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Computerized design, generation and simulation of meshing of orthogonal offset face-gear drive with a spur involute pinion with localized bearing contact
~
Pergamon
Mech. Math. TheoO" Vol. 33, No. I/2, pp. 87-102, 1998 ~'. 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain P l h S0094-114X(97)00022-0 0094-114X/98 $19.00 + 0.00
COMPUTERIZED DESIGN, GENERATION AND SIMULATION OF M E S H I N G OF O R T H O G O N A L OFFSET F A C E - G E A R D R I V E WITH A S P U R I N V O L U T E P I N I O N WITH L O C A L I Z E D B E A R I N G C O N T A C T F. L. LITVIN and A. EGELJA Gear Research Laboratory, Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60607-7022, U.S.A.
HEATH
J. T A N a n d G . McDonnell Douglas Helicopter Systems, Mesa, AZ 85205, U.S.A. (Received 19 June 1996; in revised form 12 December 1996; received for publication 1997)
Abstract--Computerized generation, localization of bearing contact and simulation of meshing and contact of an orthogonal offset face-gear drive with a spur involute pinion are considered. The authors propose an approach that enables one to: (i) localize and stabilize the bearing contact, and (ii) simplify the design of a face-gear free of undercutting and pointing. The existing theory of face-gear drives is complemented with the idea of axes of meshing that enables one to: (i) simplify the determination of conditions of tooth pointing, and (ii) apply an alternative approach for the derivation of meshing equation. Computer programs for all stages of investigation are developed. Numerical results for the test of the developed theory and its illustration are provided. The authors limit the discussions to the case of an orthogonal face-gear drive with a crossing angle of 9 0 . However, the proposed approach is easy to extend for non-orthogonal face-gear drives. © 1998 Elsevier Science Ltd.
NOMENCLATURE N ~ - n u m b e r of teeth of the sharper (i = s), pinion (i = 1), and face-gear (i = 2) Eo--nominal value of the shortest center distance between the pinion and the grinding disk (Fig. 3) Er--instantaneous value of the shortest center distance between the pinion and the grinding disk (Fig. 3) l ~ d i s p l a c e m e n t of the grinding disk in the direction of the pinion axis (Fig. 3) a~--parabola coefficient of the parabolic function Si--coordinate system u~0k~--surface parameters of the shaper 0o~--angular width of the space of the shaper on the base circle (Fig. 5) r,~position vector in system S,(i = s, 1, 2) rh~--radius of the base circle of the shaper (Fig. 5) ~o--pressure angle [Fig. 6 (b)] ~--proflle angle for determination of pointing (Fig. 7) N~--unit normal and normal (Fig. 5) to the shaper surface Mj~--matrix of coordinate transformation from system S~ to system S, E,--shaper (i = s), pinion (i = 1) or gear (i --- 2) tooth surface ~,--angle of rotation of the shaper (i = s) or face gear (i = 2) (Fig. 2) v,m~--~relative velocity vector ~c~-'~--relative angular velocity vector E--shortest center distance between the shaper and the face gear (Fig. 2) X,, Y,, Z,---coordinates of the axis of meshing m.,,--gear ratio R,--inner (i = 1) and outer (i = 2) radii of face gear free of undercutting and pointing (Fig. 8) Pj---diametral pitch A/, Aq--segments for the determination of pointing [Fig. 6 (a)] vd~---velocity of contact point that moves over the tooth surface of the shaper L,----limiting line on the shaper tooth surface (i = s) (Fig. 9), or face-gear tooth surface (i = g) L--fillet line (Fig. 10)
n,,
87
88
F.L. Litvin et
al.
0k,.--shaper parameter at the addendum cylinder r~,~--radius of the addendum cylinder of the shaper L,--limiting length of the shaper for avoidance of undercutting (Fig. 8) 7 shaft angle 05/ angles of rotation of pinion (i = l), and face gear (i = 2) being in mesh A~--transmission errors [Figs 14 (a), (b), (c)] v~(s)--velocity vector of the shaper in S~ v;2--velocity vector of the face gear in S~ o,'"--angular velocityvector of the face gear in S~ p--screw parameter 1. INTRODUCTION The face-gear drive is an invention of the Fellows Company. The theory of face-gear drives was represented in the Russian literature in works [1] and [2]. The importance of face-gear drives was recognized when such gear drives with torque splitting [3] were proposed for application in helicopter transmissions. New developments of the geometry of face-gear drives with intersected axes, based on application of the modern theory of gearing, were accomplished and summarized in [4]. Intensive research of face-gear drives is performed at the Crown Gear B.V. Company [5]. This paper represents the geometry of the offset face-gear drives and tooth contact analysis. The authors propose: (i) New and simplified approaches for the avoidance of undercutting and pointing of face-gear teeth. The solution to the problem of teeth pointing is based on the application of the concept of axes of meshing developed in [2] and [4]. (ii) Design of face-gear drives with improved conditions for the localization and stabilization of bearing contact due to application of an involute pinion with a crowned surface. The developed theory was tested by computerized simulation of meshing and contact of pinion and face-gear tooth surfaces. Numerical examples are provided.
2. GENERATION OF FACE-GEAR DRIVES WITH LOCALIZED BEARING CONTACT The pinion is a conventional involute spur one (Fig. 1). A face gear may be generated by an involute spur shaper. The process of generation is based on the simulation of shaping as meshing
Face gear
Pinion Fig. 1. Orthogonal offset face-gear drive.
89
Meshing of orthogonaloffsetface-geardrive
Xa,Xh
Xs
Zh , Zs
/
Za, Z 2 6.) 2
E
Fig. 2. Appliedcoordinate systems.
of the pinion and the face gear of the drive represented in Fig. 2. The shaper and the face gear form an offset drive that is similar to the face-gear drive to be designed, and perform rotational motions about the same axes as in the face-gear drive (Fig. 2). Misalignment of the face gear may cause separation of the pinion-gear tooth surfaces and edge contact. To avoid these defects, it is necessary to localize the bearing contact by substituting the instantaneous line contact of tooth surfaces by point contact. The localization of the bearing contact can be achieved: (i) by choosing a shaper whose tooth number N~ is slightly larger than the pinion tooth number N,, ( N s - N, = 1-3) [4]; (ii) by crowning of the pinion tooth surface as proposed in this paper. The number of teeth of the shaper, in the second case, is the same as that of the pinion. Figure 3 shows an example of crowning of the pinion tooth surface when this surface is generated by form-grinding. The axial profile of the grinding disk is the theoretical involute profile of the
90
F.L. Litvin et al.
