Hering, J.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

ANALYSIS OF COUPLING MECHANISMS J. HERI:'\G POL YTECI!!\IC C"'l \ EH~1T), BU1APbT mtroduetiou Thl' 1']''''';''l1t ,.;Iudy analy:,c,.; kill(,lllati(' and dynamic r('lation,.; of trall";-mi,.;,.;ioll of rotary mOiion lwiwe,']] twn ('O!1IW,'!t·.l ,.;haf!,.;, parallel or i1l1er-;,wetill~. III it,.; eonrft' tlw following pninb will h,· in\''',.;iigait'd : ]. Dt·~ro·,· or fre,'don1 11,·ee,.;,.;ary for coupling: nlPchani"Hl"; : 2. Y"riation,; of all~ular y,·jocitj,·,.; al](l angular a( ..... Jt'ralioll of cOl!Il('ele,l pawlld . .;haft,,: Jl10111,'nt,.; and fore(',.; a('tillg Oll Ih .. "haft,.; : torqlH'''; and jWllding J11ollwnl,. ,«;ting Ull tlw ,.;haft,.; : 4·. Ho,,' "h,,1'I";. l'aralld. int,'r::' .. etillg or ,.;ke,\". ea]] lw ('Ol1Il<·ct,·d ";0 that the all~\Ilar Yt']oeiti",.; of hoth ,.;haft::, arc "'Iual ,\1 allY in,.lant : S. Bac'it' principk" of ck:,ign of "ynchronou,; driYe.~. \Vithill th,' ,.;eop'· of tlw abo\"(' the accuracy of apprtJximat'· formulae :' pxamillt'tl alld tlll" limit,.; of th,·il' (ll'pli"ahility an' olItlillt"l. 1. Degree of freellom neces"ary for eoupling mechanisms Locat!' th,' two "haft,; tq h,· COlln,·,'t,·,l. I anti :2. in a ~y,.;t'·Ill of coor-rlill<lt,·,.; x. y. ;: ';0 Ihat th,· ""nlml lill"'; or th,' ,.;haf[:, eoillcid .. with the axi" y (Fig. 1 a). Th .. deg:n',' of fn·,·dolTI of a coupling m"clwlli:'1ll d"lltd,'''; tlw 11l1I111wr of intl"llt'ndt'Ilt rdatiy(' motion,.; allowt·d for th.· "hafts by tht' eoupling along or ahollt Ill(' coordinat .. axes. a) Shift ~haft :2 paralIelly "ith it"eli' alon~ th .. axis x to lIlt' distance SX' and alon~ tlw axi,. .::; io lIlt' di,-ull1c(' So (Fig. lh). A" a r .. "ultn\"o paralll·l :,haft:, at a dii'tanct' of s f~~"+ s~ from "Hcll oth .. r ha\',' be!'n obtain-ed. Aecordillgly. Iht' eoupling COll!ll'(·tin~ tIlt',.;!" paralld "haft:- "houlcl 1}<' :,ueh that they hay .. n·latiy,· I1I0tiOII,. aloJ1~ t"\\O direction:, perpeIHlieular to '><lch otht'1'. b) .Rota! .. "haft 2 abollt the <lxi,.- x through an angle u,,' and about the axis;: through an angle u: (Fig. le). T\\"o "haft:, lucnt"d al an angle 64 J. HERUG Q = arccos(cos CJ. x cos (lz) relatiyely to each other have been produced. Accordingly, for their coupling a mechanism is needed which allows them to rotate about two axes perpendicular to each other. Possibilities of securing the degree of freedom necessary for coupling mechanisms will be examined below. a. jZ i ~3a. ---'I,. !J Fig. Fig . . ') For parallel shafts the two motions of translation can be produced hy means a cr08s-pin inserted hetween the shafts which allows them relative motions in two directions perpendicular to each other (Fig. 2 a). Such is the mechanism of the Oldham coupling. For shafts center lines of which intersect, the two rotations can he made possible through the insertion of a cross-pin similar to that mentioned aboye, with the difference that a joint mechanism is applied instead of the solution with a link (Fig. ~b). This is the way a cardan joint is cIeyeloped. AXALYSIS OF COUPLUG .UECHAXISJIS 65 In both solutions a third member, viz. a cross-pin has been inserted between the shafts. The problem can also be solved in such a way that the two shafts secure the degree of freedom necessary through a direct contact (Fig. 3a and 3b). A special formation of the two contacting shaft ends a. 1 Fig. 3 iE required for this purpose. The simplest ways of connecting two shafts, parallel or intersecting, directly to each other, will be investigated next. 2. Transmission of rotation between parallel shafts a) Kinematical analysis In the case of parallel shafts connected directly to each other, for the angle of rotation !PI of the driving shaft L angle of rotation ff2 of the driven a Fig. 4 shaft 2 and angular deviation rh - !PI = rp, from Fig. 4a, the follov.