~ Pergamon Mech. Mach. Theory Vol. 32, No. 3, pp. 363-373, 1997 © 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved PII: S0094-114X(96)00050-X 0094-114x/97 $17.00 + 0.00

A N A L Y S I S O F M E S H I N G O F B E V E L O I D G E A R S

CARLO INNOCENTI DIEM, Department of Mechanical Engineering, University of Bologna, Viale Risorgimento, 2,

40136 Bologna, Italy

(Received 4 May 1995)

Almtraet--This paper analyses the meshing of a pair of beveloid gears mounted on intersecting or skew shafts. Precisely, the dependence of the backlash on the gear geometry and the shaft relative position are investigated. The study is preceded with the presentation of a new, basic nomenclature to describe the kinematically relevant dimensions of beveloid gears. Finally, a numerical example is reported. © 1997 Elsevier Science Ltd. All rights reserved.

1. I N T R O D U C T I O N

Beveloidt gears, also known as involute conical gears, represent the extreme generalization of involute gears [1-3]. Primarily adopted for power transmission between nearly-parallel shafts, either crossed or intersecting, they fill a niche left by bevel gearing due to limitations of conventional bevel-gear cutting machines [1, 2, 4].

Beveloid gears can be easily produced by the same CNC hobbing and grinding machines commonly employed for cylindrical involute gears. As additional advantages, beveloid gears can compensate for shaft misalignment and allow for adjustment of the gearing backlash. Although, for non-parallel shafts, the tooth contact is theoretically pointwise, the tooth bearing can be increased through surface hardening followed by gear grinding: application of beveloid gearing to the transmission of propeller-driven boats with an inboard motor is typical [4].

In spite of their favorable features, beveloid gearings have been largely neglected by the technical literature. To the author's knowledge, no paper has so far addressed the determination of the meshing backlash of a pair of beveloid gears. The most advanced papers on the subject seem to be those by Mitome [5-7], focused on hobbing of beveloid gears. In these papers, the geometric features of a beveloid gear are indirectly described in terms of the geometry of an imaginary rack that, when suitably placed and subsequently moved with respect to the gear, generates the gear. All hobbing parameters are then determined by imposing the hob to envelope the imaginary rack. In kinematics terms, the papers by Mitome analyse the meshing with zero backlash of a conventional helical involute gear--the hob--with a beveloid gear.

This paper presents the analysis of meshing of two beveloid gears. The gears have general geometry and are mounted on crossed or intersecting axes. Based on a new nomenclature that focuses exclusively on the essential features of a beveloid gear, the analysis provides the relation governing the meshing--with a prescribed amount of backlash--of beveloid gears. Only the basic conditions pertaining to the involute helicoidal portions of the tooth flanks are taken into account; any further issue such as undercutting, interference, or contact ratio evaluation is not addressed.

The proposed methodology represents the generalization of existing approaches, whose applicability is bound to the special cases of two involute spur gears mounted on parallel axes, two helical involute gears mounted on crossed or parallel axes, a beveloid gear in mesh with a helical involute gear [5-7].

A numerical example shows application of the reported results to a case study.

tBeveloid is a registered trademark of the Invincible Gear Co., Livonia, MI, U.S.A.

363

364 Carlo Innocenti

2. BASIC NOMENCLATURE

The surface representing either flank of a tooth of a beveloid gear is an involute helicoid stemming from a base cylinder coaxial with the gear (see Fig. 1). If intersected with a plane orthogonal to the gear axis, the base cylinder and the tooth flank appear as a circle and its involute [8-10]. The curve along which the involute helicoid is joined to its base cylinder is the base helix, whose tangent line at whatever point forms the angle fl (fl~] - n/2, ~/2[) with respect to the gear axis (angle fl is positive for right-handed helices, as in Fig. 1). The radius p of the base cylinder, together with the base helix angle 13 univocally define the shape of an involute helicoid. The pitch h of the helicoid is given by

h - 2ztp tgfl " (1)

and is considered as positive for right-handed helicoids. In the sequel, reference to the parameter k will be frequent, proportional to the reciprocal of pitch h according to the following definition

k = tg__ff_~ 2re p = T (2)

If the gear axis is given an orientation by a unit vector n, a tooth flank is said to be a right-hand or a left-hand flank according to whether the following quantity is positive or negative:

( P - O ) × n ' e . (3)

Here O is a point on the gear axis and e is the outward normal to the tooth flank at a generic point P (see Fig. 1, representing a left-hand tooth flank).

