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Computer simulation and stress analysis of helical gears with pinion circular arc teeth and gear involute teeth
.~iech..~¢ack. TheoFy Vol. 26. No. ~ pp. 145-1.54. 1991 Pnnted m Great Britain. All rights Rserved
00~4-1t4X,ql $3.00+0.00 CopYn8ht C 1991 Pergamon Press pie
COMPUTER SIMULATION A N D STRESS ANALYSIS OF HELICAL GEARS WITH PINION CIRCULAR ARC TEETH AND GEAR INVOLUTE TEETH C H U N G - B I A U T S A Y a n d Z. H. F O N G Department of Mechanical Engineering, National Chiao Tung University. 1001 Ta Hsueh Road, Hsinchu, Taiwan 30050. R.O.C. ( Receired 17 Norember 1989; receired for publication 24 April 1990)
Al~tract--The tooth contact analysis (TCA) technique and finite element method (FEM) have been applied to 8,car contact analysis and stress analysis. A mathematical model for helical gears with pinion circular arc teeth and gear involute teeth is assumed to be given by Litvin and Tsay. The gcomctry of the gears are described by the parameters of manufacturing. Simulation of the conditions of gear meshing. including the axes mi~lignment and center distance variation are performed. The locations of bearing contact and contact pattern of mating tooth surfaces arc determined by the TCA, Results of the TCA provide the location and direction of applied Ioadings for the computer-aided FEM stress analysis. By applying the given mathematical model and the TCA techniques, a computer-aided stress analysis model for a new pair of gearing has been propo~d. An automatic meshes generation (AMG) program has been deveh)ped for the FEM stRss analysis. A three-dimensional strcs.s analysis for this type of gearing has also been investigated and exprcs.~.'d by Von Mists stress contour distributions.
NOMENCLATURE
a~, hf--Tool .~tting of rack cuttcr T+ u r. h e - T o o l setting of rack cutter ~;r
S(/)(X(,').
J--Jacobian l~--Variable paramctcr which determines the location on rack cutter T-r [M,]--Coordinatc transformation matrix; transformation from coordinate sys. tern S~ to coordinate system S, N-~Normal vector of surface n---Unit normal vector of surface R--Position vector Y~,".Z(/))---Coordinate system where superscript i ~=F. P. I. 2 . f F denotes circular arc rack cutter. P denotes involute rack cutter, I denotes pinion, 2 denotes gear and J" denotes assemble frame. Subscript j = a, b, c, d, h,fdenote the coordinate systems defined in Figs I-7 T,--Number of teeth of pinion (i ,- !) or gear (i = 2) u+~Variablc parameter, u r = ,-'h ,~', I is represented on coordinate system S~,*~ as shown in Fig. 2
/J/--llclix anglc of rack cutter j (/ =/~: P) &c--Center distance variation Ay--Crossed angle of axes mi~lignment (in degrc~sl &~'--Kincmatic errors (in arc-second) ~:,--Gcnerated tooth surface i (i = I, 2) £j~-Generating rack cutter surface /
(j - F, P) 0 ~ V a r i a b l c parameter which determines the location of a point on circular arc rack cutter £ r A~, A~,--Lead angle of rack cuncr j (j = F, P) p~---Radius of working part of circular arc rack cutter £~ ~D,--Rotation angle of gear i when generated by rack cutter q~--Rotation angle of gear i when two gears arc meshing with each other ~/.--Pressure angle of normal section of rack cutter
INTRODUCTION Involute helical gears arc widely used in industry for parallel axes power transmission. Bearing contact of this type of gearing is a straight line when two gears are meshing with each other. Hence. they are very sensitive to the axes misalignment. If helical gears are designed with pinion circular arc teeth and gear involute teeth, the bearing contact at every instant is a point instead of a line. Thus. the bearing contact of gear tooth surface is localized and the center of bearing contact moves along tooth surface. Therefore. the gears are not so sensitive to gear axes misalignment. Due to elasticity, the contact point spread over an elliptical area when two gears are meshing with loading. 145
146
Cm.'yc~-BxauTs^Y and Z. H. F'ONG
Dimension of contact ellipse depends on the loading and curvatures of gear tooth surfaces. By choosing the radius of pinion circular arc teeth appropriately, the contact stress can be reduced substantially. In this paper, a mathematical model for helical gears with pinion circular arc teeth and gear involute teeth has been assumed to be given by Litvin and Tsay[I, 2]. Derivation of the mathematical model and further investigations have been based on the idea of Litvin [3, 4] who first proposed this type of gearing and the method for its generation. Some Russian researchers such as Reshetov [5], Dobrovolsky et al. [6] and Niemann [7] presented extensive investigations on the correction of gears generated by rack cutters. Ou and Seireg[8] analyzed circular arc gears by interactive graphics in 1986. The tooth contact analysis (TCA) technique has been developed by Litvin[4], Wilcox[9] and Gleason Works[10]. Applying the mathematical model and TCA technique, a computer-aided TCA has been investigated. The locations of bearing contact and contact pattern of mating tooth surfaces are determined by the TCA. Kinematic errors of the gear train can also be calculated in the mean time. Computer simulation of conditions of gear meshing, including axes misalignment and center distance variation of rotating shafts, has been investigated and illustrated in examples. The contact pattern of the circular arc helical pinion and involute shaped helical gear is a point contact rather than a line contact which appears in a traditional involute helical gear. Due to lower sensitivity of a gear train to the gear misalignment with point contact, it is advantageous to have a point contact pattern in case of having axes misalignment. Therefore, lower assembly precision is permitted when utilizing this type of gearing. Calculation of gear transmission errors is of interest for two main reasons: (a) it is one of the main sources of noise and vibration and (b) it predicts the precision of a gear train. In this decade, the developments of computer science and the TCA technique make it possible to determine the unk)adcd transmission errors of a gear train precisely and efficiently. The location of bearing contacts on both circular arc pinion and involute helical gear tooth surfaces were the same although the ccnter distance was changed. In case of axes misalignment, only a very small dislocation of bearing contact occurred on circular arc pinion tooth surface. With this special feature, a shorter gear tooth can be designed without reducing its contact ratio. Due to its lower sensitivity to the axes misalignment and lower requirement for assembly accuracy, this type of gearing is very suitable to apply to the power transmission devices such as trucks, helicopters and other industrial machines. Kamel and Navabi [I I] discussed the digitizing in mesh generation for computer-aided finite element model generation in 1980. Wilcox [i 2] discussed the gear tooth stress as a function of tooth contact pattern shape and position, and Coy and Hu-Chih Chao [13] investigated the finite element grid size to account for Hertz deformation for spur gears in 1982. In this paper, the computer-aided finite element method (FEM) stress analysis [14, 15] has been applied to analyze the stress distribution of the gears. An automatic meshes generation program for gear teeth has been developed to generate nodal points and meshes for FEM stress analysis. It is worth mentioning that the nodal points and meshes generated by the A M G computer program are based on the given mathematical model of gear tooth surfaces. All meshes generation are described by parameters of the rack cutters. Once the necessary parameters of rack cutters are input, the meshes and nodal points are automatically generated for stress analysis. Direction and location of the applied Ioadings are determined by the results of TCA. The results of stress analysis are plotted as Von Mises stress contours. Some computer graphs are included to highlight the results of stress analysis. A method of compensation for the dislocation of bearing contact induced by errors of manufacturing and assembly has been proposed. Some numerical examples have also been presented to illustrate the influence of the above-mentioned errors and the method of compensation for the dislocation of bearing contact. The contents of this paper covered the solutions to the following problems: (I) computer graphs for the pinion and gear: (2) computer simulation of the conditions of gear meshing and bearing cbntact; (3) investigation of the sensitivity of gears to the errors of manufacturing and assembly due to center distance variation and axes misalignment; (4) compensation for kinematic errors and dislocation of bearing contacts
Computer simulation and stress analysis of helical gears
147
and (5) applying a computer-aided FEM to analyze the stress distribution of the gears. Three-dimensional stress analysis for this type of gearing has been studied. M A T H E M A T I C A L M O D E L FOR P I N I O N A N D GEAR T O O T H S U R F A C E S
Derivation of the mathematical model for a helical gear with pinion circular arc teeth and gear involute teeth is based on the theory of gearing [3, 4] and the concept of differential geometry. The equations of circular arc pinion and involute helical gear tooth surfaces are given as follows [2]: x~" = (pn sin 0 n - bn+ rl )cos 0m + (pncos On- bncot 0n)sin 01 sin ).n. Y'L" -- (pp sin On- bn+ r~ )sin ¢~ - (pncos 0 n - bncot On)cos ck~sin ).n,
:4,=pncosOncos).n
an
cos ).n
(!)
