Design and investigation of gear drives with non-circular gears applied for speed variation and generation of functions

Comput. Methods Appl. Mech. Engrg. 197 (2008) 3783–3802

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Design and investigation of gear drives with non-circular gears applied for speed variation and generation of functions Faydor L. Litvin a, Ignacio Gonzalez-Perez b,*, Alfonso Fuentes b, Kenichi Hayasaka c a

Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, United States Department of Mechanical Engineering, Polytechnic University of Cartagena, Campus Universitario Muralla del Mar, C/ Doctor Fleming, s/n 30202 Cartagena, Spain c Gear R&D Department, Research and Development Operations, Yamaha Motor Co., Japan b

a r t i c l e

i n f o

Article history: Received 7 January 2008 Accepted 1 March 2008 Available online 10 March 2008

a b s t r a c t The paper covers design and generation of gear drives with non-circular gears: (a) a gear drive formed by an eccentric involute one and by a non-circular gear; (b) generation of functions by non-circular gears. The developed theory is illustrated with numerical examples. Ó 2008 Elsevier B.V. All rights reserved.

Keywords: Gear design Non-circular gears Generation of functions Eccentric gears

1. Introduction Development of generation of non-circular gears [12,23–28] makes the manufacture of such gears as easy as circular gears. Therefore it is not surprising that design of gear drives with noncircular gears became the subject of research of many scientists [1–37]. The contents of the paper are the following ones: (1) Design and generation of an eccentric gear drive formed by an eccentric planar or helical involute gear and conjugated non-circular gear has been developed. The main feature of the eccentric involute gear is that its center of its rotation does not coincide with the geometric center. The gear drive may be designed with helical teeth (see Fig. 1) and straight teeth, the bearing contact may be localized, and, by design with small eccentricity, the drive may be applied in reducers (as a gear drive with reduced sensitivity to misalignment). (2) Application of non-circular gears for generation of functions (Section 4) has been developed for the following cases: (i) wherein the derivative of the function is of a varied sign, (ii) the centrodes are unclosed curves, and (iii) two pair of gears are applied for generation. In case (iii), the design requires application of a functional wð/Þ ¼ f ðf ð/ÞÞ; * Corresponding author. Tel.: +34 968 326429; fax: +34 968 326449. E-mail address: [email protected] (I. Gonzalez-Perez). 0045-7825/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2008.03.001

ð1Þ

where wð/Þ is the given function, f ð/Þ is the transmission function of a pair of non-circular gears. (3) Application of modified elliptical gear (Section A.1.6) has allowed design of: (i) a gear drive with an asymmetric transmission function, and (ii) lobes (which centrodes are modified ellipses). (4) Algorithms for determination of tooth surfaces of non-circular gear generated by a shaper or by a hob are proposed. A simple approach for determination of avoidance of undercutting of a non-circular gear is developed. A functional for observation of identity of mating centrodes is proposed. The developed theory is illustrated with numerical examples and with graphs and drawings. 2. Gear drive formed by eccentric involute pinion and non-circular gear 2.1. Centrodes of eccentric gear drive 2.1.1. Introductive comments The discussed below gear drive (called for the purpose of abbreviation Eccentric drive) is formed by an eccentric involute pinion 1 and conjugated non-circular gear 2 (Fig. 1). The non-circular gears of the eccentric gear drive may be designed and generated with straight and helical teeth. The eccentric drive is a competitive one to the one formed by elliptical gears [28]. The approaches proposed in the paper allow

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Fig. 1. Eccentric involute pinion and conjugated non-circular gear: (a) for helical drive, (b) for a planar drive; O1 and O2 centers of rotation; O the geometric center.

to generate the non-circular gear by a hob (in case of a convex centrode) and by a shaper (if the centrode is a convex–concave one). The performed investigation covers the basic topics of geometry, design, and generation of eccentric drives. 2.1.2. Equations of mating centrodes Fig. 2a and b shows centrodes r1 and r2 in initial and current positions Centrode r1 is an eccentric circle of radius rp1 and is represented in polar form as (Fig. 2b) 2

1

r1 ðh1 Þ ¼ ðr 2p1  e21 sin h1 Þ2  e1 cos h1 ;

ð2Þ

where parameter h1 determines the position of r 1 ðh1 Þ with respect to polar axis O1 A1 (Fig. 2b).

Henceforth we will use representation of a centrode in terms of derivative m12 ð/1 Þ (see Section A.1.2 of Appendix), where /i ¼ hi (i ¼ 1; 2) is the angle of rotation of the centrodes; parameter hi determines location of r i ðhi Þ with respect to polar axis Oi A (see Section A.1.2). For r1 , we have r1 ð/1 Þ ¼

E : 1 þ m12 ð/1 Þ

ð3Þ

Centrode r2 is represented by two following equations m12 ð/1 Þ ; 1 þ m12 ð/1 Þ Z /1 d/1 /2 ð/1 Þ ¼ : m12 ð/1 Þ 0

r2 ð/1 Þ ¼ E

Fig. 2. For derivation of centrodes: (a) initial position; (b) current position.

ð4Þ ð5Þ

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Here, the derivative function is m12 ð/1 Þ ¼

E  r1 ð/1 Þ c1 ¼ 1 2 r1 ð/1 Þ ð1  e2 sin / Þ2  e1 cos / 1

1

ð6Þ

1

and c1 ¼ rEp1 ; e1 ¼ rep11 . Function /2 ð/1 Þ is the transmission function and its determination requires numerical integration. Centrode r1 is already a closed form curve (a circle of radius rp1 ). Centrode r2 might be a closed form curve by observation of equation Z 2p 2p d/1 ¼ ; ð7Þ /2 ð/1 Þ ¼ n m12 ð/1 Þ 0 wherein n is the number of revolutions that eccentric gear 1 performs for one revolution of the non-circular gear. Derivation of centrodes r1 and r2 enables to design as well a cam mechanism with a constant center distance between the follower as an eccentric circle and a cam for generation of functions. The cam is determined as the centrode of the non-circular gear of the eccentric drive. 2.1.3. Curvature of centrode r2 and applications 2.1.3.1. Basic derivations. Knowledge of curvature of centrode r2 is necessary for: (i) choosing the method of generation of the gears by a hob or by a shaper, and (ii) for avoidance of undercutting of tooth profiles. Considering that a centrode is represented by a polar curve rðhÞ, its curvature radius my be determined as [20,27]: h dr 2 i32 r 2 þ dh qðhÞ ¼ : ð8Þ dr 2 d2 r r2 þ 2 dh  r dh 2 The condition of convexity of the polar curve may be represented as  2 2 1 dr d r j ¼ ¼ r2 þ 2 r 2 P0 ð9Þ q dh dh or by an alternative inequality " # 2 d r 2 2 ð10Þ sin l rð2  sin lÞ  sin l 2 P 0: dh   r is formed by position vector rðhÞ and Here, angle l ¼ arctan dr=dh tangent t to the polar curve (see [27]); j is the curvature of the polar curve. In this paper, taken into account that the curvature of centrode r1 is known, determination of j2 is obtained by using Euler–Savary equation [18], that relates curvatures j1 and j2 of centrodes r1 and r2 of the eccentric gear drive (Fig. 3) as follows:  1 1 1 1 þ sin l1 : þ ¼ ð11Þ q1 q2 r 1 ð/1 Þ r2 ð/1 Þ Here 1 1 ¼ ¼ j1 ; q1 r p1

1 ¼ j2 ; q2

ð12Þ

where q2 ¼ jIA2 j (Fig. 3), r 2 ð/1 Þ ¼ E  r1 ð/1 Þ;

ðFig:3Þ:

Eq. (11) yields  1 1 þ sin l1  j1 : j2 ¼ r 1 ð/1 Þ E  r1 ð/1 Þ Convexity of centrode r2 is guarantied if  1 1 þ sin l1  j1 P 0: r 1 ð/1 Þ E  r1 ð/1 Þ

ð13Þ

ð14Þ

ð15Þ

Fig. 3. For derivation of curvature radius q2 ð/1 Þ of centrode r2 .

