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65 Claudia Mari Popa, Dinel Popa Optimisation Methods for Cam Mechanisms Abstract. In this paper we present the criteria which represent the base of optimizing the cam mechanisms and also we perform the calculations for several types of mechanisms. We study the influence of the constructive parameters in case of the simple machines with rotation cam and follower (flat or curve) of translation on the curvature radius and that of the transmission angle. As it follows, we present the optimization calculations of the cam and flat rotation follower mechanisms, as well as the calculations for optimizing the cam mechanisms by circular groove followers help. For an easier interpretation of the results, we have visualized the obtained cam in AutoCAD according to the script files generated by a calculation program. Keywords: cam, curvature radius, constructive parameters, pressure angle, circular grove. 1. Optimisation criteria The term of optimization comes from the Latin word optimus, which is the su- perlative of good, which means the best, very good, properly indicated, suited etc. According to Explaining Dictionary of Romanian Language, by optimization is under- stood technically the ensemble of scientific research (papers) that is looking for the best option in finding a solution for a problem or, according to another definition, the process of constant improving until the best solution it is reached. Mathematically, by optimization is understood the reasoning, the calculus permits in finding values of one or more parameters which correspond to the maximum of a function. In the case of a cam mechanism, one of the optimization criteria is the criteria of the curvature radius, according to that, the curvature radius of the cam in the case of a follower which is flat or has a positive curvature radius must be positive and even higher then an imposed value. ANALELE UNIVERSITII “EFTIMIE MURGU” REIA ANUL XVII, NR. 1, 2010, ISSN 1453 - 7397 66 Another criteria that must be taken into account is that of the pressure angle (noted with ) which must fulfill the condition cr < , where 030=cr at in- creasing and 060 at decreasing. On the basis of this two criteria will be obtained better condition for the com- plex cam mechanisms to function. Next we will define as a technically functional cam the cam that is obtained by classic procedures of processing of a tool machine, having the curvature radius positive and that respects the condition of the critical pressure angle ( cr < ). 2. The optimisation of simple cam mechanisms with transla- tion follower There is considered the case of a flat follower (fig. 1), case where the para- metrical coordinates of the cam are: ,sincos)( ;cossin)( 0 0 ssRy ssRx += ++= (1) and .2 sAO = (2) x O A2 O Y y s R X 0 Figure 1. Mechanism with rotation cam and flat translation follower. From (1) are deducted: ;sin)( ;cos)( 0 0 ssRy ssRx ++= ++= (3) ,cos)(sin)( ;sin)(cos)( 0 0 ssRssy ssRssx +++= +++= (4) and its also deducted the expression of the curvature radius: 67 || )()( 2/322 yxyx yxR c += ; 0 ssRRc ++= . (5) If, for example, the displacement law is: )2cos1(0 = hs . (6) Then the curvature radius is 2cos3 000 hhRRc ++= (7) and becomes minimum for 2pi = when is given by: 00min 2hRRc = . (8) In this case, for hR 20 = the minimum curvature radius becomes null and the cam has the shape from figure 2, a, and if 00 2hR < the curvature radius is canceled in the points where 2pi < ; pipi <<2 (fig. 2, b) and have negative value for 2pi < and the cam becomes nonfunctional. If we use flat follower in these conditions we can obtain cams technically functional. y x y x Figure 2. Nonfunctional cams with null or negative curvature radius. Let us consider the general case where the curvature radius at the top in the case of a flat follower becomes negative 0)( max0 <++ ssR and to determine the radius r of the circular follower so that the cam will be functionally. The equations of the followers groove (fig. 3) are ,cos;sin 22 ryrx == (8) and from the equalities: 68 ,coscossin ;sinsincos 0 rsrRyxy ryxx A A ++== == (9) are deducted the equations of the curvature family ),,()cos(cos)( );,()sin(sin)( 2 10 frsrRy frsrRx =+++= =+++= ; (10) that depends on the parameters , . O xA r 2 2 0 vA2 R + r + s y X y x Y Figure 3. Mechanism with rotation cam and flat rotation follower. The equation of the envelope checks the equation: 0 22 11 = ff ff , (11) that becomes srR s ++= 0 tg . (12) The cam equations are given by the relations (10) where checks the equa- tion (12). In the top ( 2pi = ) are fulfilled the conditions ;0 ;0 ;0 ;max <=== ssss 69 ;0 ;R s max0 =++= sr (13) ))(1();( ;0 max0max0 rsRxrsRyx ++=++== and the curvature radius is |1| || )(|| |||| )()( max 232/322 ++=== += rsR x y yx y yxyx yxR c . (14) Knowing that 0 max0 <++= srR s , (15) is deducted )1)(( )()( max0 max0 2 max0 ++ ++++= srR ssRrsRR c . (16) And next, by knowing that 01 < ; 0max cR is obtained: )( max0 2 max0 ssR sRr ++ +< . (18) 3. The optimisation of mechanisms with rotation follower For the starters, is considered the mechanism with a flat follower from figure 3 that has the lengths bRd ,, 0 and the displacement rule: )(22 = . (19) From the equations: ,sincossin ;cossincos 20 2 ++=+= +== bRyxy dyxx (20) are obtained the equations of the cam: ),cos()sin(cos)(sin );sin()cos(sin)(cos 220 220 ++= +++= bbRdy bbRdx (21) where 1 sin)(cos 2 202 += bRd . (22) 70 b x O2 2O y1 O 2x R0 1 X 2 2y Y Figure 4. Mechanism with rotation cam and flat rotation follower. The parameter represents the distance from the tangent point until the point 2O and as a result the pressure angle is given by the relation: = b arctg . (23) The curvature radius || )()( 2 3 22 yxyx yxR c += (24) and the pressure angle depends of the amplitude 02 of angle 2 as from the dimensions bR +0 and d . The numerical analyze is made tabular with the step 01= calculation the derivates by limited differences with sets of values for bR + and d. Is considered the displacement rule: )2cos1(202 pi = ; pi 2sin102 = . (25) On the basis of the relations (19) (24) is made a calculation program in Pascal. This works for different sets of values. To begin interpreting much easier the obtained results it is useful in visualizing the cam that is obtained with the varying parameters. This visualization is made in AutoCAD, based on a script file generated by the calculation program. In figure 5 is represented the cam obtained in the case 100 =R , 5=b . It is observed that in the cases where 60=d and 40=d that the obtained cams are nonfunctional and have negative curvature radius on some parts. 71 In the case where 20=d the cam is technically nonfunctional although it has a continuous aspect, at = 60 and = 240 are noticed holes, the flat follower cannot continuing the external contour of the cams groove. From the point of view of the pressure angle it is ok, at the lifting maneuver = 21.746max for =120 . R0=10; b=5; d=20 y x R0=10; b=5; d=60 y x R0=10; b=5; d=10 y x R0=10; b=5; d=40 y x Figure 5. The cams obtained in the case b = 5. In the case where 100 =R , 5=b and 20=d though it was obtained a cam technically nonfunctional, but by keeping the same dimensions can be obtained a solution technically functional by using the follower with a circular groove (fig. 6). 72 r b 1 x R0 2O O 1y 2x X d y2 2 Y Figure 6. Mechanism with rotation cam and circularly rotation. So are obtained: - the equations of the follower sin2 rdx += ; cos22 ryy o = ; (26) - the general equations ,cossincossin ;sincossincos 222220 2222 yxyrRyxy yxdyxx o +++=+= +== (27) which are deducted the equations: );cos()sin(cos)(sin );sin()cos(sin)(cos 2222 0 20 222220 +++= ++++= yxyrRdy yxyrRdx o (28) - the grabbing condition is: )1(cos)(sin )1(sin)(cos 2 0 22 0 22 22 0 22 + +++= yyrRd dyrRdtg , (29) - the equation of the cam ),cos( )cos()sin(cos)(sin );sin( )sin()cos(sin)(cos 2 2 0 22 0 20 2 2 0 22 0 20 +++= ++++= r ydyrRdy r ydyrRdx (30) where is deducted from the equation (29) - the pressure angle is: 73 ++= sincosarctg 0 2 rd ry . (31) In the numerical case are given the values considered in the previous case of cam technically nonfunctional and there more is considered bry 02 . Next is represented for different cases the cam obtained in the case of a mechanism with curve follower and different curvature radius. In figure 7 it was represented for the case 100 =R , 5=b , 20=d , with light grey the cam obtained in the case of a mechanism with flat follower and with black the cab obtained in the case of a curve follower with the curvature radius of the follower of: 10,30,50,100=r . R0=10; b=5; d=20 r=30 y x r=100 y x r=10 y x r=50 y x Figure 7. The cams obtained in the case b = 5 and d = 20. In the case of the mechanism with a flat follower (the light grey cam) there is no functional cam. In all the four cases of curve follower the obtained cam is func- tional, the pressure angle fulfilling the condition cr < as both for the lifting race and for the descend race. There is observed that by lowering the curvature radius of the follower is obtained a cam with a bigger minimum curvature radius 74 4. Conclusions In this paper are presented two major optimization criteria: the criteria of the minimum imposed curvature radius and the criteria of the pressure angle. It is studied the influence of constructive parameters over the curvature radius and the pressure angle with the help of a calculation program. For an easier interpretation of the results, the obtained cam was visualized considering the parameters that vary. This visualization is made in AutoCAD, using a script file generated by a calculation program. In the case of optimizing the cam mechanism with a circular grooved follower, keeping the same constructive measures, were obtained technically functional cams. It was studied the influence of a third parameter: the radius of the circular follower. It was considered useful in overlaying the obtained cams with flat follower with the cams with a circular follower with different curvature radius of the follower. References 1 Dudi, Fl. and Diaconescu, D., Optimizarea structural a mecanis- melor, Technical Publishing House, Bucharest, 1987. 2 Notash, L., Fenton, R.G., Mills, IK., Optimal design of flexible cam mechanisms, Eighth world congress on the theory of machines and mechanisms, pg. 695-698, Prague, Czechoslovakia, 1991. 3 Pandrea, N., Popa, D., Mecanisme. Teorie i aplicaii CAD, Technical Publishing House, Bucharest, 2000. Addresses: Prof. Dr. Eng. Claudia Mari Popa, Grup colar “Armand Clinescu”, Pitesti, Str. I.C. Brtianu, nr. 44, Piteti, claudia_mari_ Prof. Dr. Eng. Dinel Popa, University of Piteti, Str. Trgul din Vale, nr.1, Piteti, dinel_