1、 STD*AGMA 94FTMb-ENGL L797 m Ob87575 0004579 380 m 94FTM6 A Boundary Element Procedure for Predicting Helical Gear Root Stresses and Load Distribution Factors by: M. L. Clapper, Ford Motor Company and D. Houser, Ohio State Universitv American Gear TECHNICAL PAPER COPYRIGHT American Gear Manufacturer
2、s Association, Inc.Licensed by Information Handling Services STD-AGMA 74FTMb-ENGL 1797 W Ob87575 0004580 UT2 A Boundary Element Procedure for Predicting Helical Gear Root Stresses and Load Distribution Factors M. L. Clapper, Ford Motor Company and D. Houser, Ohio State University The statements and
3、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 A method to accurately predict gear root stress for parallel axis gears is explored using a combination of three analysis techniques
4、: boundary elements, elastic body contact analysis, and the moment-image method. The method is computationally faster than three-dimensional finite element programs and avoids the use of semi-empirical relationships. The three techniques are combined to determine gear root stress across the face wid
5、th and predict these stresses through the mesh cycle for both spur and helical gears. The method allows the user the flexibility to determine stress at any mesh position. This is in contrast to semi-empirical methods that typically determine stress for a single load position in the mesh cycle at a s
6、ingle stress location on the gear tooth. The root stress predictions are compared to both experimental strain gage resuits and finite element modehg techniques for verification. Results are presented for the prediction of load distribution factors as a function of misalignment and crowning types wit
7、h comparisons being made with AGMA Eactors. Copyright O 1994 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 223 14 October, 1994 ISBN 1-5558944 however, each alone does not fully describe the stress condition through a mesh cycle for stresses at any locatio
8、n across the face width. One can develop three-dimensional models of geared systems and use finite element techniques, though model development time and computational intensity can be extreme, and contact conditions are seldom modeled properly. The presented stress predictions are results of existin
9、g computer programs developed for general gear analysis. These programs include the Load Distribution Program (LDP) that predicts loading across the face width for multiple mesh positions and the post processing program GGRAPH. The stress determination technique can be applied with data from any loa
10、d prediction technique of a gear pair. This technique does not replace finite element techniques for very accurate three dimensional states of stress but provides an approach that is fast, yet not constrained by empirical limits. The following sections present the developed technique and compare the
11、 computational results with experimental strain gage results and three-dimensional finite element results. Further comparisons are made with AGMA factors and show load distribution factors and stress factors as functions of misalignment and crowning types. The study of load distribution and root str
12、ess changes with respect to types of gear errors and modifications demonstrates the sensitivity of gear stress to these factors. Background: The computational method to predict gear root stress is based on the work by Jaramillo i for thin cantilever plates with constant thickness and infinite length
13、. The series solution for bending moments is based on the deflections due to point loads at arbitrary heights on the cantilever plate. The application of the plate theory to gear stresses was proposed by Wellauer and Seireg 2 for the prediction of bending moment distribution for plates of finite len
14、gth. They proposed that the moments reflect at the edge of the tooth, causing higher summed 1 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STD-AGMA SqFTMb-ENGL 1777 Ob87575 0004582 775 bendinn moments in the affectcd region. This moment-image metho
15、d The Stress prediction Me therefore, significant error could occur by outputting stress at an incorrect radius. cm im ibo am zdo *o0 JSO *w -ml Fim 17 LDP predicted Root Stress Through Mesh Cycle, Pinion m 4mm.m 4oam.m rcmaoo ramam -.m mca0.m irmoao 1WW.m rmam am - Figure 18 CAPP Maximum Principle
16、Root Stress, Pinion The previously presented gear cases in this section show LDP to predict stresses relatively accurately for a variety of gear casts. Stresses through a mesh cycle and stresses across the face width compare well for gear sets with and without profde and lead modifications. The anal
17、ysis shows that stresses at the edges of teeth necd to be investigated further. This suggests that the imaging of the moments at the edges may not be entirely valid or load predictions may be too high at the helical edges. Load Distribution Factors: This section evaluates gear stress changes with mi
18、salignment and crowning types using LDP and also demonstrates the methods advantage in quickiy determining stress trends due to arbitrary gearing conditions. Relationships of gear stress to non-uniform loading is a difficult analytical problem due to various factors such as misalignment, elastic def
