1、I L O 95FTM14 Study of Effect of Machining Parameters on Performance of Worm Gears by: hand Narayan, Xerox Corporation, Donald Houser and Sandeep Vijayakar, Ohio State University TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesStudy of
2、Effect of Machining Worm Gears Parameters on Performance of Anand Narayan, Xerox Corporation, Ohio State University Donald Houser and Sandeep Vijayakar, nhe statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the American G
3、ear Manufacturers Association. Abstract This paper studies the effect of machining parameters on the performance of wm gears using a special purpose finite element technique. Algorithms are presented to determine the worm and gear geometries by simulating the grinding action of the a”nding wheel and
4、 the cutting action of the hob. Results are presented delineating the effect of machining parameter such as the hob oversize, hob swivel angle, profile modification etc., on performance parameters such as the sizeand location of contact zone on the gear tooth, contact stresses, root stresses, load d
5、istribution and transmission error. Results are also presented on the effect of load on transmission error of worn gears. Copyright O 1995 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Vuginia 223 14 October, 1995 ISBN: 1-55589463-4 COPYRIGHT American Gear Manufactu
6、rers Association, Inc.Licensed by Information Handling ServicesEffect of Machining Parameiem on Performance of Worm Gears hand Narayan Xerox Corporation 800 Phillips Road. 147-54A, Webster. NY 14580 Sandeep M. Vijayakar Advanced Numerical Solutions 2085 Pine Grove Lane. Columbus, OH 43232 Donald R.
7、House! Professor. Dept. of Mechanical Engineering The Ohio State University. Columbus. OH 43210 Introduction Some of the earliest work on worm gears was done by Dudley . grindins wheel and solving the equation of conjugacy. Each pomt in the worn axial section is then rotated and simultaneously trans
8、lated DOUI the worm axis (to account for the worm lead) to obtain a curve m space defined by n poinrs or nodes. These curves are known as axodes Ref II The worm geomem is thus completei! defined by these axodes Fis. 2 shows the worm obtained by this procedure. The finite element mesh is constructed
9、using a user defined template Ref. id. The geometry of the gear is obtained from the geomeE of the hob which cuts the gear and the machine motions. The axial section of the hob is 4. Rotation to account for swivel angle of hob 5. Rotation of gear “geared to the rotation of the hob The points thus tr
10、ansformed need not lie on the surface of the gear tooth For a point to lie on the surface of the gear tooth. it must satis- the equation of conjugacy .e ng.vg=O where, np is the normal vector to the gear surface and vF is the velocity vector and can be obtained by differentiating the position vector
11、 in the gear coordinate system with respect to time. Fig. 3 shows a gear tooth generated using the above procedure. The finite element mesh was generated using a user defined template Ref. 141. The information required to generate the finite element mesh is specified in a template file. The template
12、 file contains information in smctiy numeric format and hence the same template file can be used for both the worm and the gear. The geomew of the worm and gear is defined by curves in space mnning along the facewidth. These curves in space are called axodes and the points on these curves are called
13、 nodes. Each node is defined by two vectors. one defining the position and the other defming the normal to the surface at that node. Thus each curve in space is associated with two axodes. one defining the position and the other defining the normal. The geomery of the worm and the gear is defined by
14、 a number of these axodes. The template file contains information about the number of axodes used to define the surface. the number of axodes used to define the interior. number of nodes per axode, number of elements per tooth. the axode connectivity used to define each element etc. These parameters
15、 can be changed to generate the desired mesh. for example if root stresses are of importance then more elements can be used in the root region. The mesh files for the worm and gear are generated based upon these template files. Once the worm and gear geomenies have been generated. they are aligned t
16、ogether and the load and friction values are specified. The alignment information is contained in a file called the configuration file. The configuration file contains information about the load. coefficient of obtained from the grinding wheel geornetp using the procedure described above. Tne profil
17、e of the hob thus obtained is modified b!, applying a parabolic modification. The parabolic modification is applied by keeping the modification zero at the pitch point and Parabolicall VVk2 the modification to the tip and the root. Once the axial section of the hob has 3 COPYRIGHT American Gear Manu
