The curvature of conjugate profiles in points of contact

The curvature of conjugate profiles in points of contact

Citation for published version (APA):

Overdijk, D. A. (1989). The curvature of conjugate profiles in points of contact. (Memorandum COSOR; Vol. 8918). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1989

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Department of Mathematics and Computing Science

Memorandum COSOR 89-18 The curvature of conjugate profiles

in points of contact D.A. Overdijk

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513

5600 MB Eindhoven The Netherlands

Eindhoven, July 1989 The Netherlands

CONTACT

D.A. Overdijk

Eindhoven University of Technology

Depanment of Mathematics and Computing Science P.O. Box 513

5600 MB Eindhoven The Netherlands

o.

Introduction and summary

Consider a point of contact on conjugate tooth profiles of the pinion and the gear in a gear pair. The following quantities with respect to the curvature of the conjugate profiles in the point of contact are of practical interest

the principal curvatures of the conjugate profiles in points of contact;

the principal directions of the conjugate profiles in points of contact, in particular the angle between corresponding principal directions on the conjugate profiles;

the direction of the curve of contact in points of contact.

The principal curvatures at a point on a surface are the real eigenvalues of a symmetric (linear) operator on the tangent space of the surface at the point under consideration. The principal direc-tions at a point on a surface are the mutually perpendicular eigenvectors of this symmetric opera-tor. In Section 1, we give a survey of the theory of the curvature of surfaces, using elementary linear algebra. The results can be found in books on differential geometry, e.g. [1]. [2], [5], The conjugate profiles in a gear pair are in contact along a curve of contact. In Section 1 we discuss the curvature of two bodies in the points on their curve of contact. We present formulas for the angle between the corresponding prinCipal directions of the boundary surfaces of the bodies on the curve of contact and for the angle between the direction of the curve of contact and the princi-pal direction of one of the boundary surfaces corresponding to its smaller principrinci-pal curvature.

Consider a point on the curve of contact of two conjugate tooth profiles in a gear pair. The tangent spaces at this point of contact on the conjugate profiles coincide. Hence. there are two symmetric operators Bland B2 • both acting on the common tangent space of the conjugate profiles at the point of contact. The real eigenvalues of B 1 (B 2) are the principal curvatures of profile 1(2) and the corresponding eigenvectors are in the direction of the principal directions of profile 1 (2). The symmetric operator B

=

B 1 + B 2 is non-negative and has an eigenvalue zero. The eigenvector of B with eigenvalue zero is in the direction of the curve of contact. The positive eigenvalue of B is the relative curvature of the conjugate profiles in the point of contact under consideration. The above considerations constitute the starting point of a new and efficient algo-rithm for the calculation of the curvature of conjugate profiles in points of contact for the general gear pair. We describe this algorithm in Section 2.

1. The curvature of surfaces

Let S be a surface in three-dimensional space. The linear space of vectors from a point P on S parallel to the tangent plane of Sat P is referred to as the tangent space T(P) of Sat P. The unit

-?

nonnal vector on S at the point P is denoted by n(P). In every point on S there exist two unit

nor--?

mal vectors, and therefore we make a choise such that the vector nCP) depends continuously on the point P on S.

-? -?

Let 1 E T(P) be a unit tangent vector of Sat P and let C be a curve on S through P such that t is a tangent vector of C at P. Take s for the arc length parameter on C such that P corresponds to

-?

s

=

0 and such that

s

is increasing in the direction of I. We consider five vector fields on the curve C: -? I(S) = -? 18(S)

=

-? (1.1) n(s)= -? nsCs) = -? n*(s)

=

unit tangent vector of C at the point corresponding to

s

and in the direction of increasing s;

-?

d t (s)/ds;

unit normal vector on S at the point on C corresponding to s; dn(s)/ds;

principal normal on C at the point corresponding to

s.

The curvature IC(S) of C at the point on C corresponding to s, satisfies the Frenet formula

-? -?

(L2) t 8(S) = K(S) n*(s) .

Later in this section we shall show, that there exists a symmetric (linear) operator 8 on the tangent space TCP) of Sat P, such that

-+ -?

