THE WORM CONJUGATED WITH AN ORDERED CURL OF INVOLUTE CYLINDRICAL SURFACES

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1 TH AAL OF DUĂRA D JO UIVRITY OF GALAŢI FACICL V, TCHOLOGI I MACHIBUILDIG, I TH WORM COJUGATD WITH A ORDRD CURL OF IVOLUT CYLIDRICAL URFAC icolae OACA, Virgil Gariel TODOR icuşor BAROIU, Florin UAC Department of Manufacturing ngineering, Dunărea de Jos University of Galaţi, florin.susac@ugal.ro ABTRACT The involute teeth of straight or helical teethed wheels are usually machined with ho mill. The ho mills admit as primary peripheral surface a cylindrical helical surface with constant pitch. In this paper, ased on the complementary theorem of generating trajectories family, it is demonstrated that the worm conjugated with an ordered curl of involute flanks is also an involute worm. The proposed development is ased on analytical representations of the enwrapping surfaces with single point contact. In prolem solving was applied the method of intermediary surface (generating rack gear), developing the issue as a succession of enwrapping surfaces with linear contact. As application, it is presented a solution of the same issue in a graphical design environment CATIA. Keywords: graphical method, CATIA, generating trajectories family. ITRODUCTIO The involute teethed wheels are usually machined y enwrapping, using the rolling method, with rack gear tools, gear shaped tools of ho mills [, ]. The constructive profiling of the worm cutter, assumes to know the primary form of the generating worm as ase of active surface which represent the teeth of cutting tool the ho mill. The ho mill profiling can e made ased on the fundamental theorems of surfaces enwrapping, the Olivier or Gohman theorem []. The second theorem of Olivier is particularly used, as theorem for enwrapping of surfaces with point contact, with intermediary surface s method. The intermediary surface is represented y the flanks of rack gear conjugated with a wheel with straight or helical teeth. For the involute profile of the surfaces curl, at contact with a ho mill, were imagined solutions for determining the theoretical form of the worm, y uler himself and, further y Kutzach (95). These solutions were given y Olivier fundamental theorems. For the same prolem type, solutions were proposed y Oancea [4], ased on the Gohman theorem, using the method of intermediary surface. They are methods known for generating various worms types [, 3, 5] such as the Archimedes s worm and involute worm with tools which materialize a straight line generatrix or for the unfolded worm, with an in-plane surface (grinding wheel) modelling a plane tangent to a helical surface. Also, Litvin et al. [3] recommend techniques and solutions for generation of worms on type K and F. For profiling the primary peripheral surface of the worm conjugated with involute teeth were imagined solutions and were created specialized algorithms in LIP for AutoCAD [6]. Also, were made applications in CATIA regarding the generation of helical surfaces [7] using the capailities of this graphical design environment [8]. In this paper, is proposed an analytical approach for determining the worm conjugated with an ordered curl of involute cylindrical surfaces (involute straight teethed wheel), ased on a complementary theorem of surface enwrapping the family of generating trajectories [9]. The goal is to demonstrate, in an analytical way, that the worm conjugated with involute teeth is an involute worm.. IVOLUT FLAK OF TTHD WHL In figure, is presented the frontal profile of the involute tooth, associated with a reference system, XYZ, where the Z axis is overlapped y the teethed wheel s axis. At the same time, is defined the reference system associated with the rack gear s

