NASA Technical Memorandum 106176 AIAA-93-2296 Army Research Laboratory ARL-TR-158 Contact Stress Analysis of Spiral Bevel Gears Using Nonlinear Finite Element Static Analysis G.D. Bibel University of North Dakota Grand Forks, North Dakota / A/- -_>,r7 p_ o I p. i_ z _ E Z _ 0 A. Kumar and S. Reddy University of Akron Akron, Ohio and R. Handschuh U.S. Army Vehicle Propulsion Directorate Lewis Research Center Cleveland, Ohio Prepared for the 29th Joint Propulsion Conference and Exhibit cosponsored by the A/AA, SAE, ASME, and ASEE Monterey, California, June 28-30, 1993 U.5. ARMY h,- r_ NKSA National Aeronautics and Space Administration RESEARCH LABORATORY https://ntrs.nasa.gov/search.jsp?R=19930017848 2020-07-17T21:14:27+00:00Z
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NASA Technical Memorandum 106176
AIAA-93-2296
Army Research LaboratoryARL-TR-158
Contact Stress Analysis of Spiral Bevel Gears
Using Nonlinear Finite ElementStatic Analysis
G.D. Bibel
University of North Dakota
Grand Forks, North Dakota
/ A/- -_>,r7p_
o
I p. i_
z _ EZ _ 0
A. Kumar and S. Reddy
University of AkronAkron, Ohio
and
R. Handschuh
U.S. Army Vehicle Propulsion DirectorateLewis Research Center
Cleveland, Ohio
Prepared for the29th Joint Propulsion Conference and Exhibit
cosponsored by the A/AA, SAE, ASME, and ASEEMonterey, California, June 28-30, 1993
Coordinate transformations involving both rotation
and translation require mixed matrix operations of
multiplication and addition. Matrix representation of
coordinate transformations will need only
multiplication of matrices if position vectors arerepresented by homogeneous coordinates'. The
following 4 x 4 matrices are required to transform the
homogeneous coordinates of a point on the cutting
head to a point on the work piece.
Matrix [M..] transforms coordinate system St,
attached to the cutting head, into system S,, rigidlyconnected to the cradle.
[i0 0 o1IMp] = c_q ;sinq ;sinq (16)
±_qo cOSqo sTq I
Matrix [M_.] transforms coordinate system S, into
system Sm attached to the frame.
cos_b e a:sin 4_, (17)[M,J = ±sin,, cosq,
0 0
Matrix [M_] transforms coordinate system S= into system
Sp which orientates the apex of the gear beingmanufactured.
cosb 0-sin8 -L.sin6'
0 1 0 ±E.[M_.] =
sin_ 0 cos8 L,.ces8
0 0 0 1
(18)
Matrix [Mj transforms coordinate system Sp into systemS. which located the apex of the gear being manufacturerwith respect to S=
[M,p] =• sin4_,,,cos4_,,, O
-sinp 0 cosp
0 0 0
(19)
Matrix [M.,,] transforms coordinate Sa into system Swattached to the work piece.
[Mj =:_sin¢,, cos4,,, 0 0
0 0 1
0 0 0
(2o)
TABLE I - PINION AND GEAR DESIGN DATA
PINION GEAR
Number of teeth pinion 12 36
Dedendum angle, deg 1.5666 3.8833
Addendum angle, deg 3.8833 1.5666
Pitch angle, deg 18.4333 71.5666
Shaft angle, deg 90.0 90.0
Mean spiral angle, deg 35.0 35.0
Face width, mm (in) 25.4(I .0) 25.4 (I .0)Mean cone
distance, ram (in) 81.050.191) 81.05 0.191)
Inside radius of gear
blank, mm (in) 5.3 (0.6094) 3.0 (.3449)
Top land thickness, mm (in) 2.032 (0.080) 2.489 (.098)
Clearance, nun ('m) 0.762 (0.030) 0.92964 (0.0366)
TABLE II - GENERATION MACHINE SETTINGS
PINION GEAR
Concave Convex Concave Convex
Radius of cutter, r, in
Blade angle, _k, degVector sum, I._
Machine offset, E=, in
Cradle to cutter distance, s, in
Cradle angle q, deg
Ratio of roll, M=,,
Initial cutter len_.h, u, in
Initial cutter orientation, O, deg
Initial cradle orientation, Oc, deg
Xc
2.9656
161.9543
0.038499
0.154576
2.947802
63.94
0.30838513
9.59703
126.83544
-0.85813
3.0713 3.0325
24.33741 58.0
-0.051814 0.0
-0.1742616 0.0
2.8010495 2.85995
53.926 59.234203
0.32204285 0.950864
7.42534 8.12602
124.43689 233.9899
-11.38663 -0.35063
xe
A
Inside blade (convex side);
=
2.967522.0
0.0
0.0
2.85995
59.234203
0.950864
7.89156
234.9545
-12.3384
Fi_mare 1.
