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Engineering Science Engineers, Part C: Journal of Mechanical Proceedings of the Institution of Mechanical The online version of this article can be found at: DOI: 10.1243/095440605X31508 687 2005 219:Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science Hao Wang, Ce Zhang and Guanlong Chen KHV Indexing Cam Mechanism: A New Intermittent Mechanism Published by: On behalf of: Institution of Mechanical Engineers can be found at:Science Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical EngineeringAdditional services and information for Alerts: What is This? - Jul 1, 2005Version of Record at ZHEJIANG UNIVERSITY on January 5, Downloaded from KHV indexing cam mechanism: a new intermittent mechanism Hao Wang1?, Ce Zhang2, and Guanlong Chen1 1School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, Peoples Republic of China 2School of Mechanical Engineering, Tianjin University, Tianjin, Peoples Republic of China The manuscript was received on 13 September 2004 and was accepted after revision for publication on 5 April 2005. DOI: 10.1243/095440605X31508 Abstract: The KHV indexing cam mechanism is a new type intermittent mechanism that has a structure similar to the KHV planetary gear transmission (one type of the planetary drive with small teeth difference, where K indicates the sun gear, H indicates the pivoted arm, and V indicates the output mechanism). This paper focuses on the generation of a pitch curve and a cam profi le in such a new mechanism. Three types of the KHV indexing cam mechanism are compared and discussed, and the equations of cams pitch curves are derived. The cams profi le is generated by Boolean operations on the offsets of the cams pitch curve. After this, the cusps on the cam profi le are eliminated and replaced by a particular Hermite curve. The animations of all three types are illustrated, and a prototype of such mechanism is reported. Keywords: intermittent mechanism, indexing cam mechanism, planetary transmission 1INTRODUCTION Indexing cam mechanisms are widely used in the industry. Three types of such mechanisms, the paral- lel indexing cam mechanism, the Ferguson indexing cam mechanism, and the barrel indexing cam mechanism, are the traditional types well known today 18. In recent years, some new types of indexing cam mechanisms have also been reported, e.g. the synthesis of the spherical indexing cam mechanism with direct contact between cam and follower was reported by Gonzalez-Palacios and Angeles 9. They extended this concept to the spherical indexing cam mechanism including rollers 10. Gonzalez-Palacios and Angeles 11 also pro- posed a unifi ed approach aiming at the synthesis of indexing cam mechanisms with direct contact transmission. A new type of parallel indexing cam mechanism with an internal cam was reported by Nishioka and Nishimura 12. Zhang 13, 14 created a new concept of indexing cam mechanisms the planetary indexing cam mechanism and presented two types of it. In this paper, the concept is extended to the KHV indexing cam mechanism, which is an intermittent mechanism that has a layout of KHV planetary gear transmission. The new mechanism is suitable for the working conditions where a large number of stops are needed. As most of the rollers can be engaged with the cam in the mechanism, a higher strength and more compact design can be obtained. This new mechanism could expand the applicable ranges of the indexing drivers to wider industry applications. This paper focuses on the generation of the pitch curve and the profi le of the cam in such a mechan- ism. The structure of this paper is as follows. First, section 2, discusses three types of the KHV indexing cam mechanism, namely, type I, type II, and the retrogressed mechanism; then the generation of the cams pitch curve and cam profi le are explained and theanimationofthemechanism(typeIandtypeII) is illustrated in sections 3, 4, and 5, respectively, after a brief introduction of the retrogressed mechanism in section 6. Finally a prototype of the mechanism is presented in section 7. ?Corresponding author: Auto-body Manufacturing Technology Centre, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200030, Peoples Republic of China. 