- Email: [email protected]
Synthesis of plane linkages to generate functions of two variables using point-position reduction—Part 1. Rotary inputs and output
Mechanism and Machine Theory, 1972, Vol. 7, pp, 363-370.
Pergamon Press. Printed in Greet Britain
Synthesis of Plane Linkages to Generate Functions of Two Variables Using Point-Position Reduction - Part 1. Rotary Inputs and Output T. S. Mruthyunjaya*
Received 8 December 1971 Abstract The paper presents simple graphical procedures for position synthesis of plane linkage mechanisms to generate functions of two independent variables. The procedures are based on point-position reduction and permit synthesis of the linkage to satisfy up to six arbitrarily selected precision positions. 1. Introduction EARLIER investigations in the field of synthesis of function generators have largely been confined to the problem of generating functions of a single independent variable. Several methods, graphical and analytical, are available in the literature for this purpose. However, there have been relatively few attempts at synthesis of'linkage mechanisms to generate functions of two independent variables. Svoboda[1] proposed the plane, seven-link two-degree-of-freedom linkage shown in Fig. 1 for generation of functions of two independent variables. He also suggested a homographic procedure for position synthesis of this linkage. However, his procedure is rather tedious and requires an enormous amount of drafting work to arrive at a satisfactory linkage. Allen [2] developed graphical methods for position synthesis of this linkage for a maximum of six precision positions chosen such that they always correspond to three values of one of the independent variables and for which, at each of these three values, the second independent variable has not more than three arbitrarily chosen values. An additional restriction, in the case of six precision positions is that the three values of the first independent variable should be equally spaced. Allen [2] also developed a graphical method for third-order derivative synthesis of the linkage. Freudenstein and Sandor[3] presented a method, which is partly graphical, for position synthesis to satisfy five arbitrarily selected precision positions. Their method is based on computation of the Burmester points for five finitely separated positions of a plane. A fully analytical treatment of this problem for position synthesis to satisfy twelve arbitrarily selected precision positions was carried out by Phillipp and Freudenstein[4]. They considered a slightly different linkage obtained by replacing one of the three binary floating links in Fig. 1 with a ternary link. Their work showed that the calculations involved can become quite extensive and time-consuming even when programmed on a high-speed digital computer. An analytical method for fourth-order derivative synthesis of the linkage is given by Lakshminarayana and Narayana Murthy [5]. The procedures presented in this paper are graphical in nature and can be used for position synthesis to satisfy up to six precision positions. These simple and rapid pro* Lecturer, Dept. of Mechanical Engineering, Indian Institute of Science, Bangalore 12, India.
363
364
~3
B4
AI el)
Figure 1. The seven-link two-degree-of-freedom linkage. cedures place no restriction on the choice of precision positions. The procedures are based on the powerful technique of point-position reduction developed largely by Hain [6]. It may be recalled here that Goodman [7] was the first to suggest the possibility of using point-position reduction to synthesize linkage mechanisms to generate functions of two variables. However, as far as the author's knowledge goes, there has been no attempt in this direction reported in the literature, though the technique has been used to great advantage in the case of single-degree-of-freedom linkages [6, 8, 9]. 2. Formulation of the Problem The plane, seven-link, mechanism shown in Fig. 1 represents the two-degree-offreedom linkage with the minimum number of links permitting rotational inputs and output. Theoretically, it should be possible to synthesize this linkage to satisfy fourteen precision positions since an equal number of design parameters, viz. eight link ratios, three starting positions and three scale factors are involved. Let z =f(x,y) be the function to be generated with precision points at (x~,yi), i = 1,2, 3, 4 . . . . . etc. Assuming the scale factors
~x
~y
Az
K~=A--~, K~=Acb, and K~=A---~
the incremental rotations 0t2,013 . . . . . (~12, $a3 . . . . . of the input links and $12, $13. . . . of the output link corresponding to the various precision positions can be calculated. Now the dimensions of the linkage have to be determined such that these incremental rotations of input and output links are coordinated. In Fig. 1 the linkage is shown in one of its precision positions and the locations occupied by the moving pivots at other precision positions are also marked. As the input links AoA, BoB and the output link CoC rotate through incremental angles corresponding to the precision positions I, 2, 3, . . . . etc. the double joint at G passes through the positions G t , G.,, G3 . . . . . etc. Conversely, it can be stated that if the joint G is guided along positions G t, G2, Ga . . . . . etc., the links AoA, BoB and CoC will rotate through the corresponding incremental angles.
