A three-dimensional gear system between crossing and intersecting axes is considered under the assumption that the first derivative of the transmission ratio is zero in the vicinity of the point of contact. The following are obtained; (1) an equation that relates the normal curvatures of the tooth surfaces in the section that passes through the vector of relative velocity; (2) a relation between the principal curvatures and principal directions of the two tooth surfaces; and (3) new formulas for determining the reduced curvatures of the surfaces

Litvin, F. L.

English

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y
RELATIONSHIPS BETWEEN THE CURVATURES OF TOOTH SURFACES
IN THREE-DIMENSICNAL GEAR SYSTEMS
F. L. Li}vin
(NASA-TM-7513C)	 FE,LAIICNSHIPS BETWEEN THE	 N77-30475
CUEVAIURES CE TCCIH SURFACES IN
THREE-01MENSICNAL GEAF SYSTEMS (Kdnne[ (Leo)
,.	 Al^soc idtes)	 14 p HL A02 /MF A01	 CSCL 131	 UnclasG3/37 41116
Translation of "Die Beziehungen zwischen den Krummungen
der Zahnoberflachen bei raumlichen Verzahnungen," Zeitschrift
fur Arigewandte Mathematik and Mechanik, Vol. 49, No. 11,
November 1969, pp. 685-690
1
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INI'U1 BRAiH:N .V
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-°"	 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON, D.C. 20546
	 JULY 1977
N
1.	 Report No.
NASA TM 751L30
2.	 Government Accession No. 3.	 Recipient's Catalog No.
4.	 Title and Subtitle
RELATIONSHIPS BETWEEN THE CURVATURES OF
S.	 Report Dab
July 1977
6,	 Performing Organization CodeTOOTH SURFACES IN THREE-DIMENSIONAL
GEAR SySTEMS
7.	 Authors) 8.	 Performing Organization Report No.
F. L. Litvin
10.	 Work Unit No.Institute for Precision Mechanics and
Optics (Leningrad)
11.	 Contract or Grant No.
NAS	 _9.	 Performing Organization Name and Address
13.	 Type of Report and Period CoveredLeo Kanner Associates
Redwood City, California 94063
Translation12.	 Sponsoring Agency Name and Address
National Aeronautics and Space Adminis-
tration, Washington, D.C. 20546
14.	 Sponsoring Agency Coda
15,	 Supplementary Notes
Translation of "Die Beziehungen zwischen den Krummungen der
Zahnoberflachen bei raumlichen Verzahnungen," Zeitschrift
fur Angewandte Mathematik and Mee:hanik, Vol.
	 49, No. 11,
November 1969, pp.	 685- 690.
16,	 A6strnct
A three-dimensional gear system between crossing and inter-
secting axes is considered under the assumption that the
first derivative of the transmission ratio is zero in the
vicinity of the point of contact.	 The following are ob-
tained in the article:	 (a) an equation that relates the
normal curvatures of the tooth surfaces in the section that
passes through the vector of relative velocity; 	 (b) a rela-
tion between the pri.ncipal curvatures and principal direc-
tions of the two tooth surfaces;	 (c) new formulas for deter-
mining the reduced curvatures of the surfaces.
17.	 Key Words (SCICeted by Author(s)) 18,	 Distribution Statement
Unclassified-Unlimited
19. 	 security Clossif. (of this report) 20,	 Security Class[(. (of this page) 21• No. of Pape. 22.	 Price
Unclassified Unclassified
STANDARD TITLE PAGE
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RELATIONSHIPS BETWEEN THE CURVATURES OF TOOTH SURFACES
IN THREE-DIMENSIONAL GEAR SYSTEMS
F. L. Litvin
In stitute for Precision Mechanics and Optics (Leningrad)
1. Initial Equations
The surfaces of teeth are, in relative motion, mutually en-
veloping surfaces, and the following equations [1] apply at
their points of contact:
r(') —' r(25• C(1) — e(2); 	V(12) = 0; i = 1, 2
Here, r (i) Is the radius vector of surface E i , and e (i) is the
unit vect(?j normal to this surface in the spatially fixed coordi-
nate system; v (12) the velocity vector of point M, fixed with
respect to surface E l , relative to the corresponding point fixed
with respect to the surface E2.
Given that we have a three-dimensional gear system that
transmits motion between crossing shafts. The following equations
Ell apply in the vicinity of the point of contact of the tooth
surfaces:
I2
	 jIl d (gip) 00 2 ))	 0;	 (00)	 0
di
For w (l)	 const and 1(2) = 0, d(1 12 )/dt	 0, where 112
	