18、facturers Association, Inc.Licensed by Information Handling Servicesfriction between the worm and gear, number of teeth to be modeled on the worm and the gear and the configuration of the worm and gear with respect to each other. thus allowing for all possible misalI-ment between the worm and the ge
19、ar. Fig. 4 shows a worm and gear aligned together. Case Studies A 753. 1 I” CD thread-milled gear set used by Colboume6 is takenas the reference gear set for the parametric studies. The following machining parameters have been used for this reference gear set. - 0.05. I. Hob oversize 7. Hob Swivel A
20、ngle = 0.25 degrees - -0.001” 3. Hob Profile Modification The negative modification has the effect of removing material from the top and bottom of the hob, thus adding matenal to the top and bottom of the gear tooth. A gear torque of 34803 in-lbs along with a coefficient of friction of 0.012 is used
21、 for the analysis. Finite element meshes are generated for the worm and the gear, modeling five teeth each on the worm and the gear. The contact analysis program (CAPP) is run for eleven time steps to study the size and location of the contact zone, contact stresses. load distribution. root stresses
22、 and transmission error. Three machining parameters viz. hob oversize, hob swivel angle and hob profile modification are then varied individually and the effect of these parameters are presented. The effects of other hob geometry factors viz. the hob lead. hob lead angle, thread thickness of hob and
23、 pressure angle of hob are discussed. The effect of changing the load on aansmission error is also presented. Reference case Fig. 5 shows the contact pressure contours on the gear teeth for one time step. The contact pressure contours are distributed between teeth two. three and four. counting from
24、left to right. The contact pressure contours on tooth two are located at the top edge indicating that the tooth is about to lose contact. those on tooth three are located about the center of the tooth and the contours on tooth four are at the bottom of the tooth. indicating that the tooth has just c
25、ome into conract. The maximum contact pressure is found to be 77297 psi. The contact pressure obtained from CAPP can be verified using the Hertzian formula for contact stresses. Assuming a uniform load distribution, the maximum Hertzian contact pressure is found to be 50.000 psi. The difference in t
26、he two numbers can be attributed to the fact that the Hertzian equation assumes a uniform load distribution and the contact is assumed to be occurring over the whole face of the pear whereas in reality it occurs only over a portion of the , : ,.” Figure 4 : Worm and Gear in Mesh Figure 5 : Contact P
27、ressure Contours on Gear Teeth (Reference Cast 18 Figure 6 : Load Distribution on Gear Teeth (Refere 4 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesFigure 7 : Root Suess Distribution Along Face Width (Reference Case 8 9 10 II P u J -3 POD- Figure S
28、Vanation of Transmission Error with Time (Reference Case) Figure 10 : Load Distribution on Gear Teeth (Profile Mdification=O! Figure 11 : Contact Pressure Contours on GearTeeth (Profile Mod.=0.001) Figure 9 : Contact Pressure Contours on Gear Teeth (Profile Modification=O) Figure i2 : Load Distribut
29、ion on Gear Teeth (Profile Modificationa.001) 5 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesface. The edge contact on teeth two and four occur due to the negative profile modification. Fig. 6 shows the load distribution on the gear teeth. Most of t
30、he load is shared by. teeth three and four with only a small portion of the load on tooth two. Fig. 7 shows the root stress variation of tooth three of the sear. The - 1 end represents the exit side of the tooth and the +I side represents the entry side. The plot shows the stress to be increasing fr
31、om a value of zero at the exit side of the tooth to a value of about 6800 psi at +0.3 and decreasing to 2000 psi at the entry side. The location of the maximum stress is consistent with the load distribution plot in Fig. 6. Fig. 8 shows a time varying plot of aansmission emor with a peak-to-peak tra
32、nsmission error of 0.4631E-04 radians. The contact ratio for this gear set is between two and three. Figure 13 : Contact Pressure Contours on Gear Teeth (Hob Oversize. E- le . ca To study the effect of profile modification. the profile modification was changed to zero from negative 0.001 for the ref
33、erence case. The contact pressure contours. shown on Fig. 9, are distributed only on teeth three and four unlike the reference case where tooth two was also in contact. This is due to the fact that the change in profile modification from -0.001” to zero caused the removal of material from the top an
