SYNTHESIS OF TWO MECHANISMS WHICH GENERATE LUNULES OVER AN EQUILATERAL TRIANGLE S SIDES

Transcription

1 SYNTHESIS OF TWO MECHANISMS WHICH GENERATE LUNULES OVER AN EQUILATERAL TRIANGLE S SIDES Assciate Prfessr PhD Ludmila SASS, University f Craiva, Faculty f Mechanics, ludmila_sass@yah.cm Prfessr Iulian POPESCU, University f Craiva, rdicaippescu@yah.cm Abstract. A synthesis fr a mechanism generating lunules acrss the sides f an equilateral triangle is perfrmed. The mechanism cnsists f an articulated quadrilateral, prlnged with a RRR dyad. It is prved that lunules can be generate, but the speeds f pltting pints are neither equal, nr cnstant. The synthesis f anther mechanism, cmpsed f tw articulated parallelgrams is afterward perfrmed. Delayed cranks are used this time and cnsequently equal speeds are btained. Keywrds: Hyppcrates s lunules, mechanism t plt lunules 1. Intrductin A lunule represents a plane gemetric surface with a half-mn shape. It is built acrss a chrd n a circle and it is delimited by the arc frm a circle and anther arc f circle, built acrss that chrd, nw representing the diameter fr the new arc. Hyppcrates frm Chis ( B.C.) has built lunules acrss the sides f a rectangular triangle. The small surfaces frm the basis f peple fingers nails are als called lunules. [3] presents an analysis f lunules and surfaces at issceles triangles. Lunules generated alng the sides f a trapezid are studied in [4]. A histry f Hyppcrates s lunules is presented in [5], alng with their analysis when cnsidering a isscel rectangular triangle and a trapezid. The lunules aria is cmputed in [6] fr a regular triangle. Gemetrical aspects f lunules are apprached in [1], alng with details n the mdality t cnstruct. A synthesis f a mechanism used t trace lunules fr a rectangular triangle can be fund in [1]. Here is als prvided a descriptin fr a mechanism used t plt lunules alng the sides f a square. N ther similar infrmatin was fund with respect t lunules pltting in the specialty literature. 2. Synthesis f the first mechanism An equilateral triangle GHK was firstly built (Fig. 1). The angles which present interest fr ur paper are emphasized. The generated lunules are between the circle f radius OH and centre in O - described by the rad OH, and arcs f circle whse diameters are the sides f the equilateral triangle. OH is a separate leading element, nt related t the mechanism which generates the ther arcs f lunules. The mechanism s synthesis is made by nticing the fllwing particular mvements. If the rad AB is rtating and the pint B is pltting the arc crrespnding t the lunule frm the side GH, starting frm G, then the pint C, starting frm H is pltting the arc crrespnding t the lunule f the side HK. The articulated quadrilateral mechanism ABCD is built as described abve. 41

2 Fr the lunule crrespnding t the side KGM, a tracing pint F is used. F belngs t the element EF which belngs t the dyad BFE f type RRR, added t the quadrangle. In this way a mechanism DCBAFE was btained. It cnsists f a leading element and tw dyads. C H y B D R O A x K E G F Fig. 1. Mechanism with tw dyads tracing lunules The fllwing equatins can be written: KH KG GH BC BF a (1) EH GH cs 30 a cs 30 (2) O EG a sin 30 (3) 0,5a R cs 30 (4) OA OD OE Rsin 30 (5) xa OAcs30 ya OAsin 30 (6) xd xa yd ya (7) xe 0 ye OE (8) AB CD EF 0, 5a (9) 42

3 a xb xa cs 2 a yb ya sin 2 (10) a xg 2 yg OA (11) xh 0 yh a cs30 OE (12) a xk 2 yk yg (13) A prgram was built t trace the mechanism fr =55 (Fig. 2). Fig. 2. The mechanism in a certain psitin Fig. 3. Successive psitins Certain successive psitins f the mechanism are depicted by Fig. 3. As lng as the quadrilateral mechanism des nt prvide cnstant angles f the rcker CD, when rtating the rad AB at cnstant angles ne can ntice crescent values fr the angle CDH tward the zne neighbring K. When the cycling step is adpted t get mre clarity fr the drawing, a higher angle is btained tward extremity. The same phenmenn is nticed in the zne EK. One can see that fr small cycling steps, these areas are swept t. Fig. 4 depicts the lunules traced with this mechanism acrss the three sides f the triangle. 43