(1.3) ns(O)

=

-81 .

-? -?

Differentiation of the relation ( t (s), n (s»

=

0 yields

-+ -+ - + - +

(ts(s), n(s»

+

(t(s), nsCs»=O Use the Frenel fonnula (1.2) to obtain

-+ -+ - + - +

(1.4) K(s)(n*(s), n(s» =

-<

1 (s), ns(s» . Substitution of s

=

0 in (1.4) yields

---+ -+ ---+-+

(1.5) te(P;C)(n*(P;C),n(P»=(t,Br) ,

where B is the symmetric operator on T(P) in (1.3) and te(P; C)

=

curvature of C at P , ---+

n'"

(P ; C)

=

principal normal on C at P , ---+ n(P)

=

normal on Sat P . ---+ ---+

Let 8(P; C) be the angle between n*(P;C) and n(P). From (1.5) we get

-+ ---+

(1.6) te(P; C)

=

(t, B r)/cos(8(P;

C» .

-+ ---+

The plane through P parallel to the tangent vector t and the normal vector n(P) is denoted by

---+ ---+

N (P; t). The curve of intersection of the plane N (P; t) and the surface S is referred to as the

---+

normal section through P in the direction t. The curvature of the normal section at P in the

---+ ---+

direction t is called the normal curvature te(P; t) of the surface S at the point P in the direction

---+ -+

t. If the normal vector n(P) is in the direction of the center of curvature of the normal section,

---+ -+

then the normal curvature te(P; 1) is positive, otherwise negative. If the normal curvature te(P; t)

---+

is positive, then n(P) and the principal normal on the normal section at P coincide, otherwise they have opposite directions. Hence, by use of (1.6), we get

-+ - + - +

(1.7) te(P; t)

=

(t • B t) .

The linear operator Bon T(P) is symmetric. Hence, there exist two mutually perpendicular unit

-+ -+

vectors t 1 (P) e T(P) and 1 2(P) e T (P) such that

---+ ---+ -+

(1.8) tet(P) :=K(P; (1):5; K(P; t):5; K(P; t2)=: K2(P) .

The normal curvatures Kt(P) and te2(P) in (1.8) are said to be the principal curvatures of the

SUT----+ ---+

face S at the point P, and the corresponding tangent vectors t 1 (P) and t 2(P) the principal direc-tions of the surface S at the point P. The prinCipal curvatures KI (P) and K2 (P) are the real

eigen--+ ---+

values of the symmetric operator Bon T(P). The principal directions t I(P) and t2(P) are the corresponding mutually pcrpendiculareigenvectors of the symmetric operator Bon T(P).

---+

-+ -+ -+ -+ -+

(1.9) t

=

t 1 (P) COS(E>(

t»

+

t 2(P) sin(E>(

t» .

Substitution of (1.9) in (1.7) yields

-+ -+ -+

(1.10) K(P; t)=Kl(P)COS2(E>(t)+ K2(P)sin2(E>(t» .

Equation (1.10) is known as the equation of Euler. The curvature of the surface S at the point P is often described in tenns of

K(P) := K) (P)

+

K2(P). the mean curvature of S at P ,

(1.11) and

A

K(P) := K} (P)K2(P), the total (Gaussian) curvature of S at P Hence,

-K = Kl

+

K2 = trace (B) , A K

=

K} K2

=

det(B) , (1.12) - -2 .. 1-Kl

=

(K - (K - 4K)2 )/2 •

We now discuss the symmetric operator Bon T(P) referred to in (1.3). For ease of calculation we introduce a right-handed cartesian coordinate system. The corresponding coordinates are denoted by (x, y. z)

=:!.

The surface S is specified by the parametric equations, with parameters A and ~,

We suppose that the tangent vectors :!). := d!ldA and !J1 := d!ld~ are linearly independent at every point on S. Hence, the vectors !A. and !J1 are the clements of a basis in the tangent space T(P) of S at P. For ~ E T(P) we write ~

=

<l:!A.

