COMPLEMENTARY THEOREMS OF SURFACE ENWRAPPING EXPRESSED IN GRAPHICAL FORM

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1 , TECHNOLOGIES IN MACHINE BUILDING, ISSN , 2014 COMPLEMENTARY THEOREMS OF SURFACE ENWRAPPING EXPRESSED IN GRAPHICAL FORM Virgil TEODOR 1, Silviu BERBINSCHI 2, Ioan BAICU 2, Nicuşor BAROIU 1, Nicolae OANCEA 1 1 Deartment of Manufacturing Engineering, Dunarea de Jos University of Galati, 2 Mechanical Engineering Deartment, Dunarea de Jos University of Galati, Virgil.Teodor@ugal.ro ABSTRACT In this aer it is resented the issue of surface enwraing, associated with a air of rolling centrodes, based on the comlementary theorems as substituting circles family and minimum distance theorem. The rofiling issues are resented for rack gear and gear shaed tools. The solving of these roblems is based on the comlementary theorems of surface enwraing, exressed in analytical form. Using the caabilities of the CATIA design environment are resented secific methods for grahical rofiling of rack gear and gear shaed tools. Examles based on both comlementary theorems are resented, establishing the rofiles of the tools secifically for the analysed generating tyes. The examles refer to surfaces comosed of line segments and circle arcs. It is emhasized the basic quality of the grahical method the accuracy of the obtained rofile and the intuitive form in which this roblem is solved. KEYWORDS: surface enwraing, comlementary theorems, grahical methods, CATIA 1. COMPLEMENTARY THEOREMS OF SURFACE ENWRAPPING O 1,..., O i-1, O i+1 centres and, in this way, to define the family of substituting circles for the curve C Σ. In this aer, the issues of the two comlementary methods are resented. Firstly, the secific theorems for these methods will be resented: the substituting circles family theorem and the minimum distance theorem [3] The theorem of substituting circles family In case of generating rofiles associated with a air of rolling centrodes, this theorem is based on the definition for family of substituting circles. The family of circles substituting a curve associated with a centrode is the family of circles that, having the centres onto the associated centrode, are, at the same time, tangents to this curve (see figure 1), Obviously, the circle family is not unique. For ractical requirements it is ossible to imose a geometrical constraint regarding the osition of the Fig. 1. Family of substituting circles The secific theorem [3, ] is: the enwraing of a curve associated with a air of rolling centrodes is the substituting circles family, 73

2 transosed, in the rolling motion, onto the conjugated centrode. For the current circle of the family and of the C Σ rofile, (see figure 2): X X( u); C (1) Y Y ( u ); X Rr cos1 r cos ; i i C Y R r sin1 sin, r i i (2) from the tangency constraints, the following are defined: - the secific enwraing conditions, Y u sin1 X u cos1 X( u) X u Y( u) Y u (3) ; Rr kinematics for the generation by rolling: generation with rack gear or with gear shaed tools The theorem of minimum distance The secific theorem states that: the enveloing of a rofile associated with a air of rolling centrodes is the locus of the oints belonging to the rofile for which, in various rolling ositions, the distance to the gearing ole is minimum. Considering a rofile C Σ, see (1) and figure 3, where it is resented a air of rolling centrodes: C 1 a circle with radius R r and C 2 a straight line tangent to C 1 in the gearing ole, joined with the rack gear, it is defined the relative motion: ( ) r T X u R 3 1, Yu ( ) Rr 1 (6) - the value of r i radius, that determines the C Σ curves family: ri Rr cos 1 i Xu ( ) cos i Yu ( ) sin i; - the value of i angle, (4) C 1 Xu ( ) cos1 Yu ( ) sin 1 Rr ; Xu ( ) sin1 Yu ( ) cos 1Rr 1. (7) X tg u i. Y u (5) Fig. 3. Minimum distance method; xy global reference system; XY mobile reference system, joined with C Σ ; ξη mobile reference system, joined with the rack gear (C Σ ) φ1 ole: Let the following be the coordinates of gearing Fig. 2. Family s circle These elements allow to determine the enveloing of C Σ rofile, (1), based on the secific P 0; P P Rr 1, (8) 74

3 the distance from the current oint onto the C Σ is given by 2,, d u 1 u 1 R r 1 (9) In rolling, for any osition, for an increment of the rotation angle i / n (n integer), are resented ositions of the hexagonal shaft side, in XY reference system, see figure 5. of which minimum value accomlished the condition u, 1 u u, 1Rr 1 0 u, (10) reresenting the secific enwraing condition. Obviously, the distance is measured on the normal draw from the gearing ole to the C Σ rofile. The condition (10) may have a secific form at generation with gear shaed tools [3,. 9]. 2. GRAPHICAL METHOD: APLICATIONS, EXAMPLES Next, rofiling examles will be resented for rack gear and gear shaed tools, based on the comlementary theorems, solved in the CATIA design environment Rack gear recirocally enveloing with a hexagonal shaft ( minimum distance method) Fig. 5. Successive ositions in rolling In figure 4, the following are resented: - the rofile of the hexagonal shaft; - XY reference system joined with the shaft; - C 1 centrode associated with the shaft. This centrode is a circle with radius R r ; - C 2 centrode associated with the rack tool. Fig. 4. The rofile of the shaft; C 1 and C 2 centrodes; the gearing ole Fig. 6. Normal from rofile of the gearing ole; the oint onto the tool s rofile In figure 7, it is reresented a limit osition of the hexagonal shaft, for /6. In this case, the oint M is onto the rack gear centrode. In figure 8, the successive ositions of the blank are resented, for a rotation increment of 1 60/ 25, the normals from the relative ositions of the gearing ole gearing from the rack centrode to the blank s ositions, in rolling. 75