4 Fig. 4. Traced lunules Fig. 5. Successive psitins f radii Fig. 6. Arcs at a certain mment Fig. 5 depicts the successive psitins f the elements AB, CD and EF fr an peratinal cycle. Variable angles can als be nticed at the rad CD. The different speeds f the tracing pint are emphasized by Fig. 6. Here, fr the same range f variatin f the generalized crdinate, the traced curves have different lengths. Only at the end they get fulfilled. The arcs traced at a given mment are presented here. 3. Synthesis f the secnd mechanism The secnd step was t realize the synthesis f anther mechanism, relying n a sequence f articulated parallelgrams, each having tw welded cranks (Fig. 7). The mechanism is represented by dashed lines. The lunule acrss the side GH is pltted by the pint P belnging t the crank AB, welded t AB 1. The lunule acrss the side KH is traced by the pint P 1 frm the crank DP 1, welded t the crank DC, and the lunule acrss the side KG is traced by the pint C 2 frm the crank EC 2. 44

5 C y B1 P1 H P 60 D 60 O A 60 x R B2 K E G C2 Fig. 7. The secnd mechanism, built acrss a sequence f parallelgrams Based n the angle between the crank AB 1 and the abscissa, n the crdinates f the pints A, D, E (prvided by the crrespnding equatins) and n the equatins (6), (7) and (8), we get: B1 C AD (14) xc xd 0,5a cs60 (15) 0 yc yd 0,5a sin60 0 xb 1 xa 0,5a cs 60 (16) yb 1 ya 0,5a sin 60 xp xa 0,5a cs 60 (17) yp ya 0,5a sin 60 xp1 xd 0,5a cs120 (18) yp1 yd 0,5a sin120 xc 2 xe 0,5a cs 2 (19) yc 2 ye 0,5a sin 2 xb2 xd 0,5a cs (20) y yd 0,5a sin 45

6 The resulting mechanism is given by Fig. 8, which als presents bth shifted cranks. Fig. 9 depicts the successive psitins f the mechanism. Fig. 8. The mechanism in a certain psitin Fig. 9. Successive psitins The successive psitins f the pltting cranks are presented by Fig. 10. Fig. 10. Psitins f pltting cranks Fig. 11. Lunules Fig. 12. Curves at a certain mment Fig. 11 presents the lunules traced by this mechanism acrss the three sides f the equilateral triangle. This time the curves pltting is perfrmed at cnstant speeds (Fig. 12), unlike the case f articulated quadrilaterals. The curves pltted at a certain mment are als pltted and ne can see that they are equal. Cnclusins - The synthesis f tw riginal mechanisms was perfrmed. They plt lunules acrss the sides f an equilateral triangle. - The first mechanism relies n an articulated quadrilateral, prlnged with a dyad f type RRR. - The checking f the mechanism s successive psitins demnstrated its crrect peratin. 46

7 - Because at this mechanism the last driven element has angles which are variable frm ne psitin t anther (and therefre exhibits a nn-unifrm mvement), the lunules, althugh crrect, are pltted with variable speeds. - The secnd mechanism relies n tw articulated parallelgrams, with shifted cranks, welded t real cranks frm mechanism. - In this last case the lunules pltting speeds are equal and cnstant. References 1. Ppescu Iulian, Sass, L. - Lunulele lui Hipcrate. Vl. "The VII-th Editin f the Natinal Cnference with Internatinal Participatin, GRAFICA-2000, Craiva, ct. 2000", pp Ppescu Iulian, Sass, L. - Figuri gemetrice ale antichităţii, generate cu mecanisme. Vl. The VII-th Editin f the Natinal Cnference with Internatinal Participatin, GRAFICA- 2000, Craiva, ct. 2000, pp Lunules/IntrductinTLunules.htm [Accessed: 11 th Nvember 2015] [Accessed: 11 th Nvember 2015] LunuleIppcrate.htm [Accessed: 1 st December 2015] [Accessed: 11 th Nvember 2015]. 47