t-bncotOntan).nsin).n+r~dpltan).r
and
x~:~= (le cos O. - ae - r:)cos 02 - (le cos ~. - ae)cot ¢. sin ).e sin Oz. y~:J = - (le cos ~,. - ae - rz)sin 0: - (/e cos 0. - ae) cot ~. sin ).e cos 0:, "-':' " - - (ae tan ~b, - le sin ~,)cos ~.e + ( c o s ~ae sin~, + be cos ).p + r:~b: tan ).e.
(2)
si !p ) tan ).e sin ).e
where Pn represents the radius of circular arc of rack cutter Y~n; On and Ir express the surface coordinates of the pinion and gear. respectively; ).n and ).n indicate the lead angles of the pinion and gear. respectively; an and bp are tool settings of rack cutter Y~n; an and be are tool settings of rack cutter Y-r; ~ and ~z are rotation angles of the pinion and gear. respectively, when generated by rack cutters; and ~. is the pressure angle for both pinion and gear. Unit normals of circular arc pinion and involute helical gear tooth surfaces are also given as follows [2l: n~',~= sin Oncos ~l + cos Onsin ).,.sin ~,, n "~j,= sin Op sin ~l - cos 0 n sin ).nCOS ~j,
(3)
n ~ = cos Oncos ).e
and n ~ = sin ~, cos ~z - cos ¢, sin ).e sin O2, n~2~_ _ sin 0, sin 05
.~' =
cos ~'n sin ).pcos O2.
(4)
c o s ~ . cos ).,.
The transverse cross-section of pinion and gear are shown in Figs 1 and 2. Computer graphs of pinion and gear are plotted by applying equations (I) and (2), respectively. It is noticed that
i 1I
ii|*~
~*tl*
S Fig. I. Transverse cross section of circular arc pinion.
Fig. 2. Transverse cross section of involute gear.
148
CHL'~O-BL~UTSAYand Z. H. FONG
the number of teeth for both pinion and gear are 18, and modulus is chosen as M = 1.0 ram. Coefficient of tool setting e = 0.3 for the pinion and zero for the gear. COMPUTER SIMULATION OF CONDITIONS OF GEAR MESHING The TCA method has been applied to simulate the conditions of gear meshing. The basic concept of the method is that the position vectors and unit normals of both tooth surfaces should be the same at the point of contact. Therefore, representing the equations of two mating gear tooth surfaces in the same coordinate system is necessary. By applying the mathematical model and the TCA method, the kinematic characteristics of this type of gearing can be investigated. In order to simulate the conditions of gear meshing, coordinate systems S~(X~. Y~. Z~), S:(X:, Y:. Z:), SI(X r, Yr, Zr) and Sh(Xh, Yh, Z,) have been set up as shown in Fig. 3. The conditions of gear meshing can be simulated by changing the settings and the orientations of coordinate system Sh with respect to coordinate system $1. For instance, when simulating the change of center distance AC, the origin O, of coordinate system S~ may be displaced by AC with respect to the origin Or of coordinate system S r. Similarly, when simulating the crossed axes of misalignment A~,. the coordinate system Sh may be rotated about axis X~ through an angle At' with respect to fixed coordinate system S r. By applying the coordinate transformations, the equations of pinion tooth surface and gear tooth surface can be represented in fixed coordinate system S / a s follows:
{n ~"} = [g,d [M~, 1[ R'," }
(5)
{r ~:'} = [M,:]{ r',-" }
(~)
and
where superscripts "( I)'" and "(2)" denote the pinion and gear, respectively. Subscript "'f'" denotes the fixed coordinate system, and subscript "1" dcnotes the coordinate system which is attached to the pinion while subscript "2" denotes the coordinate system which is attached to the gear. It is noticed that the center distance is defined as C' = r, + r z + AC in Fig. 3; $; and ,0', are rotation angles of the pinion and gear, respectively, when two gears arc meshing with each other; and r t and r: are pitch radii of the pinion and gear, respectively. After coordinate transformations, both pinion and gear tooth surfaces are represented in fixed coordinate system S/. At the point of contact, due to tangency of two contacting tooth surfaces, the position vectors and their unit normals of both tooth surfaces should be the same. Therefore, the following equations must be observed [4]:
R~2)
(7)
n~"= n?~',
(8)
R~I~_ .[
where R[ ~ and R~-'~ indicate the tooth surfaces of pinion and gear, respectively, represented in coordinate system St; n~Uand n~-'~express the surface unit normals of pinion and gear, respectively, represented in coordinate system St.. Equation (7) expresses that pinion tooth surface and gear tooth surface own a common contact point determined by the position vectors R~u and R~ :~. Equation (8) indicates that pinion and gear have a common surface unit normal at their contact
Fig. 3. Relationship among coordinate systems S~. S,. S: and S,.
Computer simulationand stress analysis of helical gears
149
point. Since equation (8) is constrained by the relation of unit vectors In)"[ = la)'-'[ = !. equation (8) implies two independent equations only. Vector equations (7) and (8)• if considered simultaneously, yield a system of five independent equations with six unknowns: Or. lp. dp,. dp,'. dp,. and ~:'. The existence of the solutions to the system equations (7) and (8) depends on the Jacobian of the simultaneous equations in the range of the working area of two mating gears. If equations (7) and (8) are represented in the following form:
.f,(o~.
I,,~,. ~. ,/,') = x ? , - x ? , = o,
f:(o,, l,. ,/,,. ,#':. ,/,'.) = I : ? , - r ? ) = o. .f,(o,~, t,.. (I),. ~:.. ,~:.) = z ? , -
~'.:,~)=""
A(~,
f,(~, 4 , 1 . 4 , 1 ) = - " •
,.
fy
w
z f , = o. "':'
(9)
0•
- ' Iv :'=0. ,.
and the variable ~),' is considered as an input variable. The system of equations have one unique solution only if the Jacobian of system equations (9) is not equal to zero throughout the working range of the tooth surfaces: J =
u(/~.A•A.A,A) D(O~. Ip. g,,, ~,:. (b:')
#0.
