Modelng Concave Globodal Cam wth Swngng Roller Follower: A Case Study Nguyen Van Tuong, and Premysl Pokorny Abstract Ths paper descrbes a computer-aded desgn for desgn of the concave globodal cam wth cylndrcal rollers and swngng follower. Four models wth dfferent modelng methods are made from the same nput data. The nput data are angular nput and output dsplacements of the cam and the follower and some other geometrcal parameters of the globodal cam mechansm. The best cam model s the cam whch has no nterference wth the rollers when ther motons are smulated n assembly condtons. The angular output dsplacement of the follower for the best cam s also compared wth that of n the nput data to check errors. In ths study, Pro/ENGINEER Wldfre 2.0 s used for modelng the cam, smulatng motons and checkng nterference and errors of the system. Keywords Globodal cam, sweep, ptch surface, modelng. G I. INTRODUCTION LOBOIDAL cam mechansms are wdely used n ndustry. Compared to other cam-follower systems, the globodal cam-follower mechansms have many advantages, such as: compact structure, hgh loadng capacty, low nose, low vbraton, and hgh relablty. They are wdely used n machne tools, automatc assembly lnes, paper processng machnes, packng machnes, and many automated manufacturng devces. In term of the shape, globodal cam s one of the most complcated cams. Up to now, lots of efforts have been made n fndng the way to descrbe the complcated surfaces of the globodal cam. Yan and Chen [7[, [8] derved mathematcal expresson for the surface geometry of the globodal cam wth cylndrcal rollers and hyperbolod rollers based on coordnate transformaton, dfferental geometry, and theory of conugate surfaces. To analyze the transmsson error and to synthesze the tolerances, Cheng [1] also made the geometrc mathematcal models of the globodal cam surface by the conugatng theory. Tsay and Ln [6] studed the globodal cam wth concal rollers. From machnng pont of vew, they presented the sweep surfaces for the cam surfaces by means of the sweep surfaces of the tool paths. Yuan et al. [2] used computer to develop a package, whch s a combnaton of N. V. Tuong s wth the Department of Manufacturng Systems, Techncal Unversty of Lberec, Studentska 2, 461 17 Lberec 1, Czech Republc (phone: +420-776831230; fax: +420-485353656; e-mal: tuongnv@gmal.com). P. Pokorny s wth the Department of Manufacturng Systems, Techncal Unversty of Lberec, Studentska 2, 461 17 Lberec 1, Czech Republc (emal: premysl.pokorny @ tul.cz). AutoCAD R14, 3D Studo Max, and VBA, to generate the surfaces of the roller gear cam. In addton, many researchers have studed on other aspects of the globodal cam such as the contact between cam surfaces and the rollers, dynamcs, machnng on four-axs or fve-axs machne tools, etc In ths study, to llustrate the cam surfaces from angular nput and output dsplacements, some modelng methods are presented. II. THEORETICAL BACKGROUND The globodal cam rotates about ts axs and the cam drves a roller follower. There are two types of globodal cams. The frst one s the globodal cam that has a groove on ts surface and the roller follower oscllates when the cam rotates. The cam of ths type s ether convex or concave. The second one has one or more rbs on ts surface. Ths type s also called roller gear drve or Ferguson drve [5]. The two surfaces of the rb always contact wth the rollers (cylndrcal or concal) of the follower. Ths type has two subtypes: concave globodal cam wth an oscllatng follower and ndexng globodal cam wth a turret follower (Fg. 1). The rb of these cams looks lke a thread or a blade so that sometmes they can be called thread-type or blade-type globodal cam. In ths study, the sngle thread-type s the globodal cam that we wll deal wth. Fg. 2 llustrates the geometrcal relatonshps between a concave globodal cam wth a oscllatng follower. In ths fg., the development plane s the plane that s normal to the axs of the roller and located anywhere along the length of the roller. The ntersecton pont between the development plane and the axs of the roller s the ptch pont (P). Datum plane s the plane normal to the cam axs and contans the follower axs. The angular dsplacement of the roller s measured from ths plane. Followng are some parameters related to globodal camfollower system [3], [4]. - angular nput dsplacement (the rotaton angle of the cam). - angular output dsplacement from datum plane (the rotaton angle of the follower). has a relatonshp wth and t can be expressed by functon = f( ), [2]. 0 angle from datum plane to start of follower moton, measured n drecton of moton. If the start pont s encountered after the datum plane then 0 s postve. 1 angle between the axs of the upper roller wth the datum plane. At the begnnng, when the upper roller s at the 776
