GEAR SHAPED CUTTER A PROFILING METHOD DEVELOPED IN GRAPHICAL FORM

Transcription

1 HE ANNALS OF DUNAEA DE JOS UNIVESIY OF GALAI FASCICLE V, ECHNOLOGIES IN MACHINE BUILDING, ISSN , 05 GEA SHAPED CUE A POFILING MEHOD DEVELOPED IN GAPHICAL FOM Nicşor BAOIU, Virgil Gabriel EODO, Camelia Lăcrămioara POPA, Nicolae OANCEA Department of Manfactring Engineering, Dnarea de Jos University of Galati, Department of Mechanical Engineering, Dnarea de Jos University of Galati virgilteodor@galro ABSAC he classical methods for the stdy of enveloping srfaces are known as basic methods, sch as Olivier theorems, Gohman method or Willis theorem Other methods are known as specific methods, as, for example, the Nikolaev method for profiling tools which generate helical srfaces, or complementary methods for the stdy of enveloping srfaces his paper presents an alternative method the method of relative trajectories for profiling tools which generate by enveloping ordered crls of profiles associated with a pair of rolling centrodes An analytical argment is presented, for the stdy of generation sing gear shaped ctter as well as a graphical method developed in CAIA design environment, sing capabilities of this software Nmerical applications are also presented, inclding a comparison between the reslts obtained with an analytical method and the reslts obtained with the graphical method, in order to prove the qality of the proposed method KEYWODS: gear shaped ctter relative trajectories, crves enveloping, CAIA INODUCION he generation of ordered crls profiles, sch as teethed wheels flanks, teeth of side mill for ctting, polygonal shafts, or polygonal bshes, is sally machined with gear shaped ctters In this case, the profiles of the crls to be generated and, in the same time, the profiles of the tool s teeth are associated with a pair of rolling centrodes In most cases these centrodes are circles he isse is related to determination the profile for the gear shaped ctter teeth he tools generate by shaping, in the rolling process, an ordered crl of profiles associate with a centrode he problem can be solved sing the Willis basic theorem of gearing [], also, sing the Gohman theorem [], [3], sing complementary theorems, as the minimm distance method or the method of in-plane generating trajectories [], [3] Moreover, the graphical methods were realised in AtoCAD [4] In these methods the D modelling is sed and rigoros soltions are presented, based on an original software programmed in Lisp Using the CAIA graphical environment, generating mechanisms were imagined for the profile to be generated in order to find profiling soltions for the gear shaped ctter tools [5] he problems of profiles interference are examined In this paper is proposed a graphical soltion, based on the description of the generating relative trajectories (in principle, cycloid crves) of points belongs to the profile to be generated in the tool s space A new form for enveloping condition is proposed he soltion is simple, easy to apply and rigoros Comparative applications are presented verss analytical soltion in order to prove the method s qality An assembly composed of three elements is sed in order to make the graphical profiling he first element represents the tool s centrode, the circle with radis r, the axis, and the X Y Z reference system s origin he second element is the piece s centrode and the reference system joined with this Moreover, the profile to be generated is sketched here, and, also the straight lined segment perpendiclarly to this profile he third element is named S_fix and incldes the fixed reference systems his element is 9