Fig. 4 illustrates function q2 ð/1 Þ for gear drives with: (a) a convex centrode r2 ; (b) a convex–concave r2 . The location and orientation of curvature radius q2 ð/1 Þ is visualized by application of a four-bar linkage (Fig. 3) with links: (a) link 1 as O1 A1 , (b) link 2 as A1 A2 , (c) link 3 as O2 A2 , and (d) link 4 as O1 O2 , where jA1 A2 j ¼ q1 þ q2 . It is known from kinematics of four-bar linkage that point F (of intersection of extended links 1 and 3) is the instantaneous center of rotation of link 2 with respect to link 4. Vector IF is perpendicular to O1 O2 and its magnitude is determined as jIFj ¼

r 1 ð/1 Þ  q1 sin l1 jq2 sin l1  r 2 ð/1 Þj ¼ : q2 r2 ð/1 Þ q1 r1 ð/1 Þ

ð16Þ

2.1.3.2. Avoidance of undercutting. Drawings of Fig. 5 show generation of an involute spur gear with a rack-cutter of profile angle ac . The centrode of the rack-cutter is its middle line, the centrode of the gear is the pitch circle of radius rp . Undercutting of the involute gear is avoided by the installment of the rack-cutter wherein [27] m 2

sin ac

P rp :

ð17Þ

Avoidance of undercutting of non-circular gear of eccentric gear drive is based on following considerations: (i) Gear 2 of the drive has various tooth profiles, but they may be represented (approximately) as tooth profiles of respective circular gears with curvatures radii qA ; qB ; . . . ; qK (Fig. 6). ðAÞ (ii) Radius q2 of curvature of centrode r2 for point A corresponds to profiles of the tooth notified as RA . Similarly, profiles of tooth RB are represented as ones of spur gears of ðBÞ radius q2 , and so on (Fig. 6). (iii) Undercutting occurs for a tooth with the smallest radius q2 of the substituting circular gear; it is the substituting gear ðAÞ of radius q2 . (iv) Fig. 6a shows representation of tooth profiles of gear 2 with centrode r2 designed for a gear drive with n ¼ 3. Here, n is the number of revolutions of eccentric gear 1 that is performed for one revolution of driven gear 2 with centrode r2 . (v) The idea of application of substituting circular gears may be applied for avoidance of undercutting for all eccentric gear drives with centrodes r1 and r2 . For this purpose, it is necesðAÞ sary to obtain functions q2 ðe; nÞ (Fig. 6b) that represents the

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Fig. 4. Illustration of function q2 ð/1 Þ for eccentric gear drives with design parameters r p1 ¼ 22:35 mm; n ¼ 4, and (a) e ¼ 0:2, (b) e ¼ 0:7.

2

Fig. 5. Illustration of avoidance of undercutting of a spur involute pinion as the condition: m < r p sin ac .

ðAÞ

minimal curvature radius of q2 ; A is the point of centrode r2 (Fig. 6a) where q2 ¼ q2;min ; e ¼ rep (see Eq. (6)). (vi) Avoidance of undercutting for all eccentric gear drives is obtained by choosing such a module with which functions ðAÞ

q2 m

2

will be out of the square with the height 1= sin ac (Fig. 6b).

2.2. Generation of the non-circular gear by shaper and hob 2.2.1. Introduction The purpose of this section is to derive the algorithms that relate the motions of the generating tool (shaper, hob) and non-circular of the drive being generated. Such relations are represented

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Fig. 6. Illustration of: (a) representation of tooth profiles of non-circular gear 2 (m is centrode r2 and n is centrode of substituting circular gear); (b) line of undercutting for the case wherein ac ¼ 20 .

by non-linear equations and are the basis of computerized generation of the non-circular gear. The eccentric gear of the drive may be generated as a conventional involute gear, with the observation by manufacture and assemble of the location of the geometric center with respect to the center of rotation. In addition, for the purpose of localization of the bearing contact, it is necessary apply double crowning of the eccentric gear. There is a possibility of generation of the non-circular gear by a shaper that is identical to the eccentric involute gear of the drive. This allows to perform the generation of the non-circular gear observing by manufacture a constant center distance between the shaper and non-circular gear being generated. However, the identity of the shaper and the eccentric involute gear is an unfavorable limitation and has not been applied by the authors.

2.2.2. Generation of non-circular gear by a non-eccentric shaper 2.2.2.1. Derivation of surface R2 of non-circular gear generated by shaper. We remind that the shaper is not identical to the involute eccentric gear of the drive. The derivation is based on the following procedure: (1) Two coordinate systems Ss and S2 are considered rigidly connected to the shaper and the to be determined surface. (2) An involute tooth surface rs ðus ; vs Þ is considered as given for a shaper with pitch radius qs , that may differ from q1 . (3) Coordinate system Ss is rotated while coordinate system S2 is rotated and translated. Motions of systems Ss and S2 are defined by kinematic relation of their centrodes (see Fig. 7b). ðsÞ

ð2Þ

vI ¼ vI ;

ð18Þ

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(5) Non-circular gear tooth surface R2 is obtained by simultaneous consideration of matrix transformation r2 ðus ; vs ; h2 Þ ¼ M1s ðh2 Þrs ðus ; vs Þ

ð28Þ

and equation of meshing   or2 or2 or2 f ðus ; vs ; h2 Þ ¼  ¼ 0:  ous ovs oh2

ð29Þ

However, the approach proposed is based on matrix derivations (see below) that allows to computerize the derivations.

2.2.2.2. Coordinate transformation in transition from coordinate system Ss to S2 . Derivation of Eq. (28) is performed as follows: ðO2 Þ

r2 ðus ; vs ; h2 Þ ¼ M2n ðw2 ÞMnf ðxf

ðO2 Þ

; yf

ÞMfs ðws Þrs ðus ; vs Þ:

ð30Þ

Here 2

cos w2

6 6 sin w2 M2n ¼ 6 6 0 4

 sin w2

0

cos w2

0

0

1

0

0

0 2

cos ws

6 6 sin ws Mfs ¼ 6 6 0 4

 sin ws

0

cos ws

0

0

1

0

0

0

0

2

3

6 6 60 Mnf ¼ 6 60 4 0

7 07 7; 07 5 1 0

1

3

ðO2 Þ

0

0

xf

1

0

yf

0

1

0

0

0

1

ðO2 Þ

3 7 7 7 7 7 5

7 qs 7 7: 0 7 5 1

Fig. 7. For derivation of non-circular gear tooth surface R2 generated by a regular involute shaper.