Unlike conventional helical gears, parameters pr and fir defining the involute helicoid of the right-hand flank of a beveloid gear are generally different from the corresponding parameters p, and fl~ pertaining to the left-hand flank (see Fig. 2 where, incidentally, the base helix of the right-hand flank is left-handed, whereas the base helix of the left-hand flank is right-handed). Consequently, also pitches hr and h~--and parameters kr and k,--of the right-hand and left-hand flanks of the same tooth may differ.

The section of a tooth with a plane z orthogonal to the gear axis is now considered (see Fig. 3, where unit vector n is supposed to point toward the reader). On this plane, the angular base

/

/

/

/

/ i

/

/ s

, I t

n

e

Fig. 1. The involute helicoid.

Analysis of meshing of beveloid gears 365

n • ~ / l¢

• / s / / I / / S "

/ j - •

s f . ~

Fig. 2. The base helices of a tooth.

thickness of a tooth is defined by the angle tp between the radii through the starting points Er and E~ of the involutes pertaining to the right-hand and left-hand flanks respectively.

I fkr differs from k , ~0 depends on the transverse plane z. In this case, there is one plane 30, called the reference plane, for which tp = 0 (in Fig. 2, the reference plane coincides with the transverse section on which the radii of the base cylinders are dimensioned). Accordingly, if a generic transverse plane r is considered, and q is the axial displacement that superimposes 30 on r - -pos i t ive if concordant with n---the angular base thickness tp relative to z is expressed by:

¢p = ( k , - k r ) q (kl # kr). (4)

For beveloid gears characterized by k~ = k , as well as for conventional helical or spur gears, tp does not depend on 3. In these cases, the notion of the reference plane does not apply, and the value of tp at any transverse section is itself apt at representing the angular base thickness of the tooth.

3. THE PATHS OF CONTACT

When an involute helicoid rotates about a fixed axis while keeping in contact with another involute helicoid, itself free to rotate about another fixed axis, the contact point moves along a rectilinear path that is tangent to the base cylinders of the two helicoids and orthogonal to the

Fig. 3. Transverse section of a beveloid gear tooth.

366 Carlo Innocenti

ao

Fig. 4. The base cylinders of a pair of beveloid gears.

helicoids at the contact point [8-10]. This section is aimed at finding, in view of their subsequent utilization, the two paths pertaining to the contact of a pair of left-hand and right-hand flanks of a beveloid gearing. Regardless of the actual overall dimensions of the gears, the paths of contact are considered as extending from one base cylinder to the other.

Two beveloid gears mounted on crossed rotation axes are considered (see Fig. 4). Unit vectors nL and n~ are chosen along the axes in such a way that when the angular velocity of gear 1 is concordant with n~, the angular velocity of gear 2 is opposite to n2. Due to this convention, the left-hand flanks of the teeth of gears 1 touch the left-hand flanks of the teeth of gear 2 (and conversely for right-hand flanks). In the sequel, a double-index notation will be adopted for some symbols: the first index, i, will refer to the considered gear (i = 1, 2), while the second index, j , will refer to either side of contact ( / = - 1 for a left-hand flank; j = 1 for a right-hand flank). For example, fl:._~ denotes the base helix angle of the left-hand flanks of the teeth of gear 2.

A basic condition to satisfy stems from the expression of the transmission ratio of the gearing when written both in terms of the number of teeth N~ and N2 of gears 1 and 2, and in terms of the geometric parameters of corresponding helicoids on both gears[8]. Two equivalent mathematical formulations of such a condition are

N~ plglllj. N~ _ k2dvlj N2 - p2ju2j' N2 k i j v 2 j

where symbols u~j and v~j have the following meaning:

(5)

With reference to Fig. 4, the relative position of the gear rotation axes is defined by their mutual

u,j = cos fl~j; v~j = sin fl,j. (6)

Analysis of meshing of beveloid gears 367

distance a0 and by angle s0 that n2 forms with n~; s0 is measured counterclockwise with respect to vector (A2 - A0, A~ and A2 being the points of intersection of the gear axes with their common perpendicular (in Fig. 4, s0 is positive).