w(1)/w(2)
is instantaneous transmission ratio.
Moreover,
PY) — 40) 1 VMr
I The reader is referred to E73 for supp2ementary material.
ir Numbers in the margin indicate pagination in'the foreign text.
_'AIn
IHere vet) is the velocity of a point transmitting motion, in
conjunction with the surface; vTU) is the velocit y of the point
relative to the surface of the tooth.
AnalogouslY,
(1A).,..	 ,	 ,	 iuft)=.i'm-L;i;^+).=cu(ry'X.e(p.^erq.;
.q:h,
The vector of relative velocity is calculated from
Q 6)'I ` )) ) t ' i 1^v(i21 	 w^i2j X r(})_._
 
It X 1i(12)	 tniL ' ^., o,P2f =lu( t.> —a^^.lr4.
Here r (1) is the radius vector of the point of contact on the
tooth surfaces (the origin 0 of radius vector r (1) lies on the
line of application of the vector a (l) ); R is the radius vector
of an arbitrary point on the line of application of vector w(z)
originating from point 0.
If we transform equations (1,2) and (1.5), we obtain
0 -(J)
))(aF_ 2er+) + 1)( 2)^ - ism ^ e'm .j cuU -'> X e(1) ; i;i) v(+a) -f- ((,(I), 
eh) .t (t2)) } (c(t> m(}e ) } 0)) —0 ,.
The curvature of the enveloped surface in the normal section is
determined using the following equation [2]:
r(t) _
	 ia+ ) V(rt)
3 	
i
l
/686
	