-ing relationships may be derh-ed : S + T • COS !PI (2.1) .5 Periodica Polytechnica ~I III I}. 66 J. HERLYG s· sin (2.2) tucp = -'" r -'- s . cos Cfl The maximum of angular displacement can be determined through. differentiation of 2.2 with respect to rr l' which pve,:; tg Cfmax = -This yalue appears at ff2 = 90". The yariation in angular yelocity IS s r s 2-r 1- (; r and the approximate magnitude of the degree of irregularity s o'~:2-r (2.3) (2.4} Taking this into consideration and substituting tg cp r''/ cp, Taylor's expansion of 2.2 gives cp' ~, - ~ sin CPl -+ 0.5 (~) 2 sin 2 CPl (2.6} Relative error in 2.6 is R 2 ( ; r +1- CPmax (2.7} cP tg CPmax which, for s . = 0,10 0,15 and 0,20 r with ... ,..,0 f{ll1ax = ;),' 8,60 and 11,5 0 gives a relative error of R 2,5% 5,2% and 9,2% respecth·eJy. AXAL j·';;15 OF COCPLLr& JIECH.LHS.lIS 67 As can be seen, the approximate expression 2.6 yield~ results of suf-ficient accuracy only in case of s .< 0.2 (2.8) r provided En eIror of 9<j~ may still be considered as tolerable. The approximate value of the variation of relative angular velocity may be obtained through differentiation of Eq. 2.6 with respect to time. and division by 0)1 : Jw 0' . 0' ,2 (-2 J cos 2 9-\ (2.9) where An gular acceleration of the shaft 2 is [ 0' '0' 2 '1 1'2 = CUI 2 sin crI - 2 \ --:d sin 2 crI (2.10) Variatiolls of the angular deviation and angular velocity are shown ill Fig. 4b. The result 2.8 obtained by the investigalion of the accUl'acy of Formula 2.6 holds true, of course, also for Relationship 2.9 and 2.10. b) Dynamical analysis As is seen in Fig. 5a, on both shaft ends an only force P iF exerted. loading then for torsion and bending. The torque acting on the shaft 1 is lUes! = 1v1 = Po . r = P . r . cos (P (2.11) On the basis of the figure the torque affecting shaft 2 may be written as lY1es, = P [r . cos cp + s . cos CPJ = (2.12) 68 J. IIERJ.\"G Substituting the values 2.1 and 2.2 111 2.12, the relative variation 111 the torque s s r -;- S • cos crI r a I' Fig.5a In addition to the torque, the force P acts on hoth shaft ends. Its magnitude is according to the figure p M cos er r· cos T Suhstititung the value of 2.2 in 2.14 we get p = NI ______ :----______ _ r (2.13) (2.14) (2.15) The way of construction and the variation in time of the relative torque are shown in Fig. 5b, and the bending force P can he constructed according to Fig. 5c. AX.IL '·SI'; OF COCPLLYG .UECHASISJfS 270' p t Po I ~ -==::::::::==+======~::::::::::::+=======--, ,0 gO' Fig. 5b. t: I 1180' 270' 3. Transmission of rotation hetween shafts intersecting a) Kinematical analysis 69 For shafts cent"r lines which inter~('cL between the angle::: of rotation l{!1 and (h of the driying shaft 1 and driYen :,haft 2, respectiyPly, and :::haft angle a. from Fig. 6a, the following relationship may be derived: 19 ({2 = tg (rl . ('os Cl (3.1 ) Hence the angular deyiation q; = 72 - q·l may be determined as tg If tg q:dCO_S_(l ___ l) 1 -;- tg2 (rl . cos a (3.2) The angular deyiation has its large;:;t yalue at 1 (3.3) 70 .T. HERLYG this largest "'-Hlne is cos a-I tg 'fmax a The variation III angular ,-elocit...