365 The seven-link mechanism can be considered to consist of three link-pairs namely, ,4o,4-AG, BOB-BG, and CoC-CG with their ends ,40, B0 and Co pivoted to the frame and the other ends pivoted together at a common point G. Considered separately, Aod G, BoBG and CoCG each form a two-jointed open kinematic chain with the fixed link. In such a chain if the fcee end is guided along a certain path the link pivoted to the frame will swing through definite angles. Therefore the problem of synthesis of the twodegree-of-freedom linkage now reduces to one of determining the dimensions of three separate two-jointed open kinematic chains ,4oAG, BoBG and CoCG such that when their ends G are guided to move from position 1 to positions 2, 3, 4 etc. the links pivoted to the frame viz..4o,'1, BOB and CoC will rotate through angles 01~, 013. • • ~i=, ~ 3 . . . . and ~=, 013. . . . etc. respectively. It may be noted that the point positions G~, G=, G3 etc. can be arbitrarily selected. If now the three chains are pinned together at their free ends the resulting linkage will satisfy the requirements for coordinations of rotations 0, tk, and 0 for the precision positions 1,2, 3 . . . . . etc. The constituent problem of determining the dimensions of the two-jointed kinematic chains can be solved by using the principle of inversion in the case of three precision positions and inversion combined with point-position reduction in the case of four, five and six precision positions. The constructions are explained in detail in the following sections. The first step in synthesis in each case is to assume suitable scale factors and calculate the incremental angles of rotation of the input and output links for the various precision positions. 3. Synthesis for Three Arbitrarily Selected Precision Positions The locations of the point positions G~, G= and G3 are selected arbitrarily (Fig. 2). The location of the fixed pivot Co is also selected arbitrarily. The lines CoG2 and CoG3 are rotated through angles - $1~ and - ~,3 respectively to positions CoG=c' and CoG3~' (inversion). The centre of the circle passing through points G1, G2~' and G3~' gives C,, the location of the pivot C in position 1. CoC~G~ is the required two-jointed chain in position 1. The chains.4o,4 IG~ and BoB~G1 are determined by similar constructions. The resulting two-degree-of-freedom linkage is shown in position 1 (Fig. 2). In view of the many arbitrary choices made during the construction an infinite number of linkages can be obtained all of which satisfy three precision positions. Bo
B,o J
~
c,
,:,
C, p@
Ao
h
Figure2. Synthesisforthreearbitrarilyselectedprecisionpositions.
366
4. Synthesis for Four Arbitrarily Selected Precision Positions The locations of the point positions G1, Gz, G3 and G 4 a r e selected arbitrarily (Fig. 3). The location of the fixed pivot Co is chosen along one of the perpendicular bisectors of the distance between any two point positions such that these point positions subtend an angle at the pivot Co which is equal to the incremental angle of rotation of the output link CoC between these positions. Let, for example, Co be located along the perpendicular bisector e34 of G3U4 such that the angle G3CoG4 is equal to ~a4. Then the relative pole for the displacement of the link-pair CoC-CU between positions 3 and 4 will coincide with the fixed pivot Co. Thus a single point position reduction is achieved. Since there are, in all, six pairs of point positions GIGz, G2Ga, G3G4, G1G3, GIG4 and GzG4 and since for each pair the location of Co can be chosen along the corresponding perpendicular bisector on either side of the pair, there are twelve possible locations for the pivot Co. Now taking 4 as the reference position of CoCG, the lines COG3,CoG=and CoG~ are rotated through angles - ~ , ~42, a n d - ~4~ respectively to positions CoG3c', CoG.z", and CoG,~"(inversion). G3~, will, of course, coincide with G4. The centre of the circle through G4, G2~" and G~~" will give C4 the location of pivot C in position 4. CoC4G4 is the required two-jointed chain in position 4. The positions C3, Cz and CL are located as the intersections of circular arcs of radius C4G4 centered at G3, Gz and Gl respectively with the circle through C4 of radius CoC4 and centre Co. The chains A oA1GI and B oB1G~ are determined by similar constructions. The resulting two-degreeof-freedom linkage is shown in position 1 (Fig. 3). Because of the many arbitrary choices during the construction an infinite number of linkages can be obtained all of which satisfy four precision positions.
u-,.- elz
Figure 3. Synthesis for four arbitrarily selected precision positions.
367
5. Synthesis for Five Arbitrarily Selected Precision Positions In this case the G point positions cannot be selected entirely arbitrarily. At least three of them should be chosen so that they lie on a circle centered at one of the fixed pivots and subtend angles at that pivot which are equal to the respective incremental angles of rotation of the corresponding input or output link. The remaining G-point positions are determined by construction. For example, let point positions Us, G3 and G, lie on a circle of any radius centered at any point Co such that angles G=CoG~and G=CoG, are equal to 023 and 02, respectively (Fig. 4). Then the relative poles for the displacement of link-pair CoC-CG between positions 2 and 3 as well as positions 2 and 4 will coincide with the fixed pivot Co. Thus a double point position reduction is achieved. For the three point positions G2, G3 and G, the chains AoAoG= and BoB2G=in position 2 are determined using the procedure indicated earlier for three precision positions. AoA=G=BzBo now constitutes a five-bar linkage. The G-point positions GI and Gs corresponding to precision positions 1 and 5 are located giving the inputs (0=1, 6~1) and (025, 0hs) to this five-bar linkage. The problem now is to determine the dimensions of the two-jointed chain CoCG such that when end G is guided to move from position G1 to positions G~, G3, G, and G5 the link CoC will rotate through corresponding incremental angles. This can be done by inversion. Taking 2 as the reference position of the chain CoCG, the lines CoG, COG3, COG,, and CoG5 are rotated through angles -0zl, --023, --02,, and -~=5 respectively to positions COG(', CoG3c', CoG, ~' and CoGs~'. G8~=and G, ~' will, of course, coincide with G2. The centre of the circle through points G2, GI ~ and Gs ~ will give C2, the location of pivot C in position 2. CoC~G2 is the required two-jointed chain in position 2. The resulting two-degree-of-freedom linkage is also shown in position 2 (Fig. 4). In this case also an infinite number of linkages can be obtained all of which satisfy five precision positions. 6. Synthesis for Six Arbitrarily Selected Precision Positions
In this case at least four of the G-point positions should be selected such that they lie on a circle centered at one of the fixed pivots and subtend angles at that pivot which
Fioure 4. Svnthesis for five arbitrarily selected precision Dositions.