pinion. During generation, the grinding disk is plunged and the shortest distance is varied in accordance to the equation (Fig. 3) Ep Eo - a,l~ (1) ~-
Here, Eo and Ep are the nominal and instantaneous values of the shortest center distance, ld is the displacement of the grinding disk in the direction of the pinion axis, (Ep - Eo) is the plunge of the grinding disk, ad is the parabola coefficient of the parabolic function (aal~). 3. A X E S OF M E S H I N G
Litvin [2] has proven the following theorem: A gear drive with crossed axes of rotation with line contact of tooth surfaces is considered. The tooth surfaces are in line contact at every instant and one of the interacting surfaces is a helicoid. Then, there are two straight lines I-I and II-II that lie in parallel planes, have constant location and orientation during the process of meshing, and the normal to the interacting surfaces at any regular point of surface tangency passes through I-I and II-II. These lines are called the axes o f meshing. An involute spur shaper is considered as a particular case of an involute helicoid, when the screw parameter p of the helicoid approaches to infinity. The location and orientation of the axes of meshing in this particular case is as shown in Fig. 4. Axis of meshing I-I lies in plane x~~)= 0, and its orientation is determined as Z ") y~l)
N2 Ns
(2)
Id f
Grinding wheel EP
1 Pinion
t
f
Fig. 3. Crowning of the pinion tooth surface.
gO
Meshing of orthogonal offset face-gear drive
Xa Xh
91
axis of m e s h i n g s,Zh
Os Oh
Za, Z 2
Oa 02 E>oo II
II Fig. 4. Location and orientation of axes of meshing.
Axis of meshing II-II is parallel to Zh but lies in a plane x~") that approaches infinity along the negative direction of Xh.
4. EQUATION OF TOOTH SURFACES
4.1. Appliedcoordinatesystems Movable coordinate systems Ss and $2 are rigidly connected to the shaper and the face gear, respectively (Fig. 2). Coordinate systems Sh and S~ are the fixed ones.
4.2. Shapersurface This surface is represented in coordinate system Ss (Fig. 5) by the equation
I+rbs[sin(Oks+OoO--Okscos(O~-~+Oo,)]l r~(u, 0~..,)=
--rbs[COS(0~-,~
+ 0os) + 0k.~sin(0k, + 0o0] "
(3)
ffs
Here, (0ks, us) are the surface parameters; rbs is the radius of the base circle, and 0o~ determines the width of the space of the shaper on the base circle (Fig. 5). In the case of a standard shaper,
92
F.L. Litvin et al.
parameter 0o~ is represented by the equation 7~
0o~ = 2N---~- inv~o
(4)
where ~0 is the pressure angle. Equation (3) determines surfaces of both sides of the space; index k (k = 7, fl) in designations 0k~ corresponds to involute curves 7 - ~ and/3 - / 3 (Fig. 5). The upper signs in equation (3) correspond to profile 7 - 7. Unit normals to the shaper surfaces are represented in S~ as:
N~
n~= [N~I'
N~
Or~
8r~
(5)
= ~--~.~ x 0u----~
that yields
n~
= ~-cos(O~ + Oo~)] [ T-sin(~ + 0o~)]
(6)
4.3. Equations of face-gear tooth surface The face-gear tooth surface E2 is represented by the following equations r2(us, 0ks, ~bs) = M2~(q~)r~(u~, 0k~)
(7)
flus, 0~, qSs) = 0
(8)
Equation (7) represents in $2 the family of shaper tooth surfaces; M2~ is the 4 × 4 matrix that describes the coordinate transformation from S~ to $2. Equation (8) is the equation of meshing (see below). Equation (7) and (8) represent surface E2 by three related parameters. Xs
base c i r c l e 7
\
Nsr j Os
/
Fig. 5. Shaper involute profiles.
Meshing of orthogonal offset face-geardrive
93
4.4. Equation of meshing We may use two alternative approaches for the derivation of the equation of meshing. The first approach [2, 4] is based on the application of the equation N~. v~2' = f(u.~, Oks,¢~) = 0
(9)
where vls~2~= vIss~- v~2~= (to~~2~× rs) - (E × o2~21) is the relative represented in The second equation used
(10)
velocity (the velocity of sliding). The subscript "s" indicates that the vectors are coordinate system &. approach is based on the application of the axis of meshing and the respective for derivations is
X ~ - x ~ _ Y~--y~=Zs-z~ N,s N,~ N:~
(11)
Equation (l 1) describes that normal Ns passes through the axis of meshing; X,, Y~, and Z are the coordinates of the axis of meshing; xs, ys, and zs are the coordinates of a current point of the shaper tooth surface. After transformations, equation (1 l) yields the equation of meshing. Using one of the described approaches, we obtain that
f(u, Ok~,¢~) = m2sUscos(¢~ _ (0k~+ 0o0) - rb~ = 0
(k = 7, fl)
(12)
In equation (12) m2s = Ns/N2 is the gear ratio.