2.2.2.3. Matrix derivation of equation of meshing f ðus ; vs ; h2 Þ ¼ 0. Matrix transformation (30) may be expressed by consideration of cartesian coordinates as

where

q2 ðus ; vs ; h2 Þ ¼ L2n Lnf Lfs qs ðus ; vs Þ þ R:

ð2Þ

vI

ð2Þ

ð2Þ

ð2Þ

¼ vI;rot þ vI;tr1 þ vI;tr2 :

ð19Þ

(3) Coordinate system Ss is rotated on ws that is related with polar angle h2 by function ws ¼

sðh2 Þ ; qs

ð20Þ

wherein sðh2 Þ is a function of the polar angle h1 of the eccentric gear sðh2 Þ ¼ q1 ðh1  arcsinðe1 sin h1 ÞÞ:

ð21Þ

(4) Coordinate system Sn is translated with system S2 on magniðO Þ ðO Þ tudes xf 2 and yf 2 , while system S2 is rotated on the magnitude w2 . Such magnitudes may be represented as functions of polar angle h2 ðO2 Þ

xf

ðO2 Þ

yf

¼ rðh2 Þ cos l2 ;

¼ rðh2 Þ sin l2 ; p w2 ¼ h2 þ l2  : 2

ð22Þ ð23Þ ð24Þ

Functions rðh2 Þ; h2 ; and l2 may be expressed as functions of polar angle h1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 rðh2 Þ ¼ E12  rðh1 Þ ¼ E12  q1 ð25Þ 1  e21 sin h1  e1 cos h1 ; 0 1 Z h1 c1 B C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 ¼  1Adh1 ; ð26Þ @ 2 2 0 c1  1  e1 sin h1 þ e1 cos h1 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 1  e21 sin h1 A: ð27Þ l2 ðh1 Þ ¼ arctan @ e1 sin h1

ð31Þ

The 3  3 matrices L are obtained from 4  4 matrices M; R is given by 2 3 ðO Þ ðO Þ xf 2 cos w2 þ yf 2 sin w2 þ qs sin w2 6 7 7 ðO Þ ðO Þ ð32Þ R¼6 4 xf 2 sin w2  yf 2 cos w2  qs cos w2 5: 0 ðs2Þ

Relative velocity v2 ðs2Þ v2

is given as

_ ¼ q_ 2 ¼ ðL_ 2n Lnf Lfs þ L2n Lnf L_ fs Þqs þ R:

Derivations of derivatives follows:   dh2 dl2 dh1 ; w_ 2 ¼ þ dh1 dh1 dt 1 dsðh2 Þ dh1 w_ s ¼ ; qs dh1 dt ðO2 Þ

ðO Þ x_ f 2 ¼

dxf

dh1 ðO2 Þ

ðO Þ y_ f 2 ¼

dyf

dh1

ðO Þ w_ 2 ; w_ s ; x_ f 2 ;

ð33Þ and

ðO Þ y_ f 2

are obtained as

dh1 ; dt dh1 : dt ðO2 Þ

dx

ðO2 Þ

dy

2 dl2 dsðh2 Þ Derivatives dh , ; dh1 , dhf 1 ; and dhf 1 may be obtained easily. See dh1 dh1 [28] for details of derivation. Then, equation of meshing may be determined as

ðs2Þ

ðs2Þ

f2 ðus ; vs ; h1 Þ ¼ n2  v2

¼ L2s ns  q_ 2 ¼ 0

ð34Þ

or as fsð2sÞ ðus ; vs ; h1 Þ ¼ ns  vð2sÞ ¼ ns  ðLs2 q_ 2 Þ ¼ 0: s

ð35Þ

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2.2.3. Generation of the non-circular gear by a hob 2.2.3.1. Installation of hob. Generation of non-circular gear tooth surface R2 by a worm thread surface Rw is being considered. For the purpose of simplicity, an imaginary rack-cutter tooth surface Rc is considered in continuous tangency with surfaces R2 and Rw . Fig. 8a shows worm, rack-cutter and non-circular gear at their initial positions. Point Of is the common point of tangency of three surfaces R2 ; Rw ; and Rc . Worm shaft and non-circular gear shaft are crossing by angle cw2 (see Fig. 8b). Angle cw2 is given as cw2 ¼

p  bc  kw ; 2

ð36Þ

wherein bc is the helix angle of the skew rack-cutter and kw is the lead angle of the worm. Worm thread surface may be considered, for the purpose of simplicity, as a surface with two independent parameters ðhw ; mw Þ. 2.2.3.2. Generation of surface R2 of non-circular gear by worm. The derivation of R2 by a worm thread surface Rw is based on following procedure (see Fig. 9): (i) Two coordinate systems Sw and S2 are considered rigidly connected to the worm thread surface and the to be determined gear tooth surface. A fixed reference system Sf is considered for definition of motions of systems Sw and S2 . (ii) Worm thread surface Rw is considered as given by vector Rw ðhw ; mw Þ.

Fig. 9. Applied coordinate systems for derivation of non-circular gear tooth surface R2 generated by a hob: (a) systems S2 and Sc of non-circular gear and imaginary rack-cutter; (b) system Sw of worm and auxiliary system Ss .

(iii) Two sets of motions are provided to the worm: (a) Rotation /w about axis zw of the worm. (b) Translation sw along axis zf that is parallel to the axis of the gear. Coordinate system Ss is a movable coordinate system that is translated with system Sw . (iv) Rotation and translation of the worm are accompanied by rotation and translation of the non-circular gear as follows: (a) Rotation w2 about axis z2 of the gear. ðO Þ (b) Translation defined by position yf 2 along axis yf . ðO2 Þ Magnitudes w2 and yf may be determined as functions of polar angle h2 of the non-circular gear as follows: ðO2 Þ

yf

ðh2 Þ ¼ rðh2 Þ sin l2 ðh2 Þ;

w2 ðh2 Þ ¼ h2 þ l2 ðh2 Þ  l20 ; Fig. 8. For derivation of non-circular gear tooth surface R2 generated by a hob: (a) installation of non-circular gear, worm, and imaginary rack-cutter; (b) installation of worm respect to imaginary rack-cutter.

ð37Þ l20

p ¼ : 2

ð38Þ

Since rðh2 Þ; h2 and l2 are related with polar angle h1 of the eccentric gear by functions

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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 rðh2 Þ ¼ E12  rðh1 Þ ¼ E12  q1 1  e21 sin h1  e1 cos h1 ; Z

0 h1

c1 B C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1Adh1 ; @ 2 c1  1  e21 sin h1 þ e1 cos h1 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 1  e21 sin h1 A; l2 ðh1 Þ ¼ arctan @ e1 sin h1

h2 ¼

ð39Þ

1 ð40Þ

0

ð41Þ

a new function gð/w ; sw ; h1 Þ ¼ 0 has to be determined in order to relate /w and sw with polar angle h1 . (v) Coordinate transformation between systems Sw and S2 determines the family of worm thread surfaces in system S2 as r2 ðhw ; mw ; /w ; sw Þ ¼ M2w ð/w ; sw ÞRw ðhw ; mw Þ:

ð42Þ

Here, /w and sw are independent generalized parameters of motion, that means that the generation is a double enveloping process; M2w 4  4 matrix describes coordinate transformation from system Sw to system S2 . (vi) Equations of meshing ðw2Þ

ðhw ; mw ; /w ; sw Þ ¼ 0;

ð43Þ

ðw2Þ

ðhw ; mw ; /w ; sw Þ ¼ 0;

ð44Þ

f1 f2

relate parameters ðhw ; mw ; /w ; sw Þ. Simultaneous consideration of Eqs. (42)–(44) determines surface R2 of non-circular gear.