(I0)
if the Jacobian is equal to zero in the working range of tooth surfaces, there are two possible cases which may occur: (I) the system of equations have no solution at all or (2) the system of equations might have infinite number of solutions {i.e. line contact). For the latter case. constraints should be added to obtain the desired solutions. A constraint, say : )J) = 1.0 ram. will lead to only one unique solution for the discussed system of equations. The contact point is at the edge of the mating tooth surfaces if two gears are meshing with a misaligned angle. Due to discontinuity at the edge of two mating tooth surfaces, the coordinates of contact point arc the same while their unit normals may be different. NUMERICAL EXAMPLES The following examples demonstrate how the TCA technique is applied to simulate the conditions of gear meshing. Kinematic errors induced by manufacturing and assembly errors such as crossed axes misalignment and center distance variation are also investigated. Computer programs for meshing simulation and tooth contact analysis have been developed. The kinematic error of the gear train A~b~ is defined as:
A ~ i = ~ 2, - ~TI; ,
lit)
where Ti and T2 represent the number of teeth of pinion and gear, respectively; bl and b~ are rotation angles of pinion and gear, respectively, when two gears are meshing with each other. The kinematic errors and dislocation of bearing contacts of the gear train can be compensated by regrinding the pinion or gear with a modified helix angle. Some important manufacturing data used in the following examples arc given in Table I, wberc kj is a manufacturing parameter.
Case !. Ideal case It is assumed that the gear train has neither axes misalignment nor center distance variation, i.e. A3, = 0 ° and AC = 0 mm as shown in Fig. 3. The results of tooth contact analysis and kinematic errors are calculated and listed in Table 2. Table I. Some important data for pinion and ~ a r Item
Tooth No.
Normal module (rnm)
Helix angle (deg)
pitch radius(ram)
k,
Gear
54
5
30
55.88
0.5
Pinion
I8
5
30
51.96
0.5
150
Cm.:~G-BIAu TsAY
Table 2. Of (deg) 0000 5.000 I0.000 15.000 20.000 25.000
TCA
~ (deg) 0.000 1.667 3.333 5.000 6.667 8.333
and
Z.
H.
FO.~G
results and kinematic errors for ideal case
~ (deg) 0.000 5.000 10.000 15.000 20000 25.0(0
~: (deg) 0.000 1.667 3.333 5.000 6.660 8.333
Op(deg) 25000 25000 25.000 25.000 25.000 25.000
Ie (ram) 2.758 2.758 2.758 2.758 2.758 2.758
Errors arc-sac 00000 0.0000 0.0000 0.0000 0.0000 00000
In Table 2, 0F and le indicate the locations of bearing contact on pinion and gear tooth surfaces, respectively, represented in terms of surfaces coordinates of rack cutters. It is noticed that the contact points of the pinion have an identical value of 0 e = 25.000" in the process of gear meshing. The same phenomenon is observed for the gear. It means that the location of bearing contacts of the pinion and gear at every instant are the same for each cross-section. For the ideal case of gear meshing, gear train does not induce kinematic error at all. The location of contact points on pinion are shown in Fig. 4. Case 2. Center distance rariathm
If the center distance of two gear rotating shafts is different from the design value, say, AC" = I mm. The results of T C A and kinematic errors are listed in Table 3. In Table 3, it is found that no kinematic error occurred when the center distance o f two rotating shafts is changed. It is also noticed that the location of bearing contacts on pinion is not changed for the center distance variation, in other words, during the process of gear meshing, the height of contact points at every instant is the same on pinion tooth surface while the height o f contact points is changed on gear tooth surl,tce with the center distance variation. The results showed that the bending stress of the gear tooth should be carefully studied because the bending moment may be increased due to dislocation of bearing contact. Case 3. Crossed axes misaligmnent
It is assumed that rotating shafts have a c r o s ~ d axes misalignment, say A7, = 0.1', in Fig. 3. Kinematic errors of the gear train will be induced. The T C A results and corresponding kinematic errors are calculated and listed in Table 4. it is found that the location of bearing contacts has only a small variation compared with those for the ideal case. The kinematic errors induced by crossed axes misalignment can be compensated by regrinding the pinion with a modified helix angle ]J~ = 29.9249 ~. The T C A results with modified helix angle of II = 29.9249 are shown in Table 5 and Fig. 5.
L o c a t i o n of c o n t a c t
"
for ideal case ¢t'=O ° ~'= 5° ~'= 10 °
~'= 15" ¢t'= 20 ° ¢t' = 25"
Fig. 4, Locations of contact points for the ideal case.
151
Computer simulation and stress analysis of helical Bears Table 3. T C A results with center distance variation AC -- i m m (0.48% variation) Errors
~; (des)
~'. (des)
~l (des)
0.000 5.000 I0.000 15.000 20.000 25.000
0.000 1.666 3.333 5.000 6.666 8.333
0.505 5.505 10.505 15.505 20.505 25.505
COMPUTER-AIDED
¢: (deg) -0.705 0.962 2.628 4.295 5.962 7.628
FEM
Or (des)
I, (ram)
arc-scc
25.000 25.000 25.000 25.000 25.000 25.000
1.664 1.664 1.664 1.664 1.664 1.664
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
STRESS
ANALYSIS
In this section, the computer-aided FEM has been employed to gear stress analysis. The stress model applied to bending and contact stresses analysis of the gear is based on two well-known analytical approaches: TCA and FEM. The mathematical model of helical gears discussed in the previous section has been employed to define the geometry of the gear and pinion. An automatic meshes generation program has also been developed to divide the gear tooth into elements, and to generate nodal points and meshes automatically for FEM stress analysis. The TCA computer program has been developed to determine the location and direction of the applied Ioadings. The frontal method [14, 15] is employed to calculate FEM stiffness matrix. The frontal method provides easier assembly of equations and storing only the active coefficients. These active coefficients are formed as a triangle in a stiffness matrix associated with the elimination processes. Therefore, the frontal method is very suitable to apply to a small computer system. An eight-node isoparametric hexahedron element has been chosen for the three-dimensional FEM stress analysis. For simplicity, the following assumptions are made for stress analysis: (I) The stress discussed here is within the elastic range of materials. Plastic and nonlinear behaviors of the material are neglected. (2) lsotropic materials arc assumed. The properties are the same in all directions. (3) Nodal points on the boundaries are assumed to be fixed. (4) Small displacement is assumed, in this paper, eight Gaussian integration points were chosen to calculate the stiffness matrix for each element. The contours of stress distribution were plotted by applying the surface fitting technique. Due to length limitation of the paper, only stress contour diagram of the pinion are shown in this section. it is assumed that the basic manufacturing data are the same as stated in previous examples and the loading is 890 N. Figure 6 shows a single-tooth pinion with meshed elements which generated by automatic meshes generating computer program, it is noticed that nodal points on the boundaries are assumed to be fixed. Three-dimensional stress distributions of the pinion are plotted by Von Mises stress contours with different cross-section as shown in Figs 7-10. The loading is assumed to be applied at the contact point when rotation angle of the pinion is zero, ~; = 0". Table 4. T C A results and kinematic errors due to crossed misaligned angle A7h = 0.1 '
~; (des)
~2 (des)
~l (deg)
~: (des)
0.000 5.000 I 0.000 15.000 20.000 25.000
0.000 1.662 3.323 4.985 6.647 8.308
-0.322 4.674 9.670 14.667 19.663 24.659
-0.108 i.557 3.223 4.888 6.553 8.219
Op (des)
I r (ram)
Errors arc-see
25.369 25.369 25.369 25.369 25.369 25.369
2.821 2.821 2.821 2.821 2.821 2.82[
0.000 - 18.130 - 36.260 - 54.391 -72.523 -90.655
Table 5. Compensated results for the case of crossed axes misalignment with a modified helix angle/~ = 29.9249
~; (des)
~'., (deg)
~= (deg)
~: (deg)
0.000 5.000 10.000 15.000 20.000 25.000
0.000 1.667 3.333 5.000 6.667 8.333
--0.084 4.912 9.909 14.905 19.901 24.897
-0.024 1.646 3.317 4.987 6.658 8.328
Or (de S)
Ip (ram)
Errors arc-.',~-c
25.093 25.093 25.093 25.093 25.093 25.093
2.775 2.775 2.775 2.775 2.775 2.775
-0.000 -0.002 -0.004 -0.005 -0.005 -0.005
152
CHU~G-BL~u TSAY and 7' H. FONG
100.0 80.0
Hel.tx Angle
60.0 40.0
d
o 29.84.5 + 29.887 29.925 = 29.962 o 30.000
20.0 O.OI
~d - 2 o . o ~ -40.0 -60.0
-80.0 -100.0 0.0
5
10
t5
20
25
Fig. 5. Kinematic err~)rs due Io crossed axes mi.,mlignment with compen~tion.