34、d bottom of the gear tooth and thus reduced edge contact. The maximum contact pressure increased to 84907 psi. Fig. 10 shows the load distribution on teeth three and four. The maximum root stress on tooth three increased IO 12.000 psi from 6000 psi for the reference case. The peak to peak transmissi
35、on error reduced to 0.3292E-O4 radians. The contact ratio is below two. The profile modification was then increased to 0.001”. This has the effect of adding material to the top and bottom edges of the hob and thus removing material from the pear. A positive modification is usually eiven to the hob t
36、o prevent edge contact. Fig. 11 shows that the contact pressure contours are distributed on teeth three and four. with majority of the load on tooth three. This fact is confmed by the load disrribution plot on Fig. i?. The edge load which appears on the reference case has disappeared due to the posi
37、tive modification. The maximum root stress on tooth three increases to 18.000 psi due to the fact that the majonry of load is now carried by tooth three. Effect of Hob Oversize Figure II : Load Dismbution on Gear Teeth (Hob Oversiz-0) To study the effect of reduction in hob oversize, the oversize wa
38、s reduced to zero while keeping all other parameters fixed. The contact pressure contours for the same time step as the reference case is shown in Fig. I ?. The contours are shifted towards the exit side of the tooth as a result of decrease in hob oversize. The peak contact pressure increased to 158
39、,860 psi from 77297 psi for the reference case. Fig. 14 shows the load distribution on the gear teeth. The loads on the teeth are shifted towards Figure 15 : contact Pressure Contours on Gear Teetli (Ho 6 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Service
40、sFigure 16 : Load Dismbunon on Gear Teeth (Hob Oversize=O.l) Figure 19 : Contact Pressure Contours on Gear Teeth (Swivel Angle=O.T! Figure 20 : Load Distribution on Gear Teeth (Swivel hgle=0.5) Figure 17 : Contact Pressure Contours on Gear Teeth (Swivel Angle=O) Fipre 18 . Load Distribution on Gear
41、Teeth (Swivel Xngle=Oi Figure 21 : Variation of Transmission Error with Load (Profile Mod.=0.001) COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesthe leaving side of the gear teeth and there is a sharp load at the end Of tooth three. he magnitude of ma
42、ximum root sness doesnt VV significantly from the reference case though the location shifted towards the exit side of the tooth. The peak to peak transmission error was 0.7539E-O4 radians. The hob oversize was then increased to 0.1”. The contact pressure contours on Fig. 15 are shifted towards the e
43、ntry side of the gear teeth. The peak contact pressure increased to 84.343 psi. The load dimibution on Fig. 1 6 shows the loads also to be shifted towards the enny side with a sharp edge load on tooth wo as it goes out of contact. The location of the maximum root stress also shifts towards the entry
44、 side and the magnitude increased slightly. The peak to peak transmission error was 0.2649844 radians. Effect of Hob S wivela The hob swivel angle was reduced to zero from 0.25 degrees. The contact pressure contours shown on Fig. 17 are shifted towards the entry side of the gear teeth. The maximum c
45、ontact pressure increased to 122,892 psi. The load distribution plot on Fig. 18 shows a sharp load at the edge of tooth two. The maximum root stress increased to 14,000 psi and the location of maximurn stress shifted towards the entry side. The peak to peak uansmission error was 0.2 1 16E-04 radians
46、. The hob swivel angle was then increased to 0.50 degrees. ?lie contact pressure contours shown on Fig 19 are shifted to the exit side as are the load distribution plots on Fig. 20. There is a sharp load on the edge of tooth three and a small edge load on tooth two. The location of maximum root stre
47、ss also shifted towards the exit side and the peak to peak transmission error was 0.4249E-04 radians. Effect of Other Paramete- The effect of other hob parameters were also studied. Reduction in the hob lead moved the contact zone towards the bottom left side and an Increase in hob lead moved it tow
48、ards the top right side of the gear tooth. Reduction in hob pressure angle has the effect of moving the contact zone to the top side and increase in pressure angle has the effect of moving it to the bottom side. Hob thickness was found to have no effect on the contact zone and reduction in hob lead
49、angle had the same effect of pressure angle .e. reduction in hob lead angle moved the contact zone to the top side and increase in lead angle pushed the contact zone to the bottom side of the tooth. Fig. 2 I shows the variation in transmission error with roll angle for three different loads. The plot is similar to plots of variation in uansmission error with load for spur and helical gears. The load of 10.000 in-lb. gives the optimum transmission error plot with almost no variation transmission error with roll angle. r: Conclusions The parametric studies in this paper de
