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pitch the lobes becoming bevel 1 two cording directrix gears special linkage, locity As cal, and are a their other duced and and gears algorithm posed, gears motion cone was crown-rack on Refs. hob, was formulated in Ref. H2085111H20852. In particular, both base curves for the right and left involute tooth profiles were obtained through the formulation of the pitch curves and their evolutes. An interesting application gear ported noncircular H20851 cam posed conventional and paper, which is applicable to N-lobed elliptical bevel gears and the pitch surface of their conjugate crown-rack. In particular, the fundamental geometrical proprieties of the publication 2010; 201 Journal Downloaded 03 Mar 2011 to 59.72.127.140. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm of noncircular gears to the design and investigation of drives and open-loop spatial mechanisms has also been re- in Refs. H2085112,13H20852. The current state of art on the synthesis of cylindrical gears is summarized and described in Ref. 14H20852 with the aid of several interesting examples. Moreover, the generation of the ruled surfaces of pure-rolling mechanisms and the pitch surfaces of skew-gears were pro- and developed in Refs. H2085115–19H20852 through the application of a geometric approach and by means of dual algebra the principle of transference. A comprehensive methodology spherical ellipse are analyzed in order to obtain a suitable equation of the spherical ellipse in polar form with respect to one of its foci. In fact, the basic pitch elliptical cone, which has one single lobe, is obtained by means of the basic spherical ellipse as the directrix curve of the pitch conical surface, whose apex is coinci- dent with the center of the fundamental sphere for the relative motion between the pairs of elliptical bevel gears. As in the case of N-lobed elliptical cylindrical gears, a pair of basic elliptical bevel gears can mesh properly only when they are twins and ro- tate about axes passing through their focal points and the apex of the pitch cones. However, N-lobed elliptical bevel gears can be generated from a basic spherical ellipse, thereby increasing the number of speed cycles per revolution. Therefore, the proposed formulation has been extended to the general case of N-lobed elliptical bevel gears, where both pitch cones and the pitch surface Contributed by the Power Transmission and Gearing Committee of ASME for in the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 7, final manuscript received December 14, 2010; published online February 22, 1. Assoc. Editor: Prof. Philippe Velex. of Mechanical Design MARCH 2011, Vol. 133 / 031002-1Copyright ? 2011 by ASME Giorgio Figliolini DiMSAT, University of Cassino, G. Di Biasio 43, Cassino (Fr) 03043, Italy e-mail: figliolini@unicas.it Jorge Angeles Department of Mechanical Engineering and CIM, McGill University, 817 Sherbrooke Street, Montreal, QC, H3A 2K6, Canada e-mail: angeles@cim.mcgill.ca Synthesis N-Lobed A general formulation bevel gears and the two pitch cones and and combination of formulation is implemented circular bevel gears Keywords: elliptical crown-rack Introduction Elliptical bevel gears can be used to transmit motion between intersecting axes with a variable transmission ratio and ac- to a suitable motion program, which is imposed by the spherical curves of the elliptical pitch cones. This kind of are highly specialized, their application being found in some instances, e.g., when motion is generated by a four-bar and the application of interest requires that both the ve- and the acceleration of the output link vanish concurrently. this is not possible with a four-bar linkage, be it planar, spheri- or spatial, inserting an elliptical gear train between the motor the input link yield the desired result. Elliptical bevel gears an invention due to Mac Cord H208511H20852 and Grant H208512H20852, who analyzed