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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-Perezb,*, Alfonso Fuentesb, Kenichi Hayasakac aDepartment of Mechanical and Industrial Engineering, University of Illinois at Chicago, United States bDepartment of Mechanical Engineering, Polytechnic University of Cartagena, Campus Universitario Muralla del Mar, C/ Doctor Fleming, s/n 30202 Cartagena, Spain cGear R (b) generation of functions by non-circular gears. The developed theory is illustrated with numerical examples. ? 2008 Elsevier B.V. All rights reserved. 1. Introduction Development of generation of non-circular gears 12,2328 makes the manufacture of such gears as easy as circular gears. Therefore it is not surprising that design of gear drives with non- circular gears became the subject of research of many scientists 137. 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 w/ ff/;1 where w/ is the given function, f/ is the transmission function of a pair of non-circular gears. (3) Application of modifi ed elliptical gear (Section A.1.6) has allowed design of: (i) a gear drive with an asymmetric trans- mission function, and (ii) lobes (which centrodes are modi- fi ed ellipses). (4) Algorithms for determination of tooth surfaces of non-circu- lar gear generated by a shaper or by a hob are proposed. A simple approach for determination of avoidance of under- cutting 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 abbre- viation 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 0045-7825/$ - see front matter ? 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2008.03.001 * Corresponding author. Tel.: +34 968 326429; fax: +34 968 326449. E-mail address: ignacio.gonzalezupct.es (I. Gonzalez-Perez). Comput. Methods Appl. Mech. Engrg. 197 (2008) 37833802 Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: /locate/cma to generate the non-circular gear by a hob (in case of a convex cen- trode) and by a shaper (if the centrode is a convexconcave one). The performed investigation covers the basic topics of geome- try, design, and generation of eccentric drives. 2.1.2. Equations of mating centrodes Fig. 2a and b shows centrodes r1and r2in initial and current positions Centrode r1is an eccentric circle of radius rp1and is represented in polar form as (Fig. 2b) r1h1 r2 p1? e 2 1sin 2 h1 1 2? e1cosh1; 2 where parameter h1determines the position of r1h1 with respect to polar axis O1A1(Fig. 2b). Henceforth we will use representation of a centrode in terms of derivative m12/1 (see Section A.1.2 of Appendix), where /i hi (i 1;2) is the angle of rotation of the centrodes; parameter hi determines location of rihi with respect to polar axis OiA (see Sec- tion A.1.2). For r1, we have r1/1 E 1 m12/1 :3 Centrode r2is represented by two following equations r2/1 E m12/1 1 m12/1 ;4 /2/1 Z /1 0 d/1 m12/1 :5 Fig. 1. Eccentric involute pinion and conjugated non-circular gear: (a) for helical drive, (b) for a planar drive; O1and O2centers of rotation; O the geometric center. Fig. 2. For derivation of centrodes: (a) initial position; (b) current position. 3784F.L. Litvin et al./Comput. Methods Appl. Mech. Engrg. 197 (2008) 37833802 Here, the derivative function is m12/1 E ? r1/1 r1/1 c1 1 ? e2 1sin 2 /1 1 2? e1cos/1 6 and c1 E rp1;e1 e1 rp1. Function /2/1 is the transmission function and its determina- tion requires numerical integration. Centrode r1is already a closed form curve (a circle of radius rp1). Centrode r2might be a closed form curve by observation of equation /2/1 2p n Z 2p 0 d/1 m12/1 ;7 wherein n is the number of revolutions that eccentric gear 1 per- forms for one revolution of the non-circular gear. Derivation of centrodes r1and r2enables to design as well a cam mechanism with a constant center distance between the fol- lower 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 r2and applications . Basic derivations. Knowledge of curvature of centrode r2is 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 profi les. Considering that a centrode is represented by a polar curve rh, its curvature radius my be determined as 20,27: qh r2 dr dh ? ?2 hi3 2 r2 2 dr dh ? ?2 ? r d2r dh2 :8 The condition of convexity of the polar curve may be represented as j 1 q r2 2 dr dh ?2 ? r d2r dh2 P 09 or by an alternative inequality sinl r2 ? sin2l ? sin2l d2r dh2 “# P 0:10 Here, angle l arctan r dr=dh ? is formed by position vector rh and 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 