+

~:!)L' and we introduce the (2 x I)-matrix

u

corresponding to the vector!! E T(P)

(1.14)

u

:=[a,~f ,

where [a, ~]T is the transpose of [a, ~]. For!! E T(P), the matrix

u

depends on the

parameteriza-tion (A., ~) of the surface S. Let (A', ~') be another parameterization of the surface S. The parame-ters A and ~ are invertible functions of the parameters A' and

J.l',

say

A

=

A(A', ~') ,

~

=

~(A'. ~') .

The (2x I)-matrices corresponding to the vector!! E T(P) with respect to the parameterizations

(1.15) u=Su',

where the (2x2)-rnatrix S is given by

[

aA aA

1

_ aA' all' S- all all . aA' all' (1. 1 6) A

Let A be a linear operator on the tangent space

rep)

at P on S. We introduce the (2 x 2)-rnatrix A

corresponding to the linear operator A on T(P), such that (1.17) (A!!f=Au,!! E T(P) .

The (2 x 2)-rnatrices corresponding to the linear operator A on T (P) with respect to the

parame-A A

terizations (A, Il) and (A', Il') of S, are denoted by A and A' respectively. One easily verifies

(1.18) A'=S-IAS.

where the (2 x 2)-rnatrix S is given in (1.16). Consider the (2 x 2)-rnatrix Q at P on S

(1.19)

Q

~

[: : ] .

where

E=E(A, Il) :=<!A,!).) • (1.20) F = F (A, Il) := <!A' !11) •

G = G (A, Il) := <!I1' !11) .

The quantities E, F, G in (1.20) are the fundamental coefficients of the first order at the point

peA,

J..l) on

S.

It is easily vcrificd,lhal for !!,! E

rep)

(1.21)

<!!'

y) =

(uf Qv .

Let (A', Il') be another parameterization of the surface S. The corresponding fundamental coefficients of the first order are denoted by E', F', G ' , and

Q'

=

[EI

F'j.

F' G' We have

where the (2 x 2)-matrix S is given in (1.16).

-+

The components of the unit normal vector n(P) on S at P (A, Il) are !!(A, Il). Consider the (2 x2)-matrix R at P on S

(123) R

= [:

:1 ·

where

L = L(A. Il) := (!!, !uJ = -(!!~, !~) ,

0.24) M

=

M (A, Il) := (!!.

!A.iL)

=

-(!!:I.,

! .. ) ,

N = N (A.Il) := (!!, !14l)

=

-(!!Il' ! .. ) .

The quantities L, M, N in (1.24) are the fundamental coefficients of the second order at the point

peA, Il) on S. Let (A'.Il') be another parameterization of the surface S. The corresponding coefficients of the second order are denoted by L', MI, N', and

R'

=

[L'

MIl.

M' N' We have

where the (2 x 2)-matrix S is given in (1.16).

Since (,!!,!!)

=

1. we have !!~ E rep) and !!Il E rep).

Write

(1.26) !!:I.

=

O:!:I.

+

~:!Il '

!!Il

=

'Y:!:I.

+

o:!1l .

It follows from (1.26) and the definitions in (1.20) and (1.24). that

ex = (FM - GL)/(EG - F2) , (1.27) ~

=

(FL - EN)/(EG - F2) •

,,(=(FN -GM)/(EG _F2) •

0= (FM - EN)/(EG - F2) . Now introduce that (2x2)-matrix

(1.28)

B

=- [: :] .

From (1.27. 28) and (1.19). (1.23) we conclude, that (1.29)

B

=

Q-l R .

A

Let

0:.

J.1') be another parameterization of the surface S. The matrix corresponding to Bin (1.28)

A

is denoted by B'. From (1.22). (1.25) and (1.29) we get (1.30)

B'=S-IBS.

A

where the (2x2)-matrix S is given in (1.16). Hence, the matrix B in 0.28) corresponds to

a

linear operator B on the tangent space T(P) at P on S; compare 0.30) and (1.18). We shall show, that B is the symmetric operator on T (P) referred to in (1.3).

For !!'!'. E T(P) we have, by use of(1.21), (1.17), (1.29),

(!!, B!'.) =

(ul

QQ-l

Rv

=

(ulRv

=

(v)TRu

= (!:, B~) . Hence, the operator B is symmetric.