368 are equal to the respective incremental angles of rotation of the corresponding input or output link. The remaining G-point positions are determined by construction. For example, let point positions G~, G2, G3 and G4 all lie on a circle of any radius centered at any point Co such that angles G1CoG~, GICoG3 and GICoG4 are equal to ~b12,~b13and ~b14respectively (Fig. 5). Then the relative poles for the displacement of link-pair CoCCG between positions 1 and 2, 1 and 3, 1 and 4 all coincide with fixed pivot Co. Thus a triple point position reduction is achieved. For the four point positions G1, G2, G3 and G4, the two-jointed chains AoAG and BoBG in position 1 are determined following the procedure indicated earlier for four precision positions. A oAIG IBIBo now constitutes a five-bar linkage. The remaining G-point positions G5 and G6 corresponding to precision positions 5 and 6 are located by giving inputs (015, ~15) and (016, ~16) to this five-bar linkage. The problem now is to determine the dimensions of the two-jointed chain CoCG such that when G is guided to move from position G1 to positions G2, G3, G4, G5 and G~, the link CoC should rotate through the corresponding incremental angles. This is done by inversion as before. Taking 1 as the reference position, the lines COG2, COG3, COG4, CoG~ and CoGs are rotated through angles -~/12,--~bx3,--b14,-~15 and - ~ 6 respectively to positions CoG2 c', CoG3c', COG4% CoG5~' and CoGe~'. G2~, G3~' and G4c' will, of course, coincide with point G1. The centre of the circle through points G1, G5 c' and G6 ~ will give C1, the location pivot C in position I. CoC~GI is the required two-
Be
Figure 5. Synthesis for six arbitrarily selected precision positions.
369
jointed chain in position 1. The resulting two-degree-of-freedom linkage is also shown in position 1 (Fig. 5). Because of the several possible locations for the fixed pivots, a large number of linkages can be obtained all of which satisfy six precision positions. Conclusion The point-position reduction technique has been utilized to develop successful graphical procedures for the synthesis of plane linkage mechanisms to generate functions of two independent variables. The technique of point-position reduction has been known for a long time, but its application to synthesis of two-degree-of-freedom linkages is believed to be novel. The synthesized linkage can satisfy up to six arbitrarily selected precision positions. Though the procedures do not exploit the potentialities of the linkage considered to the fullest extent, it is hoped that they will be found useful by designers of mechanisms for the simple reason that they permit determination of several alternative configurations of the linkage in a short time and thus enable selection of the best linkage configuration which satisfies other important criteria like space considerations, transmission angle requirements etc. 7.
Acknowledgements-The author wishes to thank Dr. M. R. Raghavan, Assistant Professor, Department of Mechanical Engineering, Indian Institute of Science, Bangalore, for useful discussion and suggestions in the preparation of this paper. The help and encouragement given by Prof. M. A. Tirunarayanan, Professor-incharge, Department of Mechanical Engineering, Indian Institute of Science, Bangalore, is gratefully acknowledged.
References [ 1] S V O B O D A A., Computing Mechanisms and Linkages, pp. 223-299. McGraw-Hill, New York (1948). [2] A L L E N C. W., The design of linkages to generate functions of two variahles, J. Engng Ind. Trans. ASME 81B, 23-29 (1959). [3] F R E U D E N S T E I N F. and S A N D O R G. N., On the Burmester points o f a plane, J. Appl. Mech. Trans. ASME28, 41--49 (1961). [4] P H I L L I P P R. E. and F R E U D E N S T E I N F., Synthesis of two-degree-of-freexiom l i n k a g n s - A feasibility study of numerical methods of synthesis of bivariate function generators, J. Mechanisms 1, 9-21 (1966). [5] L A K S H M I N A R A Y A N A K. and N A R A Y A N A M U R T H Y R. G., Derivative synthesis of plane mechanisms to generate functions of two variables, J. Mechanisms $, 249-271 (1970). [6] H A I N K.,Applied Kinematics, Second edition, pp. 524-618. McGraw-Hill, New York (1967). [7] G O O D M A N T. P., See discussion in [2]. [8] A L L E N C. W., Point position reduction, Machine Design 31,141-145 (1959). [9] SONI A. H. and H U A N G M., Synthesis of four-link space mechanisms via extension of point-positionreduction technique, J. Engng Ind. Trans. A SME 93B, 85-89 (1971). CmlTe3 r[Jloclmx Mexal~3MOlS l'lpoll3so~xumx q ~ l'lo~o~-mdi To.~¢. q a c ~ I l e p s ~ : Y r a o s u e I
]
~syx NepeM~ ¢ I~OMOIHIOr[plll~.~ifll ~ Be~Tmero n Be~oMo~O 3area.
T. C. M p y ~ u o ~ a n P e 3 m M e ~ ~ r a pa60ra 3am~Maerc~ pazs~rHeM rpa#pH~ecm~x M C T O n O B cmrreza nnoczsx M e X a H H 3 M O B B UCJIH npOH3~eHtLq ~byHKUI4~]Inyx Hc3aBHCHMblXnepeMCHHMX y~oB~eTBOpMs3mHx npOH3So~bno sM6paaHblM TOtDIbIM noJloxelm~lM. YflOMI~HTTbI~ MCXaHH3M ~J]$1 CHHTC3a 31"O C~MH31~HHiIK C ~yM.q CTfJI~IVIMH CBO6OlIbl COCTOB.rlJnOII~C~[H3 TpOl~Horo HenO]IBR)Kaoro 3Bona AoBoCo, rpex 6saapmax zpaeonmnon AoA, BoB H CoC cny3zam~x Ka~ senynme u seno~v~e 3Beua H c TI~X 6nHapHblX ~IeyxHoBo~KoBbIX 3~H AG, BG, ~ CG, e KoTopblx TOKKa G np~/l~'aBJllleT nDyxnoBO~KOBid~ m a p s s p c nBollsym cmr3m. Ecns ~cc Macu~ra6H~le ¢[mrrop~ 21n.q Bxo~a n sMxo~a ni~6paa~ npa~H.~Ho, cmrre3 M ~ M a coxpaub~ercx ]c onpc~e~leHHW pa3MepOs 3BCH TaKHM 06pa30M ~ITO yrHbI BpameH~: 0:2, 0~a, ... a e ~ m c r o 3BCHa AoA,
~ , ~ .... eeaymero~BesaBoB s ~ , ~ ....
