x = inv = tan - . Therefore, the result of the function inv-1x = , where is an angle. For more

16、information on the calculation of this function, see AGMA 930-A05, Annex E 1. Step 4. Calculation of the minimum tight mesh center distance limit, ad minThe minimum tight mesh center distance, ad min, of the test gear with the master gear for a spur and parallel axis helical double flank mesh is: co

17、smin2cos inv inv2costidTwww-1 nn3nw minwtwwaFzadzms szza(4) where ad min is the minimum tight mesh center distance of the test gear with the master gear, mm. NOTE: For internal gears, equation 3 will actually give a minimum value result and equation 4 will give the maximum value result. When specify

18、ing tight mesh center distance limits, it is important to also include a definition of the master gears number of teeth and normal circular tooth thickness upon which the tight mesh center distance limits are based. Crossed axis helical and worm gear double flank tight mesh center distance limits Th

19、e calculation for crossed axis and worm gear double flank meshes differs from other cylindrical gear meshes because the gears see each other in a way which is analogous to two racks in mesh as opposed to two involute gears in mesh. Crossed axis helical gears include the case where the driving member

20、 is a master worm as shown in Figure 5 used to measure a helical gear at right angles. The calculations presented here are also sufficiently general to include the case where two helical gears mesh at shaft angles other than ninety degrees as well as the case in Figure 6 where a plastic test worm is

21、 meshed against a master spur gear. In the case of worm gears, the master gear would actually be a cylindrical worm mounted at a right angle to the worm gear. NOTE: The formulas presented here allow for meshing on the double flank tester at any shaft angle. 8 13FTM12 Figure 5. Master worm in double

22、flank mesh with a plastic helical gear (Courtesy of Web Gear Services Ltd.) Figure 6. Plastic test worm in double flank mesh with a master spur gear at an offset shaft angle (Courtesy of Web Gear Services Ltd.) Step 2. Calculation of the meshing shaft angle on the double flank tester, The shaft angl

23、e, , on the double flank tester for a given crossed axis helical gear or worm gear mesh is calculated as follows: w3 (5) where is the meshing shaft angle on the double flank tester, degrees or radians. NOTE: Careful adherence to the sign of each of the helix angles (i.e., right and left hand) is cru

24、cial in this calculation. Step 3. Calculation of the maximum tight mesh center distance limit, ad maxThe maximum tight mesh center distance, ad max, of the test gear with the master gear for a crossed axis helical or worm gear double flank mesh is: ()2tan 2n3 nw max n idTwd maxnss m Faa(6)9 13FTM12

25、Step 4. Calculation of the minimum tight mesh center distance limit, ad minThe minimum tight mesh center distance, ad min, of the test gear with the master gear for a spur and parallel axis helical double flank mesh is: ()2tan 2n3 nw min n idTwd minnss m Faa(7) Some software programs incorrectly cal

26、culate tight mesh center distance for crossed axis helical gears and worm gears using the parallel axis approach in the previous section instead of this method. If the sum of the normal circular tooth thicknesses between the master gear and test gear are close to the normal pitch, the calculation pr

27、ocedure detailed in the previous section may present results that are close to the actual values, however as the sum of these tooth thicknesses deviates from the normal pitch, the calculation error becomes increasingly significant. The method shown in this section is always preferred for crossed axi

28、s helical and worm gears. Test radius Test radius can be measured on a double flank tester as is demonstrated in Figure 3. Test radius is similar to tight mesh center distance in terms of setup and calibration. However it differs in that it is calculated as the tight mesh center distance of the mesh

29、 minus the test radius of the master gear as shown in the following equation: wrw d r3wzRa Rz (8) where Rrwis the instantaneous test radius of the external or internal test gear (i.e., working gear), mm; adis the instantaneous tight mesh center distance of the mesh on the double flank tester, mm; Rr

30、3is the test radius of the master gear as seen by a rack (see next section for further explanation), mm. As a result, the scale between the left side vertical axis in Figure 3 and the right side vertical axis is shifted by the magnitude of the master gear test radius. One of the reasons why test rad

31、ius is specified instead of tight mesh center distance is due to the common misconception that the master gears number of teeth and its normal circular tooth thickness have no influence on the limits of a test gears test radius. The assumption is that regardless of the master used, the test radius l

32、imits of a test gear are constant. If this statement were true, then it would obviously be an advantage in circumstances where a manufacturer may have a different master gear than the purchaser. However, in reality, careful analysis of the equations 1 through 8 shows that there is some difference in