4.5. Avoidance of pointing The top land width of a face-gear tooth is not constant (Fig. 10). There is an area where the top land width becomes equal to zero, which means that the tooth becomes pointed. Our goal is to determine the outer radius R2 (Fig. 8) of the face gear for the area of pointing B (Fig. 10). Two approaches were proposed in [2] and [4] for the determination of area of pointing. The first one requires simultaneous consideration of the equations of the surfaces of both sides of the face-gear tooth and the determination of the area where both surfaces have a common point. The other one is based on the consideration of cross-sections of tooth profiles of the shaper and the face gear. Both approaches have been applied by the authors; the results are compared but only the second one, the simpler one, is presented in this paper. This approach is based on the following considerations: (1) Drawings of Fig. 5 show that xs = 0 is the plane of symmetry of the shaper space. At a position of the shaper when ¢~ = 0, coordinate system & coincides with Sh (Fig. 2) and the axis of meshing I-I belongs to the plane x~ = 0. (2) It is evident that cross-sections of the shaper tooth surface, produced by planes that are parallel to xs axis, represent the same involute curves. Two such planes, H~ and I-I_,, are shown in Fig. 6(a). Plane Hi which is perpendicular to zs (it is parallel to x0 intersects the axis of meshing I-I at a point P~ that belongs to the axis of symmetry of the cross-section of the space of the shaper. Figures 6(b) and 7 show points P~ and P2 of the intersection of the axis of meshing I-I planes HI and I-I2, respectively. (3) A normal to the shaper tooth surface is perpendicular to the z~ - axis and therefore it lies in plane 1-I,. In accordance to the definition of the axes of meshing the common normal to the surfaces of the shaper and the face-gear passes through the axis of meshing. Considering the conditions of tangency of the cross-section profiles of the shaper and the face-gear, we can make the following conclusions: (i) such points of tangency are M, and M2 in plane HI, and N~ and N2 in plane 1-12 [figs 6(b) and 7]; (ii) the common normals to the cross-section profiles pass through these points and the points of axis of meshing Pi (i = 1, 2); (iii) the common tangents to the cross-section profiles form an angle c¢0 and ~, respectively. (4) We assume now that plane 1-I2 is the plane where the pointing of the cross-section profiles of the face gear occurs. Our investigation shows that the cross-section profiles of the face-gear deviate only slightly from the straight lines. Therefore, we assume that point K (Fig. 7) of the
94
F . L . Litvin et al. Zs
I
/FI2
I
A1
Aq /
/
Addendum l i n e of t h e face gear
t~
~g
S
r'ps
a)
I
/
$
,/3
,M 1
/
Fps
b) Fig. 6. Intersectionof shaper tooth surface and axis of meshing i-1 by planes FI, and FI:. intersection of the tangents to the profiles is the point of intersection of the real cross-section profiles of the face gear. (5) Using the consideration discussed above, we are able now to derive the equations for the determination of the outer radius R2 of the face gear for the zone of pointing. Step I: Vector equation (Fig. 7) O~A + ANI + N t K = OsK
(13)
t a n ~ 2Pdrm - -
(14)
yields that [4]: where
0o~ is represented by equation (4).
N~
_ 0o~
Meshing of orthogonal offset face-geardrive
95
S t e p 2: We consider that point P~ belongs to the pitch cylinder of the shaper [Fig. 6(a)], and the location of P2 with respect to P~ is determined with segments AI and Aq [fig. 6(a)]. Drawings of Figs 6(a), 6(b) and 7 yield
Aq = O~P: - O~P, =
rb~
rb~
cos~
cos~0
AI=
_
N~ / c o s a zo_-- cos ~']
2/'~\
cos~
(15)
/
Aq tan y~
S t e p 3: The location of plane H2 is determined with parameter
(16)
L2 [Fig. 6(a)], where
L2 = rp + Aq = Nscos ~o tan 7s 2PdCOS~ tan y~
(17)
where tan 7~ = 1/m2s. The outer radius R2 of the face gear (Fig. 8) is represented as
R: =,j-k-;+ L~
(18)
Ys
¢
rm P
=0t
A
K
/ P2 2a Fig. 7. For the derivation of determination of angle ~t. M M T 33/I-2
D
96
F.L. Litvinet al. Xh
L2
oh,o
Zs,Zh E
L1
Fig. 8. Limitationsof the face gear.
4.6. Avoidance of undercutting of face gear Undercutting can be avoided if the face-gear tooth surface is free of singularities. This can be achieved by limiting of the inner radius of the face gear. An effective approach for the determination of singularities of the generated surface is proposed in [2] and [4]. The approach is based on application of the following equations (19)
¥~ss) + ¥~s2) = 0
0fdu
0f 00ks 0fd4,s
OUs dt + dOks dt t- 0dpl d t = 0
(20)
Vectors of equation (19) are represented in coordinate system Ss; v~~is the velocity of a contact point that moves over the tooth surface of the shaper; v~~21is the relative velocity of a shaper point with respect to the face gear. Equation (20) is the differentiated equation of meshing that was represented by equation (8). Equations (19) and (20) represent a system of four linear equations in terms of two unknowns: dus/dt and dOks/dt considering dcks/dt as chosen. The system has a certain solution for the unknowns if the matrix
A =
0r~ 0r~ ~u~ OOks
of
of
0us
00k~
v~21 "] ---~
of d4, Oq~s dt
J
(21)
has the rank r = 2. The yields four determinants of the third order that are equal to zero. Our invstigation shows that the equality of two of them to zero is equivalent to the existence of
Meshing of orthogonal offset face-geardrive
97
the equation of meshing. Therefore, it is necessaey to use one of the following two equations
A2
A2 ~--
~Xs OXs 8u~ 80,
v ,.~
Oz~
8z,
v~s2~
f.,
)co,, f . , ~
Is2)
OYs ~Y~
~,.2~
~u~
80k~
- ,.s
az~ 8u~
,~z~ ~0~,~
v~_~2, -
f~,
)Co. f , . - ~
= 0
(22)
= 0
(23)
Here f,,, J~. and f~, are the partial derivatives of the equation of meshing taken with respect to us, 0,s amd ~bs, respectively, x~, ys, and z~ are the coordinates of a point of the shaper surface Es in coordinate system &; (v!?fI, vlr~I, v~]21) are the components of the relative velocity in the same coordinate system. Either equation (22) or (23) yields the same following relation
F(u~, 0k~, ~bs) = 0
(24)
We point out that the same parameters are related by the equation of meshing (8). We may now determine the set of singular points on the face gear tooth surface using the following equations. r2 = r2(us, 0,~, q~), f l u , 0,~, ~ ) = O,
F(u~, 0,.~, q~) = 0,
(k = 7, B)
(25)
Equation (25) enables us to determine the limiting line Lg of the face-gear tooth surface £2 to avoid undercutting. Similarly, we can determine the limiting line Ls of the shaper tooth surface Z~ (Fig. 9) using the equations r~ = rdu~, OkO, f(u,~, O,s, (a,,) = O,
F(U~, Ok~, (as) = O, (k = 7, fl)
(26)
Line L~ contains regular points of surface £~, but generates singular points on surface £2. Knowing one of the two limiting lines, say L,, we may also determine Ls using a coordinate transformation from, & to $2. Due to asymmetry of the face-gear tooth surfaces, both surfaces have to be checked for undercutting. The dimensions of the blank that is used for manufacturing of the face gear are determined using the larger value of the inner radius R~ (Fig. 8) that is determined from conditions
L
Ls Fig. 9. Limitingline L~on the shaper surface.