2 TH AAL OF DUĂRA D JO UIVRITY OF GALAŢI FACICL V centrode and the xyz gloal reference system, initially overlapped y XYZ reference system. Fig.. Involute profile, C and C rolling centrodes; I wheel s rotation; II - rack gear s translation The involute of circle with radius R, see figure, is defined y vectorial equation OM OT TM, () where: OT R cos i R sin j ; () sin j TM R i, (3) R cos with variale parameter, and and constant geometrical values. In this way, the involute equations, the uler equations, are defined as: X R cos R sin sin ; Y R sin R sin cos. (4) The two constants, and are deduced from equations: x y R x y R r ;, (5) R Rrcos, ( normalized, current =) and R r radius of C centrode. The rolling process kinematics for the two centrodes includes the movements: - the rotation of C centrode and, joined with this, the XYZ reference system around the z axis of the gloal reference system, T x X, (6) 3 with rotation angular parameter; - the translation of the C centrode, along the axis: Rr x a; a. (7) Rr In this way, the relative motion of the XYZ reference system regarding the system, from (6) and (7), results in form: T Rr 3 X. (8) Rr From (8) results: R cos sin sin R cos sin sin R cos R R cos sin Rr; R sin R (9) R cos cos Rr. The involutes family, in the reference system, can e restrained in form: R cos R sin R sin R ; R cos R. r r () The enwrapping of involutes family () represents the profile of rack gear in plane. 3. RACK GAR RCIPROCALLY WRAPPIG WITH IVOLUT FAMILY It is proposed the determination of enwrapping condition y the method of generating trajectories family [9]. For this, are defined the directrix parameters of the normal to the involute (the TM versor, see figure ) denoted with n : n sin i cos j. () The direction of normal to the involute (4) may e written as: X R cos R sin sin sin ; Y R R cos cos, with variale scalar value. () 3

3 FACICL V TH AAL OF DUĂRA D JO UIVRITY OF GALAŢI It is possile to define the trajectories family for the normal () in the generating movement (8), y transforming: T : a, (3) 3 or, developed: R cos sin sin Rr; R sin cos cos R. R R r (4) In the centrodes rolling process, the family of normals to the involute profile (4) must pass through the gearing pole: ; P (5) Rr. From the requirement that the equations (4) accomplish the conditions (5), it is otained an equations system, where is eliminated the parameter: R cos sin R and sin sin R R sin cos cos cos R Rr Rr. r ; (6) From the equality of the two conditions for results the specifically enwrapping condition:. (7) In this way, joining the equations which represent the involute family () with the enwrapping condition (7) results the parametrical equations of the rack gear s profile : R cos R sin Rr ; (8) R sin R cos R. r For the involute profile of circle with R radius the relations are known: tan and R Rr cos, (9) so, the equations (8) can e rought to form: x = -R j sina cosa; () h = R j sina sina, which, with notation: u R sin, () may e descried y the following equations: u cos ; () u sin, representing the profile of the generating rack gear in the plane. If we accept that the involute flank is a cylindrical surface, with frontal profile (4) and with a generatrix parallel with the Z axis, as is the case of straight teeth wheel, having the generatrix equation Z t, (t variale), (3) then, the generating rack gear s equations are: u cos ; usin ; t, (4) with u and t variales parameters, see figure. The equations (4) represent, in principle, a cylindrical surface with generatrix parallels with the axis. The cylindrical surface is reduced, in this case, to a plane parallel with the axis. Fig.. The rack gear s flank the plane, reciprocally enveloping with the involute flanks; reference systems; generating kinematics 4. AALITICAL PROFILIG OF HOB MILL The ho mill, reciprocally enwrapping with flanks or the involute teethed wheel, accepts as primary peripheral surface a cylindrical helical surface with constant pitch, which admits a contact point 33

4 TH AAL OF DUĂRA D JO UIVRITY OF GALAŢI FACICL V (characteristic point) with the involute cylindrical flank in the frontal plane of the teethed wheel, constituting a pair or enveloping surfaces. In the following is proposed the determination of conjugated worm (the primary peripheral surface of ho mill) using the complementary method of the generating relative trajectories. The method is applied in analytical form, using the principle of intermediary surface (generating rack gear) in linear contact with the involute cylindrical flank of the teethed wheel the characteristic curve etween the wheel and the rack. Also, the helical surface of the ho mill is profiled as surface reciprocally enveloping with the generating rack gear s flank. The two characteristic curves are simultaneous onto the rack gear s surface. The instantaneous intersection point of these curves represents the characteristic point the point where the ho mill generates the involute flank of the teethed wheel. In figure 3, are presented the reference system associated with the lank, the generating rack gear and the future ho mill. relative reference system joined with the A axis, which results from the decomposition of the helical movement (the worm s helix with axis V and p helical parameter); X Y Z relative reference system joined with the worm representing the primary peripheral surface of the worm, enwrapping of the involute flank of the wheel; It is considered that the generating movement of the worm the primary peripheral surface of the future ho mill (V, p) is decomposed in two movements: - translation along the generatrix of the rack gear s surface, with direction t parallel with the lank s axis the axis Z; - rotation around the A axis parallel with the V axis of the ho mill and at the distance a from this (see figure 3): a p tan. (5) o, the decomposition of the helical movement Fig. 3. Reference systems generating kinematics They are defined: xyz is the gloal reference system with z axis overlapped y the teethed wheel; x y z auxiliary reference system joined with the axis of the future ho mill (y the axis of the worm); XYZ relative reference system joined with the involute flank of the teethed wheel (initially overlapped to the xyz reference system); relative reference system, joined with the centrode associated with the rack gear, with axis parallels to xyz; the axis is overlapped y the C centrode of the rack gear. may e symolically represented: V, p T t, v A, A. (6) with v and A - movement parameters. The flank of the generating worm results as enwrapping of the generating rack gear s surface (3) in the helical motion (V, p) or in the movements assemly in which it is decomposed rotation with axis A and translation along the generatrix of t versor, see figure. 34