pair of spherical ellipses rolling on each other while fixed on foci, their free foci moving at a constant distance from each . The concept of N-lobed elliptical bevel gears was intro- without any formulation. Olsson reported on the analysis the manufacturing of noncircular bevel gears H208513H20852 while Litvin Varsimashvili considered the cutting of noncircular bevel by means of the intermittent generating method H208514H20852.An for involute and octoidal bevel gear generation was pro- where the exact spherical involute tooth profile of bevel and their crown-rack were obtained through the pure-rolling of a great circle of the fundamental sphere on the base H208515H20852. Likewise, the tooth flank surface of octoidal bevel gears obtained as the envelope of the tooth flat flank of the octoidal during the pure-rolling motion of its flat pitch surface the pitch cone. The synthesis of cylindrical elliptical gears was proposed in H208516–10H20852 by considering several methods, such as rack cutter, and shaper cutter, while the synthesis of their base curves of the Pitch Cones of Elliptical Bevel Gears of the synthesis of the pitch cones of a pair of N-lobed elliptical surface of its conjugate crown-rack is proposed. In particular, pitch surface of their crown-rack are obtained for any number and in any configuration during their pure-rolling motion. This in MATLAB; several significant examples are included, with a particular case thereof. H20851DOI: 10.1115/1.4003412H20852 gears, kinematics synthesis, noncircular pitch cones, for the synthesis and manufacture of different types of gears was also proposed in Ref. H2085120H20852 through the application of screw theory. More recently, the design of spherical multilobe-cam mecha- nisms was proposed H2085121H20852 as a possible alternative to their bevel gear counterparts for mechanical transmission systems in robotic devices. In fact, cam-roller mechanisms feature low friction, low backlash, and high strength, even if the power transmission is limited by the pressure angle, which is variable, while in involute bevel gears, the pressure angle is constant. Similar to multilobe-cam mechanisms, a variable transmission ratio can also be obtained by means of noncircular bevel gears, which can be designed based on their noncircular pitch cones, as proposed in Ref. H2085122H20852. In particular, the mathematical model of noncircular pitch cones, along with some geometrical consider- ations on the crown-rack, were formulated to generate the tooth profiles by means of a crown-rack cutter. Similar to noncircular cylindrical gears, this method allows the generation of noncircular bevel gears with convex spherical pitch curves only. The synthesis of the pitch surfaces of noncircular skew-gears was proposed in Ref. H2085123H20852, as an extension of previous work by the authors H2085119H20852. The pitch surfaces of N-lobed elliptical bevel gears were also generated by assigning the corresponding variable transmission ratio but any geometrical parameter and characteris- tic of the spherical pitch curves and cones were not taken into account. Ageneral methodology for the synthesis of the pitch surfaces of spherical cams that can produce virtually any periodic transmis- sion ratio is available in literature H2085116H20852, but this is not applicable to the synthesis of the pitch surfaces of elliptical bevel gears, or of any bevel gear for that matter, as this methodology does not yield the rack surface. Hence the motivation of the work reported in this of nation motion. several 2 are cal basic spherical face, sphere gears. cone radius equal at obtained planar origin E not form 031002-2 Downloaded 03 Mar 2011 to 59.72.127.140. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm their crown-rack can be obtained for any number and combi- of lobes, and in any configuration during their pure-rolling The proposed algorithm was implemented in MATLAB; illustrative examples are included in this paper. The Spherical Ellipse The fundamental geometrical proprieties of the spherical ellipse recalled in order to obtain a suitable expression of the spheri- ellipse in polar form with respect to one of its foci. In fact, the elliptical pitch cone is obtained by means of the basic ellipse as the generatrix curve of the conical pitch sur- whose apex is coincident with the center of the fundamental for the relative motion between the pairs of elliptical bevel Referring to Figs. 1–3, a spherical ellipse E S and its elliptical C can be generated through the fundamental sphere S with R and the planar ellipse E T with major and minor semi-axes to r M and r m , respectively. The plane T of E T is tangent to S the origin H9024 of frame F T H20849H9024,x,y,zH20850. The pitch cone C can be by sweeping a line through the center O of S along the ellipse E T . The apex O of the generated cone C is also the of the frame F S H20849O,X S ,Y S ,Z S H20850. Thus, the spherical ellipse S is obtained as the intersection curve of C with S. Thus, E S is planar. The planar ellipse E T can be expressed in Cartesian through the well-known equation Fig. 1 Axonometric view of the spherical cone Fig. 2 The xz section of the elliptical cone / Vol. 133, MARCH 2011 x Q 2 r M 2 + y Q 2 r m 2 =1 H208491H20850 where x Q and y Q are the Cartesian coordinates of point Q in frame F T . Referring to Figs. 2 and 3, r M and r m can be expressed as r m = R tan H9251 m , r M = R tan H9251 M H208492H20850 where H9251 m and H9251 M are the minimum and maximum pitch angles of C while R is the radius of S. Likewise, x Q and y Q can be expressed as x Q = R tan H9251 x , y Q = R tan H9251 y H208493H20850 where H9251 x and H9251 y are the angles that the z-axis makes with OQ x and OQ y , respectively, and defined positive in the clockwise di- rection. Points Q x and Q y are the projections of Q onto the x- and y-axes, respectively. Thus, substituting Eqs. H208492H20850 and H208493H20850 into Eq. H208491H20850, one obtains tan 2 H9251 x tan 2 H9251 M + tan 2 H9251 y tan 2 H9251 m =1 H208494H20850 which is the Cartesian equation of the spherical ellipse. The real foci F 1 and F 2 of the spherical ellipse E S are symmetri- cally located with respect to the z-axis, along the arc AB, as shown in Fig. 2. Their position is given by the apex angle H9251 F . Similar to planar ellipses, the eccentricity e of E S is defined as the ratio between the apex angle H9251 F and the maximum pitch angle H9251 M , namely, e = H9251 F H9251 M H208495H20850 The eccentricity e can be expressed as e = 1 H9251 M cos ?1 H20873 cos H9251 M cos H9251 m H20874 H208496H20850 where H9251 m and H9251 M define the particular spherical ellipse. Similar to the case of N-lobed elliptical cylindrical gears, a pair of basic elliptical bevel gears can mesh properly only when they are twins and rotate about axes passing through both their foci and the common apex of the pitch cones O. In fact, referring to Fig. 4, the position of a point P of E S can be referred to the foci F 1 and F 2 through the apex angles H9253 and H9253H11032, respectively. In particular, one obtains H9253+H9253H11032 =2H9251 M H208497H20850 which expresses the fundamental property of the basic spherical ellipses, i.e., the sum of the apex angles for any point of the spherical ellipse is constant and equal to twice the maximum pitch angle. Fig. 3 The yz section of the elliptical cone Transactions of the ASME to about ellipses. lar a gears. point through position F terclockwise, where given single-lobed Z gears, tion. can which 3 gears mulated metric Three-lobed tical gears, the among relative mental tached, and Their with Journal Downloaded 03 Mar 2011 to 59.72.127.140. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Consequently, a pair of equal spherical ellipses can be assumed be the pitch curves of elliptical bevel gears, which can rotate their foci on the pure-rolling motion between the spherical Therefore, the formulation of the spherical ellipse in po- form with respect to the fixed focus F 1 is convenient to obtain general formula for the pitch cones of N-lobed elliptical bevel Referring again to Fig. 4, the position vector r P of a generic P of E S can be referred to the frame F S H20849O,X S ,Y S ,Z S H20850 the angles H9253 and H9277, where H9277 is the angle that gives the of P with respect to the positive X S -axis. Plane X S Z S of S is coincident with the plane xz of F T , measured positive coun- as shown in Fig. 2. Angle H9253 can be expressed as a function of angle H9277, namely, tan H9253= tan 2 H9251 M ? tan 2 H9251 F tan H9251 M H208491 + tan 2 H9251 F H20850 ? tan H9251 F H208491 + tan 2 H9251 M H20850cos H9277 H208498H20850 H9251 M and H9251 m are input data of the spherical ellipse and H9251 F is by Eqs. H208495H20850 and H208496H20850. Thus, Eq. H208498H20850 expresses in polar form a spherical ellipse with respect to the axis of rotation S . Equation H208498H20850 is also useful to generate N-lobed elliptical bevel thereby increasing the number of speed cycles per revolu- In fact, the equation generating N-lobed elliptical bevel gears be obtained just by multiplying the angle H9277 of Eq. H208498H20850 by n, is the number of lobes. Synthesis of the Pitch Cones of Multilobed Gears The synthesis of the pitch cones of N-lobed elliptical bevel and the pitch surface of their conjugate crown-rack is for- in a general framework based on the fundamental geo- proprieties of the single- and N-lobed spherical ellipses. elliptical pitch curves P 1 , P 2 , and P 3 , and their ellip- pitch cones C 1 , C 2 , and C 3 , for the driving and driven bevel and their crown-rack, respectively, are sketched in Fig. 5 at starting configuration of their pure-rolling motion. The contact C 1 , C 2 , and C 3 is along the instant screw axis H20849ISAH20850 for their pure-rolling motion while the ISA intersects the funda- sphere S at point I. Moving frames F 1 H20849O,X 1 ,Y 1 ,Z 1 H20850 and F 2 H20849O,X 2 ,Y 2 ,Z 2 H20850 are at- each, to its corresponding N-lobed elliptical pitch cone C 1 C 2 , respectively, while F 3 H20849O,X 3 ,Y 3 ,Z 3 H20850 is attached to C 3 . Z-axes, or axes of rotation, are co-planar between them and the ISA, according to the Aronhold–Kennedy theorem H2085124H20852. Fig. 4 Illustration of the generation of the spherical ellipse of Mechanical Design Moreover, the ISA changes its position during the relative pure- rolling motion because of a suitable rotation about the apex O and, consequently, a variable transmission ratio is obtained. In particular, the N-lobed elliptical pitch curve P 1 of the driving bevel gear can be derived from Eq. H208498H20850, namely, tan H9253 1 = A 1 B 1 + C 1 cosH20849n 1 H9277 1 H20850 H208499H20850 where A 1 , B 1 , and C 1 are given by A 1 = tan 2 H9251 1M ? tan 2 H9251 1F , B 1 = tan H9251 1M H208491 + tan 2 H9251 1F H20850, C 1 = ? tan H9251 1F H208491 + tan 2 H9251 1M H20850H2084910H20850 In turn H9251 1M = H9253 1M +H9253 1m 2 , H9251 1F = H9253 1M ?H9253 1m 2 H2084911H20850 The subscript 1 in the variables of Eqs. H208499H20850–H2084911H20850, added to those shown in Figs. 1–4, indicates the pitch curve P 1 of the driving bevel gear with a number of lobes equal to n 1 . Thus, the position vector r 1 of a point on P 1 can be expressed in frame F 1 as r 1 = RH20851sin H9253 1 cos H9277 1 sin H9253 1 sin H9277 1 cos H9253 1 H20852 T H2084912H20850 where the apex angle H9253 1 is expressed with respect to the angle H9277 1 by means of Eqs. H208499H20850–H2084911H20850, thus obtaining vector r 1 as a function of angle H9277 1 only. The components of vector r P with respect to frame F S , which have the same meaning of the components of r 1 with respect to frame F 1 , can be readily calculated by referring to Fig. 4 and through angles H9253 and H9277. The equation of the elliptical pitch curve P 2 can be obtained by considering the kinematics of