Two Cartesian reference frames W~ and W2 are now introduced with origins at A~ and A2 respectively (see Fig. 4). The x-axis of W~ is oriented from `41 to ,42; the z-axis is parallel to n~ and the y-axis is defined according to the right-hand rule. Similarly, the x-axis of reference frame W2 points from ,42 to A~, whereas the z-axis is parallel to n2. The 4 x 4 matrix M0 for coordinate transformation from W2 to W~ is

Mo =

- 1 0 0 ao

0 -Uo -Vo 0

0 -Vo Uo 0

0 0 0 1

(7)

In equation (7), the following shorthand notation has been introduced:

uo = cos So; v0 = sin so. (8)

In order to compute the coordinates of the ending points P~j and P2j of the path of contact P~jP2j (see Fig. 5), reference frames V~j (i = 1, 2) are introduced with the following characteristics: the origin B~j of V,j lies on the z-axis of W,, on the transverse plane that contains point P,j; the z-axis of Vij is collinear with the z-axis of W~; the positive x-axis of V~j passes through point Pi~. Figure 6 illustrates a view of the reference frames on gear i (i = 1, 2), from the positive z-axis of reference frame W~. In this view, origins B~_~ and B~,~ of reference frames Vi_~ and Vi, l a r e superimposed to

--2 Fig. 5. The paths of contact.

368 Carlo Innocenti

Pi,1

Bi,_l, Bi, l, Yi

Pi,-:

Xi,-I Xi,1

Xi ~ / . / Yi,-1

J

Fig. 6. The reference frames associated with gear i (i = 1, 2).

At, whereas (x~_~, yi,-l) and (x~.~, yi.l) are the (x, y) axes of reference frames V~,_I and, respectively, V~.t.

If 0u denotes the rotation about n, that superimposes W~ on Vu (see Fig. 6), the 4 x 4 matrix Mu for coordinate transformation from Vu to IV,- is

where

Mij

cu -s~j 0 0

s u c u 0 0

0 0 1 bu

0 0 0 1

(9)

cu = cos 0u; su = sin 0u (10)

and bu is the z-coordinate of point Bu in reference frame I4",.. According to the foregoing assumptions, the homogeneous coordinates of Pu in reference frame Vij are

(Pu)wj = [Pu, O, O, 1] T (11)

[the index outside the parenthesis on the left-hand side of relation (11) indicates the reference frame

Analysis of meshing of beveloid gears 369

where the vector is to be resolved into its components]• In addition, the homogeneous coordinates in V,~ of the unit vector eg~ representing the outward perpendicular to the f lankj of a tooth of gear i at any contact point--and also the direction of the contact path PuP2j--are:

(e,~)~,j = j[O, - u,j, v,~, O]E ( 1 2 )

Since the flanks labeled on both gears by the same second index j are mutually in contact, the following relations must be satisfied:

e ~ + e2~ = 0 ( 1 3 )

( P ~ - Pu) = ~eu (14)

where aj is the length of contact path PuP2~. In these equations, quantities 0u, 0~a, bu, b~, and ~r~ are the unknowns.

In order to resolve vector equations (13) and (14) into their components in reference frame W~, the following relations are considered:

(eu)w, = m u ( e u ) v , j = j [ u , j s u , - u , j c u , v , j , 0] T (15)

(e2j)w, = MoMzj(e2a)v2j = j [ - U2,jS2j, Uobl2dC2j - - UOU24, U0/'/2jC2d "Jr" Uolfl2~i, 0] T (16)

( P i j ) w l = M 1 j ( P u ) w j = [p~jcu, p l j s l j , bl~, 1] v (17)

(Pzj)w~ = MoM:a(P~ i ) v : / = [ -p~acz~ + ao, - p ~ j u o s 2 j - bz~vo, - p z v v o s ~ + bz~uo, 1] T. (18)

Supposing v0 # 0, the second and third components of vector equation (13) provide the following values for c~j and c2j:

U2,j -~ U0UIj• UIj -~ U0/fl2j c~a= , c.,a= (19) UOIglj UOld2 d

(The case v0 = 0--corresponding to parallel shafts--should be considered separately; it will not be reported here for the sake of conciseness.)

