RACK-GEAR TOOL APPROXIMATIVE PROFILING BY BEZIER POLYNOMIALS I. CIRCLE ARC ELEMENTARY PROFILE. NICOLAE OANCEA*, IONUT POPA** and VIRGIL TEODOR*

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1 ULETINUL INSTITUTULUI PLITEHNI IN IŞI Publicat de Universitatea Tehnică Gh. sachi din Iaşi Tomul LIV (LVIII), Fasc. 1, 008 Secţia NSTRUŢII E MŞINI RK-GER TL PPRIMTIVE PRFILING Y EZIER PLYNMILS I. IRLE R ELEMENTRY PRFILE Y NILE NE*, INUT PP** and VIRGIL TER* bstract. In this paper is presented an algorithm for rack-gear tool s profiling, based on ezier polynomials for the circle arc profiles. The approximate method, use a small points number on the profile (3 or 4 points regarding the used polynomial type). They are presented numerical examples which proof the method s quality, regarding the algorithms based on the surface enveloping fundamentals theorems. Key words: surface enveloping, approximate profiling, ezier polynomials, rack-gear profiling. 000 Mathematics Subject lassification: 535, Introduction The rack-gear tool s profiling, reciprocally enveloping with an ordered surfaces curl, may be realized using representation by poles (ezier polynomials) [1], [8], [9]. The rack-gear tool enveloping with an ordered surfaces curl, associated with a rolling centrodes couple may be determined using the fundamentals theorems of the surfaces enveloping [], [6], as so as using the complementary methods [5], [7], [10]. Not in all situations meet in the industrial practice is need a very rigorous profiling of rack-gear tool s teeth flank. The worm cutter tool s form for generating the large side mill teeth, tools for spline shafts generation, allow to know not very rigorous the tool s profile.

2 0 Nicolae ancea et. al. In following is proposed a new approach of the rack-gear tool s profiling based on the method to approximate by ezier polynomials the tool s profile. The method presume to known a limited number of points (3 or 4, regarding the ezier polynomial degree) which simplify the calculus and the time to profile. The method presume a tabular algorithm, regarding the elementary profile to be generated: circle arc; involute arc; trochoidal arc, as simple profiles, which compose more complexes profiles, meet in industrial practice.. ircle arc elementary profile lgorithm In following, will be examinee the application for rack-gear tool s profile approximation methodology in case of a circle arc profile generation, see figure 1. Fig. 1- ircular profile associated with the rolling centodes couple Is defined the circle arc knowing two points on it, [, Y ] and [, Y ] as so as the value of r radius. bviously, these are enough to determine the circle center [, Y ]. re defined the circle s parametrical equations (the primary profile to be generated): ( ) cos ; (1) + r Y ( ) Y r sin, variable parameter, between limits,

3 () arccos ul. Inst. Polit. Iaşi, t. LIV (LVIII), f. 1, r and Table 1. arccos Primary profile Enwrapping condition + r cos sin + Ycos arcsin Y Y r sin λ Y r. + sin + Y cos Y + r cos r sin arcsin + r cos sin + Ycos arcsin Y Y r sin Points on the rack-gear profile cos Y sin sin + Y cos cos Y sin sin + Y cos cos Y sin sin + Y cos pproximation polynomial coefficients Is defined the circle arc family in the rolling motion of the two centrodes: [ + r cos ]cos [ Y sin ]sin ; r + (3) ( ) [ + r cos ]sin + [ Y r sin ]cos.

4 Nicolae ancea et. al. Table. Primary profile Enwrapping condition + r cos sin + Ycos arcsin Y Y r sin λ 0 1/ 3 Y Y Y r sin r cos sin + Y cos arcsin + 3 sin + Y cos Y + r cos r sin arcsin + r cos sin + Ycos arcsin Y Y r sin Points on the rack-gear profile cos Y sin sin + Y cos cos Y sin sin + Y cos pproximation polynomial coefficients / 3 cos Y sin sin + Y cos cos Y sin sin + Y cos

5 ul. Inst. Polit. Iaşi, t. LIV (LVIII), f. 1, Similarly, are defined the values of the angular parameter, in the rolling motion of the two centrodes, knowing the normal at circle in the current point, uuur (4) N :[ ( )]sin + [ Y Y( )]cos 0 and the 1 centrode equations: cos ; (5) 1 Y sin. From these, result the conditions to determine the angle value, corresponding to the characteristically points on the circle, ( )sin ( )cos (6) arcsin + Y, condition which represents the specifically enwrapping condition, see table 1, for a approximation polynomial of nd degree and, similarly, in table, for a polynomial of 3 rd degree. In tables 1 and, they are presented the calculus elements of the ezier approximation polynomials for tool profile associated with the rack-gear. The ezier polynomial for nd degree is: (7) and for 3 rd degree is: (8) ( ) ( ) λ + λ 1 λ + 1 λ ; λ + λ( 1 λ) + ( 1 λ), ( ) ( ) ( ) 3 3 λ + 3λ 1 λ + 3λ 1 λ + 1 λ ; ( ) ( ) ( ) 3 3 λ + 3λ 1 λ + 3λ 1 λ + 1 λ. 3. Numerical examples ased on the presented algorithm, is showed a numerical example for a circular profile, see figure 1, for rack-gear tool s profiling, reciprocally 87.0;4.8 and enveloping with an circle arc defined between points [ ] [ 80;0], with characteristics: r 0 mm; center coordinates [ 100,0], profile associated with a circular centrode with radius R 100 mm. rp