startng pont then 0 = 1. 2 angle between the axs of the lower roller wth the datum plane. t - dstance between the axs of the follower to the end of the roller, measured along the roller axs. e clearance between the end of the roller and the cam body. F - dstance from the axs of the follower to the ptch pont. C - dstance between the cam axs and the turret axs. R - perpendcular dstance from cam axs to the ptch pont, expressed as R = C F.cos( 1 ) (1) h dstance from the ptch pont to the datum plane. It s the heght of the pont P and presented as h = F.sn ( 1 ) (2) Obvously, the coordnates of the ptch ponts on the rollers can be calculated f the angular nput and output dsplacements are known. From these coordnates and some other nformaton, the ptch surfaces of the cam can be modeled. Datum plane Cam (a) Oscllatng follower (b) Indexng turret follower Fg. 1 Globodal cam - thread type R Development plane l Roller 0 2 1 F Follower t C Fg. 2 Globodal cam- oscllatng follower arrangement III. MODELING METHODS The globodal cam can be modeled by usng CAD software. In ths study, Pro/Engneer Wldfre 2.0 s used. There are two methods to model the globodal cam: surface-based modelng and sold-based modelng. In the frst method, all surfaces of the cam are frst modeled and then the surface model s converted to sold. A. Surface-Based Modelng In a globodal cam-follower system, when the follower 777
rotates, the locus of roller axs wll generate a ruled surface (ptch curved surface) n space [6]. The two axes of two rollers n ths case study wll generate two ptch curved surfaces. The globodal cam surfaces can be obtaned from the ptch surfaces by offsettng them a dstance that s equal to the radus of the roller. There are several ways to get the ptch surface. The followng are three ways to model the ptch surface. Model 1: Sweep a straght lne that s collnear wth the roller axs. The two end ponts of ths lne must le on two curves. One of these curves s a crcle n the datum plane. Ths crcle goes through the ntersecton pont of the roller axes and ts center s n the cam axs. The other curve s a three-dmensonal (3D) curve and t s the orgn traectory. Ths 3D curve s the locus of a pont, whch located on the roller axs (t can be the ptch pont), when the follower rotatng. The coordnates of that pont can be calculated n the cylndrcal coordnate system as - nput angular dsplacement Model 3: Sweep an opened secton whch conssts of three straght lnes. Two of them are collnear wth the axes of the two rollers. The last one connects them together. Smlarly, the constrants that are used to make model 2 (Fg. 3(b)) are also used here. In ths method, frst, the body surface s modeled, and then the two ptch surfaces are done. Make one or two offsets from the ptch surfaces to get the cam surfaces. In model 2, a boundary surface can be added to unte the two cam surfaces f they do not ntersect each other. After that, the body surface, the cam surfaces and the boundary surface are merged together. The unted surface wll be soldfed to become a sold. Last, perform some cuts to get the desred cam. B. Sold-based modelng An endmll cutter can generate the surfaces of a globodal cam. If the dameters of the cutter and the roller are equal, the moton of the cutter wll be smlar to the moton of the roller (a) Fg. 3 Constrans for model 2 and model 3 R h F. sn (3) C F. cos (4) where = 1, 2, correspondng wth the upper and the lower ptch surfaces; = 1,2,,n, correspondng wth the angular output dsplacements. Model 2: Sweep a straght lne wth three constrants: () the orgn traectory s a crcle n the datum plane and ts symmetrc axs concdes wth the cam axs, () the angle between ths lne and the datum plane vares when the cam rotates and ts value s (Fg. 3(a)), () the coordnates of a pont on ths lne satsfes formulas (3) and (4) above. The orgn traectory can be a crcle that s the ntersecton between the datum plane and the body surface of the cam. Datum graphs and sketcher relatons are used to fulfll the constrants () and () above [9]. The same procedures are done to get the two ptch surfaces of the cam. (b) n the machnng process, and of course, the cutter must rotate about ts axs (roller axs). The sweep surface of the tool path can represent the cam surface. The followng s one way to get the cam surface. Model 4: Cut the bank by a rectangular secton to form the cam surfaces f the followng constrans are performed: 1) The wdth of the secton s equal to the dameter of the roller. The length of the secton s arbtrary provdng that t s longer than the length of the roller plus the clearance e (Fg. 2). 