98
F. L. Litvin et al.
A
surface
surface
Fig. 10. Offset face-gear tooth (cross-sections of the working and fillet surface).
of undercutting for the generating profiles 7 and ft. The research results show that, for a positive offset (E > 0), the critical shaper surface that will cause undercutting in the surface with cross-section as curve fl-fl, and it is the opposite for negative offset (E < 0). The point of intersection of line L (Fig. 10) with the addendum cylinder of the shaper is critical for undercutting. The shaper parameter 0ks that corresponds to the addendum is determined by the equation 0* =
rbs
,
(27)
where r,s and rbs are the radii of the addendum and the base cylinder of the shaper. Substituting the value of 0~ into equation (3), we obtain the coordinates (x*, y*, z*) of the point of intersection of the limiting line with the addendum cylinder of the shaper. Knowing these values, the limiting inner radius of the face gear (R,) and the limiting length of the shaper (L,) (Fig. 8) are determined as
R, = ~
+ y~
L, = z,
(28) (29)
A computer program was developed for the numerical solution of the limiting value L~. The solution of non-linear equations represented above requires an initial guess that can be obtained by solving the following non-linear equation for ~ks sin 2 ~ks + rn~O~ ~ + sin ~k~-T- O~ cos ~ks cos3 G~ = 0
(30)
where Oh is the value of shaper surface parameter at the addendum cylinder, and ~ks is represented by equation ~=~s+_(O~+Oo~) k=?,/~ (31)
Meshing of orthogonal offset face-geardrive
99
5. COMPUTERIZED SIMULATION OF MESHING AND CONTACT The procedure of simulation of meshing is directed at the determination of transmission errors and the shift of the contact path caused by misalignment. The developed computer program is based on the continuous tangency of interacting surfaces being in point contact [2] and [4]. Point tangency of surfaces is achieved due to crowning of the pinion tooth surface (see Section 2). 5. I. Bearing contact The bearing contact on the face-gear tooth surface is formed as a set of instantaneous contact ellipses. The center of such an ellipse is the instantaneous point of contact. The determination of axes of the instantaneous contact ellipse and its orientation is based on the approach developed in Refs [2, 4]. The localization of the bearing contact is based on the following considerations: (i) the number of teeth of the pinion is the same as the shaper, (ii) the localization of the bearing contact is achieved by the modification of the pinion tooth surface. 5.2. Results of TCA (tooth contact analysis) Computerized simulation of meshing and contact was accomplished for a gear drive with a crowned pinion tooth surface and the following design parameters: Ns = 20, N~ = 20, N2 = 100, E = 2 (in.), 7 = 90°, a = 0.03, P~ = 10. Misalignment of the face-gear drive was simulated by the change of the center distance E and the shaft angle 7. The results of TCA are shown in Figs 11, 12 and 13. One of the benefits of the new approach with the crowned pinion is that the desired location of the bearing contact on the face-gear tooth surface may be obtained just by the axial displacement of the modified pinion. 5.3. Contact path The contact path on the face-gear tooth surface can be determined by the vector function r2(0s(~b.;), ~b~(gb~)),where 0s and ~s are the same parameters, and ~b~is the face-gear angle of rotation in progress of meshing with the pinion. By applying the driving pinion with the same number of teeth as the generating shaper, but with the modified tooth surface, the contact path is substantially stabilized. This is a big advantage in comparison with the approach when the localization of being contact is accomplished by applying different numbers of teeth of the driving pinion and generating shaper.
Fig. 11. Contact path without misalignment.
100
F.L. Litvin et al.
r
! ! !
!
/ /
f
/
J
f
Fig. 12. Contact path with applied misalignment (change of center distance AE = -0.05 ram). 5.4. Transmission errors
The transmission errors are represented by the equation A ' NI , q~2 = ~ - N2 (ok, - ~b*)
(32)
In the equation above, we designate with qS* the value of q~f that corresponds to q~* = 0, which is the initial value of ~b~ (the position error). The value of rk* can be determined from the numerical function ~bf(~b~)~.,=0. Linear function N ~ / N f f q b f - qb*) represents the theoretical angle of rotation of the face gear as a function of ~bf, the pinion angle of rotation. The results of transmission errors, for the case when the pinion with modified tooth surface is applied, show that the magnitudes of transmission errors are not zero, but they do not exceed the values of 2-3 arc seconds in the case of a misaligned gear drive. This means that the face-gear drive is not sensitive to misalignment if the localization of bearing contact is performed by application of an involute pinion with modified
Fig. 13. Contact path with applied misalignment (change of crossing angle A7 = -0.05").
I01
Meshing of orthogonal offset face-geardrive i
(arc see)
a)
O
I
I
c~ t
l -3
l
l
l
-2
- 1
0
arc
sec)
I
-1
I
1
i 2
3
I
I
I
I
0
1
2
3
I
I
I
I
I
-1
0
1
2
3
=
C a(deg)
I
b)
o
7 I
CQ
I
I
-2
-3
C2 (deg)
i
At2 (arc sec)
c)
o
7 i
,~ -3
I -2
Fig. 14. Transmission errors of face-gear drive: (a) no misalignment, (b) A7 = - 0.05°.
:
C z (deg) AE=-0.05
mm, (c)
surface and the same number of teeth as the generating shaper. This is an advantage of face-gear drives in comparison with other gear drives that are used for transformation of power between crossed axes. The transmission errors (for one cycle of meshing) are represented in Figs 14(a), 14(b) and 14(c) for an ideal case (no misalignments) and the case when misalignments exist. 6. CONCLUSIONS (1) Equations for analysis and design of an offset face-gear drive with a spur involute pinion have been derived. (2) Localization of bearing contact by application of a pinion with crowned surface has been achieved.
102
F.L. Litvin et al.
(3) A simplified a p p r o a c h for the design o f drives with face gears free o f u n d e r c u t t i n g a n d p o i n t i n g has been represented. (4) C o m p u t e r i z e d s i m u l a t i o n o f meshing a n d c o n t a c t o f the offset face-gear drive has been accomplished. Acknowledgments--The authors express their deep gratitude to the McDonnell Douglas Helicopter Systems, and personally to Mr Robert J. King, for the financial support of this research.
1. 2. 3. 4. 5.
REFERENCES Davidov, Y. S., Non-Involute Gears. Mashgis, 1950 (in Russian). Litvin, F. L., Theory of Gearing. Nauka, 1968 (in Russian). Litvin, F. L., Wang, J.-C., Chen, Y.-J., Bossier, R. B., Heath, G. and Lewicki, D. J., A.G.M.A. Paper, 92FTM, 1992. Litvin, F. L., Gear Geometry and Applied Theory. Prentice Hall, Englewood Cliffs, N J, 1994. Basstein, G. and Sijtstra, A., A.G.M.A. Technical Paper, 93FTM7, October 1993.