2.2.3.3. Derivation of function. gð/w ; sw ; h1 Þ ¼ 0. The derivation is performed as follows (see Fig. 10): (i) An imaginary rack-cutter is considered in simultaneous meshing with the non-circular gear and the worm. At the initial position, system Sc coincides with system Sf and common tangent line t  t between the three surfaces, Rw , R2 ; and Rc , is at position t 0 . (ii) Due to rotation and translation of the worm on /w and sw , the common tangent t  t will take position t2 . The location ðO Þ of system Sc in Sf is determined by xf c . (iii) Displacement of system Sc may be obtained as the sum of ðO Þ ðO Þ independent displacements Dxf 1c and Dxf 2c ðOc Þ

Dxf

ðO Þ

ðO Þ

¼ Dxf 1c þ Dxf 2c :

ð45Þ

ðO Þ

Displacement Dxf 1c ¼ Of S is caused by translation sw and is defined ðO Þ by positions t 0 and t 1 . Displacement Dxf 2c ¼ SOc is resulted by rotation /w and is defined by positions t1 and t 2 . (iv) Illustrations of Fig. 10b yield the following relations: ðO Þ

 Dxf 1c ¼ Of S ¼ tan bc sw ; ðO Þ

 Dxf 2c ¼ SOC ¼

pw cos kw /w ; cos bc

ð46Þ ð47Þ

where pw is the pitch of the worm. ðO Þ (v) Since xf c depends on polar angle h1 as follows: ðOc Þ

xf

¼ sðh2 Þ þ rðh2 Þ cos l2 ðh2 Þ;

ð48Þ

0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 1  e21 sin h1 A: l2 ðh1 Þ ¼ arctan @ e1 sin h1

function gð/w ; sw ; h1 Þ ¼ 0 is obtained finally as ðOc Þ

gð/w ; sw ; h1 Þ ¼ xf

ð49Þ ð50Þ

ðh1 Þ þ tan bc sw þ

pw cos kw /w ¼ 0: cos bc

ð52Þ

2.2.3.4. Coordinate transformation in transition from coordinate system. Sw to S2 . Derivation of Eq. (42) is performed as follows: r2 ðhw ; mw ; /w ; sw Þ ¼ M2f ð/w ; sw ÞMfs ðsw ÞMsw ð/w ÞRw ðhw ; mw Þ: 2

 cos w2

6 6  sin w 2 M2f ¼ 6 6 4 0 0 2 cos cw2 6 0 6 Mfs ¼ 6 4 sin cwg

0

0

yf

 cos w2

0

yf

0

1

0 0  sin cw2

1

0

0

cos cwg

0 0 0 cos /w  sin /w

6 sin / 6 w Msw ¼ 6 4 0 0

ðO2 Þ

sin w2

sin w2

3

7 cos w2 7 7; 7 5 0

ðO2 Þ

0

3

1

qw 7 7 7; sw 5 1 3

0

0

cos /w

0

0

1

07 7 7 05

0

0

1 ðw2Þ

ð51Þ

ð53Þ

Here

2

wherein sðh2 Þ ¼ q1 ðh1  arcsinðe1 sin h1 ÞÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 rðh2 Þ ¼ E12  q1 1  e21 sin h1  e1 cos h1 ;

Fig. 10. For derivation of function gð/w ; sw ; h1 Þ ¼ 0.

2.2.3.5. Matrix derivation of equations of meshing f1 ðhw ; mw ; ðw2Þ /w ; sw Þ ¼ 0 and f2 ðhw ; mw ; /w ; sw Þ ¼ 0. Matrix derivation of equations of meshing is applied as follows:

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(i) Vector position r2 in homogeneous coordinates is given by r2 ðhw ; mw ; /w ; sw Þ ¼ M2w ð/w ; sw ÞRw ðhw ; mw Þ 2 3 ðO Þ a11 a12 a13 sin w2 ðyf 2  qw Þ 6 7 ðO Þ 6a a22 a23  cos w2 ðyf 2  qw Þ 7 21 7Rw ðhw ; ew Þ; ¼6 6 7 4 a31 a32 a33 5 sw 0

0

0

wherein L_ 2w ¼ L_ 2f Lfs Lsw and s_ w ¼ 

1 ð54Þ

ðw2Þ

a11 ¼  cos w2 cos cw2 cos /w þ sin w2 sin /w ;

ð55Þ

a12 ¼ cos w2 cos cw2 sin /w þ sin w2 cos /w ;

ð56Þ

a13 ¼ cos w2 sin cw2 ;

ð57Þ

a21 ¼  sin w2 cos cw2 cos /w  cos w2 sin /w ;

ð58Þ

a22 ¼ sin w2 cos cw2 sin /w  cos w2 cos /w ;

ð59Þ

a23 ¼ sin w2 sin cw2 ;

ð60Þ

a31 ¼ sin cw2 cos /w ;

ð61Þ

a32 ¼  sin cw2 sin /w ;

ð62Þ

a33 ¼ cos cw2 :

ð63Þ

(ii) Vector position r2 in cartesian coordinates may be represented as 2 3 a11 a12 a13 6 7 q2 ðhw ; mw ; /w ; sw Þ ¼ 4 a21 a22 a23 5qw ðhw ; mw Þ a31 2

a32

a33 ðO Þ

3

sin w2 ðyf 2  qw Þ 6 7 6 þ 4  cos w ðyðO1 Þ  q Þ 7 w 5 2 f sw ¼ L2w ð/w ; sw Þqw ðhw ; mw Þ þ R:

ð64Þ

Here, L2w is a 3  3 matrix obtained from M2w . Matrix L2w may be obtained as L2w ¼ L2f Lfs Lsw ;

ð65Þ

whereas vector R is defined as ðO2 Þ

R ¼ ½sin w2 ðyf

 qw Þ

ðO1 Þ

 cos w2 ðyf

 qw Þ

s w T :

ð66Þ

(iii) Considering sw as constant (sw ¼ c), the relative velocity of the worm thread surface with respect to gear tooth surface may be obtained as ðw2Þ _ v2;sw ¼c ¼ q_ 2 ¼ L_ 2w qw þ R;

ð67Þ

wherein L_ 2w ¼ L_ 2f Lfs Lsw þ L2f Lfs L_ sw

ð68Þ

and cos bc ðO Þ x_ c : /_ w ¼  pw cos kw f

1 ðO Þ x_ c ; tan bc f

ð74Þ

Then, equation of meshing may be obtained as f2

wherein

ð73Þ

ðw2Þ

¼ n2  v2;/w ¼c ¼ 0;

ð75Þ

(v) Derivations of derivatives   dh2 dl2 dh1 ; þ w_ 2 ¼ dh1 dh1 dt ðO2 Þ

ðO Þ y_ f 2 ¼

dyf

dh1 ðOc Þ

ðO Þ x_ f c ¼

dxf

dh1

Derivatives for details.

ðO Þ w_ 2 , y_ f 2 ; and

ðO Þ x_ f 2

are as follows: ð76Þ

dh1 ; dt

ð77Þ

dh1 : dt dh2 dl2 , dh1 dh1

ð78Þ ;

ðOc Þ f

dx

dh1

; and

ðO2 Þ f

dy

dh1

may be obtained easily. See [28]

3. Generation of the eccentric gear by providing localized contact Generation of the eccentric gear is performed as a regular involute gear. Localization of bearing contact is achieved by double crowning of the gear. Application of a grinding disk or a grinding worm may be applied as it is described in detail in [27]. A parabola coefficient apc is applied for profile crowning while a parabola coefficient apl is applied for longitudinal crowning of the eccentric gear tooth surface. 3.1. Tooth contact analysis The eccentric gear tooth surface R1 and the non-circular gear tooth surface R2 have been previously obtained in their own rigidly connected reference systems S1 and S2 , respectively. A fixed reference system Sf is considered for investigation of tooth contact along the cycle of meshing. Algorithm for tooth contact analysis of one single pair of teeth have been developed in [27]. Such an algorithm has to be applied here for each single pair of teeth. Path of contact and function of transmission errors have been investigated. Basic gear drive data are shown in Table 1. Fig. 11 shows paths of contact on tooth number one and eleven for various values of shaft angle error. Fig. 12 shows the transmission function along a whole revolution of the eccentric gear obtained by TCA. The theoretical transmission function may be derived as

ð69Þ

Then, equation of meshing may be obtained as ðw2Þ

f1

ðw2Þ

¼ n2  v2;sw ¼c ¼ 0;

ð70Þ

wherein n2 ¼ L2w nw :

ð71Þ

Here, nw is the unit normal to the worm thread surface. (iv) Considering /w as constant (/w ¼ c), the relative velocity of the worm thread surface with respect to gear tooth surface may be obtained as ðw2Þ

_ v2;/w ¼c ¼ q_ 2 ¼ L_ 2w qw þ R;

ð72Þ

Table 1 Design parameters Number of teeth of the eccentric pinion, N 1 Number of teeth of the non-circular gear, N 2 Module, m Pressure angle, a Helix angle, b Face width Parameter of eccentricity, e1 Parabolic coefficient for profile crowning, apc Radius of grinding disk, qD Parabolic coefficient for longitudinal crowning, alc

21 63 2.0 mm 20° 15° 25 mm 0.2 0:002 mm1 125.0 mm 0:0001 mm1

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Fig. 11. Contact paths on eccentric gear drive with localized bearing contact at pair of teeth (a) 1, and (b) 11, for several values of shaft crossing angle error.