19、lections, manufacturing accuracy, tooth crowning, and profile modifications. Figure 19 graphs stress factors versus misalignment of a 9.4 pitch, 15 degree helical pinion, found in Table 5, for three LDP predicted cases and the AGMA analytical method. The plotted stress factor is the peak stress divi
20、ded by the stress for an unmodified gear pair in perfcct alignment. This is similar to AGMA bending stress Load Distribution Factor K, defined as peak load divided by average load on the instantaneous lines of contact. nie LDP predicted cases include: gear pair with no modifications, pinion lead mod
21、ified with 0.001 parabolic crown beginning at .Y inward on both face width sides, and pinion lead modifid with 0.001 circular crown across the entire face width. The presented AGMA stress factor estimation uses the analytical method for the face distribution factor for lead mismatch ,=l.O+G-e,.z.w/(
22、l.8.,.p,) (5) found in lo (G = 1,500,000 Ib/im*/(in. of Fw). Respectively, the reference pinion peak stresses were 53.16 ksi and 53.02 ksi for AGMA and LDP. In viewing Figure 19, LDP predicts linear increase in stress versus misalignment for ail three LDP cases. The two LDP crowning case predictions
23、 greatly reduce stress sensitivity to misalignment. In conclusion, the LDP methodology provides a tool to the designer for evaluating the stress sensitivity relationship to gear errors and modifications. 6 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servic
24、esSTD-AGMA 94FTMb-ENGL 1997 Ob87575 00174587 457 2.4 2.2 2 8 1.8 c o 2 1.6 1.2 1 0.8 O o.MW32 0.oOW 0.0006 0.038 0.001 0.0012 Misalignment inAn Figure 19 Stress Factor vs. Misalignment and Crowning Types Conciusions: The aim of this study was to create a general program to estimate gear root stresse
25、s for both spur and helical gears. The predicted stress results were compared to experimental and finite element cases and the following conclusions were reached: The moment-image method with a combination of two-dimensional model boundary elements estimates gear stresses many times faster than thre
26、e-dimensional model finite element cases. The method includes load sharing prediction for modified gear teeth. Tensile normal stress results match with real geared systems within expected errors for both spur and helical gears. Peak stresses at any loading condition are within 15% of measurements or
27、 finite element predictions. The stress method shows comparable results to the benchmark cases for both modified and unmodified gears. In studying stresses at the edges of helical gears, LDP may not properly model the helical edge effects. In general, the stresses at the edges of the gear were predi
28、cted higher by LDP than the benchmark case results. Edge effect stresses should be subject to further studies. Acknowledgments: The authors would like to thank the sponsors of the Gear Dynamic and Gear noise Research Laboratory and the US Army Research Laboratory (URI Grant DAA03-92-GO120; 1992-1997
29、; project monitor Dr. T.L. Doligalski) for their support of this research. Also special thanks is given to Mr. Tsuyoshi Yoshida for the helical gear strain gage data, and Ford Motor Co. for allowing Mr. Clapper to take time in this endeavor. References: i Jaramillo, T.J., “Deflections and Moments Du
30、e to a Concentrated Load on a Cantilever Plate of Infinite Length,” Journal of Applied Mechanics, March 1950, pp. 67-72. 2 Wellauer, E.J., and Seireg A., “Bending Strength of Gear Teeth by Cantilever-Plate Theory,” Journal of Engineering for Industry, 82, 1960, pp. 213-222. 3 Oda, S. and Shimatomi,
31、Y., “Study on Bending Fatigue Strength of Helical Gears, Ist report, Effect of Helix Angle on Bending Strength”, Bulletin of the JSME, Vol. 23, No. 177, March 1980. 4 Oda, S. and Shmatomi, Y., “Study on Bending Fatigue Strength of Helical Gears, Znd report, Bending Fatigue Strength of Casehardened H
32、elical Gears”, Bulletin of the JSME, Vol. 23, No. 177, March 1980. 5 Kugimaya, H., “Stresses in Helical Gear Teeth”, Bulletin of JSME, Vol. 9, No. 36, 1966. 6 Vijayakar, S.M., and Houser D.R., The Use of Boundary Elements for the Determination of the Geometry Factor”, AGMA Technical paper, 87FM4,198
33、7. 7 Houser, D.R., “Gear Noise Sources and Their Prediction Using Mathematical Models”, SAE Gear Design Manufacturing and InsDection Manual, Chapter 16, AE-15, SAE Engineers Inc., 1990. 8 Conry, T.F. and Seireg A., “A Mathematical Programming Technique for the Evaluation of Load Distribution and Opt
34、imal Modifications for Gear Systems”, ASME Journal of Engineering for Industry, Vol. 94, July 1972. 9 Vijayakar, S.M., “Finite Element Methods for Quasi-Prismatic Bodies with Application to Gears,” Ph.D. Dissertation, The Ohio State University, 1987. IO American Gear Manufacturers Association -ANSYA
35、GMA 2001- B88, “Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth”, pg. 21, Nov. 1993. 7 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesApplElldh Note: All units in tabla are English units. Table 1 Case A Table 2 Case B Table3 CaseC Table4 CaseD Table 5 Misalignment Study Helical Gear Data COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services