This example shows that how TCA technique is incorporated with computer-aided FEM for gear stress analysis. The main advantages of applying FEM incorporated with TCA technique are: (!) Easy to define the geometry of tooth surface analytically when the manufacturing data were changed. (2) Easy to determine the contact pattern, contact position and direction of the applied loading. In addition, the advantages are quite obviously when the effects of modifying some manufacturing data were investigated. CONCLUSIONS
From the results of previous sections, some important characteristics of this type of gearing may be summarized as follows: (I) The contact pattern is a point contact rather than a line contact. (2) Compared with the ideal case of gear meshing, the contact positions of this type of gearing are not changed too much when a misaligned angle of rotating shafts exists. (3) Lower assembly precision is permitted when this type of gearing is employed. ModuLe - 5 , Tooth No. ,, 18, Load - 8 9 0 N
ModuLe ,, 5, Tooth No. - 18, Load = 8 9 0 N Van Mises stress contour, q~l = 0 *
,',,\\\\\\\\\\\\\\\\~'~ Fig, 6, Meshes for a single toolh or the pinion.
Fig. 7. Von Miscs stress contour of layer I,
Computer simulation and stress analysis of helical gears
153
ModuLe = 5 , Layer = 2 , Load ,, 8 9 0 N
Tooth No.,, 18, qb,-O °
Fig. 8. Von Mises stress contour of layer 2. ModuLe-5, Load,,890
ModuLe ,, 5 , Tooth NO.,, 1 8 , L a y e r - 4
N at ~,-0"
Layer I, 3
Fig. 9. Van Mises stress contour of layer 3.
Fig. I0. Van Mises stress contour of layer 4.
(4) in case of center distance variation, the contact position on the pinion tooth surface is not changed while the contact position on the gear tooth surface is changed. Bending stress may be increased due to the dislocation of bearing contact on the gear tooth surface. (5) By regrinding the pinion with a modified helix angle, dislocation of the bearing contact will be improved and kinematic errors of the gear train will be reduced substantially when the same gear train is assembled again, Smaller bending stress and noise can be expected for the gear train with the proposed method of compensation. (6) Computer-aided FEM incorporates with TCA techniques is a very powerful tool for gear stress analysis. Ackno.'led~ement--The authors are grateful to the National Science Council of R.O.C. for their grant. Contract No. NSC-77-040I-E009-03.
REFERENCES I. F L. Litvin. C. B Tsay. J. J. Coy and R. F. Handschuh. AGMA Technical Paper 86FTM3 (1986). 2. C. B. Tsay. J. Chin. Inst. Engng 10(I). 45-51 (1987).
154
CHu.'~G-BtAu TSAYand 7_ H. FON(;
3. F. L. Litvin, Proc. ,~d WId Cong. Gearing, Paris, Mar. (1986). 4. F. L. I.atvin, Theory of Gearing, 2nd edn (in Russian~. Nauka. Moscow (1968). The new edition (in English). revised and completed (in press). 5. D. N. Rcshetov. Machine Destgn. Mir. Moscow (1978). 6. Dobrovolsky et al., Machine Elements. Mir. Moscow (c. 1960). 7 G. Niemann, Mazchinen¢lemente. Spnnger, Berlin (1965). 8. Z. Ou and A. Sclreg, Trans. A S M E JI Mech. Transm. Automn De.~. 106, 65-7l (1986). 9. L. E. Wilcox. AGMA Paper I)2,29.25 (1982). 10. Understanding Tooth Contact Analysis. Glcason Works Publication, SD3IgA (1978). If. H. A. Kamel and Z. Navabi, J..Mech. Des. 102, 552-560 (1980). 12. L. E. Wilcox. AGMA Paper P2,29.25 (1982). 13. J. J. Coy and C. Hu-Chih Chao, J. ,Mech. Des. 104, 759-766 ¢1982). 14. N. Ida and W. Lord, Int. J. numer. ,Methods Engng 20, 625-641 (1984). [5. E. Thompson and Y Shimazaki. Int. J. numer. Methods Engng 15, 889-910 (1980).
RECHNERSIMULATION UND SPANNUNGSANALYSE DES SCHNECKENGETRIEBES MIT KREISBOGENZ,g, HNEN FIDR D A S R I T Z E L U N D E V O L V E N T E N Z P . H N E N FOR DAS SCHNECKENRAD Z~mmenfas.~unl~--Das Vcrfahrcn dcr Z.ahnbcrdhrungsanaly.~(TCA) und die Finite Element Mcthodc (FEM) wcrdcn zur Analyse dcr Zahnbcr6hrung und dcr Spannung verwcndct. Es gibt Ixreits cin mathcm.',ti.~hcs M(~Icll l'i~r Schncckcngctriclx mit Krcisbogenz~hncn am Ritz¢l und Evolvcntcnzfihncn am .Schncckcnrad yon Litvin und C, B. Ts-',y. Dic Z-',hnradgcomctric wird dutch die Paramctcr der Fcrtlgung bcschricbcn. Dic Simulation dcr Vcrzahnung ¢inschliefllich dcr Achsenabwcichungcn und dcr Variicrung des Mittclpunktabstandcs wcrdcn durchgcfiihrL Die Betnebsw',:ilzlinieund die Kontaktart der verzahnlcn Zahnfl~ichen wcrdcn mit dcm TCA-Verfahren ermittelt. Die Ergcbnis.~ liefcrn Lage und Richtung der auftrctcndcn Belastungcn fiir die 5pannungsanaly.~ mittels FEM. Unter Verwendung des gegebcnen mathcmalischcn Modells und dcm TCA-Vcrfahrcn ist ein Modcll der rechnerunterstiitztcn Sp;,nnung.~naly~ fiir ncuc Z."ihncpaarungcn vorge~hlagen worden. Ein .~ll~U~itig~ Erzeugungsprogr;,mm f,'ir Vcrzahnungcn (AM(;) ist f6r die Spannungsanaly~ rail FEM enlwickell worden. Einc drci dimcnsi~malc Spannung~nalysc f(ir cincn ~)Ichcn ~lhntyp wird durchgcfiihrt und mit Von MisesVcrl.cilung (t~ Spannungsumris~s dargcstellt.