eeao~oeo ~ma
CoC ~ e ~ y
nep~uM s
nocneay~m~M~
TOtlHbIMH MP-~-~ronoJIO~HIdLqMHcornaco~asu. Kor~la 3BeHa AoA, BoB H CoC BpamawT~ tlepc3 3THe yrJlb~, TOtlKa G nBI,I~TC~I H3 HO.rlO~KeHH~GI K n o ~ o ~ m ~ 4 G=, G~, H T. ~. Hao6opoT MOXHO cKa3aTb ~ro eC~IH TOqKa G n a n p a ~ e a a e C~OeM ,~e~7~e~tm H3 n o n o ~ e n ~ G : x nono~zeHu~M G2, G3 . . . . , 31~Ha AoA, BoB H CFC 6y~yT npamaTcs qepe3 e~me ynOMSHyTble yrnbL CCMH3BeHHblI~MeXaHH3M MOX~eTpa~CMaTpHBaTbC~I
370 KaK COCTaarL~lOmKi~Cg H3 TpeX OTReJIbHbIX ~IByxlnaplttlpHMX OTKpblTblX KHHeMa.TI~eCEHX uenett AoAG, BOB(} ~ C o C G O ~ H e H H b l X e TO'V~e G. ~ y x m a p e m p u e m tIemb, x a z a ~ m e , cocroerrcs co 3Beaa rlpHCOe~tlHeHHOrO O]IJttlM XOHI/OM K CTOtVflCC C nOMOUXIO IJa/I~bbl H maOHHpOM K 3Belly Ha BTODOM KOHHe. Ocraenn~c~ ~oHeu ~ p y r o r o 3neHa cm36O~©H. EcJIH s Taxol[ ~¢IIH cnO6O~I]MIR l¢olleU Hallps.ar/eH no OHp~[O.~IeHHOMy I1y'rH, 3BCHO COC~[4H(~HHOe CO CTO~KOfl 6y~OT epatuawcs qep¢3 onpe~enermsxe yrnbl. T a Z H M o 6 p a 3 O M CHHTe3 M e X a H H 3 M a MOXL~r 6brrb HCIIO.rIHeHc~eIxyIoIIXHM MeTO~OM'. I. C Haqana rlpHHHMaCTC~ oUpeJIeJICHHH¢ nonoxea~Lq G:, G2, G3, ... romc~ G. O m ~ npo~BoynbHme c cny~ae rpex H ~ r ~ p e x To~nax n o n o x c m ~ , ao ~ BMcmero qKcna noHo~HIt~ 3TO ]le~aeTC~ KaZ OI][HCaHO BO B T O p O M l~fHrI~. 2. T p K oTnenbHble n ~ y x m a p F m p m ~ e uewa AoAG, B o B G ~ C o C G ycTaRoeneHmae Taza~ cnOCO6OM, qTO e,C,.rlR zOHel/ G ~IBi~g'CTCgH3 IIOJIO~eHI,Lq Gt K nono~eH~SM G2, G 3 , . . . , 3BeHa. AoA, BoB ~ CoC spaIRa.loTCg "[epc3 yrHH 012 , 013 . . . . . ~12, ~ t 3 . . . . H ~bx2, ~ 3 . . . . OTHOCgTe.~HO. CHHTe3 ]Ieyx~apHgpHolt tlCnH C2Xe.~aHC noMomto npHHllmIa KgHCMaTh~eCKOg HH~pCHI~ ~ T ~ X TOqHbiX rIOJIO~¢HHI~ H KHHCMaTH= tICCl¢Ol~HHBCpC~d~C rlpHae]leHlleM IIOHO~OHI~ TOqel¢ ~JLq e ~ c m e r o ,mcna TOql-lblX IIOHO~eHHI~L E c . ~ Tenepb TpH OT~e,JI]bHMe ]IByXIIIapHHpHble IXelllf COe~[4HeHbl BM(~"r~ B HX CBO60,/XHbiX KOHI/.ax, HL'I~Ka.I.OJ.I~ M~,v~aHE[3M C: ,/~ByMM c " r ~ l ~ CBO60~bl yaos, nCTnOpSeT T[~OBaHI,LqM ]]JIM KOOp~H.Ha~ yrnoe~x nepe~en~em~t 0, ~, ~ ~. 3TOT MCTO~ MO~'T 6SlTt, ecrloJl~3osatl K~! CHHT¢3a Mcxats~3Ma s ue~H y ~ o ~ e T ~ O p e m ~ ~ x c ~ a . r ~ u o m e ~ r ~ n p o ~ B o m ~ o ~ 6 p a ~ m , ~ ~O~m~M n o ~ o x e ~ R ~ . B ~ a ~ o M ~o~mo noa~ 6 o a ~ u o e ~mcao a a ~ p ~ a ~ m . ~ x pemeu~a ~ 6yny~©ro n ~ 6 o p a ~a o ~ o ~ ~pyr~x zp~repm~ za~ n e p e ~ a T o ~ n a e yrJx~, n.rm n p o c r p a ~ c r s e t m s ~ e Tpe6onah-HS.