2) Sweep ths secton and the axs of the secton must follow two 3D curves. These curves are loc of two ponts, whch are on the roller axs, when the follower rotates. One of these curves s the orgn traectory. The curves whch are used for model 1 can be appled here. 3) Ths secton plane s always normal to the orgn traectory. In ths method, frst a sold body s modeled. Then, two cuts are performed to get the cam surfaces. Last, some cuts are 778
done to get the desred cam. IV. APPLICATION EXAMPLE A. Input Data and Pre-Calculatons The angle between two axes of the rollers s 60 0. The ncrement of the nput angle of the cam s 0.2 0, starts from 0 and ends at 360 0. To observe easly, the relatonshp between the angular nput and output dsplacements s showed n Fg. 4. Some selected dsplacements are presented n the appendx. Followng are some other parameters of the system, whch are showed n Fg. 2: d = 25.5 mm, l = 16 mm, C= 107.8 mm, t = 58.7 mm, 0 = 7.49 0, e = 2.3 mm. There are some calculatons that must be done before makng the models as follows: 1) Calculatng the angular outputs ncluded 0. 2) Calculatng the angle 1 and 2 wth. 1 0 (5) 2 60 (6) 3) Calculatng the coordnates of two ptch ponts for each ptch surface. The ptch ponts are located at the dstance F= 59.7 mm on the roller axes. All the calculatons are done n Mcrosoft Excel 2003. 1 Angular output, degree 50 45 40 35 30 25 20 15 10 5 0 0 40 80 120 160 200 240 280 320 360 B. Modelng Results and Checkng Interference In Fg. 5 are four types of the cam modeled from precedng data. In ths fg., the ptch surfaces are n transparent state to see the cam surfaces easly. All cams look great and smlar. To choose the best model, the nterference between the cam and the roller must be checked. In order to check the nterference, frst, an assembly of the cam and the follower s made. After that, use the Mechansm Desgn module to defne the geometrcal relatonshps of the system, make t move and analyze ts moton also. Last, use a knematcs analyss to obtan nformaton on nterference between components When the globodal cam rotates, the follower wll stay or rotate dependng on the locaton of the rollers on the cam surfaces. The follower wll not move when the rollers stll contact wth the cam surfaces n the dwell perods. To get a moton for the follower, a pont on one roller axs has to trace along a 3D curve on the ptch surface. The ptch pont now Angular nput, degree Fg. 4 Angular nput/output dsplacement can be used for ths purpose. Ths 3D curve s avalable on the model 1 and model 4. Ths curve must be drawn on the others also. Among the four models, model 1 has no nterference between the cam and the roller when the cam rotates one revoluton, whle models 2, 3 and 4 have nterferences (Fg. 6). Model 2 and 3 have 51 and 35 postons of nterference, respectvely. Model 4 has a lot of postons where nterferences occur. There are totally 1800 postons checked for a full revoluton of the cam. The angle between two postons (called frames n Mechansm Desgn module) s 0.2 0. Ths value s smlar to the ncrement of the nput angle of the cam. In comparson wth model 2 and model 3, model 4 has bgger nterference volumes and they can be seen n the graphc wndow. The result s that model 1 s the best one. The model 2 can also be a good one f the part accuracy s set from 0.0012 (the default value) to 0.0005. In ths case, there s no nterference between the cam and the roller. 779