687 C16204# #IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science at ZHEJIANG UNIVERSITY on January 5, Downloaded from 2STRUCTURE OF THE DEVICE The structure of a KHV indexing cam mechanism is similar to a cycloid speed reducer, whose scheme is depicted in Fig. 1. A cycloid speed reducer is com- posed of an epitrochoid planet gear g, an input shaft H (pivoted arm) with an eccentricity, and a number of rollers fi xed in sun gear b. For pure rotational motion of the output shaft V, the eccentric wobble of the planet gear g is fi ltered out by device W, which is a parallelogram mechanism. As the layout of a KHV indexing cam mechanism is similar to the cycloid speed reducer, Fig. 1 is also used as the scheme of the new mechanism. In such a mechanism, the pair of planet gears g and the roll- ers in sun gear b are replaced by a camroller pair to implement the intermittent motion. In the motion process, the input shaft H rotates at a constant speed, while the rotation of the planet gear g is intermittent. The rotational motion of the planet gear g is also fi ltered out by device W to output shaft V. TheKHVindexingcammechanismcanbe designed as two different types according to the layout of the camrollers pair, namely, type I, in which planet gear g is a cam and the rollers are fi xed in sun gear b, and type II, in which the rollers are fi xed in the planet gear g and the sun gear b is an internal cam. When the output shaft is the sun gear b, and V is fi xed, the mechanism is no longer a planetary mechanism, but an ordinary gear train retrogressed from the planetary mechanism (Fig. 2). Because such a retrogressed mechanism could also fulfi ll the task of an intermittent motion, it is catalo- gued as the retrogressed type of the KHV indexing mechanism. Research work has been reported earlier on type II of the mechanism by Zhang 13. In this paper, attention is focused on the synthesis of all three types. 3PITCH CURVE OF THE CAM 3.1Equation of the pitch curve On the basis of the layout of the mechanism mentioned earlier and the relative motion of its com- ponents, the coordinate system and the parameters in type I are constructed as in Fig. 3. Consider three bodies: g, playing the role of the planet cam, b being the sun gear with rollers, and H being the input shaft with an eccentricity. Oband Ogdenote the centre of sun gear b and planet cam g, respect- ively. The fi xed coordinate system ObXbYbis rigidly fi xed on the sun gear b, and the input shaft H rotates around the point Obwith a constant angular velocity. A relative coordinate system OgXgYgrigidly con- nected to the planet cam g is also set up, and Og also represents the rotational centre of the planet cam g. We assume that e represents the eccentricity in the input shaft, and the radius of the sun gear b (the distances between Oband the centre of rollers Mi) is Rz. The number of rollers in the sun gear b is z. The angular displacement of the input shaft H isuH, and that of the planet cam g isug. Let Mi (i 1,.,z) represent the centre of the rollers, respectively. Three position vectors, Rzi, H, and RTi, are also shown in Fig. 3, where Rzi is fi xed on the sun gear b and oriented towards the centre of the roller Mi with the origin Ob , H is fi xed on the input shaft and oriented towards the centre of the planet cam Og with the origin Obalso, and RTirepresents the position of the cam pitch curve with the origin Og. Fig. 1Scheme of a KHV indexing cam mechanism (type I and type II) Fig. 2Scheme of a KHV indexing cam mechanism (the retrogressed mechanism)Fig. 3Coordinate system of type I 688Hao Wang, Ce Zhang, and Guanlong Chen Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering ScienceC16204# #IMechE 2005 at ZHEJIANG UNIVERSITY on January 5, Downloaded from Their relationship yields RTi(t) Rzi H Rzej(ai?ug(t)? e ej(uH(t)?ug(t) t 0, T?,i 1, 2,.,z(1) For type II, the coordinate system and the par- ameters are as constructed in Fig. 4. In such type, g plays the role of the planet roller gear, b is an internal cam, and H is the input shaft with an eccentricity. Oband Ogdenote the centre of sun gear b and roller gear g, respectively. The fi xed coordinate system ObXbYbis also rigidly connected to the internal cam b, and the input shaft H rotates about the point Obwith a constant angular velocity. The moving coordinate system OgXgYgis rigidly con- nected to the roller gear g, and Ogis the rotation centre of the roller gear g. Also, e represents the eccentricity, and the radius of the roller gear g is Rz. The number of rollers in the sun gear b is z. The angular