the relative pure-rolling motion be- tween driving and driven elliptical bevel gears. Referring to Fig. 5, angle H9253 12 between the axes of rotation Z 1 and Z 2 of C 1 and C 2 , respectively, is given by H9253 12 =H9253 1 +H9253 2 H2084913H20850 where H9253 1 and H9253 2 are the variable angles that the ISA makes with Z 1 and Z 2 , respectively. The transmission ratio between driving and driven bevel gears can be expressed in the form H9275 1 H9275 2 =? sin H9253 1 sin H9253 2 , with H9275 1 =H9272˙ 1 and H9275 2 =H9272˙ 2 H2084914H20850 i.e., H9275 1 and H9275 2 are the angular velocities of the N-lobed elliptical pitch cones C 1 and C 2 , respectively. Fig. 5 Starting configuration of three-lobed elliptical pitch cones and the pitch surface of their crown-rack MARCH 2011, Vol. 133 / 031002-3 the dif where sumed the H9272 which through n which expresses cones. whose H9253 12 = tan ?1 H20875 ? LH11006 H20881 L 2 ?4H20849H ? MH20850H20849I ? MH20850 2H20849H ? MH20850 H20876 H2084921H20850 Fig. cones 031002-4 Downloaded 03 Mar 2011 to 59.72.127.140. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Thus, combining Eqs. H2084913H20850 and H2084914H20850, the angle of rotation H9272 2 of pitch cone C 2 for the driven bevel gear can be expressed in ferential form as dH9272 2 dH9277 1 = tan H9253 1 sin H9253 12 ? cos H9253 12 tan H9253 1 H2084915H20850 dH9277 1 =?dH9272 1 since the position vector r I of I has been as- attached to the fixed frame F 0 H20849O,X,Y,ZH20850 of Fig. 6 while moving frame F 1 rotates counterclockwise through an angle 1 . Substituting Eqs. H208499H20850–H2084911H20850 into Eq. H2084915H20850, one obtains H9272 2 = H20885 0 H9277 1 A 1 dH9277 1 H20849B 1 sin H9253 12 ? A 1 cos H9253 12 H20850 + C 1 sin H9253 12 cosH20849n 1 H9277 1 H20850 H2084916H20850 leads to H9272 2 =? 2A 1 n 1 H20881 H20849B 1 sin H9253 12 ? A 1 cos H9253 12 H20850 2 ? H20849C 1 sin H9253 12 H20850 2 tan ?1 H11003H20873H20881 B 1 sin H9253 12 ? A 1 cos H9253 12 ? C 1 sin H9253 12 B 1 sin H9253 12 ? A 1 cos H9253 12 + C 1 sin H9253 12 tan n 1 H9272 1 2 H20874 H2084917H20850 The coefficients A 1 , B 1 , and C 1 of Eq. H2084917H20850 can be determined angles H9253 1M and H9253 1m , which are the usual input data with 1 while the angle H9253 12 is related to the equation H9272 2 H11569 =? n 1 n 2 2H9266 H2084918H20850 expresses the rotation H9272 2 H11569 of C 2 for H9272 1 =2H9266. Equation H2084918H20850 the correct meshing between the lobes of the pitch From Eq. H2084917H20850, for H9272 1 =2H9266, and Eq. H2084918H20850, one has H20849H ? MH20850tan 2 H9253 12 + L tan H9253 12 + H20849I ? MH20850 =0 H2084919H20850 coefficients H, I, L, and M are given by H = B 1 2 ? C 1 2 , I = A 1 2 , L =?2A 1 B 1 , M = H20875 n 2 H20849tan 2 H9251 1M ? tan 2 H9251 1F H20850 n 1 H20876 2 H2084920H20850 Thus, angle H9253 12 can be expressed from Eqs. H2084919H20850 and H2084920H20850 as 6 General configuration of three-lobed elliptical pitch and their crown-rack / Vol. 133, MARCH 2011 The positive sign in Eq. H2084921H20850 corresponds to the case of external gears, while the negative sign corresponds to the case of internal gears. Thus, the angle of rotation H9272 2 of the driven pitch cone can be obtained by substituting Eq. H2084921H20850 into Eq. H2084917H20850, which becomes dependent on angle H9272 1 only. Similar to P 1 , the equation of the elliptical pitch curve P 2 of the driven pitch cone C 2 can be expressed through a position vector r 2 in frame F 2 as a function of angles H9253 2 and H9277 2 , as shown in Fig. 4 for a general case. In particular, the variable angle H9253 2 is obtained as H9253 2 =H9253 12 ?H9253 1 H2084922H20850 where H9253 12 is constant and H9253 1 is a function of H9277 1 , as per Eqs. H208499H20850–H2084911H20850. In addition H9277 2 =H9266?H9272 2 H2084923H20850 because the position of point I in frame F 2 is on the negative X 2 -axis at the starting configuration of C 2 . Angle H9253 2