p11G
i
The equations of the enveloping surfaces are usually signif-
icantly more complex than the equations of the enveloped sur-
face. For this reason, it is desirable to express the curvature
y
of the enveloping surface in the normal section by means cf the
curvature of the enveloped surface and the parameters of relative
motion 137:
_... t('1.iQ)	 -^i,iA ..I	 ^p(tc)	 P^ ^titl 'f'
x
2
•	 ti
3T'.c
ly
I
In equations (1.7) and (1.8), the sign of the curvature
is positive if the surface's radius of curvature and the normal
unit vector are in the same direction. The vectors er'- ) and v(i)
are collinear in the principal direction of the surface.
I
2. The Relationship Between the Curvatures cf the Mutually En-
veloped Surftw in the Common Normal °ection Passing Through
the Vector v.
L. V. I{orostelev [4] and A. M. Pavlov [5] suggested deter-
mining the curvatures of the mutually enveloping surfaces of the
	 ((
teeth in the normal section that passes through the vector of
relative velocity v (12)	 In this section, the vectors v (1) , vr(,2)
i
and v (12) are collinear.
We now seek a relationship between the normal: curvatures in
the section that passes through vector v(12) . The vectors er1),
vri), v (12) and m (12) xe (1) lie in the plane that is tangent to
the surfaces of the teeth at their point of contact. We resolve
these vectors in the two directions containing the unit vectors
it and in corresponding to the direction of v (12) and the
direction perpendicular to it.
Curvature in the normal section passing through i t is de-
termined by the equation
,,pl(2,1)
	 xirl -
	 umi	 (' _ ) ) ^)
a
On the basis of equations (1.6)-(2.1) we obtain
f?,2)	 (xQ)	 x^e1) u^ir) _ xiA rlia) _ (oap '), CM,r) = U
(2.3)	 ((p(l), cu(12), il) — x1 11 'ri `11 1)(1) + u1'') (cu( ' ) , ( .(q , ii)	 (cU) cup:>, r (,r)) _ 0
(
By factoring v (l) out of equations (2.2) and (2.3), we obtain
I
(2.9)	 ((i) — xi-7) ^uir_>(y,U> au), ) — (u,0.), (;U>>'!'^'})I -
_ y(l) X(•.')(1,14' )9 — (a4) + R(P) 1,( 12)(W	e(l) i,) -' ((002), CM, 1,)	 0.
Relation (2.4) connects the curvatures of the tooth surfaces in
the normal section passing through vector v (12) . This relation
can be viewed as a generalization of the Euler-Savary equation
for three-dimensional gear systems. We know that this equation
relates the tooth profiles in planar gear systems when the gears
of the mechanism have parallel axes of rotation. The normal
curvature K (2) of the enveloping surface can be determined from
relation (2. 4) if the normal curvatures K(1 ) of the enveloped sur-
face, the coordinates of the point of contact and the parameters
of relative motion and transmitted motion are considered known..
In the case in which a motion is transmitted between parallel
or intersecting gear axes with the aid of a mechanism, the rela-
tive motion can be reduced to a rotation about the instantaneous
axis. For the point of contact of the tooth surfaces that lies
`a
on the instantaneous axis of rotation, we have
pr
N
where unit vector it coincides, in direction, with vector v(l) .	 /687rt
Relation (2.4) assumes the following form for.the case under
consideration:	 H
((uU-2), CO),.j,y
In planar gear systems (with parallel axes of rotation for the 	 'w
gears), unit vector it coincides in direction with the unit vec-
tor of the tangent to the profiles. In the case of intersecting
axes, unit vector i t can have any direction.
I
j
4	
r,
i	
f
tf
N	
1
3. The Relationships Between Principal Curvatures in the Case of
1
t
kf
Point and Linear Contact Between Surfaces
} 1 y
Let us first consider the case in which the tooth surfaces j
make contact with one another at one point at every instant.
	 We
wish to adopt the convention of calling the totality of points of
contact on the tooth surface the "line of operation." f#
>	 r^ We now determine, for the moving point of contact, a relation
between the direction of the tangent to the line of operation, the
principal curvatures and the principal
directions of the surfaces in contact
f
_	 tr	 with one another.
	 Let us designate theIB	
unit vectors in the principal directions
_.,	 on surface E 1 as iI and III ; the unit I^
vectors in the principal directions on
1
"(
d	 surface E 2 as iIII and i IV ; the angle j
it	 between directions I and III (Fig. 1)
as a; the principal curvatures of sur-
face E l as KI and K
	 and the principal
curvatures of surface E 2
 as KIII and K IV . Let us turn to equation
(1.6) and resolve the vectors in this equation in the directions
of unit vectors 1 I and il =.	 We observe (Fig. 1) that
uiz1' —Ili4I[I COS Q - 1?,IV Sill Qi--. V(,%	 U( L)11' Sill U + 1) (11 . Cos Qi.ri UI J
C(2)	 Sill Q. `^'. Cir\. CUS Q' CfII[ WS Q — 6V Sill Qr	 [rill .-
f-: Let us also take into consideration that the following appliesti
t	
t
o he principal directions:
I
Ui'; ) = - xl 0i'r i ;	 In —. ]	 (l = I, II)	 a
ni = 2	 (i = III, IV) .
a
r
1
f	 1	 5
j
sAfter several trans formations, we obtain a system of linear
equations of the following ypc,:
(111 eg1'+ (112
 bet 	 L1
(121 ^+1.+ 1122 b2 = G4
(3.1)	 l On `kl +Onf =b9,
1111 _ 7 H_.+ —53 ^(xlit ' H1Y) -^' (x11I —: X1V) CUs 2 uj i
1119.=. (Mill — xA')sill 2 U;
(192 ^ x1I ^- 2 RPM + xlv) — kix — x1V) cos 2 uJ	
_
, 1199 „(e(1)^ W(i'iIf EII) — xIY 11I12)
` b l = (0), [U(11), 11)	 :.j V( 2) ((MITI _I_ x1P) +(x111 — x 12) COs 2 6^ — ^UfL-1 (x171	 H1V) 5111 2 ?;
b^	 ^g(i)1 (v(I2) 
ill) — 1 Uf1Y) (x llI — x1V) FI II $ O — (/4_ jY)2 I	 2 [(Hitt I xlv) — (x^Ir — xlv) cos2o]i
b^	
,64) (((0(2)	 Jill)) —. (&1) X P(.2)))
In the case of point contact of the
must possess certain values. For this r
linear equations must be consistent. As
gebra, this requires that
1111 1119(312) 	
1121 1122
tl9l U
surfaces, v (l) and v(l)rII /688
Cason, the system of
is known from linear al-
Ll
L, =U,
b9
D9
By solving expression (3.2), we obtain a relationship of the fol-
lowing type (the parameters of motion are assumed to be given):
(3.3)
	 i ( rr, xn, Hill, iv,'a) = i) r
Relation (3.3) applies for arbitrary directions of the tan-
gent to the line of operation (for arbitrary ratio v' :v r1 ) . The
following problem frequently arises: It is necessary to determine
KIII , KIV and o, it being assumed that K  and K II are known at the
point of contact between the surfaces. We can make use of system
of equations (3.1) for this if we initially specify the ratio
v (1) : v (1)	 once we have elimLnated v (l) and v (1) we obtain a
rI	 rII'	 rI	 rII,
system of two independent equations that relate KIII and K_V and
o. The missing third equation is obtained if the dimensions of
the deformed contact surfaces are given, which we know to be
functions of the "magnitude" of osculation of the surfaces, their
principle curvatures and angle o [6].
In the case of linear contact between the tooth surfaces,
the system of linear equations must be valid for various direc-
tions of v (1) . The system must be consistent, but the values
r
VU) and vr ( 1)  become indeterminate. This is posFible if the
rank of the matrix
a,, a,, b,
az, ^rdz IrLII
is equal to one.
From this it follows that
rht Lr= !—,12 = rr.D 	 azz _ Lz
,: 1
If we note that a21 = a12, we can write the derived equations in
the form
Only two independent
(3.4), since it must
b1=11^,—ur
rrJ1 — u l x _ xl_n _ b1
dal ` b,
as _ ba
equations can be obtained from equations
be taken into consideration that
r41- T ii" > (8u	 1'^	 n:u — 11(1 1") (8a — ViI'P a._.
	