Pergamon Press. Printed in Greet Britain
Synthesis of Plane Linkages to Generate Functions of Two Variables Using Point-Position Reduction - Part 1. Rotary Inputs and Output T. S. Mruthyunjaya*
Received 8 December 1971 Abstract The paper presents simple graphical procedures for position synthesis of plane linkage mechanisms to generate functions of two independent variables. The procedures are based on point-position reduction and permit synthesis of the linkage to satisfy up to six arbitrarily selected precision positions. 1. Introduction EARLIER investigations in the field of synthesis of function generators have largely been confined to the problem of generating functions of a single independent variable. Several methods, graphical and analytical, are available in the literature for this purpose. However, there have been relatively few attempts at synthesis of'linkage mechanisms to generate functions of two independent variables. Svoboda[1] proposed the plane, seven-link two-degree-of-freedom linkage shown in Fig. 1 for generation of functions of two independent variables. He also suggested a homographic procedure for position synthesis of this linkage. However, his procedure is rather tedious and requires an enormous amount of drafting work to arrive at a satisfactory linkage. Allen [2] developed graphical methods for position synthesis of this linkage for a maximum of six precision positions chosen such that they always correspond to three values of one of the independent variables and for which, at each of these three values, the second independent variable has not more than three arbitrarily chosen values. An additional restriction, in the case of six precision positions is that the three values of the first independent variable should be equally spaced. Allen [2] also developed a graphical method for third-order derivative synthesis of the linkage. Freudenstein and Sandor[3] presented a method, which is partly graphical, for position synthesis to satisfy five arbitrarily selected precision positions. Their method is based on computation of the Burmester points for five finitely separated positions of a plane. A fully analytical treatment of this problem for position synthesis to satisfy twelve arbitrarily selected precision positions was carried out by Phillipp and Freudenstein[4]. They considered a slightly different linkage obtained by replacing one of the three binary floating links in Fig. 1 with a ternary link. Their work showed that the calculations involved can become quite extensive and time-consuming even when programmed on a high-speed digital computer. An analytical method for fourth-order derivative synthesis of the linkage is given by Lakshminarayana and Narayana Murthy [5]. The procedures presented in this paper are graphical in nature and can be used for position synthesis to satisfy up to six precision positions. These simple and rapid pro* Lecturer, Dept. of Mechanical Engineering, Indian Institute of Science, Bangalore 12, India.
363
364
~3
B4
AI el)
Figure 1. The seven-link two-degree-of-freedom linkage. cedures place no restriction on the choice of precision positions. The procedures are based on the powerful technique of point-position reduction developed largely by Hain [6]. It may be recalled here that Goodman [7] was the first to suggest the possibility of using point-position reduction to synthesize linkage mechanisms to generate functions of two variables. However, as far as the author's knowledge goes, there has been no attempt in this direction reported in the literature, though the technique has been used to great advantage in the case of single-degree-of-freedom linkages [6, 8, 9]. 2. Formulation of the Problem The plane, seven-link, mechanism shown in Fig. 1 represents the two-degree-offreedom linkage with the minimum number of links permitting rotational inputs and output. Theoretically, it should be possible to synthesize this linkage to satisfy fourteen precision positions since an equal number of design parameters, viz. eight link ratios, three starting positions and three scale factors are involved. Let z =f(x,y) be the function to be generated with precision points at (x~,yi), i = 1,2, 3, 4 . . . . . etc. Assuming the scale factors
~x
~y
Az
K~=A--~, K~=Acb, and K~=A---~
the incremental rotations 0t2,013 . . . . . (~12, $a3 . . . . . of the input links and $12, $13. . . . of the output link corresponding to the various precision positions can be calculated. Now the dimensions of the linkage have to be determined such that these incremental rotations of input and output links are coordinated. In Fig. 1 the linkage is shown in one of its precision positions and the locations occupied by the moving pivots at other precision positions are also marked. As the input links AoA, BoB and the output link CoC rotate through incremental angles corresponding to the precision positions I, 2, 3, . . . . etc. the double joint at G passes through the positions G t , G.,, G3 . . . . . etc. Conversely, it can be stated that if the joint G is guided along positions G t, G2, Ga . . . . . etc., the links AoA, BoB and CoC will rotate through the corresponding incremental angles.
365 The seven-link mechanism can be considered to consist of three link-pairs namely, ,4o,4-AG, BOB-BG, and CoC-CG with their ends ,40, B0 and Co pivoted to the frame and the other ends pivoted together at a common point G. Considered separately, Aod G, BoBG and CoCG each form a two-jointed open kinematic chain with the fixed link. In such a chain if the fcee end is guided along a certain path the link pivoted to the frame will swing through definite angles. Therefore the problem of synthesis of the twodegree-of-freedom linkage now reduces to one of determining the dimensions of three separate two-jointed open kinematic chains ,4oAG, BoBG and CoCG such that when their ends G are guided to move from position 1 to positions 2, 3, 4 etc. the links pivoted to the frame viz..4o,'1, BOB and CoC will rotate through angles 01~, 013. • • ~i=, ~ 3 . . . . and ~=, 013. . . . etc. respectively. It may be noted that the point positions G~, G=, G3 etc. can be arbitrarily selected. If now the three chains are pinned together at their free ends the resulting linkage will satisfy the requirements for coordinations of rotations 0, tk, and 0 for the precision positions 1,2, 3 . . . . . etc. The constituent problem of determining the dimensions of the two-jointed kinematic chains can be solved by using the principle of inversion in the case of three precision positions and inversion combined with point-position reduction in the case of four, five and six precision positions. The constructions are explained in detail in the following sections. The first step in synthesis in each case is to assume suitable scale factors and calculate the incremental angles of rotation of the input and output links for the various precision positions. 3. Synthesis for Three Arbitrarily Selected Precision Positions The locations of the point positions G~, G= and G3 are selected arbitrarily (Fig. 2). The location of the fixed pivot Co is also selected arbitrarily. The lines CoG2 and CoG3 are rotated through angles - $1~ and - ~,3 respectively to positions CoG=c' and CoG3~' (inversion). The centre of the circle passing through points G1, G2~' and G3~' gives C,, the location of the pivot C in position 1. CoC~G~ is the required two-jointed chain in position 1. The chains.4o,4 IG~ and BoB~G1 are determined by similar constructions. The resulting two-degree-of-freedom linkage is shown in position 1 (Fig. 2). In view of the many arbitrary choices made during the construction an infinite number of linkages can be obtained all of which satisfy three precision positions. Bo
B,o J
~
c,
,:,
C, p@
Ao
h
Figure2. Synthesisforthreearbitrarilyselectedprecisionpositions.