Let C be a curve on S through P with parametric equations

(1.31) :! =:!(s) =:!(A(s), Il(S» .

The parameter s in (1.31) is the arc length parameter on C such that P corresponds to s = O. The

~ ~ ~ ~

components of the vectors n(s)

=

n(A.(s), Il(s» and ns(s)

=

dn(s)/tis, compare (1.1), are !!(s) and !!s(s) respectively. The vector

0.32)

_1.

=

:!). (O)ds dA. (0)

+

:!f1 (0) dll tis (0)

is a unit tangent vector of Cat P, and clearly (1.33)

i

= [ :

(0),

d:

(O)]T •

From (1.26)

(1.34) _!!s(0)

₌

_!!).(O) dA. _{ds (0)}

₊

_!!11(0) dll _{ds (0)}

₌

dA. ~ dA. dJ.1

=

(0: ~(O)

+

Y ds (O»:!l.

+

(y ~(O)

+

0 -;tS(0»:!11 .

A A

(1.35) ns(O)

=

-Bf

In view of (1.17) we conclude, that (1.36) ~(O) =

-B!.. '

which proves that B is the symmetric operator on T(P) referred to in (1.3).

Using (1.12), (1.19), (1.23) and (1.27,28.29). we conclude for the mean curvature K(P) and the

"

total curvature K(P) of Sat P (A, 11)

(1.37) K(A, 11)

=

trace(B)

=

- ( l - 0

=

(EN

+

GL - 2FM)/(EG - F2) ,

~(A,

11) = det(B)

=

del(R)/det(Q) = (LN - M2)/(EG - F2) .

Consider two rigid bodies L 1 and L2 with boundary surfaces S 1 and S 2. The bodies LI and L2 are in contact along a curve C of contact. The inner unit nonna! vector on S 1 and S

z

is denoted

....,. ....,. ....,. ....,.

by n 1 and nz respectively. Hence, the nonna! vectors n 1 and n2 have opposite directions on C,

....,. -+

i.e. n 1

=

-nz on C. We consider a point on the curve C of contact. As a point on S 1 we denote it by P, as a point on S2 by Q. The tangent spaces T 1 (P) and T 2(Q) of S 1 and S2 coincide and we write T(P, Q) := T 1 (P) = T 2(Q). The symmetric operators B 1 (P) and _{B 2(Q)} referred to in (1.3)

....,.

both act on the common tangent space T(P, Q). Let teE T(P, Q) be a unit tangent vector of the

.... -+

curve C of contact at P

=

Q. Since n) (P)

=

-n2(Q), we conclude from (1.3) that .... -+

(1.38) BI(P) 'e=-B2(Q) Ie.

-+ -+

Suppose, there exists a unit vector t E T(P, Q) such that the nonna! curvature K) (P; t) of S I at

....,. - + . . . . , .

P in the direction t is negative. Then the nonna! curvature KZ(Q; t) of S

z

at Q in the direction t

must be positive and since the rigid bodies LI and L

_z

do not penetrate each other, we must have

-+ ....,. 0< l/KZ(Q; t)$; -lIKI(P; t) ,

and therefore (compare (1.7»

-+ -+....,. ....,. KI(P; t)+KZ(Q; t)=(t,(Bt(p)+Bz(Q» r)~O. Hence ....,. -+ -+ -+ (t, B I(P) t)

<

0:::> (t, (B I(P) +Bz(Q» r)~ 0 , and similarly

-+

From the above considerations we may infer, that for t E T(P, Q) we have

(1.39) _{(t,(B 1(P)+B 2(Q»} t)~ 0 . Consider the symmetric operator

on the common tangent space T(P, Q). By virtue of (1.38) and (1.39) we may conclude that

-+

B (P, Q) is non-negative and has an eigenvalue zero. For the unit vector t E T(P, Q) the relative -+

curvature v:AP, Q; t) of the bodies L 1 and L2 at P

=

Q on the curve C of contact in the direction

-+

t is defmed by

-+ -+ -+

(1.41) K,(P, Q; t)=(t,B(P, Q) t) .