displacement of the input shaft H is uH, and that of the planet cam g isug. Let Mi (i 1,.,z) represent the centre of the rollers, respectively. Three position vectors, Rzi, H, and RTi, are also used here and have the same denotion as with type I. In type II, they yield a different relationship as RTi(t) H Rzi eejuH(t) Rzej(aiug(t) t 0, T?,i 1,2,.,z(2) 3.2Relations between the input and the output of the mechanism From equations (1) and (2), it is seen that the pitch curve of either type is composed of several curves that are determined by every equation in the set correspondingly. Every curve has to be connected to each other to make sure that the pitch curve of the cam is continuous. Then, the parameters in equations (1) and (2) should satisfy RTi(T) RTi1(0)(3) RTn(T) RT1(0)(4) From equations (3) and (4), for type I, the following expression is obtained n z(5) iHg ?(z ? 1)(6) Then, the displacement of the input shaft and the output shaft are defi ned by the following equation uH(t) uH(0) 2(n ? 1)pt nT (7) ug(t) ? 2pS n (8) For type II, another expression is obtained n z(9) iHg ?z(10) the displacement of the input shaft and the output shaft in type II has a different defi nition uH(t) uH(0) 2pt T (11) ug(t) ? 2pS n (12) Figures 5 and 6 provide examples of the pitch curve of type I and type II, respectively. In both types, Rz 100, n 12, modifi ed sine motion is used to generate the pitch curve. Figure 5 also shows the connection point between every piece of curve, with an asterisk, with d 0.83 and K1 1.2, and in Fig. 6, d 1 and K1 1.92. Fig. 4Coordinate system of type IIFig. 5Pitch curve of type I KHV indexing cam mechanism689 C16204# #IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science at ZHEJIANG UNIVERSITY on January 5, Downloaded from 4PROFILE OF THE CAM 4.1Boolean operation Although the pitch curve of the planet cam is a closed curve, it is not a simple plane curve because of its self-intersection (Figs 5 and 6). Given the analytical plane curve RTias the generator curve, the offsets to RTiat distance rz are the curves defi ned by RIi(t) RTi(t) ? rzni(t) ROi(t) RTi(t) rzni(t) t 0, T?, i 1, 2,. ,z (13) where niis the unit normal vector to RTi, and RIi and ROirepresent the interior and exterior offset, respectively. If the pitch curve of the planet cam was a simple curve, the profi le of the planet cam would be the interior offset curve RIi(in type I), or the exterior offset curve ROi(in type II). Because self-intersection occurs in some pieces of the pitch curve, the profi le of the planet cam is composed of certain pieces on both the interior and exterior offsets. To determine the profi le of the planet cam, a Boolean algorithm on areas enclosed by simple curves is introduced. Let AC1be the area enclosed by curve C1, and AC2be that of curve C2. Here, three types of Boolean operation are of interest: union of AC1and AC2, denoted as AC1AC2; and subtraction of AC1and AC2, denoted as AC1AC2. An algorithm called LOBO (loops of Boolean operation) is employed to calculate the profi le of the cam. The LOBO algorithm applied here was reported by Rohmfeld 15. By employing the Boolean operation, we derive the expressions of the cam profi le for each type. As the self-intersection also occurs in the offsets of the pitch curve, we fi rst divide the offsets into two parts:oneistheareaenclosedbytheself- intersection pieces, we called as SI(in the interior offset curve) and SO(in the exterior offset curve) theotheristheareaenclosedbythenon- self-intersection pieces on the offsets, we called AI (in the interior offset curve) and AO(in the exterior offset curve). For type I, the area enclosed by the cam profi le TI is expressed by the subtraction of the union of SI and SOfrom the intersection of AIand AO. TI (AI AO)n(SI SO)(14) For type II, the area enclosed by the cam profi le TII is expressed by the union of AI, AO, SI, and SO. TO AI AO SI SO(15) Figures 7 and 8 show the cam profi le of type I and type II generated, respectively, from Figs 5 and 6 by the Boolean operation mentioned earlier. In Fig. 7, the radius of the rollers is calculated with Kz 0.6 and in Fig. 8, Kz 0.5. Fig. 6Pitch curve of type II Fig. 7 Cam profi le (type I) Fig. 8 Cam profi le (type II) 690Hao Wang, Ce Zhang, and Guanlong Chen Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering ScienceC16204# #IMechE 2005 at ZHEJIANG UNIVERSITY on January 5, Downloaded from 4.2 Smoothness