: and A solution to equations (19) can exist in terms of angles 0~j and 0zj if both quantities e u Czj are not greater than one. By manipulating either of equations (19), it can be shown that this requirement corresponds to satisfying the following inequality:

2 v2o - v u - v~j - 2UoV~jV2~ > 0. (20)

Since uu and u2j are both positive (/~j e ] - re/2, x/2[, i = 1, 2), the first scalar equation of vector equation (13)

u l j s l j = u2js2j (21)

guarantees that both quantities su and s2j have the same sign. By also noting that the first component of unit vector (eu)w~ is to be positive, the following conditions are obtained [see also relation (15)]:

2 (22) = j , / 1 - e j; =j /l - c2j.

Relations (19) and (22) together allow unambiguous determination of angles 0~j and 02j. Finally, vector equation (14) represents a set of three linear equations in the unknowns b~j, b2j,

and aj. By also considering expressions (19), (21) and (22), the solution of system (14) can be expressed in the following terms:

b~j = aoezj - p : j - - p i j (UoS ldS2d -k- CkiCeJ)•, b: j = aocu - P u - p2j(UoS~jS2j + CljCZd) (23) UoS2j UoSIj

• ao - p~dc~j - p2de2d o-, = j (24)

~LiSL/

Relations (23), together with relations (19) and (22), allow determination of the coordinates of points Pij and P2j with respect to reference frame Wj [see equations (17) and (18)]. Moreover, condition (24) directly provides the length aj of contact path P u P 2 j .

370 Carlo Innocenti

4. BEVELOID GEARS IN MESH

In the previous section, only one type of contact was considered at one time: either between left-hand flanks or between right-hand flanks. The results that have been separately obtained for either case will be merged in the present section in order to evaluate the backlash of a beveloid gearing.

In this paper, the following definition for the gearing backlash H will be adopted:

H = NiAgl = N2AT2 (25)

where NI and A?i (i = 1, 2), represent the number of teeth and the angular backlash [8] of gear i. As a first step, it is convenient to focus the attention on the special case of two beveloid gears

engaging with zero backlash (tight meshing). Extension to the case of non-zero backlash will be straightforward: it will suffice to recognize that the presence of any gearing backlash H can be simulated by supposing, for example, gear 1 in tight meshing with a fictitious gear, completely similar to gear 2 apart from the angular base thickness of its teeth, which is increased by the angular backlash A'y2.

Regarding first the case of zero backlash, two flanks facing each other on opposite sides of the same tooth space on gear 1 are considered: the left-hand flank of a tooth, LI, and the right-hand flank of the adjacent tooth, RI. Flanks LI and R1 are both destined to come into contact, respectively, with the left-hand flank, L2, and right-hand flank, R2, of the same tooth of gear 2. The angular position of gear 1 is initially chosen in such a way that the base helix of flank L, passes through point PI,-1, one of the ending points of contact path P1.-IP2.-1. At this position, helicoids LI and L2 touch each other at P1.-1. Now a sequence of four imaginary maneuvers is considered.

First, both gears are simultaneously rotated to move the point of contact of LI with L2 from PL-I to P~.-1, along the contact path PI , -1P2 , - I . The corresponding rotation of gear l, 71,, is clearly given by

0"_ I 71~ - - - (26)

,01.-lUl.-I

where tr_l is provided by equation (24) fo r j = - 1. The value of 71~ is positive because it represents a counterclockwise rotation about unit vector nl.

After this maneuver, the considered tooth of gear 2 has its left-hand flank L2 passing through point P2.-l. Now both gears are given a second rotation, opposite to the previous one, in order to make the base helix of flank R2 pass through point P2.,. The value of the necessary rotation 3'2~ of gear 2 can be computed by first projecting the base helices of gear 2 onto a cylinder with unitary radius, coaxial with gear 2; then by cutting the cylinder along a generator and developing it on a plane. On the developed cylinder, the base helices appear as lines bounding the teeth.