-+

The relative curvature K,(P, Q; t) reaches a maximum in the direction perpendicular to the direc-tion of the curve C of contact. This maximum is referred to as the relative curvature K,(P, Q) of the bodies Ll and L2 at P

=

Q on the curve C of contact. The principal curvatures Kll (P) S K12 (P) of Slat P are the real eigenvalues of the symmetric operator B 1 (P) and the prin-cipal curvatures K21 (Q) S K22(Q) of S 2 at Q are the real eigenvalues of the symmetric operator B 2(Q). Hence, the relative curvature K,(P, Q) of the bodies Ll and L2 at P

=

Q on C equals

-

-(1.42) K,(P, Q)

=

K)) (P) + K)2(P) + K2) (Q) + K22(Q)

=

K) (P) + K2(Q) ,

-

-where Kl (P) and K2(Q) are the mean curvatures of S 1 and S 2 at P and Q respectively. Suppose that P is an umbilical point of S ) • i.e.

-+ -+

Let t21(Q) and t22(Q) be unit tangent vectors in T(P, Q) in the principal directions of S2 at Q

-+

corresponding to the principal curvatures K21 (Q) and K22(Q) respectively. We take t 21 (Q) and

-+

t22(Q) as the elements of an orthonormal basis in the common tangent space T(P, Q). The

A A

matrix representations B 1 (P) and B2(Q) of the operators B 1 (P) and B2(Q) have the form

Since B (P, Q) is non-negative and singular, we have A A trace (B I (P)

+

B 2(Q»~ 0 , (l.45) A A det(B I(P)

+

B_2(Q»

=

0 . Hence, (1.46) ~ ~ K'21(Q)

<

K'zz(Q)=> te = tZI(Q) , ~

where teE T(P, Q) is a unit tangent vector of Cat P = Q.

Suppose that Q is an umbilical point of S z' i.e.

~ ~

Let t II (P) and t 12(P) be unit tangent vectors in T(P, Q) in the principal directions of S I at P corresponding to the principal curvatures K'11 (P) and K'12 (P) respectively.

We have, compare (1.46),

K'll (P)

=

-K'2(Q)

(1.48) ~ ~

K'll (P)

<

K'12(P)

=>

t e

=

t 11 (P) .

Now suppose that P and Q are not umbilical points of S I and S 2 respectively, i.e.

~ ~

We take t u(P) and 112(P) as the elements of an orthonormal basis in the common tangent

~ ~

space T(P, Q). Choose -rcJ2

<

S~ 1112 such that the vector t 11 cos(S)

+

t 12 sin(S) is in the principal direction of Sz at Q corresponding to the principal curvature K'21(Q) of S2 at Q. The

A "

matrix representations B 1 (P) and B 2(Q) of the operators B 1 (P) and B 2(Q) have the form

B

,(P)

=

[x~'

(1.50)

A [K'21 cos

2

(S)

+

K'22 sin2(S) ,cos(S) sin(S)(K'21 - K'ZZ)] B2(Q)=

cos(S) sin(S)(K'21 - K'zz) • K'Zl sin2(S)

+

K'22 cos2(S)

A

(1.51)

From (1.51) and (1.42) we conclude lCl1(P)+1(22(Q)~0 • lC21 (Q)

+

lC12(P) ~ 0 , (1.52) lCil (P)

+

lC21 (Q) S; 0 • 8=0 ~ lCll(P)+lC21(Q)=0 , 9 = 7tl2 ~ lCU (P)

+

lC22(Q) = 0 or lC21 (Q)

+ lCI2(P)

= 0 . -+ -+

Choose -7tl2

<

11 S; 7tl2 such that the vector t i l COS(ll)

+

t 12 sin(ll) is in the direction of the curve

-+ -+

C of contact at P =

Q.