Yc
Oc
Xc
Outside blade (concave side);
o =
Cutting head cone surfaces and attached coordinate system.
Xw
Toe
Cleorcnce
7
Zw
Figure 2. Projection of gear tooth with XZ plane.
XCONVEX
CONCAVE
CONCAVE
CONVEX
Y
APEX
X
// PINION
_, \\XX
XX\ ', Gear afLer 190 deg
-__ rotation about Zinion after -90 deg.
rotation about Y
PINION
GEAR
Figure 3. Orientation of gear and pinion based on solution of the system of equations, and afterrotations required for mesh.
8
tt"-'-"-x
Figure 4. Complete 3D model of gear and pinion in mesh.
Figure 5. Distorted gear after connecting gap elements.
Fig'ure6. Seventoothmodelusedfor finite element analysis of mesh.
-17629.
-57455.
-97281.
-137107.
-176933.
-216759.
°256584.
\
-296410.
Fibre 7. Stress contours in pinion tooth.
10
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REPORT DOCUMENTATION PAGE OMB No. 0704-0188
Public reportingburden for this cotlectionof infon-nationis astm_atedto average 1 hour per response, includingthe time for reviewinginstructions,searchingexistingdata sources,gatheringand maintanng ¢,e aala needed, and compis_ngand reviewingthe col|ectionof information, Send commentsregarding this burdenestimateor any other aspect of thiscollectionof information,includingsuggestionsfix reduc/ng this burden, to WashingtonHeadquarters Se_dces, Directoratetor Informatio_Operationsand Repocts,1215 JeffersonDavis Highway,Suite 1204, Arlington,VA 22202.4302, and to the Othce of Management and Budget, PaperworkReductionProje<:t(0704-0188), Washington, DC 20503.
1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATF.,_ COVERED
May 1993 Technical Memorandum
4. TITLE AND SUBTITLE 5. FUNDING NUMBERS
Contact Stress Analysis of Spiral Bevel Gears Using Nonlinear Finite Element
Static Analysis
6. AUTHOR(S)
G.D. Bibel, A. Kumar, S. Reddy, and R. Handschuh
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
NASA Lewis Research Center
Cleveland, Ohio 44135-3191
and
Vehicle Propulsion Directorate
U.S. Army Research Labo_tory
Cleveland, Ohio 44135-3191
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Washington, D.C. 20546-0001
and
U.S. Army Research Laboratory
Adelphi, Maryland 20783-1145
WU-505-62-10
1L162211A47A
PERFORMING ORGANIZATION
REPORT NUMBER
E-7876
10. SPONSORING/MONITORINGAGENCY REPORT NUMBER
NASA TM-106176
ARL-TR-158
AIAA-93-2296
11. SUPPLEMENTARY NOTES
Prepared for the 29th Joint Propulsion Conference and Exhibit cosponsored by the AIAA, SAE, ASME, and ASEE, Monterey, California, June 28-30,
1993. G.D. Bibel, University of North Dakota, Grand Forks, North Dakota 58201; A. Kumar and S. Reddy, University of Akron, Akron, Ohio 44325;
and R. Handschuh, U.S. Army Vehicle Propulsion Directorate, NASA Lewis Research Center. Responsible person, R. Handschuh, (216) 433-3969.
12L DISTRIBUTIONIAVAILABIUTY STATEMENT 12b. DISTRIBUTION CODE
Unclassified - Unlimited
Subject Category 37
13. ABSTRACT (Malximum 200 words)
A procedure is presented for performing three-dimensional stress analysis of spiral bevel gears in'mesh using the finiteelement method. The procedure involves generating a finite element model by solving equations that identify tooth surface
coordinates. Coordinate transformations are used to orientate the gear and pinion for gear meshing. Contact boundary
conditions are simulated with gap elements. A solution technique for correct orientation of the gap elements is given.
Example models and results are presented.
14. SUBJECT TERMS
Gears; Gear teeth; Mechanical drives
17. SECURITY CLASSIFICATION
OF REPORT
Unclassified
18. SECURITY CLASSIFICATIONOF THIS PAGE
Unclassified
lg. SECURITY CLASSIFICATION
OF ABSTRACT
Unclassified
15. NUMBER OF PAGES
1216. PRICE CODE
A0320. LIMITATION OF ABSTRACT
NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)