1、97FTMSl I Coordinate Measurement and Reverse of ZK Type Worm Gearing by: Xiaogen Su and Donald R. Houser, Ohio State University TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services* Coordinate Measurement and Reverse Engineering of ZK Type
2、Worm Gearing Xiaogen Su and Donald R Houser, Ohio State University The statements and opinions contained herein are those of the author and should not be construed as an offcial action or opinion of the American Gear Manufacturers Association. Abstract A comprehensive model for the measurement, insp
3、ection, performance prediction and reverse engineering of ZK type of worm gearing is developed. The measurements and the best fit processes both for the worm thread and for the gear tooth flank are discussed in detail. A CMM measurement strategy free of tip compensation which applies to many types o
4、f tools and parts in the gear industry is proposed. A real case of ZK type of worm gearing with the parabolic profile modification on the hob is studied to illustrate the reverse engineering process. Copyright O 1997 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Vir
5、ginia, 22314 November, 1997 ISBN: 1-55589-710-X COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STD-AGMA 77FTMSL-ENGL L997 Ob87575 0005050 748 Coordinate Measurement and Reverse Engineering of ZK Type Worm Gearing X. Su(Research associate) D. R. House
6、r(Professor) Department of Mechanical Engineering The Ohio State University April 20, 1997 Nomenclature E, hob lead E, worm lead C, the hob surface. $c rotational curvilinear ordinate of hob thread. C, the worm surface. $J, rotational curvilinear ordinate of worm thread. 1 C, the worm gear surface.
7、p, screw parameter of the hob (P, = 1,/27r ). Eh center distance between the worm-like pw screw parameter of the worm(p, = 1,/27r ). hob axis and gear axis while hobbing XI, swivel angle while hobbing( the axes of the tool and the the worm are crossed forming the angle yw, which equals the lead angl
8、e on the worm pitch cylinder. In reality, the tool rotates about its axis to provide the necessary cutting speed, but this rotation will be ignored in our geometry model. Some gashes are applied to the tools surface to produce the blades if the tool is a milling cutter. Litvin7 presents the equation
9、s of the worm sur- face(the hob surface equations have the same format) when the tool has straight generatrix. It resembles spur gears in profile modification. We think the worm and/or the gear do not have a profile modification if the grinders generatrix is straight. Most likely, some kind of profi
10、le modification is applied to the tool to produce a better contact pattern, for example, to lo- cate the contact point, or to eliminate edge contact or cornering. A model with a parabolic profile modi- fication is presented here. Referring to Fig. 2, in the coordinate system at- tached to the grinde
11、r, a point on its cone surface with straight generatrix can be expressed as I u ms(a)ms(e) u sin(a) - a r = U cos(a)sin(O) (1) The inward normal is (2) is . Transforming the point on the grinder surface to the system attached to the worm and applying trans- formations to r0 and no where P10 is the p
12、oint transformation matrix from the grinder system to the system 1. N10 is the nor- mal transformation matrix from the grinder system to the system 1. P, is the point transformation matrix for the screw motion, N, is the normal transformation matrix for the screw motion. They are If E, and pw are se
13、t to be O, then P10 and P, become N10 and Ns,-,ew. where $, is the rotation parameter of the worm screw thread. To generate a point on the worm surface, the fol- lowing equation must be satisfied. where no is the normal in the grinder system, and vwo is the velocity at a certain point of the worm re
14、lative to the grinder. The expanded form of the above eauation is pre- sented in p. In the case of a paraboli; profile modi- With the parabolic profile modification, the straight fication, insert the above derived new equations of rw generatrix is replaced by a parabola with its apex and n, in the e
15、quations of (18.7.7-8)q, we arrive at at u = um, the amount of modScation(material re- moved in the inward normal direction) is c2 (u - u,). Now the point on the modified pseudo-cone surface eb“()sin(?)f -k dl - 9 ,/cos(o,) =0.249975“ a:ea = 3.61339“ Another parameter needs to be calculated is the u