The vector t i l COS(ll)

+

t 12 sin(ll) is an eigenvector of B (P, Q) having eigenvalue zero. Hence, use (1.50),

8=0::;:'11=0 ,

(1.53) E> = 7tl2 and lClI (P)

+

lC22(Q) = 0::;:. 11 = 0 ,

E>

=

7tl2 and lC21 (Q)

+

lC12(P) = 0::;:'11 = 7tl2 ,

and if 9 ~ 0 and E> ~ 7tl2, then

(1.54)

[

1(21 (Q)cos2(E»

+

1(22(Q) sin2(9)

+

lCU (P)

1

11

=

arctan

cos(E» sin(E» (lC22(Q) - lC21 (Q»

-+ -+

The sib'll of E> depends on the orientation of the vectors t 11 (P) and t 12(P). We can choose the

-+ -+

orientation of the vectors tIl (P) and t 12(P) such that OS; E>S; 7tl2.

2. The curvature of conjugate profiles in points of contact

We consider the general gear pair. The smaller of the two gears is called the pinion and the larger the gear. The axes of the pinion and the gear are fixed in space. The angular velocity vector of the

-+ -+

pinion and the gear is denoted by IDI and ID2 respectively; see Figure 2.1. The angular velocity ratio U of the gear pair is constant and equals

(2.1) U

=

IDI/roz ,

-+ -+

where IDt

=

I IDll and roz

=

I ID21.

Choose the point 01 on the piruon axis and the point

Oz

on the gear axis such that the line 01

0z

is

0₁

=

Oz

= O. We introduce a right-handed cartesian coordinate system which is fixed in space. The corresponding coordinates arc denoted by (x. y. z) =:!. The point 01 on the pinion axis is the ori-gin of the coordinate system. The coordinate axes of the coordinate system are shown in Figure 2.1. The distance of the pinion axis and the gear axis is denoted by a. The angle between the

- t - t

angular velocity vectors (j) I and (j)2 of the pinion and the gear is 6. -1t < 0 S 1t. The sign of 6 is

- t

chosen such that the components W:2 of the angular velocity vector (j)2 of the gear are given by W:2 = 'LIz (cos(o), 0, sin(6»; see Figure 2. I.

z

gear axis

x

y

Figure 2.1. The pinion axis an the gear axis in the space-fixed coordinate system.

The tooth profiles on the mating gear teeth are conjugate, i.e. they are shaped such as to produce a constant angular velocity ratio during meshing. Let the tooth profIle on a tooth of the pinion in the reference position be specified by the parametric equations, with parameters A. and ~

with respect to the coordinate system in Figure 2.1. Funhermore, let +(A., Il) be the angle of

rota--+

lion of the pinion around its axis from the reference position, = 0 in the orientation of WI. such thal the point P (A.. Il) on the pinion profile (2.2) is in contact with the conjugate gear profile. The angle of rotation 41(1.., Il) can be calculated from the equation

where

with

and Xl, Y 1 , Z 1 as specified in (2.2) .

For a proof of equation (2.3) the reader is referred to [3, § 1.4] or [4].

Let P be a point on the pinion profile with parametric equations (2.2) in the reference position

q,

=

O. The position in space of the pinion point P at the moment of contact in P with the conju-gate gear profile, is called the contact position IP of P. Let P be the pinion point corresponding to the parameters (A., J.I.) in (2.2). The coordinates of the pinion point P = P (A., J.I.) in the reference position

q,

= 0 are ! I (A., J.I.) in (2.2). The coordinates !,(A., J.I.) of the contact position IP = IP (A., J.I.) can be written in the fonn

where

q,(A., J.I.) as calculatcd from cquation (2.3) ,

the antisymmetric matrix

[

0 0 0]

ill

=

00-1 .

o

1 0

For a description of rotations by means of the exponential function with matrix-argument, we refer to [3, § 1.3] or [4]. The coordinates x(A.. J.I.) in (2.4) of the contact position IP(A.. J.I.) are independent of the chaise of the reference position

q,

= O. The equations (2.4) are the parametric equations. with parameters A. and Il. of the so called surface of action of the gear pair, i.e. the sur-face containing all contact positions.

