Journal of Machine Engineering, Vol. 15, No. 4, 2015

Transcription

1 Jounal of Machine Engineeing, Vol. 15, No. 4, 2015 Received: 09 July 2015 / Accepted: 15 Octobe 2015 / Published online: 10 Novembe 2015 Vigil TEODOR 1* Vioel PAUNOIU 1 Silviu BERBINSCHI 2 Nicuşo BAROIU 1 Nicolae OANCEA 1 in-plane geneating tajectoies, CATIA, olling geneating tools THE METHOD OF IN-PLANE GENERATING TRAJECTORIES FOR TOOLS WHICH GENERATE BY ENVELOPING - APPLICATION IN CATIA The method of in-plane geneating tajectoies is a method designed to study the enveloping pofiles associated with a pai of olling centodes. This method assumes knowledge the analytical fom of the in-planes tajectoies descibed by the points onto the pofiles to be geneated. These tajectoies ae known in the efeence system of the geneating tool, which may be a ack-gea, a gea-shaped o a otay cutte tool. The envelope of these tajectoies epesents the pofile of the geneating tool. An oiginal method fo detemining the enveloping condition is pesented in this pape. Application fo odeed culing of non-involute pofiles wee developed based on the specific enveloping condition. A new solution is poposed. This solution uses the capabilities of the CATIA gaphical envionment. Gaphical solutions ae poposed vesus analytical solutions in ode to validate the poposed method. 1. INTRODUCTION The issue of ack-gea tool pofiling is solved based the Olivie theoem I [4] and, also, based on the Gohman theoem [4],[5],[8]. N v 0 (1) In equation (1), N is the nomal diection to the pofile to be geneated, in the suface s own efeence system and v is a vecto with the same diection as the velocity in the elative motion between the ack-gea tool and the suface to be geneated. The analytical solution of this poblem is univesal and leads to igoous esults. Complementay theoems wee elaboated concening this issue [5]: the method of the family of the substitution cicles and the method of the minimum distance. These 1 Univesity Dunaea de Jos of Galati, Depatment of Manufactuing Engineeing, Romania 2 Univesity Dunaea de Jos of Galati, Mechanical Engineeing Depatment, Romania * vteodo@ugal.o

2 70 Vigil TEODOR, Vioel PAUNOIU, Silviu BERBINSCHI, Nicuso BAROIU, Nicolae OANCEA methods, developed at Dunaea de Jos Univesity of Galaţi, expessed the enveloping condition in specific foms. These foms may epesent intuitive images of the enveloping pocess. The in-plane tajectoies method [8] was developed at Dunaea de Jos Univesity of Galaţi, by a eseach team diected by pofesso Nicolae Oancea and associate pofesso Vigil Teodo and was pesented as pat of doctoal thesis of Vigil Teodo. This method is a specific solution fo which the autho pesent applications based on a specific enveloping condition. The issue of geneation with tools associated with a pai of olling centodes is a pemanent concening as is pesented in specialized liteatue. An oiginal enveloping condition fom is poposed in this pape. The enveloping condition is detemined stating fom the pocess of slotting with the ack-gea tool. 2. GENERATION BY ROLLING WITH RACK-GEAR TOOL The olling centodes, the efeence systems and the geneated pofile ae pesented in Fig. 1. Fig. 1. Geneation with ack-gea tool; C 1 and C 2 pai of olling centodes The efeence systems ae defined: xy is the global efeence system; XY elative efeence system, joined with the geneated pofile and with the C 1 centode; ξη elative efeence system joined with the C 2 centode. The geneating pocess kinematics includes the absolute movements of the XY elative efeence system,

3 The Method of In-Plane Geneating Tajectoies fo Tools which Geneate by Enveloping 71 and of the ξη elative efeence system, T 3 x X (2) R x A, A R. The elative motion is detemines fom the absolute motions (2) and (3). The elative motion descibes the movement of a point belongs to the XY space egading the ξη space. The coodinates of point fom the C pofile in the XY efeence system ae (3) X X u ; C Y Y u, with u vaiable paamete. If the diecto paametes of the nomal diection to the C pofile ae defined as (4) o i j k N C X u Yu 0 (5) N C i Y u j X, (6) u then, the pependicula dawn fom the cuent point to the C pofile, has the equations: N C X X u Y ; Y Y u X, with δ vaiable paamete. The family of nomals is detemined fom (4) and (7) when the XY efeence system is moved egading the ξη efeence system u u (7) N C, X u Y u cos Y u X u sin R ; X u Y u sin Y u X u cos R. (8) Fo δ = 0, the equations (8) epesents the tajectoies family of points fom the C pofile, egading the ξη efeence system, namely the in-plane geneating tajectoies. Accoding to the Willis theoem [4], the necessay and sufficient condition fo the pofile C admits an enveloping cuve is that the pependiculas to the C pofile pass though the geaing pole. The geaing pole is the tangency point between the two centodes C 1 and C 2, see Fig. 1. The coodinates of geaing pole in the ξη efeence system ae, see Fig. 2:

4 72 Vigil TEODOR, Vioel PAUNOIU, Silviu BERBINSCHI, Nicuso BAROIU, Nicolae OANCEA P P P 0; R. (9) Fig. 2. The geaing pole and the u N C nomal diection to the in-plane tajectoy C The condition that the family of nomals to the C pofile pass though the geaing pole is give by the equations assembly, see (8) and (9): X u Y u cos Y u X u sin R 0; X u Y u sin Y u X u cos R R, (10) The equations system (10), allows to establish a dependency between the u and φ vaiables paametes: u u. (11) The dependency (11) epesents the enveloping condition between the C tajectoies and the C S pofile of the ack-gea tool. Also, the equation (11) allows detemining the δ scala value which epesents the distance fom the cuent point of the C tajectoy to the geaing pole. In the olling pocess of the two centodes, at a cetain moment, which means a defined value fo the φ paamete, the pofiles: C associated with the C 1 centode; the family of in-plane tajectoies geneated by the point fom the C cuve and the pofile of the futue ack-gea tool, admit a common nomal which pass though the geaing pole, P. The specific enveloping condition is detemines fom the equations assembly (10), emoving the paamete δ:

5 The Method of In-Plane Geneating Tajectoies fo Tools which Geneate by Enveloping 73 X cos Y sin u u u u, (12) R X u X Y u Y The ack-gea tool s pofile is defined as give by the assembly of equations (8) and (12). The ack-gea tool s pofile may be defined as the enveloping of the assembly of cuves which epesents the in-plane geneating tajectoies of the points fom the geneated pofile. These tajectoies ae descibed in the ack-gea tool s own efeence system, ξη: T u cos sin X u R. (13) sin cos Y u R The distances fom the T(u) φ tajectoies to the geaing pole should have minimum values, fo vaious olling positions. Indeed, the δ scala value is measued onto the nomal diection to the geneating tajectoy of a point fom the pofile C Σ. This nomal diection passes though the geaing pole and, as a consequence, the value of the δ paamete epesents the minimum distance fom the tajectoy to this pole. 3. APLICATIONS In the following, applications of the in-plane geneating tajectoies method ae pesented fo the detemination of the ack-gea pofile. A gaphical methodology is developed in connection with these applications RACK-GEAR TOOL FOR GENERATION OF A SHAFT WITH HEXAGONAL FRONTAL PROFILE The pofile of the hexagonal shaft, the olling centodes assembly and the efeence systems ae pesented in Fig. 3 (xy is the wold efeence system; XY - efeence system joined with the Σ suface and C 1 centode; ξη - efeence system joined with the ack-gea and the C 2 centode). The absolute movements of the efeence systems joined with the C 1 and C 2 centodes ae give by equations (2) and (3). The elative motion is defined by: T 3 X A. (14) The equation (14) epesents the elative motion of the XY space egading the ξη space.

6 74 Vigil TEODOR, Vioel PAUNOIU, Silviu BERBINSCHI, Nicuso BAROIU, Nicolae OANCEA Fig. 3. Hexagonal shaft; efeence systems and the in-plane geneating tajectoy of the point M The paametical equations of the C Σ pofile, with u vaiable paamete, ae: X b; C Y u. (15) Theeby, the nomal vecto to C Σ cuve has fom: N C 1 i 0 j. (16) The nomal in the cuent point to the C Σ cuve has equations, see (7): N C X b 1 b ; Y u 0 u. (17) The family of nomals, in elative motion egading the ξη space, has the equations: N C, b cos u sin R ; b sin u cos R. (18) If, a membe of family (18) is constaint to pass though the geaing pole is obtained the fom: b cos u sin R ; b sin u cos 0. The specific enveloping condition is detemined by emoving the δ paamete: (19) u R sin. (20)