00~4-1t4X,ql $3.00+0.00 CopYn8ht C 1991 Pergamon Press pie
COMPUTER SIMULATION A N D STRESS ANALYSIS OF HELICAL GEARS WITH PINION CIRCULAR ARC TEETH AND GEAR INVOLUTE TEETH C H U N G - B I A U T S A Y a n d Z. H. F O N G Department of Mechanical Engineering, National Chiao Tung University. 1001 Ta Hsueh Road, Hsinchu, Taiwan 30050. R.O.C. ( Receired 17 Norember 1989; receired for publication 24 April 1990)
Al~tract--The tooth contact analysis (TCA) technique and finite element method (FEM) have been applied to 8,car contact analysis and stress analysis. A mathematical model for helical gears with pinion circular arc teeth and gear involute teeth is assumed to be given by Litvin and Tsay. The gcomctry of the gears are described by the parameters of manufacturing. Simulation of the conditions of gear meshing. including the axes mi~lignment and center distance variation are performed. The locations of bearing contact and contact pattern of mating tooth surfaces arc determined by the TCA, Results of the TCA provide the location and direction of applied Ioadings for the computer-aided FEM stress analysis. By applying the given mathematical model and the TCA techniques, a computer-aided stress analysis model for a new pair of gearing has been propo~d. An automatic meshes generation (AMG) program has been deveh)ped for the FEM stRss analysis. A three-dimensional strcs.s analysis for this type of gearing has also been investigated and exprcs.~.'d by Von Mists stress contour distributions.
NOMENCLATURE
a~, hf--Tool .~tting of rack cuttcr T+ u r. h e - T o o l setting of rack cutter ~;r
S(/)(X(,').
J--Jacobian l~--Variable paramctcr which determines the location on rack cutter T-r [M,]--Coordinatc transformation matrix; transformation from coordinate sys. tern S~ to coordinate system S, N-~Normal vector of surface n---Unit normal vector of surface R--Position vector Y~,".Z(/))---Coordinate system where superscript i ~=F. P. I. 2 . f F denotes circular arc rack cutter. P denotes involute rack cutter, I denotes pinion, 2 denotes gear and J" denotes assemble frame. Subscript j = a, b, c, d, h,fdenote the coordinate systems defined in Figs I-7 T,--Number of teeth of pinion (i ,- !) or gear (i = 2) u+~Variablc parameter, u r = ,-'h ,~', I is represented on coordinate system S~,*~ as shown in Fig. 2
/J/--llclix anglc of rack cutter j (/ =/~: P) &c--Center distance variation Ay--Crossed angle of axes mi~lignment (in degrc~sl &~'--Kincmatic errors (in arc-second) ~:,--Gcnerated tooth surface i (i = I, 2) £j~-Generating rack cutter surface /
(j - F, P) 0 ~ V a r i a b l c parameter which determines the location of a point on circular arc rack cutter £ r A~, A~,--Lead angle of rack cuncr j (j = F, P) p~---Radius of working part of circular arc rack cutter £~ ~D,--Rotation angle of gear i when generated by rack cutter q~--Rotation angle of gear i when two gears arc meshing with each other ~/.--Pressure angle of normal section of rack cutter
INTRODUCTION Involute helical gears arc widely used in industry for parallel axes power transmission. Bearing contact of this type of gearing is a straight line when two gears are meshing with each other. Hence. they are very sensitive to the axes misalignment. If helical gears are designed with pinion circular arc teeth and gear involute teeth, the bearing contact at every instant is a point instead of a line. Thus. the bearing contact of gear tooth surface is localized and the center of bearing contact moves along tooth surface. Therefore. the gears are not so sensitive to gear axes misalignment. Due to elasticity, the contact point spread over an elliptical area when two gears are meshing with loading. 145
146
Cm.'yc~-BxauTs^Y and Z. H. F'ONG
Dimension of contact ellipse depends on the loading and curvatures of gear tooth surfaces. By choosing the radius of pinion circular arc teeth appropriately, the contact stress can be reduced substantially. In this paper, a mathematical model for helical gears with pinion circular arc teeth and gear involute teeth has been assumed to be given by Litvin and Tsay[I, 2]. Derivation of the mathematical model and further investigations have been based on the idea of Litvin [3, 4] who first proposed this type of gearing and the method for its generation. Some Russian researchers such as Reshetov [5], Dobrovolsky et al. [6] and Niemann [7] presented extensive investigations on the correction of gears generated by rack cutters. Ou and Seireg[8] analyzed circular arc gears by interactive graphics in 1986. The tooth contact analysis (TCA) technique has been developed by Litvin[4], Wilcox[9] and Gleason Works[10]. Applying the mathematical model and TCA technique, a computer-aided TCA has been investigated. The locations of bearing contact and contact pattern of mating tooth surfaces are determined by the TCA. Kinematic errors of the gear train can also be calculated in the mean time. Computer simulation of conditions of gear meshing, including axes misalignment and center distance variation of rotating shafts, has been investigated and illustrated in examples. The contact pattern of the circular arc helical pinion and involute shaped helical gear is a point contact rather than a line contact which appears in a traditional involute helical gear. Due to lower sensitivity of a gear train to the gear misalignment with point contact, it is advantageous to have a point contact pattern in case of having axes misalignment. Therefore, lower assembly precision is permitted when utilizing this type of gearing. Calculation of gear transmission errors is of interest for two main reasons: (a) it is one of the main sources of noise and vibration and (b) it predicts the precision of a gear train. In this decade, the developments of computer science and the TCA technique make it possible to determine the unk)adcd transmission errors of a gear train precisely and efficiently. The location of bearing contacts on both circular arc pinion and involute helical gear tooth surfaces were the same although the ccnter distance was changed. In case of axes misalignment, only a very small dislocation of bearing contact occurred on circular arc pinion tooth surface. With this special feature, a shorter gear tooth can be designed without reducing its contact ratio. Due to its lower sensitivity to the axes misalignment and lower requirement for assembly accuracy, this type of gearing is very suitable to apply to the power transmission devices such as trucks, helicopters and other industrial machines. Kamel and Navabi [I I] discussed the digitizing in mesh generation for computer-aided finite element model generation in 1980. Wilcox [i 2] discussed the gear tooth stress as a function of tooth contact pattern shape and position, and Coy and Hu-Chih Chao [13] investigated the finite element grid size to account for Hertz deformation for spur gears in 1982. In this paper, the computer-aided finite element method (FEM) stress analysis [14, 15] has been applied to analyze the stress distribution of the gears. An automatic meshes generation program for gear teeth has been developed to generate nodal points and meshes for FEM stress analysis. It is worth mentioning that the nodal points and meshes generated by the A M G computer program are based on the given mathematical model of gear tooth surfaces. All meshes generation are described by parameters of the rack cutters. Once the necessary parameters of rack cutters are input, the meshes and nodal points are automatically generated for stress analysis. Direction and location of the applied Ioadings are determined by the results of TCA. The results of stress analysis are plotted as Von Mises stress contours. Some computer graphs are included to highlight the results of stress analysis. A method of compensation for the dislocation of bearing contact induced by errors of manufacturing and assembly has been proposed. Some numerical examples have also been presented to illustrate the influence of the above-mentioned errors and the method of compensation for the dislocation of bearing contact. The contents of this paper covered the solutions to the following problems: (I) computer graphs for the pinion and gear: (2) computer simulation of the conditions of gear meshing and bearing cbntact; (3) investigation of the sensitivity of gears to the errors of manufacturing and assembly due to center distance variation and axes misalignment; (4) compensation for kinematic errors and dislocation of bearing contacts
Computer simulation and stress analysis of helical gears
147
and (5) applying a computer-aided FEM to analyze the stress distribution of the gears. Three-dimensional stress analysis for this type of gearing has been studied. M A T H E M A T I C A L M O D E L FOR P I N I O N A N D GEAR T O O T H S U R F A C E S
Derivation of the mathematical model for a helical gear with pinion circular arc teeth and gear involute teeth is based on the theory of gearing [3, 4] and the concept of differential geometry. The equations of circular arc pinion and involute helical gear tooth surfaces are given as follows [2]: x~" = (pn sin 0 n - bn+ rl )cos 0m + (pncos On- bncot 0n)sin 01 sin ).n. Y'L" -- (pp sin On- bn+ r~ )sin ¢~ - (pncos 0 n - bncot On)cos ck~sin ).n,
:4,=pncosOncos).n
an
cos ).n
(!)