(a) Ptch surfaces and cam model 1 (b) Ptch surfaces and cam model 2 (c) Ptch surfaces and cam model 3 (d) Cam - model 4 Fg. 5 Four models of globodal cam. The clearances n assembly between cam surfaces of model 1 and ther rollers must be checked to ensure that they are small enough. If these clearances are large, the errors of output angular dsplacements wll be large, too. These clearances can be measured n assembly standard mode or n mechansm mode. Some selected errors are presented n Table I n the appendx. In general, these clearances are always less than 0.2 mcrometer. In one revoluton of the cam, the bggest gaps are on the rse and return perods. These gaps also cause errors n the output angular dsplacements but these errors are (a) Model 2 (and 3) (b) Model 4 Fg. 6 Interferences (n red colour) occur between components very small and they can be gnored. In Mechansm Desgn module, the angular output dsplacements of the follower can be measured and the result can be exported to Mcrosoft Excel. By comparng the measured angular output dsplacements wth the requred nput data, the errors of the model wll be evaluated. In general, there are errors n rotaton angles of the follower when the cam moves a full revoluton. These errors vary from -0.000,000,42 degree to 0.000,000,43 degree (see Table II n the appendx). They are very small and can be omtted. 780
V. CONCLUSION In ths study, four models of the concave globodal cam are developed from the same nput data by usng the software Pro/Engneer Wldfre 2.0. The result s that the model, whch ts ptch surfaces are obtaned by sweepng straght lnes along two curves (loc of two ponts on roller axes), s the best one. Wth ths model, no nterference between components s found when the system s workng and the errors of angular rotaton of the follower are very small. Ths s a real example but ts modelng procedures can be appled for other stuatons when the angular nput/output are known. The result of ths study s very useful n terms of modelng and manufacturng globodal cam. ACKNOWLEDGEMENT The work upon whch ths paper s based s a part of MSM- 4674788501 research program, grant from the Techncal Unversty of Lberec, Czech Republc. APPENDIX TABLE I EXAMPLE OF SOME SELECTED CLEARANCES BETWEEN CAM SURFACES AND ROLLERS [deg] e 1 [mm] e 2 [mm] [deg] e 1 [mm] e 2 [mm] 0 0.0000000020 0.0000000013 195 0.0000000000 0.0001421750 15 0.0000000049 0.0000000053 210 0.0000000000 0.0000000000 30 0.0000000049 0.0000000053 225 0.0000000000 0.0000000000 45 0.0000000049 0.0000000053 240 0.0001681920 0.0000000000 60 0.0000000048 0.0000000051 255 0.0000992641 0.0000000000 75 0.0000000048 0.0000000051 270 0.0000000048 0.0000000051 90 0.0000000048 0.0000000051 285 0.0000000048 0.0000000051 105 0.0000992641 0.0000000000 300 0.0000000048 0.0000000051 120 0.0001681920 0.0000000000 315 0.0000000049 0.0000000053 135 0.0000000000 0.0000000000 330 0.0000000049 0.0000000053 150 0.0000000000 0.0000000000 345 0.0000000049 0.0000000053 165 0.0000000000 0.0001421750 360 0.0000000020 0.0000000013 180 0.0000003888 0.0000003815 Note: e 1: clearance between cam surface and upper roller (model 1). e 2 : clearance between cam surface and lower roller (model 1). TABLE II EXAMPLE OF SOME SELECTED ANGULAR INPUT/OUTPUT DISPLACEMENTS, UNIT: DEGREE * * 0.00 0.00000000 8.17575E-08 0.00000008 179.60 44.99616011 44.99616012 0.00000001 0.20 0.00000000-1.70511E-10 0.00000000 179.80 44.99904001 44.99904002 0.00000001 0.40 0.00000000-1.70083E-10 0.00000000 180.00 45.00000000 44.99999999-0.00000001. 180.20 44.99904001 44.99904001 0.00000000 105.00 0.00000000-1.54018E-10 0.00000000 180.40 44.99616011 44.9961601-0.00000001 105.20 0.00000680 6.79984E-06 0.00000000 105.40 0.00005418 5.41793E-05 0.00000000 213.00 23.59739548 23.59739506-0.00000042 213.20 23.39136475 23.39136433-0.00000042 147.00 23.59739548 23.5973959 0.00000042 213.40 23.18530279 23.18530237-0.00000042 147.20 23.80337666 23.80337707 0.00000041 213.60 22.97922791 22.97922749-0.00000042 147.40 24.00928999 24.00929042 0.00000043 213.80 22.77315844 22.77315802-0.00000042 147.60 24.21511717 24.21511759 0.00000042 214.00 22.56711268 22.56711226-0.00000042 147.80 24.42083990 24.42084032 0.00000042 148.00 24.62643991 24.62644033 0.00000042 359.60 0.00000000-1.7008E-10 0.00000000 148.20 24.83189893 24.83189935 0.00000042 359.80 0.00000000-1.70081E-10 0.00000000 360.00 0.00000000 8.17575E-08 0.00000008 Note: : theoretc angular output dsplacement of the follower *: real angular output dsplacement of the follower (model 1). : error of the angular output dsplacement of the follower (model 1). 781
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