33、 the test radius results depending on the master gears number of teeth and normal circular tooth thickness. An illustrative example is shown in Table 1 where master gears A and B have different numbers of teeth and normal circular tooth thicknesses resulting in significantly different test radius li

34、mits on the same test gear. Table 1. Numerical example of the effect of master gear tooth circular tooth thickness and number of teeth on the test radius Master gear A Test gear Master gear B Module, mm 1.0 1.0 1.0Number of teeth 38 20 50 Pressure angle, degrees 20 20 20 Total composite tolerance m

35、- 96 - Normal circular tooth thickness, mm 50% of circular pitch1.5708 0.000 mm 40% of circular pitch1.2566 0.020 mm 60% of circular pitch1.8850 0.000 mm Test radius limits, mm 9.539 0.080 9.568 0.075 10 13FTM12 Hence, there is no practical advantage in specifying test radius instead of tight mesh c

36、enter distance. In both cases, the master gears number of teeth and normal circular tooth thicknesses must be defined to make the specification valid. It is common to report either tight mesh center distance or test radius but not necessarily both. Tight mesh center distance has greater internationa

37、l usage as compared to test radius. Most North American electronic versions of double flank testers available will report tight mesh center distance and test radius, however European or Asian equipment often do not include test radius results with their equipment. Test radius of the master gear, Rr3

38、In order to calculate the result in equation 8, the test radius of the master gear, Rr3, must be determined. Unfortunately, there are several methods by which the test radius of a master gear is defined, all having potentially different results, creating even more confusion in the industry. The prac

39、tical issue in the definition is that the test radius may be different depending on whether the master gear is defined by its action against a rack or itself (see Figure 7), or against another cylindrical gear. Furthermore, if using a definition based on its action against another cylindrical gear,

40、a single master gear may have many different test radius values depending on the cylindrical gear it mates with. Hence, based on equation 8 the test radius of a test gear will change depending on the test radius of the master. Therefore if a master gear has an ambiguous test radius definition, the p

41、art gear test radius will also be inherently ambiguous. Specifying tight mesh center distance as opposed to test radius will remove this ambiguity. However, if test radius must be used, the most common practice to avoid ambiguity is to define the test radius of a master gear by the tight mesh radial

42、 distance between the master gear center and the pitch line of a mating rack whose pitch line is defined as the location where the tooth thickness is equal to its space width. In making such a standardized definition, a master gear will have a single test radius regardless of the gear it mates with.

43、 The equation for the test radius of a master gear, Rr3, as seen by a rack is as follows: (0.5)2cos 2tann3 n3 nr33nmz s mR(9)where Rr3 is the test radius of the master gear as seen by a rack, mm. Figure 7. Test radius of a master gear (in black) against a rack (in red) or a similar master (in blue)

44、11 13FTM12 Test radius limits of a test gear The test radius of a test gear is related to tight mesh center distance by the following equations: wrw max d max r3wzRa Rz (10)and wrw min d min r3wzRa Rz (11) where Rrw maxis the maximum test radius limit of the test gear, mm; Rrw minis the minimum test

45、 radius limit of the test gear, mm. Eccentricity (double flank runout) In electronic (and computer driven) gages, it is possible to use a Fourier transform calculation to extract the first order sinusoidal wave component from the measured double flank data. The first order component is shown as the

46、green sinusoidal wave in Figure 3. By using this technique, the magnitude and orientation of the test gears eccentricity can be established. In the Figure 3 example, the runout result (i.e., the peak-to-peak amplitude) is reported as 0.008 mm. This data may be useful in identifying how to improve ne

47、t shape gears (such as plastic, powder metal, or die cast gears) where the location of the mounting datum (i.e., bore or journal) to the gear geometry can sometimes be adjusted through tooling changes. The Figure 3 example would therefore report that the gears datum as mounted on the double flank te

48、ster is eccentric from the gears teeth (as a single set) by 1/2 of the runout, or in this case 0.004 mm. The term runout is a misnomer when it is derived by this double flank method. To be more precise, this is a double flank runout which should not be confused with the runout result one would obtai

49、n by actually inserting a pin or ball between the flanks of the teeth and comparing the maximum and minimum result of the individual readings. The two methods may yield slightly different results. When using double flank runout methods, the test reports should indicate the identifier double flank runout instead of just runout. Master gear design considerations Master gears used in double flank composite me