5 FACICL V TH AAL OF DUĂRA D JO UIVRITY OF GALAŢI We have to notice that, in the motion T t, v, the generating rack gear is self-generated. o, the characteristic curve of the surface (4) will not depend on this component of movement, ut only on the rotation around the A axis. This approach reduces the calculation effort, making easy to determine the characteristic curve etween the rack s surface and the worm which constitutes the primary peripheral surface of ho mill. The generating process kinematics includes: - the translation of the rack gear: R r x a; a ; (7) - the rotation of worm around the V axis: T x X ; (8) - the relative position of fixed reference systems: Rr a x x a ; a ; (9) - the relative motion etween the moile reference systems (the relative motion of the rack gear regarding the reference system joined with ho mill): X a a, (3) where: cos sin, (3) sin cos cos sin, (3) sin cos and p cos, (see (8) and fig. 3), (33) angle etween V axis and the frontal plane of the involute teethed wheel. In this way, the matrix form of the surface generated y the rack gear flank is deduced regarding the ho mill s reference system: X cos sin Y Z sin cos cos sin sin cos (34) ucos a usin p cos t. After developments, results the parametrical equations of the rack gear s movement regarding the V axis (helical movement with p helical parameter): X u cos a cos t sin cos usin p cos sin sin ; Y p cos usin t sin ; Z u cos a sin t cos cos usin p cos cos cos. (35) We have to notice that the translation of the reference system (joined with the rack gear) is defined in correlation with the translation along the V axis in the helical motion: p cos. (36) The enwrapping of the surfaces family regarding the helical surface s reference system represents the primary peripheral surface of the future ho mill. The enwrapping of the family, (35), is determined ased on the complementary theorem of the generating trajectories [9]. The versor of the normal to the surface is calculated: i j k n cos sin sin i cos j (37) and, in this way, the direction of the normal to, in the current point, can e determined, (4) and (37): u cos ksin ; u sin k cos ; t, where k is a variale scalar value. (38) 35

6 TH AAL OF DUĂRA D JO UIVRITY OF GALAŢI FACICL V The A axis is parallel with V axis and at distance a from this (a is measured along the x x axis), see figure 3. The normal equations, (38), are written in the reference system joined with A and with axis parallels to X Y Z y transformations: cos sin (39) sin cos. From (38) and (39) results the form of the normal, in the reference system with axis overlapped to A axis (see figure 3): u cos ksin ; usin k cos cos t sin ; usin k cos sin t cos. (4) By rotating the normal, expressed in the reference system, around the A axis, with angle : cos sin sin cos (4) u cos ksin u sin k cos cos t sin, u sin k cos sin t cos is determined the normal s family at the rack gear s flank, in the reference system: u cos ksin cos u sin k cos sin t cos sin ; u usin k cos cos t sin ; u cos ksin sin sin k cos sin t cos cos. (4) According to the complementary theorem of the generating trajectories [9], is imposed the intersect condition that the normals family the A axis, which, in the reference system has equations: ; A. (43) If it is eliminated the k parameter from (4) and (43) equations assemly, the specific enwrapping condition is determined. By equalizing the coordinate from (4) and (43), results: ucos cosusin sin sin k sin coscos sin sin (44) t cossin sin cos cos sin sin and, similarly, for : ucos sinusin sin cos k sin sincos sin cos (45) t coscos. sin sin cos sin cos By equalizing the equations (44) and (45), results the condition tan t u, (46) sin representing the enwrapping specific condition, which, associated with rack gear s flanks family, in the relative motion regarding the X Y Z reference system (35) determines the primary peripheral surface of the future ho mill: X u cos a cos u tan sin cos sin usin p cos sin sin ; Y p cos usin cos u tan sin ; sin Z u cos a sin usin p sin cos (47) u tan coscos. sin The equations (47) represent a cylindrical helical surface with Y axis and p helical parameter. 5. HOB MILL HLICAL URFAC FORM The worm s form is analyzed as helical surface reciprocally enwrapping with a curl of cylindrical surfaces with involute profile in frontal plane (straight teethed wheel). From (47), for, (48) the generatrix of the helical surface is determined: 36