By making use of (3.5) and the equations	 r.
	
ae1	 a2a	 (lag
we obtain three relations of the following type for determining
KIII' KIV and o:
2r(3,7)	 Can 2 a
(3.8)	 xu1 1 xn 	 xl'-1- XI1 -1- S,
(• \
	
x1—X11+G3 9 Kill ' — rn —	 co1 '2 a
where	 /689
F	 (III t1:12
G	 Ir	 a
G=
ba + u^ia) Gat '^' tiS [a) aaa
S l,a + U^IY) fla l '^' IT 03-	 -	 1
4. Reduced Normal Curvature	
7
1
Reduced normal curvature refers to the difference between
t	 the normal curvatures of the two surfaces El and E 2 in their	 j
8
icommon section. We shall designate it as
We now seek an expression for reduced normal curvature. For
this purpose we designate the angle formed by the vector v (2) and
unit vector i I in the principal direction on surface E l as q. As
in the preceding section, we resolve the vectors in equation (1.6)
in the principal directions containing unit vectors i = and i21.
We take into consideration here the fact that
U, r))( + p(us)
After several transformations, we obtain a system of two
linear equations of the following type:
j1,3)	 U31 ^ F ((&z h2 = Ira
a45. Sj + U49 Sa . 	6d r
where	 q
'rT = Or)I , b2 = pi l)f r .(141 =. Sill q, (142 =	 cus q;
b4 = Uii" ) cos q— IV) sin q
The coefficients of the unknowns, a31 and a32 , and the free term
b 3 have already been discussed in Section 3•
y
If we make use of equations (4 . 3), we obtain
Vol = b, _F (4sz (^(12)
	 u^ r '> tan (l).
(d,A)	 U,( -I- Uze tan q. 
Consequently,
{9b)	 Uri +1)(12) = Ir
r 	U^ruf)? f <l:ix!riu")
f Uar	 On tan (I
We thus obtain the following for normal curvature:
iI
'.
?At
a
i	 +;F
ra
S^
1#
q
i
as
di
To transform expression (4.6), we make use of the following
relations:
Dr^ = Orl^  + x[12) r V( )I = (I/ili + p^12
))
)) tali q:
(4.7)	 9; I) — B; I) + (W(12), g(1), iI) _ — xi ^:I  f (W(12), L'(1), il);
..	
III7 II rII ^" (	 II)
We also note that
(co02), e(1) ,
 i l) _ =031 + xI bS12));
..:Cb.. " :". „ ..	
_	
([N(12Sr L'(I) , 
_ II!) _ — (a39 t X11 Vi 'P)
If we substitute relations (4.7) and (4.8) into expression (4.6)
and transform the latter, we obtain
(4 ,9) 	 (all + ar a tan (1) 3 u1s2 qf13 + a31 VVI) f (132 itt'i
The reduced curvature reaches extreme values in two mutually
perpendicular directions [7]:
a) in the direction coinciding with the tangent T to the
line of contact of surfaces T and T
b) in the direction perpendicu
In the first direction, K(p)
Because of this, we obtain
II:II(4.111)	 Lan q,
l	 1132
1	 2'
Lar to T.
K (p) = 0 since K (1) = K(2)
min
.-_-(fU (I ), (:(1Jr 111) ^^' fiI I'll
For the direction m perpendicular to tangent T,
U"(4.11) too qn:_— Cot q,=...a31
10
1A
b -.
^j.
:_ter substituting (4.11) into (4.9), we obtain
	