t-bncotOntan).nsin).n+r~dpltan).r
and
x~:~= (le cos O. - ae - r:)cos 02 - (le cos ~. - ae)cot ¢. sin ).e sin Oz. y~:J = - (le cos ~,. - ae - rz)sin 0: - (/e cos 0. - ae) cot ~. sin ).e cos 0:, "-':' " - - (ae tan ~b, - le sin ~,)cos ~.e + ( c o s ~ae sin~, + be cos ).p + r:~b: tan ).e.
(2)
si !p ) tan ).e sin ).e
where Pn represents the radius of circular arc of rack cutter Y~n; On and Ir express the surface coordinates of the pinion and gear. respectively; ).n and ).n indicate the lead angles of the pinion and gear. respectively; an and bp are tool settings of rack cutter Y~n; an and be are tool settings of rack cutter Y-r; ~ and ~z are rotation angles of the pinion and gear. respectively, when generated by rack cutters; and ~. is the pressure angle for both pinion and gear. Unit normals of circular arc pinion and involute helical gear tooth surfaces are also given as follows [2l: n~',~= sin Oncos ~l + cos Onsin ).,.sin ~,, n "~j,= sin Op sin ~l - cos 0 n sin ).nCOS ~j,
(3)
n ~ = cos Oncos ).e
and n ~ = sin ~, cos ~z - cos ¢, sin ).e sin O2, n~2~_ _ sin 0, sin 05
.~' =
cos ~'n sin ).pcos O2.
(4)
c o s ~ . cos ).,.
The transverse cross-section of pinion and gear are shown in Figs 1 and 2. Computer graphs of pinion and gear are plotted by applying equations (I) and (2), respectively. It is noticed that
i 1I
ii|*~
~*tl*
S Fig. I. Transverse cross section of circular arc pinion.
Fig. 2. Transverse cross section of involute gear.
148
CHL'~O-BL~UTSAYand Z. H. FONG
the number of teeth for both pinion and gear are 18, and modulus is chosen as M = 1.0 ram. Coefficient of tool setting e = 0.3 for the pinion and zero for the gear. COMPUTER SIMULATION OF CONDITIONS OF GEAR MESHING The TCA method has been applied to simulate the conditions of gear meshing. The basic concept of the method is that the position vectors and unit normals of both tooth surfaces should be the same at the point of contact. Therefore, representing the equations of two mating gear tooth surfaces in the same coordinate system is necessary. By applying the mathematical model and the TCA method, the kinematic characteristics of this type of gearing can be investigated. In order to simulate the conditions of gear meshing, coordinate systems S~(X~. Y~. Z~), S:(X:, Y:. Z:), SI(X r, Yr, Zr) and Sh(Xh, Yh, Z,) have been set up as shown in Fig. 3. The conditions of gear meshing can be simulated by changing the settings and the orientations of coordinate system Sh with respect to coordinate system $1. For instance, when simulating the change of center distance AC, the origin O, of coordinate system S~ may be displaced by AC with respect to the origin Or of coordinate system S r. Similarly, when simulating the crossed axes of misalignment A~,. the coordinate system Sh may be rotated about axis X~ through an angle At' with respect to fixed coordinate system S r. By applying the coordinate transformations, the equations of pinion tooth surface and gear tooth surface can be represented in fixed coordinate system S / a s follows:
{n ~"} = [g,d [M~, 1[ R'," }
(5)
{r ~:'} = [M,:]{ r',-" }
(~)
and
where superscripts "( I)'" and "(2)" denote the pinion and gear, respectively. Subscript "'f'" denotes the fixed coordinate system, and subscript "1" dcnotes the coordinate system which is attached to the pinion while subscript "2" denotes the coordinate system which is attached to the gear. It is noticed that the center distance is defined as C' = r, + r z + AC in Fig. 3; $; and ,0', are rotation angles of the pinion and gear, respectively, when two gears arc meshing with each other; and r t and r: are pitch radii of the pinion and gear, respectively. After coordinate transformations, both pinion and gear tooth surfaces are represented in fixed coordinate system S/. At the point of contact, due to tangency of two contacting tooth surfaces, the position vectors and their unit normals of both tooth surfaces should be the same. Therefore, the following equations must be observed [4]:
R~2)
(7)
n~"= n?~',
(8)
R~I~_ .[
where R[ ~ and R~-'~ indicate the tooth surfaces of pinion and gear, respectively, represented in coordinate system St; n~Uand n~-'~express the surface unit normals of pinion and gear, respectively, represented in coordinate system St.. Equation (7) expresses that pinion tooth surface and gear tooth surface own a common contact point determined by the position vectors R~u and R~ :~. Equation (8) indicates that pinion and gear have a common surface unit normal at their contact
Fig. 3. Relationship among coordinate systems S~. S,. S: and S,.
Computer simulationand stress analysis of helical gears
149
point. Since equation (8) is constrained by the relation of unit vectors In)"[ = la)'-'[ = !. equation (8) implies two independent equations only. Vector equations (7) and (8)• if considered simultaneously, yield a system of five independent equations with six unknowns: Or. lp. dp,. dp,'. dp,. and ~:'. The existence of the solutions to the system equations (7) and (8) depends on the Jacobian of the simultaneous equations in the range of the working area of two mating gears. If equations (7) and (8) are represented in the following form:
.f,(o~.
I,,~,. ~. ,/,') = x ? , - x ? , = o,
f:(o,, l,. ,/,,. ,#':. ,/,'.) = I : ? , - r ? ) = o. .f,(o,~, t,.. (I),. ~:.. ,~:.) = z ? , -
~'.:,~)=""
A(~,
f,(~, 4 , 1 . 4 , 1 ) = - " •
,.
fy
w
z f , = o. "':'
(9)
0•
- ' Iv :'=0. ,.
and the variable ~),' is considered as an input variable. The system of equations have one unique solution only if the Jacobian of system equations (9) is not equal to zero throughout the working range of the tooth surfaces: J =
u(/~.A•A.A,A) D(O~. Ip. g,,, ~,:. (b:')
#0.