7 FACICL V TH AAL OF DUĂRA D JO UIVRITY OF GALAŢI X u cos a; tan G Y u sin cos u sin ; (49) sin tan Z u sin sin u cos, sin representing a straight line for which the directrix parameters can e defined: l cos ; sin Gm sin cos ; (5) sincos sin n sinsin. sin At the same time, the directrix parameters of the V j axis are defined in the X Y Z reference system: V l ; m ; n. (5) The relative position of the two straight lines V and G is examined. The minimum distance etween the two lines can e calculated in vectorial form, see figure 4. The angle etween the directions G and V is defined as: GV tan. (5) GV The vectorial product G V is calculated: i j k sin sin GV cos sin cos sin sin sin cos sin (53) or (56), results the definition of the angle etween G and sin G V sin sin V: i cos k (54). sin sin cos sin sin The modulus of the GV vectorial product is: sin tan. (57) sin sin GV cos sin sin sin. (55) sincos sincos The distance etween the two straight lines is Also, is calculated the scalar product defined y equation: sin GV sincos. (56) d, (58) sincos GV In this way, from (5) and the definitions of where is the mixed product r, G, V or the vectorial product modulus (55) and scalar product a sin sin cos sin cos sin sin sin cos sin (59) sin a sinsin (6) sin and r is the vector which link two points elonging to the lines G and V, see figure 4: r a i. (6) o, the minimum distance etween the two lines, G and V, is sin a sinsin sin d. (6) sin sincos sincos 37

8 TH AAL OF DUĂRA D JO UIVRITY OF GALAŢI FACICL V We have to notice that the minimum distance (6) represents the radius of a cylinder which admits V as axis and also admits as tangent direction the helical surface s straight line generatrix, (47), see figure 4. The expressions (57) and (6) can e processed, see Appendix, so result, from (57), cos tan. (63) tan sin Also, from (6), see Appendix and figure 5, results p cos d. (64) tan sin Fig. 5. The unfold of helix with p parameter on the cylinder with radius d It is oviously that, (66) so, the G line is tangent to the worm helix on the cylinder with radius d and axis V. o, the worm conjugated with the involute teethed wheel is an involute worm. 6. COCLUIO Fig. 4. The G and V lines, the minimum distance d and the angle From (64), results d cos tan, (65) p tan sin see figure 5 and Appendix. It is proved, y the complementary method of the generating trajectories, that the conjugated worm of a straight involute teethed wheel is a cylindrical helical surface with constant pitch an involute worm. The involute worm ruled worm, allows a straight line generatrix tangent to the helix onto the cylinder with radius R rt of the ho mill. The application demonstrates that the worm s generatrix is on the distance d, see (6), from the axis of ho mill and at angle from this axis. The angle represents the angle of the helix onto the cylinder with radius R rs, condition which represents the definition of the involute worm. As a consequence, the only helical surface reciprocally enwrapping with an involute teethed wheel is an involute worm accepted as peripheral primary surface of a generating ho mill. A rigorous ho mill may e constituted only if the cutting edges of the teeth are curves which elong to the involute worm. 38