7 The Method of In-Plane Geneating Tajectoies fo Tools which Geneate by Enveloping 75 In the same time, the scala value δ is detemined: R u R cos b. (21) 2 2 The δ scala epesents the distance measued along NC fom the geneating tajectoy to the geaing pole, P. Fo δ=0, the (18) equations assembly epesents the inplane geneating tajectoies family, is space ξη: T u b cos u sin R ; b sin u cos R. (22) The enveloping of the family (22) is the pofile of the ack-gea which geneates the hexagonal shaft. The hexagonal shaft, the geneating tajectoies and the ack-gea tool s pofile ae pesented in Fig. 4 and Fig. 5. The olling adius is 32 mm. Fig. 4. Geneating tajectoies; ack-gea tool s pofile and hexagonal shaft pofile Fig. 5. Coodinates of points fom the ack-gea tool s pofile

8 76 Vigil TEODOR, Vioel PAUNOIU, Silviu BERBINSCHI, Nicuso BAROIU, Nicolae OANCEA The 3D model of the hexagonal shaft is pesented in Fig. 6 (R =32 mm). A mechanism is constucted accoding to the kinematics of geneating pocess. Fig. 6. The 3D model of the hexagonal shaft The diffeences between the coodinates of points obtained by this method and the coodinates obtained by an analytical method ae pesented in Table 1. Ct. no. Table 1. Eo of gaphical method Analytical method Gaphical method ξ [mm] η [mm] ξ [mm] η [mm] Eo [mm] It is obviously that the two methods lead to identical pofiles RACK-GEAR FOR A CIRCULAR PROFILE It is consideed the pofile pesented in Fig. 7. The paametical equations of the pofile ae: X R cos ; 0 C Y sin. (23)

9 The Method of In-Plane Geneating Tajectoies fo Tools which Geneate by Enveloping 77 Fig. 7. Cicula pofile; C 1 and C 2 centodes The family of nomals to the C Σ pofile has equations: N C, R0 1 cos cos 1 sin sin R ; R0 1 cos sin 1 sin cos R. (24) The constaint that the nomal pass though the geaing pole leads to the specific enveloping condition: R0 sin acsin. (25) 1 R Fo δ=0, fom (24), the geneating tajectoies family is obtained: T R cos cos R ; 0 R sin sin R. 0 (26) The enveloping of the geneating tajectoies family T epesents the ack-gea tool s pofile. Fo the 1 2 MM ac, the constant values ae: R = 50 mm; R 0 = 48 mm; = 6.5 mm and lim = 60. The fom and coodinates of the ack-geas pofile ae given in Fig. 8, Fig. 9 and Table 2.

10 78 Vigil TEODOR, Vioel PAUNOIU, Silviu BERBINSCHI, Nicuso BAROIU, Nicolae OANCEA Fig. 8. Rack-gea s pofile fo the ac MM 1 2 Fig. 9. Coodinates of points fom the ack-gea tool s pofile Ct. no. Analytical method Table 2. Rack-gea s pofile Gaphical method ξ [mm] η [mm] ξ [mm] η [mm] Eo [mm]

11 The Method of In-Plane Geneating Tajectoies fo Tools which Geneate by Enveloping GRAPHICAL APPLICATION In ode to solve the geneating issue, a mechanism was constucted, epoducing the geneating movements when machining with ack-gea tool. The mechanism is composed fom thee elements: tool, which is defined as fixed element; piece, which has the elative motion egading the tool and base which assue the elative position of the two peviously pesented elements. The joint between tool and piece is on type ack, composed fom a pismatic joint (between tool and base) and a evolute joint (between base and piece). The mechanism simulation was done in the DMU Kinematics module of the CATIA softwae, using the simulation command. Consequently, this simulation was compiled ( compile simulation command) and was saved fo futhe eplay. The tajectoies of points fom the piece s pofile wee dawn using the tace command. The points which ae neaest to the geaing pole and belong to each tajectoy wee identified in the Geneative Shape Design module, using the extemum pola command. These points ae tangency points between the cuves family and its enveloping. The tool s pofile was obtained as a spline cuve which admits as contol points the peviously detemined points. The identity between these points and those obtained by analytical way is obviously. 4. CONCLUSION The method of geneating tajectoies is based on the geaing basic theoem the Willis theoem. The family of tajectoies is geneated by the points of the geneating pofile, in the elative motion egading the ack-gea. The tool s pofile esults as enveloping of a geneating tajectoies family. A gaphical method was pesented. This method is developed in the CATIA design envionment. It is simple and easy to apply due to the capabilities of the CATIA softwae. The gaphical method is based on the capability of this softwae to daw tajectoies of points which belongs to specific mechanism. ACKNOWLEDGMENTS The wok has been funded by the Sectoal Opeational Pogam Human Resouces Development of the Ministy of Euopean Funds though the Financial Ageement POSDRU/159/1.5/S/

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