366
4. Synthesis for Four Arbitrarily Selected Precision Positions The locations of the point positions G1, Gz, G3 and G 4 a r e selected arbitrarily (Fig. 3). The location of the fixed pivot Co is chosen along one of the perpendicular bisectors of the distance between any two point positions such that these point positions subtend an angle at the pivot Co which is equal to the incremental angle of rotation of the output link CoC between these positions. Let, for example, Co be located along the perpendicular bisector e34 of G3U4 such that the angle G3CoG4 is equal to ~a4. Then the relative pole for the displacement of the link-pair CoC-CU between positions 3 and 4 will coincide with the fixed pivot Co. Thus a single point position reduction is achieved. Since there are, in all, six pairs of point positions GIGz, G2Ga, G3G4, G1G3, GIG4 and GzG4 and since for each pair the location of Co can be chosen along the corresponding perpendicular bisector on either side of the pair, there are twelve possible locations for the pivot Co. Now taking 4 as the reference position of CoCG, the lines COG3,CoG=and CoG~ are rotated through angles - ~ , ~42, a n d - ~4~ respectively to positions CoG3c', CoG.z", and CoG,~"(inversion). G3~, will, of course, coincide with G4. The centre of the circle through G4, G2~" and G~~" will give C4 the location of pivot C in position 4. CoC4G4 is the required two-jointed chain in position 4. The positions C3, Cz and CL are located as the intersections of circular arcs of radius C4G4 centered at G3, Gz and Gl respectively with the circle through C4 of radius CoC4 and centre Co. The chains A oA1GI and B oB1G~ are determined by similar constructions. The resulting two-degreeof-freedom linkage is shown in position 1 (Fig. 3). Because of the many arbitrary choices during the construction an infinite number of linkages can be obtained all of which satisfy four precision positions.
u-,.- elz
Figure 3. Synthesis for four arbitrarily selected precision positions.
367
5. Synthesis for Five Arbitrarily Selected Precision Positions In this case the G point positions cannot be selected entirely arbitrarily. At least three of them should be chosen so that they lie on a circle centered at one of the fixed pivots and subtend angles at that pivot which are equal to the respective incremental angles of rotation of the corresponding input or output link. The remaining G-point positions are determined by construction. For example, let point positions Us, G3 and G, lie on a circle of any radius centered at any point Co such that angles G=CoG~and G=CoG, are equal to 023 and 02, respectively (Fig. 4). Then the relative poles for the displacement of link-pair CoC-CG between positions 2 and 3 as well as positions 2 and 4 will coincide with the fixed pivot Co. Thus a double point position reduction is achieved. For the three point positions G2, G3 and G, the chains AoAoG= and BoB2G=in position 2 are determined using the procedure indicated earlier for three precision positions. AoA=G=BzBo now constitutes a five-bar linkage. The G-point positions GI and Gs corresponding to precision positions 1 and 5 are located giving the inputs (0=1, 6~1) and (025, 0hs) to this five-bar linkage. The problem now is to determine the dimensions of the two-jointed chain CoCG such that when end G is guided to move from position G1 to positions G~, G3, G, and G5 the link CoC will rotate through corresponding incremental angles. This can be done by inversion. Taking 2 as the reference position of the chain CoCG, the lines CoG, COG3, COG,, and CoG5 are rotated through angles -0zl, --023, --02,, and -~=5 respectively to positions COG(', CoG3c', CoG, ~' and CoGs~'. G8~=and G, ~' will, of course, coincide with G2. The centre of the circle through points G2, GI ~ and Gs ~ will give C2, the location of pivot C in position 2. CoC~G2 is the required two-jointed chain in position 2. The resulting two-degree-of-freedom linkage is also shown in position 2 (Fig. 4). In this case also an infinite number of linkages can be obtained all of which satisfy five precision positions. 6. Synthesis for Six Arbitrarily Selected Precision Positions
In this case at least four of the G-point positions should be selected such that they lie on a circle centered at one of the fixed pivots and subtend angles at that pivot which
Fioure 4. Svnthesis for five arbitrarily selected precision Dositions.