r1is known, determination of j2is obtained by using EulerSavary equation 18, that relates curvatures j1and j2of centrodes r1and r2of the eccentric gear drive (Fig. 3) as follows: 1 q1 1 q2 1 r1/1 1 r2/1 ? sinl1:11 Here 1 q1 1 rp1 j1; 1 q2 j2;12 where q2 jIA2j (Fig. 3), r2/1 E ? r1/1;Fig:3:13 Eq. (11) yields j2 1 r1/1 1 E ? r1/1 ? sinl1? j1:14 Convexity of centrode r2is guarantied if 1 r1/1 1 E ? r1/1 ? sinl1? j1P 0:15 Fig. 4 illustrates function q2/1 for gear drives with: (a) a convex centrode r2; (b) a convexconcave r2. The location and orientation of curvature radius q2/1 is visu- alized by application of a four-bar linkage (Fig. 3) with links: (a) link 1 as O1A1, (b) link 2 as A1A2, (c) link 3 as O2A2, and (d) link 4 as O1O2, where jA1A2j q1 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 perpendic- ular to O1O2and its magnitude is determined as jIFj r1/1 ? q1sinl1 q1r1/1 jq2sinl1? r2/1j q2r2/1 :16 . Avoidance of undercutting. Drawings of Fig. 5 show genera- tion of an involute spur gear with a rack-cutter of profi le 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 sin2ac P rp:17 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 profi les, but they may be represented (approximately) as tooth profi les of respec- tive circular gears with curvatures radii qA;qB;.;qK(Fig. 6). (ii) Radius qA 2 of curvature of centrode r2for point A corre- sponds to profi les of the tooth notifi ed as RA. Similarly, pro- fi les of tooth RBare represented as ones of spur gears of radius qB 2 , 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 of radius qA 2 . (iv) Fig. 6a shows representation of tooth profi les of gear 2 with centrode r2designed for a gear drive with n 3. Here, n is the number of revolutions of eccentric gear 1 that is per- formed 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 r1and r2. For this purpose, it is neces- sary to obtain functions qA 2 e;n (Fig. 6b) that represents the Fig. 3. For derivation of curvature radius q2/1 of centrode r2. F.L. Litvin et al./Comput. Methods Appl. Mech. Engrg. 197 (2008) 378338023785 minimal curvature radius of qA 2 ; A is the point of centrode r2 (Fig. 6a) where q2 q2;min; e e rp (see Eq. (6). (vi) Avoidance of undercutting for all eccentric gear drives is obtained by choosing such a module with which functions qA 2 m will be out of the square with the height 1=sin2ac (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 re- late the motions of the generating tool (shaper, hob) and non-cir- cular of the drive being generated. Such relations are represented Fig. 4. Illustration of function q2/1 for eccentric gear drives with design parameters rp1 22:35 mm; n 4, and (a) e 0:2, (b) e 0:7. Fig. 5. Illustration of avoidance of undercutting of a spur involute pinion as the condition: m 0. In the case wherein the derivative y0x is of a varied sign in the interval of derivation, the following process of generation is ap- plied (Fig. 16): (a) Instead of function yx assigned for generation, is generated function y1x yx bx;x16 x 6 x2;81 and this allows to observe the requirement of y01x P 0: (b) Function yx assigned for generation will be obtained by the approach illustrated by Fig. 17: (i) Function y2x bx82 is subtracted from y1x 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;/3are 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); sim- ilarly, angle of rotation of gear 4 is proportional to func- tion (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)Thedifferentialprovidesthefollowingrelations between the angles of rotation of the carrier c and gears I and II (Fig. 17) 27 /I /II 2/c:83 Angle of rotation /cis equivalent to /2, and angle of rotation /IIis equivalent to /4. Eq. (83) yields that /I 2/c? /II;/c? /2;/II? /4;84 Fig.14. Illustrationofgenerationoffunctionyx 1 x, 1 6 x 6 3;/1;max /2;max 5p. Fig. 15. Schematic illustration of generation of function wa g2g1a? by two pairs of non-circular gears that generate respectively: b g1a, d g2b. Fig. 16. Illustration of functions: y1x sinx bx;y2x x, yx asinx. 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. F.L. Litvin et al./Comput. Methods Appl. Mech. Engrg. 197 (2008) 378338023793 where /2is proportional to y1x represented by function (81) and /4is proportional to y2x represented by function (82). We assign for the design that /4 2b/3;/3? /1y2x 2x:85 (v) Eqs. (84) and (85) yield that angle /Iof rotation of gear I of the differential will be obtained as proportional to function yx assigned for design /I 2/2? /4 2yx bx ? 