Let

Q

be the point of contact on the conjugate gear profile corresponding to the point P on the pinion profile. The image of the contact position IP under the rotation over the angle

~

-k4>O.,

Il),

k

=

JIV. around the gear axis in the orientation of {,t)2. is the position of the gear point

Q in the reference position IP

=

O. Hence. the coordinates :!2 (A. Il) of the gear point Q

=

Q (A., Il) in the reference position IP = 0 can be written in the form

where

:!(A, Il) see (2.4). k

=

1/U,

!!

=

(0, a, 0) are the coordinates of the point

Oz

on the gear axis; compare Figure 2.1, the anti symmetric matrix

[

0 -sineS) 0 ]

O2 = sin(o) 0 -cos(S)

o

cos(S) 0

The equations (2.5) are the parametric equations, with parameters A and Il, of the conjugate gear profile in the reference position ¢I

=

O.

~

Let n 1 (P) be the unit inner normal vector on the pinion profile at the pinion point p, Le. the

vec-~

tor n 1 (P) is in the direction of the pinion tooth at P. The components !! 1 (A. Il) of the normal

vee-~

tor n 1 (P). P = P (A., Jl), in the reference position 4> = 0 can be calculated by use of the parametric equations (2.2). The components n(A, Jl) of the common unit normal vector on the pinion profile and the gear profile at the moment of contact in the pinion point P

=

P (A, Jl), can be written in the form

The vector ~(A, Il) is independent of the choice of the reference position 4> = O.

~

Let n2(Q) be the unit inner normal vector on the gear profile at the point Q of contact

~

corresponding to the pinion point P. The components !!2(A, Il) of the normal vector n2(Q),

Q

= Q ("A, jl), in the reference position 4>

=

0 can be calculated from (2.6). We get

We give formulas for the fundamental coefficients of the conjugate profiles in points of contact. By use of (2.4-7) we get

(2.8) :!.J,l = exp(-$0t)<! .. - cI>lOt!) , :!.J,,,, = exp(-$OJ)<!" - «1>",OJ!) , !!I.l =CXP(-$OI)(~l-cI>lOl~) , !!J." =exp(-$Ol)(~",-«1>",Ol!!) , :!.Z,l = exp(-k«1>0Z)<!l - k«1>102<! - !!» •

:!.z,,,,

= exp(-k«1>Oz)<!", -

k«1>"Oz<! -

!!» .

!!Z.l = -exp(-kI\lOZ)(!!l -

kcl>lOz!!) •

!!z,'"

= -exp(-k«1>Oz)(!!" -

k«1>",Oz!!) .

The fundamental coefficients are calculated by use of the definitions in (1.20) and (1.24) and the relations (2.8). Furthennore. we use the onhogonality of the matrix exp(A). provided that the matrix A is antisymmetric. The fundamental coefficients E 1 •.••• N 1 of the pinion profile at the

point P

=

P (A, Jl) can be calculated by use of the fonnulas

(2.9)

E 1 = <! I,l. :!.J,l) = <!l - cI>lOl!.!l - cI>lOl!) •

F 1 = <!),l. :!.1,1l) = <!l - cI>l0l!.!" - c!>",Ol!) •

G I = <!l.". :!.l,,,) = <!11 - c!>jl.Ol!'!1l - IP"Ol!) • LI =-<!!I,l.!I, .. )=-(!!1. -cI>lOI!!.!l-cI>lOI!) • M) =-(!!I .... :!.I.")=-(!! ..

-1\l ..

Ol~.!",-I\l"OI!) • NI =-<!!I.",.!I.I1)=-(!!"-I\l"Ol!!.!,, -c!>jl.OJ!) .

The fundamental coefficients £2 •...• N2 of the gear profile at the point Q

=

Q(A, Jl) of contact corresponding to the pinion point P = P (A. Jl). can be calculated by usc of the fonnulas

(2.10)

£ 2 =

<!z, .. ,

!Z.l)

=

<! .. - kcl>10 2<! - !!),:! .. -

kcl>lOz<! -

!!» • F 2 =

<!z ... , !z.,,)

= <!l - kcl>l0 2<! - !!).:!", - kl\l,,02<! - !!» • G Z = <!2,,,. :!.2.11) = <!" - kl\l",02<! - !!),!" - kl\l",02<! - !!» •

L2 = -<!!2,A.. :!.2 ... ) = ~A. - kcl>l0_{2!!.:! .. - kcl>lOze! - !!» •} M 2

=

-<!!2 .... !2.,,) =

C!! .. -

kcl>1 0 2!!.!", - kl\l,,02<! -

!!» .