Fig. 12. Transmission function at an eccentric gear drive with localized bearing contact.

Fig. 13. Function of transmission errors of an eccentric gear drive with localized bearing contact.

/2t  /20 ¼

Z

0 /1 /10

1

c1 B C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1Ad/1 : @ 2 c1  1  e2 sin /1 þ e1 cos /1

ð79Þ

Fig. 13 shows the function of transmission errors along a whole revolution of the eccentric gear, obtained as D/21 ¼ ð/2  /20 Þ  ð/2t  /20 Þ:

ð80Þ

F.L. Litvin et al. / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3783–3802

3793

4. Generation of functions by non-circular gears 4.1. Introduction The contents of this section covers generation of a function by a gear drive formed by: (i) a pair of non-circular gears (see Fig. 14), and (ii) two pairs of non-circular gears 1, 2, 3, 4 (see Fig. 15). The angles of rotation of gears 1 and 4 are proportional to the independent variable x and function yðxÞ, respectively. Application of two pairs of gears allows the pressure angle in each pair to be reduced. It follows from the kinematics of gear generation that function yðxÞ; x1 6 x 6 x2 to be generated has to be a monotonous increasing function which derivative y0 ðxÞ > 0. In the case wherein the derivative y0 ðxÞ is of a varied sign in the interval of derivation, the following process of generation is applied (Fig. 16): (a) Instead of function yðxÞ assigned for generation, is generated function y1 ðxÞ ¼ yðxÞ þ bx;

x1 6 x 6 x2 ;

ð81Þ

Fig. 16. Illustration of functions: y1 ðxÞ ¼ sin x þ bx; y2 ðxÞ ¼ x, yðxÞ ¼ a sin x.

and this allows to observe the requirement of y01 ðxÞ P 0: (b) Function yðxÞ assigned for generation will be obtained by the approach illustrated by Fig. 17: (i) Function y2 ðxÞ ¼ bx

ð82Þ

is subtracted from y1 ðxÞ by application of a gear differential. Two gear mechanisms formed: (a) by non-circular gears 1 and 2, and (b) circular gears 3 and 4, are applied. (ii) Gears 1 and 3 are mounted on the same shaft and angles /1 ; /3 are proportional to the variable x. The performed design of centrodes of gears 1 and 2 provides that angle of rotation of gear 2 is proportional to function (81); similarly, angle of rotation of gear 4 is proportional to function (82). (iii) Rotation of gears 2 and 4 are provided: (a) to the carrier c of the satellite s of the gear differential, and (b) to gear II of the differential, respectively. (iv) The differential provides the following relations between the angles of rotation of the carrier c and gears I and II (Fig. 17) [27] /I þ /II ¼ 2/c :

ð83Þ

Angle of rotation /c is equivalent to /2 , and angle of rotation /II is equivalent to /4 . Eq. (83) yields that /I ¼ 2/c  /II ; Fig. 14. Illustration /2;max ¼ 5p.

of

generation

of

function

yðxÞ ¼ 1x,

/c  /2 ;

/II  /4 ;

ð84Þ

1 6 x 6 3; /1;max ¼

Fig. 15. Schematic illustration of generation of function wðaÞ ¼ g 2 ½g 1 ðaÞ by two pairs of non-circular gears that generate respectively: b ¼ g 1 ðaÞ, d ¼ g 2 ðbÞ.

Fig. 17. Structure of gear mechanism formed by a bevel gear differential (level gears I and II, satellite s, and carrier c), non-circular gears 1 and 2, and circular gears 3 and 4.

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F.L. Litvin et al. / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3783–3802

where /2 is proportional to y1 ðxÞ represented by function (81) and /4 is proportional to y2 ðxÞ represented by function (82). We assign for the design that /4 ¼ 2b/3 ;

/3  /1

ðy2 ðxÞ ¼ 2xÞ:

ð85Þ

(v) Eqs. (84) and (85) yield that angle /I of rotation of gear I of the differential will be obtained as proportional to function yðxÞ assigned for design /I ¼ 2/2  /4 ¼ 2ðyðxÞ þ bxÞ  2bx ¼ 2yðxÞ:

ð86Þ

Variation of function yðxÞ by the magnitude and sign will cause variation of angle /I of rotation of gear I of the differential. This means that gear I will be rotated with varied angular velocity in two directions.

The derivative y0 ðxÞ is varying its sign and therefore we apply the scheme of generation represented by Fig. 17. Non-circular gears 1 and 2 have to be designed for generation of function 0 6 x 6 2p:

ð87Þ

Design of non-circular gears 1 and 2 (Fig. 17) has to cover determination of their centrodes by application of the following procedure. (i) The angles of rotation of gears 1 and 2 are represented by the equations /1 ¼ k1 ðx  x1 Þ;

/2 ¼ k2 ½y1 ðxÞ  y1 ðx1 Þ;

x1 ¼ 0:

k1 ¼

/1;max ; x2  x1

k2 ¼

/2;max ; y1 ðx2 Þ  y1 ðx1 Þ

ð88Þ

x2 ¼ 2p;

x1 ¼ 0:

ð89Þ

Taking that gears 1 and 2 will perform in the process of generation turns on /1;max ¼ /2;max ¼ 2p, we obtain that k1 ¼ 1;

k2 ¼

1 : b

ð90Þ

(ii) The coefficient b may be determined by observation of the following conditions: (a) y0 ðxÞ > 0, and (b) centrodes of gears 1 and 2 have to be the convex ones. Condition (a) is observed by b > 1. Detailed derivations for observation of condition (b) is represented in [25]. The final result is that blim P 1:707:

4.2. Problem 1: generation of function yðxÞ ¼ a sin x; 0 6 x 6 2p

y1 ðxÞ ¼ sin x þ bx;

Here, k1 and k2 are scale coefficients determined as

ð91Þ

Observation of condition blim ¼ 1:707 means that centrode 1 will have a point with curvature j1 ¼ 0. (iii) Eqs. (88) with coefficients (90) yield the following transmission function /2 ð/1 Þ ¼ /1 þ

1 sin /1 ; b

0 6 /1 6 2p:

ð92Þ

The derivative function is m12 ð/1 Þ ¼

d/1 1 ¼ : d/2 1 þ 1b cos /1

ð93Þ

(iv) Equations (A.5) and (A.6), (A.7) yield the following equations for centrodes r1 and r2 of non-circular gears 1 and 2 (Fig. 17): For r1 , we have

Fig. 18. Illustration of: (a) centrodes of non-circular gears for generation of function y1 ðxÞ with coefficient b ¼ 1:707; (b) transmission function /2 ð/1 Þ wherein coefficient b ¼ 1:707.