(4. 12) 	XO') _ _	 (IJI "1 IIJ9. ..	 ...MAX
In the case in which a rotary motion is transmitted between
parallel or intersecting axes and the point of contact lies on
the instantaneous axis of rotation,
(cuu-) X ^n)lt( 1.13) xinn< 
_ — (w(i)^(1?, rt,,) (i _ 1, 1) .fi
Let us now consider the case of point contact between sur-
faces E1 and E 2 . It is desirable here to write the expression for
reduced curvature in the form of Euler's equation, which relates
the normal curvature with the principal curvatures of the surface:
	
(4. 14)
	7t(T) o p(l) — g(''^') a Y.I COs- 11 + xll Sill- (I — h',1[ c()42 (Q — (7) + my sin" (q — R)1
Here q, as before, is in the plane of contact of the angle be-
tween the principal direction containing unit vector i I and the
direction in which ic (p) is considered.
The reduced curvature also reaches extreme values in two mu- p
tually perpendicular directions in the case of point contact
between the surfaces [7].	 It is apparent that K (p) = Kmax if
dK (p) /dq = 0.
	 From this it follows that
(9.n) Lin Zq ° •	 In	 %I%,) sin 1 0 ^ a,, -_
(xit	 Hl) -1	 (xlu — xlv) cos ? a	 (41 — (1"
j
r-
11 t
C
r
k
^f
e
REFERENCES
7
1. Litvin, F. L.,	 "Investigation of meshing conditions in the
neighborhood of a given contact point," Trudy seminara po
teorii mashin i mekhanizmov 100 [Proceeding of the Seminar
on the Theory of Machines and Mechanisms], Nauka Press, 1964.
_}
2. Rashevskiy, P. K., Differentsial ' naya geometrin [Differential
Geometry], Gostekhizdat, 195 • ^,!
3. Litvin, F. L., "Application of a kinematic method to determine
the relationship between the curvatures of mutually enveloped
surfaces and conditions for the absence of tooth under-
cutting," Trudy seminars. po teorii mashin i mekhanizmov 103
[Proceedings of the Seminar on the Theory of Machines and
Mechanisms], Nauka Press, 1964.
4. Korostelev, L.	 I.,	 "Curvature of mutually enveloped surfaces
in three-dimensional gear-tooth systems," in the collection+
Teoriya peredach v mashinakh [Theory of Gear Systems in
Machines , Mashgit, 19 3.
"S
5. Pavlov, A. M., "Determination of the curvature of mutually en-
veloped surfaces in three-dimensional gear-tooth systems,"
Trudy LPI im. Kalinina 1960/211.
6. Lur'ye, A. I., Prostranstvennyye zadachi teorii uprugosti
[Three-Dimensional Problems in Elasticity Theory v
Gost-ekhizdat,	 1955.
7. Litvin, F. L., Teoriya zubch -tykh zatsepleniy [Theory of
Meshing-Gear Systems , 2nd . ;dition, chief editorial staff
for physics and mathematics lit., Nauka Press, Moscow, 1968.
i
x
l


Relationships between the curvatures of tooth surfaces in three-dimensional gear systems

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