6 4 Nicolae ancea et. al. In tables 3 and 4 and figure, are presented the profile form and coordinates for approximation polynomial of nd and 3 rd degree, regarding the profile calculated by one of the fundamentals methods [], [7], [10]. Fig. pproximated and theoretically rack-gear profile Table 3. λ pproximated tool profile Theoretical tool profile Error [mm] [rad] [mm] [mm] [mm] [mm]

7 ul. Inst. Polit. Iaşi, t. LIV (LVIII), f. 1, Table 4. λ pproximated tool profile Theoretical tool profile Error [mm] [rad] [mm] [mm] [mm] [mm] In following is showed another numerical example for a circular profile, for rack-gear tool reciprocally enveloping with an circle arc defined between 95; ;5, with characteristics: r 15 mm; points [ ] and [ ] profile associated with a circular centrode with radius R rp 100 mm. 4. onclusions 1. The presented method, although approximately, assure a good representation of the rack-gear tool s profile reciprocally enveloping with a circular profile, which may be part of a composed profile to be generated.. The method is simple, used a limited points number on the profile to be generated, so the calculus time is reduced. 3. Software dedicated to this algorithm is an instrument which helps to apply this method. 4. In order to increase the profiling precision is necessary to increase the ezier polynomials degree. cknowledgement: This paper was developed under the research programs IEI, grants I_656, funded by the Romanian Ministry of Education and Research * unărea de Jos University of Galaţi Manufacturing Science and Engineering epartment ** l. I. uza University of Iaşi Faculty of omputer Science

8 6 Nicolae ancea et. al. ccepted: 30 th pril 008 Technical University Gheorghe sachi epartment of Machine Manufacturing Technology Iaşi, Romanin, R E F E R E N E S 1. Favrolles, J. P., Les surfaces complexes, 1998, Hermes, ISN a;. Litvin, E. L., Theory of Gearing, Reference Publication 11, NS, Scientific and Technical Information ivision, Washington.., 1984; 3. Huston, R., Mavoripolis,., swald, F., Liu, Y. S., basis for solid modeling of gear teeth with application in design manufacture, In: Mechanism and Machine Theory, vol. 9, Issue 5, July, 1994, pag ; 4. Litvin, F. L., Fuentes,., Gonzales-Perez, I., arnevali, L., Sep, T., New version of Novicov-Wildhaber helical gears: omputerized design, simulation of meshing and stress analysis, available on-line, nov. 00; 5. Minciu,., roitoru,., Ilie, S., Numeric pplications Regarding etermination of the omplementary Profiles in Gear, In: IMS, 006, Romanian cademy Publishing House, ISSN , pag ; 6. ancea, N., Surface generation through winding, Volume I, Fundamental Theorems, 004, unărea de Jos University publishing house, ISN , ISN ; 7. ancea, N., Surface generation through winding, Volume II, omplementary Theorems, 004, unărea de Jos University publishing house, ISN , ISN ; 8. ancea, N., Frumuşanu, G., ura, G., lgorithms for representation by poles as a way to approximate wrapping curves of profiles associated to rolling centrodes, 006, Proceedings of the international conference on manufacturing systems, ISSN , ucharest, pag ; 9. ancea, N., Teodor, V., Popa, I., Rack Gear Tool pproximate Profiling Method, 008, WSES E 08 onference, Malta, Greece, in press; 10. Teodor, V., ancea, N., ima, M., Tool s profiling by analytical methods, 006, unărea de Jos University publishing house, ISN (10) , ISN (13) ; PRFILRE PRIMTIVĂ PRIN PLINME EZIER SULEI-REMLIERĂ (Rezumat) În lucrare, se prezintă un algoritm de profilare a sculei-cremalieră, în baza reprezentării prin polinoame ezier, destinat generării prin înfăşurare prin metoda rulării a profilurilor în arc de cerc. Metoda aproximativă, utilizează un număr redus de puncte de pe profil (3 sau 4 puncte în funcţie de gradul polinomului utilizat). Sunt prezentate exemple numerice care atestă calitatea metodei, în comparaţie cu algoritmi bazaţi pe teoremele fundamentale ale înfăşurării suprafeţelor.