F.L. Litvin et al. / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3783–3802

r 1 ð/1 Þ ¼

1 þ 1b cos /1 E ¼E : 1 þ m12 ð/1 Þ 2 þ 1b cos /1

ð94Þ

4.3. Problem 2: Generation of function yðxÞ ¼ 1x, x1 6 x 6 x2

ð95Þ

4.3.1. Introduction The specific features of centrodes applied for generation are: (i) centrodes r1 and r2 are represented as unclosed curves, (ii) they are identical, (iii) the gears may perform rotation on angles / > 2p (Fig. 14), while performing simultaneously axial translation.

For r2 , we have r 2 ð/1 Þ ¼ E

1 ; 2 þ 1b cos /1

/2 ð/1 Þ ¼ /1 þ

1 sin /1 : b

Center distance E is just a scale coefficient. Centrodes r1 and r2 determined with coefficients b ¼ 1:707 and b ¼ 1:400, are represented in Figs. 18 and 19. Centrode 1 in Fig. 19 is a convex–concave one. (v) We assign for the design that /4 ¼ 2/3 ;

/3 ¼ /1 :

3795

ð96Þ

(iv) Function yðxÞ ¼ a sin x will be obtained from gear I of the differential as   1 2 /I ¼ 2/c  /II ¼ 2/2  /4 ¼ 2 /1 þ sin /1  2/1 ¼ sin /1 ; b b ð97Þ

4.3.2. Centrodes and transmission function The angles of rotation /1 and /2 of centrodes 1 and 2 are proportional to variable x and function yðxÞ, respectively. Thus we have /1 ¼ k1 ðx  x1 Þ; /2 ¼ k2 ðy1  yÞ ¼ k2



 1 1 ;  x1 x

ð98Þ ð99Þ

where k1 and k2 are scale coefficients determined as k1 ¼

/1;max ; x2  x1

k2 ¼

ðx1  x2 Þ/2;max : x2  x1

ð100Þ

The derivative function is

wherein 2b ¼ a.

m12 ¼

Non-circular gears with the developed centrodes may be generated by the enveloping method by using a hob for centrodes shown in Figs. 18 and 19.

where

ða3 þ a4 /1 Þ2 ; a2 a3

a2 ¼ k2 ;

a3 ¼ k1 x21 ;

ð101Þ

a4 ¼ x1 :

ð102Þ

Fig. 19. Illustration of: (a) centrodes of non-circular gears for generation of function y1 ðxÞ with coefficient b ¼ 1:400; (b) transmission function /2 ð/1 Þ wherein coefficient b ¼ 1:400.

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F.L. Litvin et al. / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3783–3802

The centrodes are represented as follows 1 a2 a3 ¼E r1 ð/1 Þ ¼ E ; 1 þ m12 ð/1 Þ a2 a3 þ ða3 þ a4 /1 Þ2 r2 ð/1 Þ ¼ E  r1 ð/1 Þ ¼ E

ða3 þ a4 /1 Þ2

; a2 a3 þ ða3 þ a4 /1 Þ2 a2 /1 /2 ð/1 Þ ¼ Fð/1 Þ ¼ : a3 þ a4 /1

Here g 1 ðaÞ  bðaÞ; g 2 ðaÞ  dðcðaÞÞ are the transmission functions respectively of the two pairs of gears of the drive (Fig. 15). ð103Þ ð104Þ ð105Þ

The centrodes are identical ones by observation the conditions /1;max ¼ /2;max , and satisfaction of the functional (see Section A.1.7) that is represented for discussed example of design as a2 ð/1;max  Fð/1 ÞÞ ¼ /1;max  /1 : a3 þ a4 ð/1;max  Fð/1 ÞÞ

ð106Þ

The conditions above are satisfied and the identical centrodes are represented by Fig. 14. 4.4. Generation of function by gear drive by application of two pairs of non-circular gears 4.4.1. Introduction Generation of given function wðaÞ; a  /1 , by a gear drive with two pairs of gears (1, 2) and (3, 4) (instead of one pair) has the following advantages: (i) a larger variation of derivative ow may be provided. oa (ii) lesser pressure angle of each of the two pairs of non-circular gears may be obtained. Fig. 15 shows schematic of the gear drive formed by the centrodes of the gears of the drive. Each of the centrodes 1, 2, 3, 4 performs rotation about point Oi (i ¼ 1; 2; 3; 4). The relative motion of each of centrode i with respect to the mating centrode of a pair of centrodes (1, 2) and (3, 4) is pure rolling. Depending on the type of function wðaÞ to be generated, the centrodes might be closed curves or unclosed ones. We may consider initial and final positions of the centrodes that correspond to the beginning of motion (where /i ¼ 0, i ¼ 1; 2; 3; 4), and the end of motion (where /i ¼ /i;max ). In the case of centrodes as closed curves, we have: /1;max ¼ /2;max ¼ /3;max ¼ /4;max ¼ 2p. In the case of unclosed curves, we have: /1;max ¼ /2;max ¼ /3;max ¼ /4;max < 2p. The maximal angles of rotation of gears with unclosed centrodes may be chosen /i;max > 2p if in the process of motion the gears perform correlated axial motions, in addition to rotation (see Fig. 14). 4.4.2. Functional of gear drive with two pairs of gears We notify as functional an equation that correlates function wðaÞ to be generated with the transmission functions of the two pairs of gears that form the gear drive (Fig. 15). We may consider two cases: (i) Centrodes (1, 3) are designed as identical ones. Centrodes (2, 4) (that differ from (1, 3)) are designed as identical ones as well. Then the functional of the gear drive may be represented as wðaÞ ¼ f ðf ðaÞÞ;

ð107Þ

where f ðaÞ is the transmission function of each pair of noncircular gears. Here, a  /1 is the parameter used for representation of a centrode as a parametric curve. (ii) There are no identical centrodes among of (1, 2, 3, 4) of the gear drive (Fig. 15) and the functional of the drive is represented as wðaÞ ¼ g 2 ðg 1 ðaÞÞ:

ð108Þ

In case (i), considering as given function wðaÞ, it is necessary to determine function f ðaÞ. Usually, it might be done by an iterative process of computation. The first guess for such a process have been proposed by Litvin [25–27] and Kislitzin [19]. Litvin’s approach is based on Mean-Value Theorem that is proposed by Lagrange and is represented as f ða þ hÞ  f ðaÞ ¼ f 0 ða þ hhÞ; h

0 < h < 1:

ð109Þ

The approach provides the following approximate solutions of functional (107) [25–27] f  ðaÞ ¼

pffiffiffiffiffiffiffiffiffiffiffi wðaÞ þ a w0 ðaÞ pffiffiffiffiffiffiffiffiffiffi ffi : 1 þ w0 ðaÞ

ð110Þ

In case (ii) (for missing of application of identical centrodes), observation of functional (108) may be satisfied by: (i) choosing one transmission function, of the pair of functions g 2 ðaÞ and g 1 ðaÞ, say g 1 ðaÞ; (ii) then g 2 ðaÞ may be obtained from functional (108) considering as known g 1 ðaÞ and wðaÞ. In this paper it is proposed to choose g 1 ðaÞ ¼ f  ðaÞ (see (110) for f  ðaÞ). This allows obtain close to each other: dimensions of centrodes (1, 3); respectively, dimensions of (2, 4). Considering such a particular case wherein the gear drive of Fig. 15 is formed by two pair of elliptical gears with the same eccentricity we obtain the following expressions: h  a i wðaÞ ¼ 2 arctan a tan ; ð111Þ 2 h pffiffiffi a i  : ð112Þ g 1 ðaÞ ¼ f ðaÞ ¼ g 2 ðcðaÞÞ ¼ 2 arctan a tan 2 Eq. (112) yields that eccentricity of elliptical gears of the drive is reduced and pressure angle of each pair of elliptical gears is also reduced. 4.4.3. Functional (108): derivation of centrodes of gear drive shown in Fig. 15 (1) The derivations are illustrated with the example of generation of function y ¼ ln x;