N 2 = -(!!2.",. :!.2.",)

=

(!!'" - kq,,,,02!!,:!Jt - k I\l", O2 e! - !!» .

Finally. we summarize the steps in the algorithm for the calculation of the curvature of conjugate pro1i1es in points of contact:

1. Introduce the constants

~ == (0,

a,

0), where

a

is the distance of the pinion axis and the gear axis; compare Fig-ure 2.1;

the antisymmetric matrices

[

0 0 01

[ 0

-sin(S)

0 1

o}

== 0 0 -1 and 02 == sin(S) 0 -cos(S) ,

o

1 0 0 cos(S) 0

where 0 is the angle between the pinion axis and the gear axis; compare Figure 2.1;

k == lIU. where U is the angular velocity ratio of the gear pair; compare (2.1).

2. Specify the tooth profile on a tooth of the pinion by means of its parametric equations

:! ==:! I (A. 11) in the reference position ~ == 0 and with respect to the cartesian coordinate sys-tem in Figure 2.1; compare (2.2)

3. Determine the angle of rotation ~(A.Il) by use of equation (2.3).

4. Determine the coordinates :!(A, 11) of the contact position IP (1.,11) of the pinion point P == P (1.,11) by use of (2.4).

5. Determine the components!!} (A. 11) of the unit inner normal vector on the pinion profile at the point P == P (A, 11) in the reference position ~ == 0 by use of the parametric equations (2.2).

6. Determine the components ~(A, 1.1) of the common unit normal on the pinion profile and the gear profile at the moment or contact in the pinion point P == P (1..11) by usc of (2.6).

7. Determine the fundamental coefficients E 1 ••.•• N 1 of the pinion profile at the pinion point

P == P (A. 1.1) by use of (2.9).

8. Determine the fundamental coefficients E 2, ••• , N 2 of the gear profile at the gear point

Q

= Q

(A, J.I.) of contact corresponding to the pinion point P = P (A, J.I.) by use of (2.10).

-

-9. Determine the mean curvatures K} (P) and K2(Q) of the pinion profile and the gear profile at

A A

10. Detennine the total curvatures Kl (P) and K2(Q) of the pinion profile and the gear profile at the points

P

and Q of contact by use of (1.37).

11. Detcnnine the relative curvature K,(P, Q)

=

Kl (P)

+

KZ(Q) of the conjugate profi1es in the points P and Q of contact; compare (1.42). If K,(P, Q) turns out to be negative. then P and

Q are not real points of contact since undercuning has occured; compare (1.39).

12. Detennine the principal curvatures Kll (P)S KIZ(P) of the pinion profile at the pinion point P = P(A,~) by use of (1.12).

13. Detennine the principal curvatures K21(Q)S KZZ(Q) of the gear profile at the gear point

Q

=

Q (A, ~) of contact corresponding to the pinion point P

=

P (A., ~) by use of (1.12). 14. Detennine the angle

e

between the corresponding principal directions of the conjugate

profiles at the points P and Q of contact by use of (1.51).

15. Detennine the angle T\ between the curve of contact and the principal direction of the pinion profile corresponding to Jell at the points P and Q of contact by use of (1.53. 54).

References

[1] L.P. Eisenhart: A treatise on the differential geometry of curves and surfaces. Dover publi-cations. New York. 1909.

[2] J. Haantjes: Inleiding tot de differentiaalmeetkunde. P. NoordhoffN.V .• 1954.

[3] D.A. Overdijk: De driedimensionale constructie van Reuleaux, Dissertatie. T.U. Eindho-ven, 1988.

[4] D.A. Overdijk: Conjugate profiles on mating gear teeth, Memorandum COSOR 89-01, T.U. Eindhoven. 1989.