368 are equal to the respective incremental angles of rotation of the corresponding input or output link. The remaining G-point positions are determined by construction. For example, let point positions G~, G2, G3 and G4 all lie on a circle of any radius centered at any point Co such that angles G1CoG~, GICoG3 and GICoG4 are equal to ~b12,~b13and ~b14respectively (Fig. 5). Then the relative poles for the displacement of link-pair CoCCG between positions 1 and 2, 1 and 3, 1 and 4 all coincide with fixed pivot Co. Thus a triple point position reduction is achieved. For the four point positions G1, G2, G3 and G4, the two-jointed chains AoAG and BoBG in position 1 are determined following the procedure indicated earlier for four precision positions. A oAIG IBIBo now constitutes a five-bar linkage. The remaining G-point positions G5 and G6 corresponding to precision positions 5 and 6 are located by giving inputs (015, ~15) and (016, ~16) to this five-bar linkage. The problem now is to determine the dimensions of the two-jointed chain CoCG such that when G is guided to move from position G1 to positions G2, G3, G4, G5 and G~, the link CoC should rotate through the corresponding incremental angles. This is done by inversion as before. Taking 1 as the reference position, the lines COG2, COG3, COG4, CoG~ and CoGs are rotated through angles -~/12,--~bx3,--b14,-~15 and - ~ 6 respectively to positions CoG2 c', CoG3c', COG4% CoG5~' and CoGe~'. G2~, G3~' and G4c' will, of course, coincide with point G1. The centre of the circle through points G1, G5 c' and G6 ~ will give C1, the location pivot C in position I. CoC~GI is the required two-
Be
Figure 5. Synthesis for six arbitrarily selected precision positions.
369
jointed chain in position 1. The resulting two-degree-of-freedom linkage is also shown in position 1 (Fig. 5). Because of the several possible locations for the fixed pivots, a large number of linkages can be obtained all of which satisfy six precision positions. Conclusion The point-position reduction technique has been utilized to develop successful graphical procedures for the synthesis of plane linkage mechanisms to generate functions of two independent variables. The technique of point-position reduction has been known for a long time, but its application to synthesis of two-degree-of-freedom linkages is believed to be novel. The synthesized linkage can satisfy up to six arbitrarily selected precision positions. Though the procedures do not exploit the potentialities of the linkage considered to the fullest extent, it is hoped that they will be found useful by designers of mechanisms for the simple reason that they permit determination of several alternative configurations of the linkage in a short time and thus enable selection of the best linkage configuration which satisfies other important criteria like space considerations, transmission angle requirements etc. 7.
Acknowledgements-The author wishes to thank Dr. M. R. Raghavan, Assistant Professor, Department of Mechanical Engineering, Indian Institute of Science, Bangalore, for useful discussion and suggestions in the preparation of this paper. The help and encouragement given by Prof. M. A. Tirunarayanan, Professor-incharge, Department of Mechanical Engineering, Indian Institute of Science, Bangalore, is gratefully acknowledged.
References [ 1] S V O B O D A A., Computing Mechanisms and Linkages, pp. 223-299. McGraw-Hill, New York (1948). [2] A L L E N C. W., The design of linkages to generate functions of two variahles, J. Engng Ind. Trans. ASME 81B, 23-29 (1959). [3] F R E U D E N S T E I N F. and S A N D O R G. N., On the Burmester points o f a plane, J. Appl. Mech. Trans. ASME28, 41--49 (1961). [4] P H I L L I P P R. E. and F R E U D E N S T E I N F., Synthesis of two-degree-of-freexiom l i n k a g n s - A feasibility study of numerical methods of synthesis of bivariate function generators, J. Mechanisms 1, 9-21 (1966). [5] L A K S H M I N A R A Y A N A K. and N A R A Y A N A M U R T H Y R. G., Derivative synthesis of plane mechanisms to generate functions of two variables, J. Mechanisms $, 249-271 (1970). [6] H A I N K.,Applied Kinematics, Second edition, pp. 524-618. McGraw-Hill, New York (1967). [7] G O O D M A N T. P., See discussion in [2]. [8] A L L E N C. W., Point position reduction, Machine Design 31,141-145 (1959). [9] SONI A. H. and H U A N G M., Synthesis of four-link space mechanisms via extension of point-positionreduction technique, J. Engng Ind. Trans. A SME 93B, 85-89 (1971). CmlTe3 r[Jloclmx Mexal~3MOlS l'lpoll3so~xumx q ~ l'lo~o~-mdi To.~¢. q a c ~ I l e p s ~ : Y r a o s u e I
]
~syx NepeM~ ¢ I~OMOIHIOr[plll~.~ifll ~ Be~Tmero n Be~oMo~O 3area.
T. C. M p y ~ u o ~ a n P e 3 m M e ~ ~ r a pa60ra 3am~Maerc~ pazs~rHeM rpa#pH~ecm~x M C T O n O B cmrreza nnoczsx M e X a H H 3 M O B B UCJIH npOH3~eHtLq ~byHKUI4~]Inyx Hc3aBHCHMblXnepeMCHHMX y~oB~eTBOpMs3mHx npOH3So~bno sM6paaHblM TOtDIbIM noJloxelm~lM. YflOMI~HTTbI~ MCXaHH3M ~J]$1 CHHTC3a 31"O C~MH31~HHiIK C ~yM.q CTfJI~IVIMH CBO6OlIbl COCTOB.rlJnOII~C~[H3 TpOl~Horo HenO]IBR)Kaoro 3Bona AoBoCo, rpex 6saapmax zpaeonmnon AoA, BoB H CoC cny3zam~x Ka~ senynme u seno~v~e 3Beua H c TI~X 6nHapHblX ~IeyxHoBo~KoBbIX 3~H AG, BG, ~ CG, e KoTopblx TOKKa G np~/l~'aBJllleT nDyxnoBO~KOBid~ m a p s s p c nBollsym cmr3m. Ecns ~cc Macu~ra6H~le ¢[mrrop~ 21n.q Bxo~a n sMxo~a ni~6paa~ npa~H.~Ho, cmrre3 M ~ M a coxpaub~ercx ]c onpc~e~leHHW pa3MepOs 3BCH TaKHM 06pa30M ~ITO yrHbI BpameH~: 0:2, 0~a, ... a e ~ m c r o 3BCHa AoA,
~ , ~ .... eeaymero~BesaBoB s ~ , ~ ....