of the cam profi le In the cam profi le, the point that connects two pieces of the offsets, namely, the intersection point of RIi and ROi, became a cusp. Such a cusp is a weakness in the profi le, for its curvature is infi nite and easily broken. To eliminate the cusps, a Hermite curve is employed to replace the curves near the cusps. Asseveralrollersareengagingwiththecam simultaneously, the replacement will not affect the output of the mechanisms. Figure 9 shows the locality of the areas near one of the cusps, in which P0and P1are the points located in different sides of the cusp P. Given the end point P0and P1, and the tangent vector of the cam profi le in these points, a Hermite curve can be readily defi ned. Thus, the curve P0P and PP1 are removed from the cam profi le, and the Hermite curve is embedded in it instead. Each cusp on the profi le can be eliminated and replaced by a particular piece of the Hermite curve. After this procedure, the cam profi le is a simple, smooth, and continuous curve. Figure 10 shows the cam profi le after the procedure mentioned earlier. 5MOTION ANIMATION OF THE MECHANISM Figures 11 and 12 show the animations of the plane- tary indexing mechanism (type I and type II, respect- ively); (a) shows the beginning of an intermittent motion period; (b), (c), and (d) are the phases in the motion period; (e) is the end of motion period and also the beginning of the dwell period; (f) and (g) are in the dwell period; (h) is the end of this intermittent motion period and the beginning of the next one. Fig. 11Animation of the mechanism (type I) Fig. 9Locality of a cuspFig. 10 Cam profi le after the smoothing procedure KHV indexing cam mechanism691 C16204# #IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science at ZHEJIANG UNIVERSITY on January 5, Downloaded from 6THE RETROGRESSED MECHANISM Considering that V is fi xed (Fig. 2), and the output shaft is the sun gear b, the mechanism is a retro- gressed mechanism of the planetary mechanism (Fig. 1). Compared with an internal parallel indexing mechanism, the input shaft in such a retrogressed mechanism is H rather than the cam g, and there is no angular displacement of the cam g in any motion period. In the motion process, the input shaft H rotates at a uniform speed, there is no angular displacement on planet cam g, the rotation of the roller gear b is an intermittent motion that yields the same law as the planetary mechanism. Figure 13 shows the animation of the retrogressed mechanism of type I (Fig. 11) in a motion period, in which (a) shows the beginning of an intermittent motion period; (b), (c), and (d) are sequences of the motion period; (e) is the end of motion period and the beginning of the dwell period; (f) and (g) are in the dwell period; (h) is the end of this intermittent motion period and the beginning of the next one. Fig. 12Animation of the mechanism (type II) Fig. 13Animation of the retrogressed mechanism 692Hao Wang, Ce Zhang, and Guanlong Chen Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering ScienceC16204# #IMechE 2005 at ZHEJIANG UNIVERSITY on January 5, Downloaded from 7PROTOTYPE OF THE MECHANISM A prototype of the mechanism (type I) has been made to test the feasibility of the mechanism. Figure 14 shows the photographs of the planetary cam,assembledpairofcamandrollers,and appearance of the prototype, respectively. The cam is manufactured in a linear cutting machine, not an NC milling, as there are some grooves on the cam profi le. The manufacture of the cam by linear cutting is not very accurate, but it is enough to test the mechanisms feasibility. The prototype achieved a speed of 250 r/min driven by motor-belt chain. This prototype proved that the concept of the KHV indexing cam mechanism is feasible and suc- cessful. Another prototype has been manufactured to test the performance of this mechanism such as the positioning accuracy, highest speed, vibration, and noise. The report of the second prototype performance test will be presented later. 8CONCLUSIONS The layout of the KHV indexing cam mechanism employs a structure similar to a cycloid speed reducer. The mechanism can be designed into two different types (type I and type II) according to the differentlayoutofthecamrollerpair.From converting the output shaft V to a fi xed element, a retrogressed type can be obtained, such a mechan- ism is also an intermittent mechanism. Because of se