16、m(u value along the hob grinders generatrix at parabolas apex). The real value is 8.21514 - T&an(a,) = 8.20604“. The following parameters are the same for both the virtual and real surfaces. They include all cen- ter distances, both grinders profile angles, the angles between axes both for hobbing a
17、nd for grinding, the lead values of the worm and the hob, and finally, the pitch diameter values. 8 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services6.2 Transmission error prediction unit surface normals at the contact points. For case After we obtain t
18、he parameters of the worm and the 4, the fact that the contact point is on the end edge gear, the meshing process is simulated. During the produces one equation. The last equation comes from the fact that the gears end edge must be tangent to rotated to bring it into contact with the worm thread. ot
19、her cases, the other two equations can be written There are seven different contact possibilities. They in similar way. We will not list them here to save the are: space. Process, at a certain worm rotation phase, the gear is the worII1 thre& surface at the contact point. For 1. Worm thread surface
20、to gear tooth surface con- tact: This is the most basic contact type. This type of contact must be guaranteed to produce a good contact pattern and reduce transmission error. No further assumptions are made for the simula- tion. This is the digital description of the real mesh- ing process. We will
21、check the clearance from all the discretized grid points to the worm thread surface to make sure the found contact point is the correct one(the clearance at all grid points should be posi- 2. Worm thread surface to gears throat edge con- tive). If not, a new contact point will be found and tact. The
22、 contact point lies on arc BC of Fig. verified in a logical way. 8. The worm is rotating from -270“ to 180“ by steps of 5“. Ninety one gear rotation angles are found. 3. Worm thread surface to gears outside edge con- The transmission found by the tact- The contact point lies On One Of the top real r
23、otation angles with the theoretical ones. Fig. edges AB and CD of Fig. 8. 13 shows the transmission error(in thick line). Now the thick line is shifted both backwards and forwards which is the the shape of the transmission error curve. In our case, the maximum error is -0.32. 5. Worm thread surface
24、to gear teeth corner con- Fig. 13 also tells us the gear tooth is in real con- tact. The contact point is exactly at one of the tact with the worm thread while the worm rotates four points A, B, C, D of Fig. 8. through -250 to llOo. After llOo, the next tooth will take over the contact. The trace of
25、 all contact 4. Worm thread surface to gears end edge contact. The contact point lies on one of the two end multiP1es Of 3600 to Produce the edges AE and DF of Fig. 8. 6. Worm threads top edge to gear tooth surface points is shown in i,. 14. contact. The contact happens on the worm threads top edge.
26、 6.3 Contact pattern prediction - 7. Worm threads top edge to gears end edge con- tact. The contact happens between the worm threads top edge and the gears end edges. The minimum clearance at all points of the gear tooth grid are calculated. Now the worm is rotated and the gear is rotated to follow
27、it with the transmission error For each case, six non-linear equations with six specified by the upper boundary of Fig. 1%. ChJ- unknowns must be solved simultaneously. The six late the clearance(normd distance from Points unknowns are to the worm thread surface) at all grid points of the gear tooth
28、, keep the minimum value at each point. 2 curvilinear coordinats for worm thread surface.The contours with different clearance values is plot- 2 curvilinear coordinates for gear tooth surface. tad in Fig. 14. This figure also shows the minimum gear rotation angle. clearance curves at top and bottom
29、edges. The con- hobbing rotation angle. tact pattern on the worm thread is also shown in Fig. 13b. The six non-linear equations include three com- mon equations to match the point coordinates of the points contacting each other. Another common equa- 7 Conclusion tion is the hobbing equation between
30、the hob and the gear being hobbed. The other t.wo equations vary In this paper, a comprehensive model for the mea- from case to case. For case l, the left two equations surement, inspection and reverse engineering of the are derived from matching the two components of the 9 COPYRIGHT American Gear M