Pergamon
Mech. Math. TheoO" Vol. 33, No. I/2, pp. 87-102, 1998 ~'. 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain P l h S0094-114X(97)00022-0 0094-114X/98 $19.00 + 0.00
COMPUTERIZED DESIGN, GENERATION AND SIMULATION OF M E S H I N G OF O R T H O G O N A L OFFSET F A C E - G E A R D R I V E WITH A S P U R I N V O L U T E P I N I O N WITH L O C A L I Z E D B E A R I N G C O N T A C T F. L. LITVIN and A. EGELJA Gear Research Laboratory, Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60607-7022, U.S.A.
HEATH
J. T A N a n d G . McDonnell Douglas Helicopter Systems, Mesa, AZ 85205, U.S.A. (Received 19 June 1996; in revised form 12 December 1996; received for publication 1997)
Abstract--Computerized generation, localization of bearing contact and simulation of meshing and contact of an orthogonal offset face-gear drive with a spur involute pinion are considered. The authors propose an approach that enables one to: (i) localize and stabilize the bearing contact, and (ii) simplify the design of a face-gear free of undercutting and pointing. The existing theory of face-gear drives is complemented with the idea of axes of meshing that enables one to: (i) simplify the determination of conditions of tooth pointing, and (ii) apply an alternative approach for the derivation of meshing equation. Computer programs for all stages of investigation are developed. Numerical results for the test of the developed theory and its illustration are provided. The authors limit the discussions to the case of an orthogonal face-gear drive with a crossing angle of 9 0 . However, the proposed approach is easy to extend for non-orthogonal face-gear drives. © 1998 Elsevier Science Ltd.
NOMENCLATURE N ~ - n u m b e r of teeth of the sharper (i = s), pinion (i = 1), and face-gear (i = 2) Eo--nominal value of the shortest center distance between the pinion and the grinding disk (Fig. 3) Er--instantaneous value of the shortest center distance between the pinion and the grinding disk (Fig. 3) l ~ d i s p l a c e m e n t of the grinding disk in the direction of the pinion axis (Fig. 3) a~--parabola coefficient of the parabolic function Si--coordinate system u~0k~--surface parameters of the shaper 0o~--angular width of the space of the shaper on the base circle (Fig. 5) r,~position vector in system S,(i = s, 1, 2) rh~--radius of the base circle of the shaper (Fig. 5) ~o--pressure angle [Fig. 6 (b)] ~--proflle angle for determination of pointing (Fig. 7) N~--unit normal and normal (Fig. 5) to the shaper surface Mj~--matrix of coordinate transformation from system S~ to system S, E,--shaper (i = s), pinion (i = 1) or gear (i --- 2) tooth surface ~,--angle of rotation of the shaper (i = s) or face gear (i = 2) (Fig. 2) v,m~--~relative velocity vector ~c~-'~--relative angular velocity vector E--shortest center distance between the shaper and the face gear (Fig. 2) X,, Y,, Z,---coordinates of the axis of meshing m.,,--gear ratio R,--inner (i = 1) and outer (i = 2) radii of face gear free of undercutting and pointing (Fig. 8) Pj---diametral pitch A/, Aq--segments for the determination of pointing [Fig. 6 (a)] vd~---velocity of contact point that moves over the tooth surface of the shaper L,----limiting line on the shaper tooth surface (i = s) (Fig. 9), or face-gear tooth surface (i = g) L--fillet line (Fig. 10)
n,,
87
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F.L. Litvin et
al.
0k,.--shaper parameter at the addendum cylinder r~,~--radius of the addendum cylinder of the shaper L,--limiting length of the shaper for avoidance of undercutting (Fig. 8) 7 shaft angle 05/ angles of rotation of pinion (i = l), and face gear (i = 2) being in mesh A~--transmission errors [Figs 14 (a), (b), (c)] v~(s)--velocity vector of the shaper in S~ v;2--velocity vector of the face gear in S~ o,'"--angular velocityvector of the face gear in S~ p--screw parameter 1. INTRODUCTION The face-gear drive is an invention of the Fellows Company. The theory of face-gear drives was represented in the Russian literature in works [1] and [2]. The importance of face-gear drives was recognized when such gear drives with torque splitting [3] were proposed for application in helicopter transmissions. New developments of the geometry of face-gear drives with intersected axes, based on application of the modern theory of gearing, were accomplished and summarized in [4]. Intensive research of face-gear drives is performed at the Crown Gear B.V. Company [5]. This paper represents the geometry of the offset face-gear drives and tooth contact analysis. The authors propose: (i) New and simplified approaches for the avoidance of undercutting and pointing of face-gear teeth. The solution to the problem of teeth pointing is based on the application of the concept of axes of meshing developed in [2] and [4]. (ii) Design of face-gear drives with improved conditions for the localization and stabilization of bearing contact due to application of an involute pinion with a crowned surface. The developed theory was tested by computerized simulation of meshing and contact of pinion and face-gear tooth surfaces. Numerical examples are provided.
2. GENERATION OF FACE-GEAR DRIVES WITH LOCALIZED BEARING CONTACT The pinion is a conventional involute spur one (Fig. 1). A face gear may be generated by an involute spur shaper. The process of generation is based on the simulation of shaping as meshing
Face gear
Pinion Fig. 1. Orthogonal offset face-gear drive.
89
Meshing of orthogonaloffsetface-geardrive
Xa,Xh
Xs
Zh , Zs
/
Za, Z 2 6.) 2
E
Fig. 2. Appliedcoordinate systems.
of the pinion and the face gear of the drive represented in Fig. 2. The shaper and the face gear form an offset drive that is similar to the face-gear drive to be designed, and perform rotational motions about the same axes as in the face-gear drive (Fig. 2). Misalignment of the face gear may cause separation of the pinion-gear tooth surfaces and edge contact. To avoid these defects, it is necessary to localize the bearing contact by substituting the instantaneous line contact of tooth surfaces by point contact. The localization of the bearing contact can be achieved: (i) by choosing a shaper whose tooth number N~ is slightly larger than the pinion tooth number N,, ( N s - N, = 1-3) [4]; (ii) by crowning of the pinion tooth surface as proposed in this paper. The number of teeth of the shaper, in the second case, is the same as that of the pinion. Figure 3 shows an example of crowning of the pinion tooth surface when this surface is generated by form-grinding. The axial profile of the grinding disk is the theoretical involute profile of the
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F.L. Litvin et al.