eeao~oeo ~ma
CoC ~ e ~ y
nep~uM s
nocneay~m~M~
TOtlHbIMH MP-~-~ronoJIO~HIdLqMHcornaco~asu. Kor~la 3BeHa AoA, BoB H CoC BpamawT~ tlepc3 3THe yrJlb~, TOtlKa G nBI,I~TC~I H3 HO.rlO~KeHH~GI K n o ~ o ~ m ~ 4 G=, G~, H T. ~. Hao6opoT MOXHO cKa3aTb ~ro eC~IH TOqKa G n a n p a ~ e a a e C~OeM ,~e~7~e~tm H3 n o n o ~ e n ~ G : x nono~zeHu~M G2, G3 . . . . , 31~Ha AoA, BoB H CFC 6y~yT npamaTcs qepe3 e~me ynOMSHyTble yrnbL CCMH3BeHHblI~MeXaHH3M MOX~eTpa~CMaTpHBaTbC~I
370 KaK COCTaarL~lOmKi~Cg H3 TpeX OTReJIbHbIX ~IByxlnaplttlpHMX OTKpblTblX KHHeMa.TI~eCEHX uenett AoAG, BOB(} ~ C o C G O ~ H e H H b l X e TO'V~e G. ~ y x m a p e m p u e m tIemb, x a z a ~ m e , cocroerrcs co 3Beaa rlpHCOe~tlHeHHOrO O]IJttlM XOHI/OM K CTOtVflCC C nOMOUXIO IJa/I~bbl H maOHHpOM K 3Belly Ha BTODOM KOHHe. Ocraenn~c~ ~oHeu ~ p y r o r o 3neHa cm36O~©H. EcJIH s Taxol[ ~¢IIH cnO6O~I]MIR l¢olleU Hallps.ar/eH no OHp~[O.~IeHHOMy I1y'rH, 3BCHO COC~[4H(~HHOe CO CTO~KOfl 6y~OT epatuawcs qep¢3 onpe~enermsxe yrnbl. T a Z H M o 6 p a 3 O M CHHTe3 M e X a H H 3 M a MOXL~r 6brrb HCIIO.rIHeHc~eIxyIoIIXHM MeTO~OM'. I. C Haqana rlpHHHMaCTC~ oUpeJIeJICHHH¢ nonoxea~Lq G:, G2, G3, ... romc~ G. O m ~ npo~BoynbHme c cny~ae rpex H ~ r ~ p e x To~nax n o n o x c m ~ , ao ~ BMcmero qKcna noHo~HIt~ 3TO ]le~aeTC~ KaZ OI][HCaHO BO B T O p O M l~fHrI~. 2. T p K oTnenbHble n ~ y x m a p F m p m ~ e uewa AoAG, B o B G ~ C o C G ycTaRoeneHmae Taza~ cnOCO6OM, qTO e,C,.rlR zOHel/ G ~IBi~g'CTCgH3 IIOJIO~eHI,Lq Gt K nono~eH~SM G2, G 3 , . . . , 3BeHa. AoA, BoB ~ CoC spaIRa.loTCg "[epc3 yrHH 012 , 013 . . . . . ~12, ~ t 3 . . . . H ~bx2, ~ 3 . . . . OTHOCgTe.~HO. CHHTe3 ]Ieyx~apHgpHolt tlCnH C2Xe.~aHC noMomto npHHllmIa KgHCMaTh~eCKOg HH~pCHI~ ~ T ~ X TOqHbiX rIOJIO~¢HHI~ H KHHCMaTH= tICCl¢Ol~HHBCpC~d~C rlpHae]leHlleM IIOHO~OHI~ TOqel¢ ~JLq e ~ c m e r o ,mcna TOql-lblX IIOHO~eHHI~L E c . ~ Tenepb TpH OT~e,JI]bHMe ]IByXIIIapHHpHble IXelllf COe~[4HeHbl BM(~"r~ B HX CBO60,/XHbiX KOHI/.ax, HL'I~Ka.I.OJ.I~ M~,v~aHE[3M C: ,/~ByMM c " r ~ l ~ CBO60~bl yaos, nCTnOpSeT T[~OBaHI,LqM ]]JIM KOOp~H.Ha~ yrnoe~x nepe~en~em~t 0, ~, ~ ~. 3TOT MCTO~ MO~'T 6SlTt, ecrloJl~3osatl K~! CHHT¢3a Mcxats~3Ma s ue~H y ~ o ~ e T ~ O p e m ~ ~ x c ~ a . r ~ u o m e ~ r ~ n p o ~ B o m ~ o ~ 6 p a ~ m , ~ ~O~m~M n o ~ o x e ~ R ~ . B ~ a ~ o M ~o~mo noa~ 6 o a ~ u o e ~mcao a a ~ p ~ a ~ m . ~ x pemeu~a ~ 6yny~©ro n ~ 6 o p a ~a o ~ o ~ ~pyr~x zp~repm~ za~ n e p e ~ a T o ~ n a e yrJx~, n.rm n p o c r p a ~ c r s e t m s ~ e Tpe6onah-HS.