Figure 7 represents, on the developed cylinder, the same tooth of gear 2 at two different positions, just prior to and after rotation 72b. The axis z2 corresponds to the z-axis of the reference frame I4"2 (see Figs 4 and 5), while the horizontal axis represents the angular coordinate 62 about unit vector n: (&2 = 0 identifies the generator of the unitary-radius cylinder that crosses the positive x2 axis). The base helices of flanks L2 and R2 before rotation 72b is executed correspond, in Fig. 7, to lines ~e2_1 and ~2.-1. These lines intersect at a point on the trace z2 = q2 of the reference plane of gear 2 (see Section 2). Due to rotation 72~, lines ~2.-1 and ~2,-1 move to the new positions ~2.1 and ~2.1, also shown in Fig. 7. The projections of points P2.-, and P2.1 on the unitary-radius cylinder of gear 2 correspond to points ~'2.-1 and ~2.~, whose (&2, z2) coordinates are (02,-1, b2._ 1) and (02.1, b2.1) respectively.

With the aid of Fig. 7, by recognizing that the slope of both lines ~2.-1 and ~2., (~2.-~ and ~2.~) with respect to the z2 axis is k2.-l(k2.1) [see relation (2)], the value of 72~ can be easily computed as

~)2b = 02,1 - - 0 2 - 1 "3 I- ( ~ 2 , - I - - (b2.j - b2,_~)k2.1. (27)

Quantities 0zl, 02_ 1, b2.1, and b : _ 1 have already been determined in the previous section, whereas parameter ~2.-~ is the angular base thickness of gear 2 at the transverse section through point P : _ i. In case k2._~ and k2.1 differ, q~2.-1 is given by [see equation (4)]

q~2,-1 = (k2.- . - k2,1)(b2,_l - q2). (28)

Analysis of meshing of beveloid gears 371

Otherwise, for k2,_~ = k2A, ~o2,-i is a basic geometric parameter of gear 2. Owing to the adopted convention for the choice of unit vectors n~ and n:, the rotation ?lb of gear

1 that corresponds to rotation y2b of gear 2 is given by

N2 y,b = - yz~ N--~" (29)

At the end of the previous maneuver, flanks Rl and R2 come to contact at P2,1. Now the third maneuver is executed. It is aimed at shifting the point of contact of RI with R2 from P2A to P~.l along the contact path P2,mP~,t. The corresponding counterclockwise rotation angle of gear 1 about unit vector n~ is given by

~- 0"1 ( 3 0 ) Yle pI , IUIA '

Following this rotation, the base helix of flank R~ passes through point P).~. The fourth--and last--maneuver consists of a rotation of both gears with the purpose of

repositioning gear 1 at the orientation it had prior to the first maneuver, i.e. when the base helix of flank L1 passed through point P,_~. The corresponding rotation y)~ of gear 1 can be determined by the same approach as the one adopted for rotation y,~.

On the developed unitary-radius cylinder coaxial with gear 1, the projections ~l.~ and ~.~ of flanks L1 and R, at the end of the third maneuver are considered (see Fig. 8). Clearly, the projection ~).~ of point P~.l belongs to line ~).i. In Fig. 8, the projection £~_~ of flank L~ at the end of the fourth maneuver is also reported as passing through projection ~)_) of point P~_~. The value of rotation 7)b can directly be inferred by inspection of Fig. 8:

~)ld = - - [ 0 l , l - - 01 , - ] 21- (~I ,-1 - - ( b l . l - - hi, 1)kl,1 - - 2X/Nl]. (31)

Parameter ~pi,-i appearing on the right-hand side of this equation represents the angular base thickness of gear 1 at the transverse plane passing through point P~, i. Whenever k)_~ and k~,~ are different, ~p~_~ is given by [see equation (4)]

go~_~ = (k,_~ - kl .~) (bk- i - q,) . (32)

Otherwise, for k~_~ = k),~, ~O~_l is a basic geometric parameter of gear 1. Finally, the sought-for relation governing the tight meshing of two beveloid gears is

71a + ~lb + ~)lc + ? ld ~--- 0 (33)

~2,-1

0 2 1 )-

atan k2, ~

b2,. 1 - "-2,1 ~ x \ \ \ ~ ) " _

I

02,1 62 Y2b .

v l

Fig. 7. Developed unitary-radius cylinder of gear 2.

372 Carlo Innocenti

Z 1

bl , i

O1 1

'~ld

_

~ 1 1 ~ ~ ~ ~ I~ 1,-1

01,1

2n/N1 81

Fig. 8. Developed unitary-radius cylinder of gear 1.

where all quantities appearing on the left-hand side have already been given an expression in terms of dimensions and relative placement of gears 1 and 2.