pinion. During generation, the grinding disk is plunged and the shortest distance is varied in accordance to the equation (Fig. 3) Ep Eo - a,l~ (1) ~-
Here, Eo and Ep are the nominal and instantaneous values of the shortest center distance, ld is the displacement of the grinding disk in the direction of the pinion axis, (Ep - Eo) is the plunge of the grinding disk, ad is the parabola coefficient of the parabolic function (aal~). 3. A X E S OF M E S H I N G
Litvin [2] has proven the following theorem: A gear drive with crossed axes of rotation with line contact of tooth surfaces is considered. The tooth surfaces are in line contact at every instant and one of the interacting surfaces is a helicoid. Then, there are two straight lines I-I and II-II that lie in parallel planes, have constant location and orientation during the process of meshing, and the normal to the interacting surfaces at any regular point of surface tangency passes through I-I and II-II. These lines are called the axes o f meshing. An involute spur shaper is considered as a particular case of an involute helicoid, when the screw parameter p of the helicoid approaches to infinity. The location and orientation of the axes of meshing in this particular case is as shown in Fig. 4. Axis of meshing I-I lies in plane x~~)= 0, and its orientation is determined as Z ") y~l)
N2 Ns
(2)
Id f
Grinding wheel EP
1 Pinion
t
f
Fig. 3. Crowning of the pinion tooth surface.
gO
Meshing of orthogonal offset face-gear drive
Xa Xh
91
axis of m e s h i n g s,Zh
Os Oh
Za, Z 2
Oa 02 E>oo II
II Fig. 4. Location and orientation of axes of meshing.
Axis of meshing II-II is parallel to Zh but lies in a plane x~") that approaches infinity along the negative direction of Xh.
4. EQUATION OF TOOTH SURFACES
4.1. Appliedcoordinatesystems Movable coordinate systems Ss and $2 are rigidly connected to the shaper and the face gear, respectively (Fig. 2). Coordinate systems Sh and S~ are the fixed ones.
4.2. Shapersurface This surface is represented in coordinate system Ss (Fig. 5) by the equation
I+rbs[sin(Oks+OoO--Okscos(O~-~+Oo,)]l r~(u, 0~..,)=
--rbs[COS(0~-,~
+ 0os) + 0k.~sin(0k, + 0o0] "
(3)
ffs
Here, (0ks, us) are the surface parameters; rbs is the radius of the base circle, and 0o~ determines the width of the space of the shaper on the base circle (Fig. 5). In the case of a standard shaper,
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F.L. Litvin et al.
parameter 0o~ is represented by the equation 7~
0o~ = 2N---~- inv~o
(4)
where ~0 is the pressure angle. Equation (3) determines surfaces of both sides of the space; index k (k = 7, fl) in designations 0k~ corresponds to involute curves 7 - ~ and/3 - / 3 (Fig. 5). The upper signs in equation (3) correspond to profile 7 - 7. Unit normals to the shaper surfaces are represented in S~ as:
N~
n~= [N~I'
N~
Or~
8r~
(5)
= ~--~.~ x 0u----~
that yields
n~
= ~-cos(O~ + Oo~)] [ T-sin(~ + 0o~)]
(6)
4.3. Equations of face-gear tooth surface The face-gear tooth surface E2 is represented by the following equations r2(us, 0ks, ~bs) = M2~(q~)r~(u~, 0k~)
(7)
flus, 0~, qSs) = 0
(8)
Equation (7) represents in $2 the family of shaper tooth surfaces; M2~ is the 4 × 4 matrix that describes the coordinate transformation from S~ to $2. Equation (8) is the equation of meshing (see below). Equation (7) and (8) represent surface E2 by three related parameters. Xs
base c i r c l e 7
\
Nsr j Os
/
Fig. 5. Shaper involute profiles.
Meshing of orthogonal offset face-geardrive
93
4.4. Equation of meshing We may use two alternative approaches for the derivation of the equation of meshing. The first approach [2, 4] is based on the application of the equation N~. v~2' = f(u.~, Oks,¢~) = 0
(9)
where vls~2~= vIss~- v~2~= (to~~2~× rs) - (E × o2~21) is the relative represented in The second equation used
(10)
velocity (the velocity of sliding). The subscript "s" indicates that the vectors are coordinate system &. approach is based on the application of the axis of meshing and the respective for derivations is
X ~ - x ~ _ Y~--y~=Zs-z~ N,s N,~ N:~
(11)
Equation (l 1) describes that normal Ns passes through the axis of meshing; X,, Y~, and Z are the coordinates of the axis of meshing; xs, ys, and zs are the coordinates of a current point of the shaper tooth surface. After transformations, equation (1 l) yields the equation of meshing. Using one of the described approaches, we obtain that
f(u, Ok~,¢~) = m2sUscos(¢~ _ (0k~+ 0o0) - rb~ = 0
(k = 7, fl)
(12)
In equation (12) m2s = Ns/N2 is the gear ratio.