(I0)
if the Jacobian is equal to zero in the working range of tooth surfaces, there are two possible cases which may occur: (I) the system of equations have no solution at all or (2) the system of equations might have infinite number of solutions {i.e. line contact). For the latter case. constraints should be added to obtain the desired solutions. A constraint, say : )J) = 1.0 ram. will lead to only one unique solution for the discussed system of equations. The contact point is at the edge of the mating tooth surfaces if two gears are meshing with a misaligned angle. Due to discontinuity at the edge of two mating tooth surfaces, the coordinates of contact point arc the same while their unit normals may be different. NUMERICAL EXAMPLES The following examples demonstrate how the TCA technique is applied to simulate the conditions of gear meshing. Kinematic errors induced by manufacturing and assembly errors such as crossed axes misalignment and center distance variation are also investigated. Computer programs for meshing simulation and tooth contact analysis have been developed. The kinematic error of the gear train A~b~ is defined as:
A ~ i = ~ 2, - ~TI; ,
lit)
where Ti and T2 represent the number of teeth of pinion and gear, respectively; bl and b~ are rotation angles of pinion and gear, respectively, when two gears are meshing with each other. The kinematic errors and dislocation of bearing contacts of the gear train can be compensated by regrinding the pinion or gear with a modified helix angle. Some important manufacturing data used in the following examples arc given in Table I, wberc kj is a manufacturing parameter.
Case !. Ideal case It is assumed that the gear train has neither axes misalignment nor center distance variation, i.e. A3, = 0 ° and AC = 0 mm as shown in Fig. 3. The results of tooth contact analysis and kinematic errors are calculated and listed in Table 2. Table I. Some important data for pinion and ~ a r Item
Tooth No.
Normal module (rnm)
Helix angle (deg)
pitch radius(ram)
k,
Gear
54
5
30
55.88
0.5
Pinion
I8
5
30
51.96
0.5
150
Cm.:~G-BIAu TsAY
Table 2. Of (deg) 0000 5.000 I0.000 15.000 20.000 25.000
TCA
~ (deg) 0.000 1.667 3.333 5.000 6.667 8.333
and
Z.
H.
FO.~G
results and kinematic errors for ideal case
~ (deg) 0.000 5.000 10.000 15.000 20000 25.0(0
~: (deg) 0.000 1.667 3.333 5.000 6.660 8.333
Op(deg) 25000 25000 25.000 25.000 25.000 25.000
Ie (ram) 2.758 2.758 2.758 2.758 2.758 2.758
Errors arc-sac 00000 0.0000 0.0000 0.0000 0.0000 00000
In Table 2, 0F and le indicate the locations of bearing contact on pinion and gear tooth surfaces, respectively, represented in terms of surfaces coordinates of rack cutters. It is noticed that the contact points of the pinion have an identical value of 0 e = 25.000" in the process of gear meshing. The same phenomenon is observed for the gear. It means that the location of bearing contacts of the pinion and gear at every instant are the same for each cross-section. For the ideal case of gear meshing, gear train does not induce kinematic error at all. The location of contact points on pinion are shown in Fig. 4. Case 2. Center distance rariathm
If the center distance of two gear rotating shafts is different from the design value, say, AC" = I mm. The results of T C A and kinematic errors are listed in Table 3. In Table 3, it is found that no kinematic error occurred when the center distance o f two rotating shafts is changed. It is also noticed that the location of bearing contacts on pinion is not changed for the center distance variation, in other words, during the process of gear meshing, the height of contact points at every instant is the same on pinion tooth surface while the height o f contact points is changed on gear tooth surl,tce with the center distance variation. The results showed that the bending stress of the gear tooth should be carefully studied because the bending moment may be increased due to dislocation of bearing contact. Case 3. Crossed axes misaligmnent
It is assumed that rotating shafts have a c r o s ~ d axes misalignment, say A7, = 0.1', in Fig. 3. Kinematic errors of the gear train will be induced. The T C A results and corresponding kinematic errors are calculated and listed in Table 4. it is found that the location of bearing contacts has only a small variation compared with those for the ideal case. The kinematic errors induced by crossed axes misalignment can be compensated by regrinding the pinion with a modified helix angle ]J~ = 29.9249 ~. The T C A results with modified helix angle of II = 29.9249 are shown in Table 5 and Fig. 5.
L o c a t i o n of c o n t a c t
"
for ideal case ¢t'=O ° ~'= 5° ~'= 10 °
~'= 15" ¢t'= 20 ° ¢t' = 25"
Fig. 4, Locations of contact points for the ideal case.
151
Computer simulation and stress analysis of helical Bears Table 3. T C A results with center distance variation AC -- i m m (0.48% variation) Errors
~; (des)
~'. (des)
~l (des)
0.000 5.000 I0.000 15.000 20.000 25.000
0.000 1.666 3.333 5.000 6.666 8.333
0.505 5.505 10.505 15.505 20.505 25.505
COMPUTER-AIDED
¢: (deg) -0.705 0.962 2.628 4.295 5.962 7.628
FEM
Or (des)
I, (ram)
arc-scc
25.000 25.000 25.000 25.000 25.000 25.000
1.664 1.664 1.664 1.664 1.664 1.664
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
STRESS
ANALYSIS
In this section, the computer-aided FEM has been employed to gear stress analysis. The stress model applied to bending and contact stresses analysis of the gear is based on two well-known analytical approaches: TCA and FEM. The mathematical model of helical gears discussed in the previous section has been employed to define the geometry of the gear and pinion. An automatic meshes generation program has also been developed to divide the gear tooth into elements, and to generate nodal points and meshes automatically for FEM stress analysis. The TCA computer program has been developed to determine the location and direction of the applied Ioadings. The frontal method [14, 15] is employed to calculate FEM stiffness matrix. The frontal method provides easier assembly of equations and storing only the active coefficients. These active coefficients are formed as a triangle in a stiffness matrix associated with the elimination processes. Therefore, the frontal method is very suitable to apply to a small computer system. An eight-node isoparametric hexahedron element has been chosen for the three-dimensional FEM stress analysis. For simplicity, the following assumptions are made for stress analysis: (I) The stress discussed here is within the elastic range of materials. Plastic and nonlinear behaviors of the material are neglected. (2) lsotropic materials arc assumed. The properties are the same in all directions. (3) Nodal points on the boundaries are assumed to be fixed. (4) Small displacement is assumed, in this paper, eight Gaussian integration points were chosen to calculate the stiffness matrix for each element. The contours of stress distribution were plotted by applying the surface fitting technique. Due to length limitation of the paper, only stress contour diagram of the pinion are shown in this section. it is assumed that the basic manufacturing data are the same as stated in previous examples and the loading is 890 N. Figure 6 shows a single-tooth pinion with meshed elements which generated by automatic meshes generating computer program, it is noticed that nodal points on the boundaries are assumed to be fixed. Three-dimensional stress distributions of the pinion are plotted by Von Mises stress contours with different cross-section as shown in Figs 7-10. The loading is assumed to be applied at the contact point when rotation angle of the pinion is zero, ~; = 0". Table 4. T C A results and kinematic errors due to crossed misaligned angle A7h = 0.1 '
~; (des)
~2 (des)
~l (deg)
~: (des)
0.000 5.000 I 0.000 15.000 20.000 25.000
0.000 1.662 3.323 4.985 6.647 8.308
-0.322 4.674 9.670 14.667 19.663 24.659
-0.108 i.557 3.223 4.888 6.553 8.219
Op (des)
I r (ram)
Errors arc-see
25.369 25.369 25.369 25.369 25.369 25.369
2.821 2.821 2.821 2.821 2.821 2.82[
0.000 - 18.130 - 36.260 - 54.391 -72.523 -90.655
Table 5. Compensated results for the case of crossed axes misalignment with a modified helix angle/~ = 29.9249
~; (des)
~'., (deg)
~= (deg)
~: (deg)
0.000 5.000 10.000 15.000 20.000 25.000
0.000 1.667 3.333 5.000 6.667 8.333
--0.084 4.912 9.909 14.905 19.901 24.897
-0.024 1.646 3.317 4.987 6.658 8.328
Or (de S)
Ip (ram)
Errors arc-.',~-c
25.093 25.093 25.093 25.093 25.093 25.093
2.775 2.775 2.775 2.775 2.775 2.775
-0.000 -0.002 -0.004 -0.005 -0.005 -0.005
152
CHU~G-BL~u TSAY and 7' H. FONG
100.0 80.0
Hel.tx Angle
60.0 40.0
d
o 29.84.5 + 29.887 29.925 = 29.962 o 30.000
20.0 O.OI
~d - 2 o . o ~ -40.0 -60.0
-80.0 -100.0 0.0
5
10
t5
20
25
Fig. 5. Kinematic err~)rs due Io crossed axes mi.,mlignment with compen~tion.