2 HE ANNALS OF DUNAEA DE JOS UNIVESIY OF GALAI FASCICLE V designed to ensre the relative positioning between piece and tool he gearing pole is defined in this third reference system temporary applied constraints of this point, so it belongs simltaneosly to the normal and to the profile Sbseqently, these constraints are deleted in order not to modify the position of this point Fig he three elements of the assembly A gear type mechanism is defined in the assembly shown above his mechanism has, as fixed part, the tool and, as driving element, the rotation angle of the reference system joined with the tool Here is measred and keep the distance from gearing pole to the normal line drawn to the profile he following algorithm is applied in order to identify the contact points between piece and tool: ) he position where the normal to the profile pass throgh the gearing pole is identified simlating the mechanism In practice this identification is made monitoring the vale of the above mentioned distance and finding the mechanism position when this distance is below a certain vale ( mm) Fig Distance between gear pole and normal followed elements ) With the mechanism in this position, the sketch of the piece s profile is opened and the intersection point between the normal and the piece s profile is fixed For a correct positioning there are Fig 3 Intersection point between normal and profile temporary constraints 3) he piece draw is opened and the normal to the profile is moved into a new position of the crrent point Step is resmed with the new position of the normal and is contined with steps and 3 After an adeqate nmber of points are identified on the tool s profile, this may be materialized by drawing a spline crve throgh these points GENEAION OF ELAIVE AJECOIES IN OLLING POCESS WIH GEA SHAPED CUE he pair of centrodes is presented in figre 4 hese centrodes are circles, with r and r radii, for the rotary tool and blank he reference systems are defined as: - xy is the global reference system, joined with the revoltion axis of the blank and the C centrode - x 0 y 0 global reference system, joined with the C centrode s axis - XY mobile reference system, joined with the profile to be generated, initially overlapped on the xy reference system - X Y mobile reference system, joined with the ftre tool he rolling process kinematics for generation with gear shaped ctter assmes to accept the eqivalence: r r () Eqation () represents the rolling motion between the two centrodes, C and C he global motions are also known: and x 3 X () 0

3 FASCICLE V HE ANNALS OF DUNAEA DE JOS UNIVESIY OF GALAI x0 3 X, (3) of the two centrodes and, at the same time, of the spaces associated with XY and X Y Fig 4 olling centrodes reference systems C profile he position of the global reference systems is known: A x0 xa a A r r (4) 0 Now, the relative motion of the XY space regarding the X Y space can be dedced, X = 3 3 X-a (5) he ordered crl of profiles is defined as: X X() C Y Y(), or r X() iy() j,(6) so the directrix parameters of the normal to the profile are defined, i j k N C X Y 0 Y i X j (7) 0 0 or N C N X i+n Y j N YN X X Y (8) X X() N X Y Y() N Y (9) he trajectories of normal to the C profile can be defined, regarding the gear shaped ctter reference system, X Y : and N C X X Y cos Y X sin A cos Y X Y sin Y X cos A sin / (0) i, () r r or with i gear ratio For 0, eqations (0) represent the trajectories of points on the C in the relative motion regarding the gear shaped ctter reference system X Y : X X() cosi Y() sini A cosi ( ) () Y X() sini Y() cos i A sin i he enveloping of the in-plane trajectories family ( ), given by eqations (), represents the profile of the gear shaped ctter, reciprocally enveloping with the C ordered crl of profiles Enveloping condition he specific enveloping condition is obtained from constraint that the normals family (0) pass throgh the gearing pole (Willis theorem) [] he coordinates of the gearing pole (figre ), in the X Y Z reference system, are: Xrcosrcos i P Y r sin r sin i (3) In this way, the coincidence condition between the (N C ) normals family and gearing pole P, if the directrix parameters of the N C normal, given by (7), are: In this way, the vector of normal to C, in the XY space, has the form:

4 HE ANNALS OF DUNAEA DE JOS UNIVESIY OF GALAI FASCICLE V X() Y cos i Y() X sini A cosi r cosi X() Y sin i Y() X cos i A sini r sini, (4) 3 IN-PLANE GENEAING AJECOIES IN HE GENEAING POCESS WIH INENAL GEA SHAPED CUES In figre 5 are presented the rolling centrodes in the generating process of the internal teeth-rolling centrode for internal gearing which, removing the λ scalar parameter, determines the specific enveloping condition: X()cosi Y()sini rcosi Y cosi X sini (5) X()sini Y()cosi rsini X cosi Y sini where we consider the definition of the A vale, the distance between the centrodes axis, A r r Finally, eqation (5) can be broght to form: X X YY X cos Y sin (6) In this way, the () and (5) eqation assemblies allow to determine the enveloping of the in-plane generating trajectories by removing one of the independent parameters or φ Essentially, this allows determining the profile of the gear shaped ctter tooth s flank In principle, condition (5) can be expressed in the form, (7) and eqations () have the form: X X(), C S Y Y () he gearing line r (8) Obviosly, the eqations of the gearing line can be determined between the crves C (6) and C S (8), associating the enveloping condition with a trajectory in one of the global reference systems, as the example in (), reslting the form x cos sin X() (9) y sin cos Y() to which is associated condition (7) Fig 5 Internal gearing centrodes C and C C profile to be generated he reference systems are defined: - xy and x 0 y 0 are global reference systems joined with axis of C and respectively C centrodes - XY mobile reference system, joined with the generated profile, C - X Y mobile reference system, joined with the internal gear shaped ctter he distance between axes O and O is defined as: A r - r (0) he profile of the ordered crl of profiles is known in the XY mobile reference system, X X() C () Y Y(), with the variable parameter he generating process kinematics incldes the absolte motions: x 3 X () the revolving motion of the C centrode x0 3 X (3) the revolving motion of the C centrode he rolling correlation can be established between the anglar parameters φ and φ, in form: r r or i r, (4) r

5 FASCICLE V HE ANNALS OF DUNAEA DE JOS UNIVESIY OF GALAI where i is the gearing ration he relative position between the reference systems can be established: A x0 xa a, (5) 0 with A give by relation (7) he relative motion between the mobile reference systems, similarly with motion (5), is described by the transformation reslting ( ) X = 3 3 X-a, (6) () X X() cos i Y() sini A cosi Y X() sin i Y() cosi A sin i (7) he enveloping condition and the generating trajectories family (7) represent, by removing one of the or φ parameters, the C S profile of the gear shaped ctter, in principle, in the form: X X(), C S Y Y () 4 GEA SHAPED CUE FO GENEAING A SPLINED SHAF (3) An application related to profiling the gear shaped ctter for machining a splined shaft is presented he shaft profile is shown in figre 6 and the flank eqations are: X C (33) Y a, with variable he normals family to C can be written, starting from the eqation (0): (N ) C X X() Y cosi Y() X sini A cosi (8) Y X() Y sini Y() X cosi A sini he coordinates of the gearing pole are: XP rcos i YP rsini (9) Eqations (9) and (8) allow determining the enveloping condition: (N ) C Y() X sini r X() Y sini Y() X cosi X Y cos i A cos i cos i A sin i sin i, r (30) by removing the λ parameter from the eqation assembly (30), X( )cosi Y( )sini +rcosi X sini Y (3) cos i X()sini Y()cosi sini X cosi Y sini Fig 6 Splined shaft reference systems he variation limits for parameter are: a a (34) min i max e he nitary vector to the flank normal has eqation N j (35) he eqation of the normal to the C is N C i a j (36) In this way, the normals family eqations: (N ) C has 3

6 HE ANNALS OF DUNAEA DE JOS UNIVESIY OF GALAI FASCICLE V (N ) C X cos i (a )sin i A cos i Y sin i ( a )cos i A sin i (37) with i (38) From the condition that the normals family (37) pass throgh the gearing pole, P, P Xrcos i Y r sin i, reslts the enveloping condition, (39) arccos (40) r he in-plane generating trajectories family reslts from (37) for 0 : X cos i asin i A cos i (4) Y sin i acos i A sin i Eqations (4) and (40) determine the gear shaped tool s profile 4 Nmerical application A nmerical application is presented for a splined shaft with e = 65 mm i = 56 mm z = 0 splines b=45 mm radis of gear shaped tool r = 35 mm gearing ratio i = able and Figre 7 present the profile of the gear shaped tool and the coordinates for the profile with one tooth, in the X Y reference system able Coordinates of the flank profile X Y X Y Fig 7 Profile of the gear shaped tooth 4 Graphical soltion in CAIA For the above shown profile it was applied the algorithm presented in section he comparison between the reslts obtained with the graphical method and those obtained with the analytical one is shown in table hese reslts prove that the points fond onto the tool s profile are identical he tool s profile is presented in Figre 7 able Comparison between the two methods Analytical method Graphical method X Y X Y GEA SHAPED OOL FO A POLYGONAL BUSH here are technologies for machining holes with sqared or hexagonal sections by shaping with gear shaped ctting tool he isse soltion for gear shaped tool s profiling is proposed, sing the relative generating trajectories method (figre 5) In figre 8, the bsh shape is presented together with the crossing section, the pair of rolling centrodes and the reference systems: 4