4.5. Avoidance of pointing The top land width of a face-gear tooth is not constant (Fig. 10). There is an area where the top land width becomes equal to zero, which means that the tooth becomes pointed. Our goal is to determine the outer radius R2 (Fig. 8) of the face gear for the area of pointing B (Fig. 10). Two approaches were proposed in [2] and [4] for the determination of area of pointing. The first one requires simultaneous consideration of the equations of the surfaces of both sides of the face-gear tooth and the determination of the area where both surfaces have a common point. The other one is based on the consideration of cross-sections of tooth profiles of the shaper and the face gear. Both approaches have been applied by the authors; the results are compared but only the second one, the simpler one, is presented in this paper. This approach is based on the following considerations: (1) Drawings of Fig. 5 show that xs = 0 is the plane of symmetry of the shaper space. At a position of the shaper when ¢~ = 0, coordinate system & coincides with Sh (Fig. 2) and the axis of meshing I-I belongs to the plane x~ = 0. (2) It is evident that cross-sections of the shaper tooth surface, produced by planes that are parallel to xs axis, represent the same involute curves. Two such planes, H~ and I-I_,, are shown in Fig. 6(a). Plane Hi which is perpendicular to zs (it is parallel to x0 intersects the axis of meshing I-I at a point P~ that belongs to the axis of symmetry of the cross-section of the space of the shaper. Figures 6(b) and 7 show points P~ and P2 of the intersection of the axis of meshing I-I planes HI and I-I2, respectively. (3) A normal to the shaper tooth surface is perpendicular to the z~ - axis and therefore it lies in plane 1-I,. In accordance to the definition of the axes of meshing the common normal to the surfaces of the shaper and the face-gear passes through the axis of meshing. Considering the conditions of tangency of the cross-section profiles of the shaper and the face-gear, we can make the following conclusions: (i) such points of tangency are M, and M2 in plane HI, and N~ and N2 in plane 1-12 [figs 6(b) and 7]; (ii) the common normals to the cross-section profiles pass through these points and the points of axis of meshing Pi (i = 1, 2); (iii) the common tangents to the cross-section profiles form an angle c¢0 and ~, respectively. (4) We assume now that plane 1-I2 is the plane where the pointing of the cross-section profiles of the face gear occurs. Our investigation shows that the cross-section profiles of the face-gear deviate only slightly from the straight lines. Therefore, we assume that point K (Fig. 7) of the
94
F . L . Litvin et al. Zs
I
/FI2
I
A1
Aq /
/
Addendum l i n e of t h e face gear
t~
~g
S
r'ps
a)
I
/
$
,/3
,M 1
/
Fps
b) Fig. 6. Intersectionof shaper tooth surface and axis of meshing i-1 by planes FI, and FI:. intersection of the tangents to the profiles is the point of intersection of the real cross-section profiles of the face gear. (5) Using the consideration discussed above, we are able now to derive the equations for the determination of the outer radius R2 of the face gear for the zone of pointing. Step I: Vector equation (Fig. 7) O~A + ANI + N t K = OsK
(13)
t a n ~ 2Pdrm - -
(14)
yields that [4]: where
0o~ is represented by equation (4).
N~
_ 0o~
Meshing of orthogonal offset face-geardrive
95
S t e p 2: We consider that point P~ belongs to the pitch cylinder of the shaper [Fig. 6(a)], and the location of P2 with respect to P~ is determined with segments AI and Aq [fig. 6(a)]. Drawings of Figs 6(a), 6(b) and 7 yield
Aq = O~P: - O~P, =
rb~
rb~
cos~
cos~0
AI=
_
N~ / c o s a zo_-- cos ~']
2/'~\
cos~
(15)
/
Aq tan y~
S t e p 3: The location of plane H2 is determined with parameter
(16)
L2 [Fig. 6(a)], where
L2 = rp + Aq = Nscos ~o tan 7s 2PdCOS~ tan y~
(17)
where tan 7~ = 1/m2s. The outer radius R2 of the face gear (Fig. 8) is represented as
R: =,j-k-;+ L~
(18)
Ys
¢
rm P
=0t
A
K
/ P2 2a Fig. 7. For the derivation of determination of angle ~t. M M T 33/I-2
D
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F.L. Litvinet al. Xh
L2
oh,o
Zs,Zh E
L1
Fig. 8. Limitationsof the face gear.
4.6. Avoidance of undercutting of face gear Undercutting can be avoided if the face-gear tooth surface is free of singularities. This can be achieved by limiting of the inner radius of the face gear. An effective approach for the determination of singularities of the generated surface is proposed in [2] and [4]. The approach is based on application of the following equations (19)
¥~ss) + ¥~s2) = 0
0fdu
0f 00ks 0fd4,s
OUs dt + dOks dt t- 0dpl d t = 0
(20)
Vectors of equation (19) are represented in coordinate system Ss; v~~is the velocity of a contact point that moves over the tooth surface of the shaper; v~~21is the relative velocity of a shaper point with respect to the face gear. Equation (20) is the differentiated equation of meshing that was represented by equation (8). Equations (19) and (20) represent a system of four linear equations in terms of two unknowns: dus/dt and dOks/dt considering dcks/dt as chosen. The system has a certain solution for the unknowns if the matrix
A =
0r~ 0r~ ~u~ OOks
of
of
0us
00k~
v~21 "] ---~
of d4, Oq~s dt
J
(21)
has the rank r = 2. The yields four determinants of the third order that are equal to zero. Our invstigation shows that the equality of two of them to zero is equivalent to the existence of
Meshing of orthogonal offset face-geardrive
97
the equation of meshing. Therefore, it is necessaey to use one of the following two equations
A2
A2 ~--
~Xs OXs 8u~ 80,
v ,.~
Oz~
8z,
v~s2~
f.,
)co,, f . , ~
Is2)
OYs ~Y~
~,.2~
~u~
80k~
- ,.s
az~ 8u~
,~z~ ~0~,~
v~_~2, -
f~,
)Co. f , . - ~
= 0
(22)
= 0
(23)
Here f,,, J~. and f~, are the partial derivatives of the equation of meshing taken with respect to us, 0,s amd ~bs, respectively, x~, ys, and z~ are the coordinates of a point of the shaper surface Es in coordinate system &; (v!?fI, vlr~I, v~]21) are the components of the relative velocity in the same coordinate system. Either equation (22) or (23) yields the same following relation
F(u~, 0k~, ~bs) = 0
(24)
We point out that the same parameters are related by the equation of meshing (8). We may now determine the set of singular points on the face gear tooth surface using the following equations. r2 = r2(us, 0,~, q~), f l u , 0,~, ~ ) = O,
F(u~, 0,.~, q~) = 0,
(k = 7, B)
(25)
Equation (25) enables us to determine the limiting line Lg of the face-gear tooth surface £2 to avoid undercutting. Similarly, we can determine the limiting line Ls of the shaper tooth surface Z~ (Fig. 9) using the equations r~ = rdu~, OkO, f(u,~, O,s, (a,,) = O,
F(U~, Ok~, (as) = O, (k = 7, fl)
(26)
Line L~ contains regular points of surface £~, but generates singular points on surface £2. Knowing one of the two limiting lines, say L,, we may also determine Ls using a coordinate transformation from, & to $2. Due to asymmetry of the face-gear tooth surfaces, both surfaces have to be checked for undercutting. The dimensions of the blank that is used for manufacturing of the face gear are determined using the larger value of the inner radius R~ (Fig. 8) that is determined from conditions
L
Ls Fig. 9. Limitingline L~on the shaper surface.
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F. L. Litvin et al.