31、anufacturers Association, Inc.Licensed by Information Handling ServicesZK type of worm gearing is developed. The strate- References gies for evaluating the parameters and the machine setups are presented. Following aspects have been I William L- Janninck Chtact Surface TOPOlOgY Of addressed. Worm Ge
32、ar Teeth AGMA paper 87 FTM 14 1. Both the worm and the gear can be measured 2 J. R. Colbourne The Use of Oversize Hobs to Cut and inspected quantitatively rather than func- Worm Gears AGMA paper 89 FTM 8 are available. tionally if their parmeters and cutting setups 3 F. L. Litvin Simulation of Meshi
33、ng, Transmission Emrs and Bearing Contact for Single Enveloping 2. The developed model can be used effectively Drives AGMA paper FTM in reverse engineering. No inspection tip com- 141 1. H. seal and F. L. Litvin Computerized De- pensation is required for surface inspection and sign, G and Simulation
34、 of Meshing and This to many Contact of Modified Involute, Klingelnberg and types of tools and parts in the gear industry. Flender Type Wom-Gear Drives Power llans- 3. The prediction of transmission error and con- misssion and Gearing conference, ASME 1996 tact pattern are made without approximate a
35、s- pl K. Yoshimura and M. oya and H. T A sumptions and are closer to the reality. New Method of Inspecting an Hourglass Worn. 4. The model can be used to determine the nec- JSME International Conference on Motion and Powertransmissions,MPTSl essary corrections of machine setups in order to obtain a
36、certain contact pattern and/or to reduce transmission error. The model is also suring the worn wneels of Cylindrical very for the worm gear var- Gears International Gearing Conference, 1994 ious mismatches and machine setup errors can be simulated and studied. 7 F. L. Litvin Gear Geometry and Applie
37、d Theory 6 J. Hu and J.A.Pennel1 A Practical Method of Mea- Prentice Hall 1994 Some problems deserves further research. 8 Earle Buckingham and Henry H. Ryffel Design 1. The two curvilinear coordinates of the gear tooth of Worm and Spiral Gears Buckingham Asso- surface is the pair of (vh, uC) Unfortu
38、nately, the coates,INC. 1973 lines of constant oh, u, are approximately par- allel to each other around the mid-face of the 1 gear tooth flank(F1g. 15). This causes seri- ous problems while we want to iteratively find a certain point on the gear flank. In our case, the measured points around the mid
39、dle face are excluded from the gear tooth flank best fit pro- cess. A better algorithm has to be found so we can use a measured point from any position for the best fit. 2. We do not take the assembly errors of the worm and the gear into consideration for performance prediction. The effect of assemb
40、ly errors should be studied. 3. More different profile modifications should be modeled, such as circular and straight ones, in order to get better result in reverse engineering. 10 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesj rudace n I gear axis
41、Figure 3: Generation of bottom land surface of reve Figure 1: Installation of grinding cone lution *i 22 / 11 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesJ aCtuaUy measured surface i 5: dd virtual equidistant tooth flank Figure 7: Deviation of virt
42、ual grinders generatrix from a perfect parabola Rea! grtnder Gear loolh(4X15 discrelked) 12 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesi 0.71 2flW.63.31.15 , &.h65 0.227 02275 0.65 mal ir Figure 12: Comparison of hob axial profile ground by Figure
43、 10: Result of best fit of measured worm thread grinders of different sizes 13 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services(a) Predlcted lraNmss1on error (b) Worm thread contact patlern(0.001) . . 0.2 -1 -500 O 500 Warn Rotauon Angle(degree) Figure
44、 13: Prediction of transmission error(ZK typc Contact Pattern AM/SIS(ZK type) 2.5 L?Fg, 2.4 Figure 15: Contours of two curvilinear coordinates 2.3 Il , on the gear tooth -0.5 -0.4 -0.3 -0.2 -0.1 O 0.1 0.2 0.3 0.4 0.5 5 loor tip boundary dearame o -0.5 -0.4 -03 -0.2 -0.1 O 0.1 0.2 0.3 0.4 0.5 bonm boundary dearance : Ad, -0 -0.5 -0.4 -0.3 -02 -0.1 O 0.1 O2 0.3 0.4 0.5 gear lace width(inch) Figure 14: Prediction of the contact pattern(ZK type) 14 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services