7 FASCICLE V HE ANNALS OF DUNAEA DE JOS UNIVESIY OF GALAI - C is the centrode joined with the bsh, a circle with radis r - C centrode joined with the gear shaped tool, a circle with radis r - i gearing ratio, i r r - xy global reference system - XY mobile reference system, joined with centrode C - X Y mobile reference system joined with centrode C he eqations of the C profile are: X a C (4) Y min he variation limits for parameter are: a and max a he normal to the C profile is: i j k N C 0 0 i (43) 0 0 he in-plane generating trajectories family (7) has eqations: X acosi sini A cosi (44) Y asini cosi A sin i he enveloping condition (3) is: arcsin r, (45) with A r he (44) and (45) eqation assemblies represent the profile of the gear shaped tool Fig 8 Profile of the gear shaped ctting tool 5 Graphical soltion in CAIA he same problem was solved sing the capabilities of the CAIA design environment he comparative reslts are presented in table 4 and the tool s profile is shown in Figre 9 able 4 Coordinates of the gear shaped tool s profile obtained by analytical and graphical methods Analytical method Graphical method X Y X Y Nmerical application A nmerical application for a sqared bsh with a = 40 mm and i = 4/3 is presented In able 3 and Figre 9 are presented the coordinates and the form of the gear shaped ctting tool able 3 Coordinates of the gear shaped tool s profile X Y X Y Fig 9 Profile of the gear shaped tooth 5

8 HE ANNALS OF DUNAEA DE JOS UNIVESIY OF GALAI FASCICLE V 6 CONCLUSIONS his paper proposes a graphical soltion for the isse of profiling tools which generate by enwrapping he soltion is based on the description of the relative trajectories obtained in the generating process hese are the trajectories of the point belonging to the profile to be generated in their motions in the tool s space A new form for the enveloping condition is proposed he obtained reslts prove that the soltion is simple, easy to apply and rigoros ACKNOWLEDGEMEN his work was spported by a grant of the omanian National Athority for Scientific esearch and Innovation, CNCS UEFISCDI, project nmber PN- II-U-E EFEENCES [] Litvin, FL, heory of Gearing eference Pblication, NASA, Scientific and echnical Information Division, Washington DC, 984 [] Oancea, N, Srfaces generation by enwrapping, Vol I, II, III, Complementary heorems, Dnărea de Jos University Pblishing Hose - Galaţi, ISBN , 004 [3] eodor, VG, Contribtion to elaboration method for profiling tools which generate by enveloping, Lambert Academic Pblishing, ISBN , 00 [4] Baic, I, Oancea, N, A Grafic Variant to Profile ools for the Generation by Winding of Srfaces Profiling of otative Ctter, he Annals of Dnărea de Jos University of Galati, Fascicle V, Volme XIX, ISSN , pp 3-39, 00 [5] Berbinschi, S, eodor, V, Oancea, N, A Graphical Method Developed in the CAIA Design Environment for the Modeling of Generation by Enveloping, he Annals of Dnărea de Jos University of Galati, Fascicle V, echnologies in Machine Bilding, ISSN -4566, pp 9-35, 0 6