As anticipated, provision for non-zero backlash is now straightforward: it suffices to add the angular backlash Ay2 to the angular base thickness of gear 2. Accordingly, equation (27) takes the following more general form

~2b = 02,1 - - 0 2 , - I + ((/02,-1 + A])2) - - (BzA - - B 2 , - 1) / (2 ,1 . ( 3 4 )

If backlash is present, equation (29) still holds as it is, provided that the new value for y2o given by equation (34) is considered therein.

In the most general case, characterized by kl ,_~ 4: kl,~ and k2.-1 ~ k2.1, once new quantities M, SL, and $2 are introduced as defined by

M NI 1 . . . . - l U l -1 p l , lU l I p 2 , - l U 2 , - t /92 lU2,1

(35)

S , = N t [ k , , _ l ( q , - b, ._~) - k , . l (q , - b, , i) + 0, ._ , - 0,.1] (i = 1, 2), (36)

equation (33) can be written as

H = M + $1 + $2 + 2n (37)

where H is the gearing backlash as defined by relations (25). Equation (37) represents the key results of the present study. It can be employed both in predicting the backlash of a given gearing and in computing the correct axial placement of the gears in order to realize a prescribed backlash.

5. NUMERICAL EXAMPLE

This section shows application of the foregoing results to determination of the backlash of a beveloid gearing. Numerical data are purposely expressed by a large number of significant digits in order to enable the reader to accurately check the reported computations.

The data pertaining to gear 1 are: Nt =41 , pl,-~ = 84.388018mm, fl~_~ =23.773441 °, PLI = 90.334402mm, flL~ = 28.622610 °, q~ = 70mm. Gear 2 is characterized by: N2 = 93, pz-t = 211.278738 ram, fl2,-1 = 33.992067 °, p2,1 = 194.993259 ram, fl2,1 = 22.718607 °, q2 = - 40.5 mm.

Analysis of meshing of beveloid gears 373

Paramete rs k,j (i = 1, 2 ; j = - 1, 1), are determined by equat ion (2): kL-j = 0.005219923 m m -~, kL~ = 0.006041217 m m -l, k2,_1 = 0.003191552 m m -~, k2,~ = 0.002147206 m m -~. Gears 1 and 2 are effectively ap t at engaging because the second o f condit ions (5) is satisfied for bo th left-hand and r ight-hand flanks.

The gears are supposed to be moun ted on two crossed shafts whose relative placement is character ized by (see Fig. 4): a0 = 300 m m , s0 = - 5 9 °. According to the procedure previously explained, the evaluat ion o f the backlash evolves th rough the following steps.

First, angles 0~ ( i = 1 , 2 ; j = - 1 , 1) are computed by equat ions (19) and (22): 01._~ = -0 .21303383 rad, 02,_~ = -0 .23553615 rad, 0L~ = 0.57130759 rad, 02,~ = 0.54051341 rad. Subsequently, quanti t ies btj and aj are evaluated by equat ions (23) and (24): b~,_ ~ = -9 .545324 mm, b2_~ = 14.679555 mm, ~r_, = 62.404865 mm, b,,~ = 35.961332 ram, b2,~ = 14.162913 mm, a~ = 119.698859 mm.

Finally, equat ion (37 ) - - t h rough condit ions (35) and (36)- -provides the sought- for value for the gearing backlash: H = 0.1049456. The cor responding values for the angular backlashes of the two gears are: A7~ = 0.1466570 °, A72 = 0.0646553 °.

6. C O N C L U S I O N S

The solut ion to the p rob lem o f determining the meshing backlash of the beveloid gearing has been presented. It is based on a new nomencla ture that focuses only on the essential geometr ic features o f the beveloid gearing. Consequent ly , the formal complexi ty o f the resulting analytical expressions is kept to a min imum.

The p roposed app roach can be part icularized to simpler cases, such as hobbing or grinding of beveloid gears by convent ional tools, meshing of crossed-axis helical gears, meshing of parallel-axis spur gears.

Finally, a numerical example per ta ining to a pair o f general -geometry beveloid gears moun ted on crossed axes has been reported.

Acknowledgement--The financial support of Italy's MURST is gratefully acknowledged.

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10. F. L., Litvin, Gear Geometry and Applied Theory. Prentice Hall, 1994.