This example shows that how TCA technique is incorporated with computer-aided FEM for gear stress analysis. The main advantages of applying FEM incorporated with TCA technique are: (!) Easy to define the geometry of tooth surface analytically when the manufacturing data were changed. (2) Easy to determine the contact pattern, contact position and direction of the applied loading. In addition, the advantages are quite obviously when the effects of modifying some manufacturing data were investigated. CONCLUSIONS
From the results of previous sections, some important characteristics of this type of gearing may be summarized as follows: (I) The contact pattern is a point contact rather than a line contact. (2) Compared with the ideal case of gear meshing, the contact positions of this type of gearing are not changed too much when a misaligned angle of rotating shafts exists. (3) Lower assembly precision is permitted when this type of gearing is employed. ModuLe - 5 , Tooth No. ,, 18, Load - 8 9 0 N
ModuLe ,, 5, Tooth No. - 18, Load = 8 9 0 N Van Mises stress contour, q~l = 0 *
,',,\\\\\\\\\\\\\\\\~'~ Fig, 6, Meshes for a single toolh or the pinion.
Fig. 7. Von Miscs stress contour of layer I,
Computer simulation and stress analysis of helical gears
153
ModuLe = 5 , Layer = 2 , Load ,, 8 9 0 N
Tooth No.,, 18, qb,-O °
Fig. 8. Von Mises stress contour of layer 2. ModuLe-5, Load,,890
ModuLe ,, 5 , Tooth NO.,, 1 8 , L a y e r - 4
N at ~,-0"
Layer I, 3
Fig. 9. Van Mises stress contour of layer 3.
Fig. I0. Van Mises stress contour of layer 4.
(4) in case of center distance variation, the contact position on the pinion tooth surface is not changed while the contact position on the gear tooth surface is changed. Bending stress may be increased due to the dislocation of bearing contact on the gear tooth surface. (5) By regrinding the pinion with a modified helix angle, dislocation of the bearing contact will be improved and kinematic errors of the gear train will be reduced substantially when the same gear train is assembled again, Smaller bending stress and noise can be expected for the gear train with the proposed method of compensation. (6) Computer-aided FEM incorporates with TCA techniques is a very powerful tool for gear stress analysis. Ackno.'led~ement--The authors are grateful to the National Science Council of R.O.C. for their grant. Contract No. NSC-77-040I-E009-03.
REFERENCES I. F L. Litvin. C. B Tsay. J. J. Coy and R. F. Handschuh. AGMA Technical Paper 86FTM3 (1986). 2. C. B. Tsay. J. Chin. Inst. Engng 10(I). 45-51 (1987).
154
CHu.'~G-BtAu TSAYand 7_ H. FON(;
3. F. L. Litvin, Proc. ,~d WId Cong. Gearing, Paris, Mar. (1986). 4. F. L. I.atvin, Theory of Gearing, 2nd edn (in Russian~. Nauka. Moscow (1968). The new edition (in English). revised and completed (in press). 5. D. N. Rcshetov. Machine Destgn. Mir. Moscow (1978). 6. Dobrovolsky et al., Machine Elements. Mir. Moscow (c. 1960). 7 G. Niemann, Mazchinen¢lemente. Spnnger, Berlin (1965). 8. Z. Ou and A. Sclreg, Trans. A S M E JI Mech. Transm. Automn De.~. 106, 65-7l (1986). 9. L. E. Wilcox. AGMA Paper I)2,29.25 (1982). 10. Understanding Tooth Contact Analysis. Glcason Works Publication, SD3IgA (1978). If. H. A. Kamel and Z. Navabi, J..Mech. Des. 102, 552-560 (1980). 12. L. E. Wilcox. AGMA Paper P2,29.25 (1982). 13. J. J. Coy and C. Hu-Chih Chao, J. ,Mech. Des. 104, 759-766 ¢1982). 14. N. Ida and W. Lord, Int. J. numer. ,Methods Engng 20, 625-641 (1984). [5. E. Thompson and Y Shimazaki. Int. J. numer. Methods Engng 15, 889-910 (1980).
RECHNERSIMULATION UND SPANNUNGSANALYSE DES SCHNECKENGETRIEBES MIT KREISBOGENZ,g, HNEN FIDR D A S R I T Z E L U N D E V O L V E N T E N Z P . H N E N FOR DAS SCHNECKENRAD Z~mmenfas.~unl~--Das Vcrfahrcn dcr Z.ahnbcrdhrungsanaly.~(TCA) und die Finite Element Mcthodc (FEM) wcrdcn zur Analyse dcr Zahnbcr6hrung und dcr Spannung verwcndct. Es gibt Ixreits cin mathcm.',ti.~hcs M(~Icll l'i~r Schncckcngctriclx mit Krcisbogenz~hncn am Ritz¢l und Evolvcntcnzfihncn am .Schncckcnrad yon Litvin und C, B. Ts-',y. Dic Z-',hnradgcomctric wird dutch die Paramctcr der Fcrtlgung bcschricbcn. Dic Simulation dcr Vcrzahnung ¢inschliefllich dcr Achsenabwcichungcn und dcr Variicrung des Mittclpunktabstandcs wcrdcn durchgcfiihrL Die Betnebsw',:ilzlinieund die Kontaktart der verzahnlcn Zahnfl~ichen wcrdcn mit dcm TCA-Verfahren ermittelt. Die Ergcbnis.~ liefcrn Lage und Richtung der auftrctcndcn Belastungcn fiir die 5pannungsanaly.~ mittels FEM. Unter Verwendung des gegebcnen mathcmalischcn Modells und dcm TCA-Vcrfahrcn ist ein Modcll der rechnerunterstiitztcn Sp;,nnung.~naly~ fiir ncuc Z."ihncpaarungcn vorge~hlagen worden. Ein .~ll~U~itig~ Erzeugungsprogr;,mm f,'ir Vcrzahnungcn (AM(;) ist f6r die Spannungsanaly~ rail FEM enlwickell worden. Einc drci dimcnsi~malc Spannung~nalysc f(ir cincn ~)Ichcn ~lhntyp wird durchgcfiihrt und mit Von MisesVcrl.cilung (t~ Spannungsumris~s dargcstellt.