1 6 x 6 100

ð113Þ

by application of two pairs of non-circular gears (1, 2, 3, 4) (Fig. 15). The centrodes of the gears are not closed curves, the maximal angles of rotation are assigned in the numerical example as amax ¼ bmax ¼ cmax ¼ dmax ¼

10p : 6

ð114Þ

(2) The centrodes are represented as polar curves. (3) Centrode of gear 1 is represented by the equation r1 ðaÞ ¼

E ; 1 þ m12

m12 ¼

1 ; g 01 ðaÞ

a  /1 ;

E ¼ r1 ðaÞ þ r 2 ðaÞ: ð115Þ

Here (see details of derivation in Appendix and references [25,26]): is the derivative of the transmission function /2 ð/1 Þ  g 1 ðaÞ 

1 g 01 ðaÞ 

f ðaÞ (see Eq. (110)); E is the center distance of centrodes 1 and 2; wðaÞ  dðaÞ is the function assigned for generation by the gear drive shown in Fig. 15. The derivation of wðaÞ in terms of x and ln x is based on following relations a ¼ k1 ðx  x1 Þ ¼ k1 ðx  1Þ;

dðxÞ ¼ k2 ðln x  ln x1 Þ ¼ k2 ln x

ð116Þ

F.L. Litvin et al. / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3783–3802

Coefficients k1 and k2 are determined as amax 10p=6 ¼ 0:052889; ¼ 99 x2  x1 dmax 10p=6 k2 ¼ ¼ 1:136980: ¼ ln x2  ln x1 ln 100

k1 ¼

Acknowledgements ð117Þ ð118Þ

(4) Centrode of gear 2 (Fig. 15) is represented as follows (see Appendix and references [27,28]) r 2 ðaÞ ¼ E

m12 1 ; ¼E 1 þ g 01 ðaÞ 1 þ m12

/2 ð/1 Þ ¼ g 1 ðaÞ:

ð119Þ

(5) The derivation of centrodes of gears 3 and 4 is based on the following equations 1 r3 ¼ E ; /3 ¼ /2 ¼ g 1 ðaÞ; ð120Þ 1 þ m34 ðaÞ m34 ðaÞ ; /4 ¼ wðaÞ: r4 ¼ E ð121Þ 1 þ m34 ðaÞ Here, /3 ðaÞ ¼ /2 ðaÞ ¼ g 1 ðaÞ;   a þ1 ; /4 ðaÞ ¼ wðaÞ ¼ k2 ln k1 d/3 g 01 ðaÞ m34 ¼ ¼h  i0 : d/4 k ln a þ 1 2

3797

ð122Þ ð123Þ ð124Þ

k1

Centrodes of gears 1, 2, 3, 4 are represented by Fig. 20. The drawings confirm that the dimensions of centrodes 1 and 3 are close each to other. Similarly, may be mentioned for the dimensions of centrodes 2 and 4. Such relations of dimensions is the result of application of Eq. (110).

The authors express their deep gratitude to Yamaha Motor Company and the Spanish Ministry of Education and Science (project reference DPI2007-63950 financed jointly by FEDER) for the financial support of respective research projects. The authors express as well their deep gratitude to Kenji Yukishima (Yamaha Marine Company) for his participation in the project of non-circular gears.

Appendix A. Tutorial aspects and ideas of design A.1. Centrodes A.1.1. Basic concepts Centrodes of a gear drive with conjugated non-circular gears are planar curves r1 and r2 (Fig. 21a). The centrodes perform rotation about O1 and O2 with the following conditions: (i) The center distance jO1 O2 j ¼ E is constant. (ii) The transmission function /2 ð/1 Þ is a monotonous increasing function; the derivative function is identified as 1 m12 ¼ d/ . d/2 (iii) Centrodes r1 and r2 are in tangency at any instant at a point I that belongs to center distance O1 —O2 , and moves along O1 —O2 in the process of motion. (iv) Centrodes r1 and r2 rolls over each other. (v) Point I is the instantaneous center of rotation in relative motion.

Fig. 20. The gear mechanism for generation of function y ¼ ln x is based on two pairs of non-circular gears (Fig. 15): (a) shows centrodes 1 and 2; (b) shows centrodes 3 and 4; (c) shows functions bðaÞ and dðaÞ.

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Fig. 21. Illustration of: (a) centrodes r1 and r2 and angles /i and hi (i ¼ 1; 2); (b) and (c) polar axis Oi Ai , position vectors r i ðhi Þ and angles li ði ¼ 1; 2Þ.

Assuming that on of the centrodes, say r1 , is held at rest, centrode r2 is rotated about I with angular velocity xð21Þ ¼ xð2Þ  xð1Þ :

ð125Þ

Parameters /1 and /2 represent the angles of rotation of centrodes r1 and r2 (gears 1 and 2) about centers O1 and O2 , respectively (Fig. 21a). Centrodes r1 and r2 may be represented in polar form with parameters hi (i ¼ 1; 2, see Fig. 21a). Fig. 21a and b show polar axes O1 A1 and O2 A2 and location of position vectors ri ðhi Þ (i ¼ 1; 2) in fixed coordinate system. Orientation of common tangent t to centrodes r1 and r2 is determined by angle li (Fig. 21b and c), as tan li ¼

ri ðhi Þ ; dr i =dhi

ði ¼ 1; 2Þ;

l1 þ l2 ¼ p:

ð126Þ

Derivations of (126) is illustrated with drawings of Fig. 22.

vðiÞ ¼ xðiÞ  Oi I;

ði ¼ 1; 2Þ:

Here vð1Þ ¼ vð2Þ , and the relative velocity vð12Þ ¼ 0; vð21Þ ¼ 0, since the centrodes are rolling over each other. Eq. (126) yield that the derivative m12 may be represented as m12 ¼

xð1Þ r2 ð/1 Þ E  r1 ð/1 Þ ¼ : ¼ r 1 ð/1 Þ xð2Þ r1 ð/1 Þ

ð128Þ

Then we obtain the following equations of centrodes: (i) For r1 , we have r 1 ð/1 Þ ¼

E : 1 þ m12 ð/1 Þ

ð129Þ

(ii) Centrode r2 is determined by the following system of equations r 2 ð/1 Þ ¼ E

m12 ð/1 Þ ; 1 þ m12 ð/1 Þ

A.1.2. Equations of centrodes Fig. 23 shows in fixed coordinate systems position vectors r1 ð/1 Þ ¼ jO1 Ij; and r2 ð/2 Þ ¼ jO2 Ij of centrodes at point I. Linear velocities vð1Þ and vð2Þ in rotation about O1 and O2 are

Fig. 22. For derivation of angle l formed by extended position vector rðhÞ and tangent t to polar curve.

ð127Þ

Fig. 23. For derivation of centrodes r1 and r2 .

ð130Þ

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F.L. Litvin et al. / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3783–3802

/2 ð/1 Þ ¼

Z 0

/1

d/1 : m12 ð/1 Þ

ð131Þ

A.1.3. Transmission function /2 ð/1 Þ Function /2 ð/1 Þ is determined by Eq. (131). It has to be a monotonous increasing function. Determination of /2 ð/1 Þ may need numerical integration. Although function /2 ð/1 Þ relates angles of rotation of rolling centrodes r1 and r2 , it may serve as well as a transmission function of the gear drive wherein rotation is performed due to contact of conjugated teeth with which are provided the gears of the drive. Conjugation of gear teeth is obtained due to enveloping process of gear generation wherein the centrode rc of the generating tool rolls over the centrodes r1 and r2 . An analytical approach for determination of /2 ð/1 Þ for elliptical gears is represented in [28].