A
surface
surface
Fig. 10. Offset face-gear tooth (cross-sections of the working and fillet surface).
of undercutting for the generating profiles 7 and ft. The research results show that, for a positive offset (E > 0), the critical shaper surface that will cause undercutting in the surface with cross-section as curve fl-fl, and it is the opposite for negative offset (E < 0). The point of intersection of line L (Fig. 10) with the addendum cylinder of the shaper is critical for undercutting. The shaper parameter 0ks that corresponds to the addendum is determined by the equation 0* =
rbs
,
(27)
where r,s and rbs are the radii of the addendum and the base cylinder of the shaper. Substituting the value of 0~ into equation (3), we obtain the coordinates (x*, y*, z*) of the point of intersection of the limiting line with the addendum cylinder of the shaper. Knowing these values, the limiting inner radius of the face gear (R,) and the limiting length of the shaper (L,) (Fig. 8) are determined as
R, = ~
+ y~
L, = z,
(28) (29)
A computer program was developed for the numerical solution of the limiting value L~. The solution of non-linear equations represented above requires an initial guess that can be obtained by solving the following non-linear equation for ~ks sin 2 ~ks + rn~O~ ~ + sin ~k~-T- O~ cos ~ks cos3 G~ = 0
(30)
where Oh is the value of shaper surface parameter at the addendum cylinder, and ~ks is represented by equation ~=~s+_(O~+Oo~) k=?,/~ (31)
Meshing of orthogonal offset face-geardrive
99
5. COMPUTERIZED SIMULATION OF MESHING AND CONTACT The procedure of simulation of meshing is directed at the determination of transmission errors and the shift of the contact path caused by misalignment. The developed computer program is based on the continuous tangency of interacting surfaces being in point contact [2] and [4]. Point tangency of surfaces is achieved due to crowning of the pinion tooth surface (see Section 2). 5. I. Bearing contact The bearing contact on the face-gear tooth surface is formed as a set of instantaneous contact ellipses. The center of such an ellipse is the instantaneous point of contact. The determination of axes of the instantaneous contact ellipse and its orientation is based on the approach developed in Refs [2, 4]. The localization of the bearing contact is based on the following considerations: (i) the number of teeth of the pinion is the same as the shaper, (ii) the localization of the bearing contact is achieved by the modification of the pinion tooth surface. 5.2. Results of TCA (tooth contact analysis) Computerized simulation of meshing and contact was accomplished for a gear drive with a crowned pinion tooth surface and the following design parameters: Ns = 20, N~ = 20, N2 = 100, E = 2 (in.), 7 = 90°, a = 0.03, P~ = 10. Misalignment of the face-gear drive was simulated by the change of the center distance E and the shaft angle 7. The results of TCA are shown in Figs 11, 12 and 13. One of the benefits of the new approach with the crowned pinion is that the desired location of the bearing contact on the face-gear tooth surface may be obtained just by the axial displacement of the modified pinion. 5.3. Contact path The contact path on the face-gear tooth surface can be determined by the vector function r2(0s(~b.;), ~b~(gb~)),where 0s and ~s are the same parameters, and ~b~is the face-gear angle of rotation in progress of meshing with the pinion. By applying the driving pinion with the same number of teeth as the generating shaper, but with the modified tooth surface, the contact path is substantially stabilized. This is a big advantage in comparison with the approach when the localization of being contact is accomplished by applying different numbers of teeth of the driving pinion and generating shaper.
Fig. 11. Contact path without misalignment.
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F.L. Litvin et al.
r
! ! !
!
/ /
f
/
J
f
Fig. 12. Contact path with applied misalignment (change of center distance AE = -0.05 ram). 5.4. Transmission errors
The transmission errors are represented by the equation A ' NI , q~2 = ~ - N2 (ok, - ~b*)
(32)
In the equation above, we designate with qS* the value of q~f that corresponds to q~* = 0, which is the initial value of ~b~ (the position error). The value of rk* can be determined from the numerical function ~bf(~b~)~.,=0. Linear function N ~ / N f f q b f - qb*) represents the theoretical angle of rotation of the face gear as a function of ~bf, the pinion angle of rotation. The results of transmission errors, for the case when the pinion with modified tooth surface is applied, show that the magnitudes of transmission errors are not zero, but they do not exceed the values of 2-3 arc seconds in the case of a misaligned gear drive. This means that the face-gear drive is not sensitive to misalignment if the localization of bearing contact is performed by application of an involute pinion with modified
Fig. 13. Contact path with applied misalignment (change of crossing angle A7 = -0.05").
I01
Meshing of orthogonal offset face-geardrive i
(arc see)
a)
O
I
I
c~ t
l -3
l
l
l
-2
- 1
0
arc
sec)
I
-1
I
1
i 2
3
I
I
I
I
0
1
2
3
I
I
I
I
I
-1
0
1
2
3
=
C a(deg)
I
b)
o
7 I
CQ
I
I
-2
-3
C2 (deg)
i
At2 (arc sec)
c)
o
7 i
,~ -3
I -2
Fig. 14. Transmission errors of face-gear drive: (a) no misalignment, (b) A7 = - 0.05°.
:
C z (deg) AE=-0.05
mm, (c)
surface and the same number of teeth as the generating shaper. This is an advantage of face-gear drives in comparison with other gear drives that are used for transformation of power between crossed axes. The transmission errors (for one cycle of meshing) are represented in Figs 14(a), 14(b) and 14(c) for an ideal case (no misalignments) and the case when misalignments exist. 6. CONCLUSIONS (1) Equations for analysis and design of an offset face-gear drive with a spur involute pinion have been derived. (2) Localization of bearing contact by application of a pinion with crowned surface has been achieved.
102
F.L. Litvin et al.
(3) A simplified a p p r o a c h for the design o f drives with face gears free o f u n d e r c u t t i n g a n d p o i n t i n g has been represented. (4) C o m p u t e r i z e d s i m u l a t i o n o f meshing a n d c o n t a c t o f the offset face-gear drive has been accomplished. Acknowledgments--The authors express their deep gratitude to the McDonnell Douglas Helicopter Systems, and personally to Mr Robert J. King, for the financial support of this research.
1. 2. 3. 4. 5.
REFERENCES Davidov, Y. S., Non-Involute Gears. Mashgis, 1950 (in Russian). Litvin, F. L., Theory of Gearing. Nauka, 1968 (in Russian). Litvin, F. L., Wang, J.-C., Chen, Y.-J., Bossier, R. B., Heath, G. and Lewicki, D. J., A.G.M.A. Paper, 92FTM, 1992. Litvin, F. L., Gear Geometry and Applied Theory. Prentice Hall, Englewood Cliffs, N J, 1994. Basstein, G. and Sijtstra, A., A.G.M.A. Technical Paper, 93FTM7, October 1993.