9 FACICL V TH AAL OF DUĂRA D JO UIVRITY OF GALAŢI APPDIX xpressions of the angle and d minimum distance * 4 sin cos cos cos sin sin cos sin cos sin sin sin sin cos cos sin tan sin. tan tan sin a sin a acossin a sin sin sin cos sin. sin sin sin tan ** *** (67) (68) o: cos tan sin tan sin tan sin tan sin cos sin tan sin cos sin cos cos sin cos tan sin tan sin (69) sin sin tan cos tan sin cos cos cos cos cos tan sin cos tan sin. sin sin tan tan sin tan sin cos cos tan cos tan sin cos. (7) tan sin tan sin APPDIX d tanv. (7) p sin a cos sin sinsin a sin d tan. (7) GV sin sin sin cos sin cos sin cos sin cos a sin p cos d. (73) tan sin tan sin d a sin pcos cos tan. (74) p p tan sin p tan sin tan sin It is oviously that the angle etween the G andv vectors and the V angle the angle of helix onto the cylinder with radius d of the worm are equals, see figure 5. 39

10 TH AAL OF DUĂRA D JO UIVRITY OF GALAŢI FACICL V APPDIX 3 Geometrical values of the teeth and the dimensions of the ho mill - circular pitch of the teeth straight teethed wheel Rr pc ; (75) zd z d is the numer of teeth of the wheel; - normal pitch of the generating rack gear, Rr pn p ; rack c (76) zt - angle of ho mill s helix onto the cylinder with radius R rs : pax _ worm tan (77) R where p is the axial pitch of the ho mill and R rt is the rolling radius of the ho mill; ax _ worm - a p tan the relation (6) may e rewritten as:, (78) where is the angle etween the generatrix of the helical surface and the axis of ho mill; - from equality condition etween the normal pitch of the rack gear and the ho mill: Rr Rrt tan cos, (79) z t results rp sin R R rt z t (8) - the helical parameter of the ho mill pax _ worm Rrp p z t cos ; (8) - the a distance Rrp Rrp Rrt a zt Rrt. z sin z R (8) t s t rp rt ACKOWLDGMT This work was supported y a grant of the Romanian ational Authority for cientific Research and Innovation, CC UFICDI, project numer P-II-RU-T RFRC [] Litvin, F.L., Theory of Gearing, Reference Pulication AA, cientific and Technical Information Division, Washington D.C., 984. [] Radzevich,.P., Kinematics Geometry of urface Machining, CRC Press, London, 8, IB [3] Litvin, F.L., Fuentes, A., Fan, Q., Handschuh, R.F., imulation of Meshing and Contact and tress Analysis of Face Milled Formate Generated piral Bevel Gears, Mechanism and Machine Theory, 37, pp ,. [4] Oancea,., urfaces generation y enwrapping, Vol. I, II, Dunărea de Jos University Foundation Pulishing House, IB , Galaţi, 4. [5] emencenko, I.I., Matiuşin, V.M., aharov, G.., Proektirovanie metallorejuşcih instrumentov, Gosudarstvennoe naucino tehnicescoeizdate listvo maşinostroitelinoi literaturî, Moskva, 96. [6] Baicu, I., Oancea,., Profilarea sculelor prin modelare solidă, d. Tehnica - Info, Chişinău, IB X,. [7] Berinschi,., Teodor, V.G. Oancea,., 3D graphical method for profiling tools that generate helical surfaces, The International Journal of Advanced Manufacturing Technology, May, Volume 6, Issue 5-8, pp. 55-5,. [8] Berinschi,., Teodor, V., Baroiu,., Oancea,., nwrapping urfaces with Point Contact - Comparisson Between CATIA Method and Analytical One, The Annals of Dunărea de Jos University of Galati, Fascicle V, Volume II, pp. 7-, I -4566,. [9] Baroiu,., Teodor, V., Oancea,., A new form of in-plane trajectories theorem. Generation with rotary cutters, Bulletin of the Polytechnic Institute of Iasi, Tome LXI (LXV), Fascicle 3, section Machine Construction, I -855, pp. 9-36, 5. [] Litvin, F.L., Ignacio, G.P., Alfonso F., Generalized concept of meshing and contact of involute crossed helical gears and its application, lsevier Comput.Methods Appl. Mech. ng., 94, pp , 5. 4