4 2.2. Gear shaed tool for generating a slined shaft (the substituting circles family method) The slined shafts with arallel flanks have the geometry resented in figure 9. The radii of centrodes C 1 and C 2 are defined. Fig. 7. The M(φ i ) oint from the rack s rofile, for φ=π/6 Obviously, the rack rofile is obtained by mirror for the oosite segment. The rack gear itch is Rr. 3 In table 1, the coordinates of the rack rofile, in the ξη reference system are resented, for the hexagonal shaft with side R r = 50 mm. Fig. 8. The rofile of the generating rack gear Table 1. Coordinates of the rack gear s rofile Fig. 9. Slined shaft with arallel flanks; associated centrodes C 1 and C 2 Obviously, the R r2 radius of the gear shaed tool is determined from the condition that the length of circle 2 Rr2 be equal with an integer number of circular itch of the teeth - C : 2 Rr2 zsc Pc (11) For grinding grooves it is ossible to design gas for entering of the grinding wheel. In the considered examle only the issue of the arallel flank of the groove is studied. The distance between the axes of the two centrodes is A12 Rr1 Rr2. (12) The substituting circles have centres onto the C 1 centrode, in oints O 1,.., O i, O i+1. These oints have the angle to the centre i, arbitrarily selected, see figure

5 Fig. 10. Substituting circles family In figure 11 there is a detail resenting the tangency oints between the family s circles and the rofile of the groove flank, C Σ. On the Ci circles the tangency oints, T i, T i+1,..., with the groove flank are determined. Fig. 12. Circles family transosed onto C 2 ; the rofile of the gear shaed tool Fig. 13. Gear shaed tool s rofile In table 2, the coordinates of the gear shaed tool are resented for a slined shaft with dimensions R e = 56 mm; R i = 51 mm; a = 16 mm; R r2 = 39.2 mm. Fig. 11. Tangency oints of the C i circle with the C Σ rofile In figure 12, the circles family is resented transosed by rolling onto the centrode associated with the gear shaed cutting tool, C 2. This centrode is a circle with radius R r2. The assembly of tangency oints, T i, T i+1,..., after the transosing of the circle s family reresents the tooth flank of the gear shae, see figure 13. Table 2. Coordinates of the gear shaed tool s rofile 1-7, , , , , , , , , , , , , , , , , , , ,

6 3. CONCLUSIONS The comlementary methods entitled substituting circles family and minimum distance allow one to aroach a variety of issues regarding rofiles (or surfaces) joined with a air of rolling centrodes. Profiling examles were resented for rack gear and gear shaed tools for generating a whirl of external surfaces. The grahical methods develoed in CATIA allow obtaining rigorous results regarding the form and dimensions of the rofile generated by enwraing. The grahical method is very intuitive, avoiding the rofiling errors for these tools. The grahical method develoed in CATIA is characterised by the fact that it is very recise (the geometrical features are determined with a recision of mm, which exceeds the determining recision of surfaces in the generating rocess by numerical methods). More, the grahical methods are easy to aly, because they are based on the caabilities of the grahical design environment. ACKNOWLEDGEMENT The work has been funded by the Sectoral Oerational Program Human Resources Develoment of the Ministry of Euroean Funds through the Financial Agreement POSDRU/159/1.5/S/ REFERENCES [1] Litvin, F.L., Theory of Gearing. Reference Publication 1212, NASA, Scientific and Technical Information Division, Washington D.C, 1984; [2] Radzevich, S.P., Kinematic Geometry of Surface Machining, CRC Press, Boca Raton, London, New York, 2008; [3] Oancea, N., Surfaces generation by enwraing, Vol. II, Comlementary theorems, Editura Fundaţiei Universitare Dunărea de Jos - Galaţi, ISBN , 2004; [4] Frumuşanu, G., Berbinschi, S., Oancea, N., The Ring-Tool Profiling in CATIA Grahical Environment, Based on the Family of Substituting Circles Method, Alied Mechanics and Materials Vol. 656, , 2014; [5] Berbinschi, S., Teodor, V., Oancea, N., 3D Grahical Method for Profiling Gear Hob Tools, The International Journal of Advanced Manufacturing Technology, Volume 64, Issue 1-4, , 2013; [6] Poa, C.L., Teodor, V.G., Baroiu, N., Berbinschi, S., Cutting Tools for Generating of Rotors Comressors, Academic Journal of Manufacturing Engineering - AJME, Vol. 12, , 2014; [7] Oancea, N., Méthode numérique our l étude des surfaces enveloées, Mech. Mach. Theory., Vol. 31, 7, , 1996; [8] Teodor, V.G., Contribution to elaboration method for rofiling tools which generate by enveloing, Lambert Academic Publishing, ISBN , 2010; [9] Baicu, I., Oancea, N., Proiectarea sculelor rin modelare solidă (Tool s design using solid modeling), Ed. Tehnica Info, Chişinău, 2002, ISBN X; [10] Teodor, V., Păunoiu, V., Berbinschi, S., Oancea, N., The rofiling of Rack-Gear Tool for Genration of the Helical Surfaces, Proceedings of the International Conference on Advanced Manufacturing Engineering and Technologies NewTech 2013, Sweden,.63-72, ISBN ; [11] Berbinschi, S., Teodor, V., Frumuşanu, G., Oancea, N., Grahical Method for Profiling the Tools which Generate Internal Surface by Rolling, Academic Journal of Manufacturing Engineering - AJME, Vol. 12, ,