2bx 2yx:86 Variation of function yx by the magnitude and sign will cause var- iation of angle /Iof rotation of gear I of the differential. This means that gear I will be rotated with varied angular velocity in two directions. 4.2. Problem 1: generation of function yx asinx;0 6 x 6 2p The derivative y0x 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 y1x sinx bx;0 6 x 6 2p:87 Design of non-circular gears 1 and 2 (Fig. 17) has to cover determi- nation 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 k1x ? x1;/2 k2y1x ? y1x1?;x1 0:88 Here, k1and k2 are scale coeffi cients determined as k1 /1;max x2? x1 ;k2 /2;max y1x2 ? y1x1 ;x2 2p;x1 0:89 Taking that gears 1 and 2 will perform in the process of generation turns on /1;max /2;max 2p, we obtain that k1 1;k2 1 b :90 (ii) The coeffi cient b may be determined by observation of the following conditions: (a) y0x 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 fi nal result is that blimP 1:707:91 Observation of condition blim 1:707 means that centrode 1 will have a point with curvature j1 0. (iii) Eqs. (88) with coeffi cients (90) yield the following transmis- sion function /2/1 /1 1 b sin/1;0 6 /16 2p:92 The derivative function is m12/1 d/1 d/2 1 1 1 b cos/1 :93 (iv) Equations (A.5) and (A.6), (A.7) yield the following equations for centrodes r1and r2of 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 x with coeffi cient b 1:707; (b) transmission function /2/1 wherein coeffi cient b 1:707. 3794F.L. Litvin et al./Comput. Methods Appl. Mech. Engrg. 197 (2008) 37833802 r1/1 E 1 m12/1 E 1 1 b cos/1 2 1 b cos/1 :94 For r2, we have r2/1 E 1 2 1 b cos/1 ;/2/1 /1 1 b sin/1:95 Center distance E is just a scale coeffi cient. Centrodes r1and r2 determined with coeffi cients b 1:707 and b 1:400, are repre- sented in Figs. 18 and 19. Centrode 1 in Fig. 19 is a convexconcave one. (v) We assign for the design that /4 2/3;/3 /1:96 (iv) Function yx asinx will be obtained from gear I of the dif- ferential as /I 2/c? /II 2/2? /4 2 /1 1 b sin/1 ? ? 2/1 2 b sin/1; 97 wherein 2 b a. Non-circular gears with the developed centrodes may be gener- ated by the enveloping method by using a hob for centrodes shown in Figs. 18 and 19. 4.3. Problem 2: Generation of function yx 1 x, x1 6 x 6 x2 4.3.1. Introduction The specifi c features of centrodes applied for generation are: (i) centrodes r1and r2are represented as unclosed curves, (ii) they are identical, (iii) the gears may perform rotation on angles / 2p (Fig. 14), while performing simultaneously axial translation. 4.3.2. Centrodes and transmission function The angles of rotation /1and /2of centrodes 1 and 2 are pro- portional to variable x and function yx, respectively. Thus we have /1 k1x ? x1;98 /2 k2y1? y k2 1 x1 ? 1 x ? ;99 where k1and k2 are scale coeffi cients determined as k1 /1;max x2? x1 ;k2 x1? x2/2;max x2? x1 :100 The derivative function is m12 a3 a4/12 a2a3 ;101 where a2 k2;a3 k1x2 1; a4 x1:102 Fig. 19. Illustration of: (a) centrodes of non-circular gears for generation of function y1 x with coeffi cient b 1:400; (b) transmission function /2/1 wherein coeffi cient b 1:400. F.L. Litvin et al./Comput. Methods Appl. Mech. Engrg. 197 (2008) 378338023795 The centrodes are represented as follows r1/1 E 1 1 m12/1 E a2a3 a2a3 a3 a4/12 ;103 r2/1 E ? r1/1 E a3 a4/12 a2a3 a3 a4/12 ;104 /2/1 F/1 a2/1 a3 a4/1 :105 The centrodes are identical ones by observation the conditions /1;max /2;max, and satisfaction of the functional (see Section A.1.7) that is represented for discussed example of design as a2/1;max? F/1 a3 a4/1;max? F/1 /1;max? /1:106 The conditions above are satisfi ed 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 wa;a ? /1, by a gear drive with two pairs of gears (1, 2) and (3, 4) (instead of one pair) has the fol- lowing advantages: (i) a larger variation of derivative ow oa may be provided. (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 cen- trodes of the gears of the drive. Each of the centrodes 1, 2, 3, 4 per- forms rotation about point Oi(i 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 wa to be generated, the cen- trodes might be closed curves or unclosed ones. We may consider initial and fi nal positions of the centrodes that correspond to the beginning of motion (where /i 0, i 1;2;3;4), and the end of motion (where /i /i;max). In the case of centrodes as closed curves, we have: /1;max /2;max /3;max /4