In the case of a misaligned gear drive formed by non-circular gears, determination of transmission function /2 ð/1 Þ needs application of TCA (Tooth Contact Analysis). A.1.4. Centrodes as closed form curves A.1.4.1. Representation of closed form centrodes. A centrode as a closed form curve has to satisfy the following requirements: rð/Þ/¼0 ¼ rð/Þ/¼2p ;

ðiÞ ðiiÞ

ð132Þ

lð/Þ/¼0 ¼ lð/Þ/¼2p :

ð133Þ

Here: rð/Þ is the position vector of centrode represented as polar curve; lð/Þ is the angle formed by centrode position vector and tangent t to the centrode (Fig. 21b and c). Centrode r1 of gear 1 (see Eq. (129)) will satisfy requirements (132) and (133) with the following condition: (i) derivative m12 ð/1 Þ is a periodic function of period /1 ¼ 2p=n1 ; and n1 is an integer number. Centrode r2 is a closed form curve by observation of following requirements: (a) T1 T2 ¼ n1 n2

ð134Þ

and (see Eq. (131)) (b) Z n2p 1 2p r1 ð/1 Þ : ¼ /2 ð/1 Þ ¼ n2 E  r 1 ð/1 Þ 0

ð135Þ

Here: (i) n1 and n2 are integer numbers; and (ii) the proper value of center distance E has to be determined by application of Eq. (135). Let us consider the following practical case. The gear drive is formed by: (i) an elliptical gear with major axis 2a and eccentricity e, and (ii) conjugated gear 2 that performs one revolution for n revolutions of gear 1. The solution of Eq. (135) yields that E ¼ 1 þ ½1 þ ðn2  1Þð1  e2 Þ0:5 a

ð136Þ

as the requirement for obtaining r2 as a closed form curve. A.1.5. Examples of closed form centrodes The examples cover gear drives formed: (i) by an elliptical gear 1 and conjugated gear 2 with n ¼ 2 and n ¼ 3; (ii) two identical oval gears; (iii) a gear drive with an oval gear 1 and conjugated gear 2 wherein gear 2 performs one revolution for n ¼ 4 revolutions of oval gear 1. An example of identical unclosed centrodes r1 and r2 is represented in Section 4.3.2 as the result of generation of function yðxÞ ¼ 1x by non-circular gears. A.1.6. Principle of modification of elliptical centrodes and applications The principle of modification of elliptical centrodes is illustrated as follows (Fig. 24): (i) The current position vector Oi M of conventional ellipse is given as (Fig. 24a) O1 M ¼ rð/1 Þ;

0 6 /1 6 p: 

Fig. 24. Illustration of modification of elliptical centrodes: (a) and (b) for representation of coefficients mI and mII ; (c) for representation of conventional and modified ellipse.

ð137Þ

(ii) The respective point M of the modified centrode is determined as   /1 O1 M  ¼ r 1 ð138Þ ; 0 6 /1 6 p; jr1 j ¼ jr1 j: mI

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Fig. 25. Illustration of: (a) elliptical and modified centrodes; (b) asymmetric gear ratio function: I and II are the running and working parts of a cycle.

(iii) The same principle of modification is applied for the lower part of the ellipse; the modification coefficient is mII 6¼ mI . Fig. 25a shows a pair of conventional ellipses and the modified ellipses. Fig. 25b shows that application of modified elliptical centrodes allows to obtain for design an asymmetric transmission function m21 ð/1 Þ. Oval gears (Fig. 26a) may be considered as obtained by modification of a conventional ellipse. Fig. 26b shows a gear drive

formed by a conventional elliptical centrode and a conjugated centrode 2. The gear drive is designed to provide n ¼ 3 revolutions of gear 1 for one revolution of gear 2. It is important to recognize that centrode 2 is formed by 3 arcs of modified ellipses. We introduce in this paper the application of the idea of modification of an ellipse to the design of gear drives with lobes (Fig. 27). The tooth surfaces of lobes are arcs of modified ellipses. The design of gears with lobes is based on the following equations

Fig. 26. Illustration of: (a) oval centrodes; (b) a gear drive formed by a conventional elliptical centrode 1 and a conjugated 2 one.

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p a1 ð1  e2I Þ r 1 ð/1 Þ ¼ ¼ ; 1  eI cos mI /1 1  eI mI /1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ¼ a1 1 þ 1 þ ðn2  1Þð1  e2I Þ ;

ð139Þ ð140Þ

r 2 ð/1 Þ ¼ E  r1 ð/1 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ðn2  1Þð1  e2I Þ þ eI mII /2 mI /1 ¼ tan : tan 2 nðI  eI Þ 2

ð141Þ ð142Þ

Here: 2a1 and e1 are the major axis and the eccentricity of a conventional ellipse; n ¼ mmIII is the ratio of lobes of gears 1 and 2. A.1.7. Functional of identical centrodes There is only a small number of examples of identical centrodes that have been discussed in the literature [35]. The new examples of identical centrodes that are represent in this paper cover gear drives: (i) with elliptical gears, (ii) of lobes with an equal number of lobes (Fig. 27a), and (ii) mating non-circular gears applied for generation of function yðxÞ ¼ 1x (Section 4.3). The purpose of discussion below is to develop a functional for analytical recognition of existence of identical centrodes.

The concept considerations:

of

the

functional

is

based

on

following

(i) Fig. 28 shows a transmission function /2 ð/1 Þ for a cycle of meshing of a drive with identical centrodes 1 and 2. Points Ao and Bo correspond to the beginning and the end of the cycle. (ii) Assume that at the positions of the centrodes that correspond to Bo , rotation is provided from centrode 2 to centrode 1 in direction that is opposite to the previous one. The transformation function /1 ð/2 Þ is an inverted one with respect to /2 ð/1 Þ if the centrodes are identical indeed. (iii) It is provided by design that /2;max ¼ /1;max . (iv) Transmission function /2 ð/1 Þ consists of two symmetrical parts: Ao Am and Bo Bm where Am  Bm is the mean point of the transmission function /2 ð/1 Þ. (v) Representation of coordinates of point B by U2 ¼ d and U1 ¼ b; and q and c provide relations between parameters ð/1 ; /2 Þ and (U2 ; U1 ). (vi) The derivation of the functional is based on following equations (Fig. 28): /1 ¼ U2 ¼ b;

ð143Þ

/2 ¼ U1 ¼ d;

ð144Þ

c ¼ /1;max  U1 ¼ /1;max  d ¼ /1;max  /2 ¼ /1;max  Fð/1 Þ; ð145Þ where Fð/1 Þ ¼ /2 ð/1 Þ. Finally, it is obtained that c ¼ /1;max  Fð/1 Þ:

ð146Þ

(vii) Transformation of q is performed as q ¼ /2;max  U2 ¼ /1;max  b ¼ /1;max  /1 :

ð147Þ

(viii) Parameters q and c are related as q ¼ FðcÞ ¼ Fð/1;max  Fð/1 ÞÞ: Fig. 27. Illustration of gear drives with lobes.

(see Eq. (146))

Fig. 28. For derivation of functional for identical centrodes.

ð148Þ

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F.L. Litvin et al. / Comput. Methods Appl. Mech. Engrg. 197 (2008) 3783–3802

(ix) Eqs. (147) and (148) yield Fð/1;max  Fð/1 ÞÞ ¼ /1;max  /1 :

ð149Þ

Eq. (149) is the sought for expression of the functional for a gear drive with identical centrodes.

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