-1 s. _ W:!IJ1ax - W2min u - ~-- - --~--- - cosa Q. z (Jl 1 --= __ ::::..::::~;';ciq~~;;-~~~ __ ·w 1 Fig. 6 90' and the approximate "'-Hlue of the degree of irregularity /y ~ 2 (l - cos CI.) (3.4) (3.5) b I ~ {80' 360' (3.6) Taking thi" into consideration and ;:uhstituting tg 'f ::" 9', Taylor's expansion of 3.2 gin';: rp' ~ b' \ - 1 4 6' .. (6' )2 -.. t/ sin 2 If I + 0.5 4 sin 4 'fI (3.7) Relatin' error ill 3.7 1:" R '1" 1 - cos Cl . --- (1 - cos a)2 -;- --'-- 1 - -..:....::=.:-1 cos Cl tg 'fmax (3.8) ,-LY,IL YSIS OF COCPLLYG ,UECHA,YIS.11S which, for a with q.'max gives a relative prror of R respectively. ')~O ~;) 2.8 0 3,5% .5,9% and 30° and 4,)0 ,-and 9,1% 71 Thus the approximate formula 3.7 has a satisfactory accuracy onlv 1n case of a < 30° (3.9) agam provided an error of 9% may still be tolerated. The approximate \-ariation of angular velocity is given by the first ,derivative with respect to time of Eq. 3,7 as (3.10) and the approximate yariation in the angular velocity of shaft 2 o [ " 1'1 6' 1 . ') 0 - SN • 4 J 8 2 ~~ UJi 0, :- 4" Slll:' Cfl - ,.) u - S1l1 ' ifl (3.11) The variation:- of angular deyiation and relatin' angular yelocity are shov,'11 in Fig. 6b. b) Dynamical analysis l\tIoment is transmitted from shaft 1 to shaft 2 through a couple of forces perpendicular to shaft 2. As is seen in Fig. 7a, torques and bending ,moments acting on the shafts are (P . tg 8) a }VI . tg 8 P , sin 6 Ales._, =c a . sin ° ~= ~\'I --'-cos 8 cos {} P . AI", = -- a· coso cos {) JJ cos 6 cos {} (3.12) (3.13) (3.14) (3.15) 72 J. HERI.YG The angle {j denotes the angle of inclination of the two forks' planes: According to Fig. 7a, its magnitude is tg {} sin a sin (crI - 90°) ------''-'---=.--~ = - tgacoscrl cos a The magnitude of angle b, again according to Fig. 7a is cos /j = sin Cl. cos (crI - 90°) = sin Cl. sin crI 1 Fig. 7b (3.16) (3.17) 270' 360' 'f; AJ-AL" SIS OF COFPLLYG JIECHAYIS_1IS 73 By substituting 3.16 and 3.17 in 3.13, 3.14 and 3.15 the variations of magnitude of the moments with respect to h are obtained (3.18) (3.19) (3 20) The method of construction for each moment a;;: ·well as their variations are shovv-u in Fig. 7b. 4. How to ohtain a uniform transmission of rotation A common objectionable feature of the couplings described in the paragraphs aboy", is that the angular yelocity of th(> "haft is yariable, eYt'l1 Fi~. 8 if the driving shaft 1 has a constant angular yclocity. This state is disach-an-tageous in regard to the inertia force;;:. Next will be ;;:hown how these yarin-bilities can be done away with for connected ~haft;;: a) parallel, b) inter;;:ecting and c) skew. Put the two shafts (1 and 2) side by side parallelly and connect a third shaft, 3' to 1 through a coupling K 13 , of any chosen design provided it leaves two degrees of freedom. Let the shaft angle of 1 and 3' be 0.. Connect a shaft 3" to 2 through a coupling Kl3" which is the opposite-hand view of K 13,. Thus the systems 1-3' and 2-3" form a symmetrical layout (Fig. 8). Rotate shaft 3' through an angle Cf3" and shaft 3" through an angle Cf3" = - Q'3' about their own center lines. As a con;;:equence of the symmet-rical layout, the angular displacements of shafts 1 and 2 ,rill be rh = -7-'2 (4.1 ) 74 J. HERLI'G a) Let a space coordinate system as shown in Fig. 8 be taken. Rotate the system 2-3" about the axis )' through 180°, then shift it parallelly in the direction - y to a distance YO' Connecting the two shaft ends in this new position to each other rigidly (Fig. 9), the angnlar displacements of the parallel shafts 1 and 2 will be equal, viz. (4.2) As is seen a uniform transmission of rotation between parallel shafts can be obtained in case the layout is symmetrical in respect to some center O. Let us enlarge thc link mechanism shown in Fig. 4a by using symmetry with respect to some center (Fig. lOa). The disadvantage with this device Fig. 9 is that also the link 3 has to be mounted in bearing;;:. In order to avoid this, a link mechanism with s = r (4.3) will be examined. From Fig. lOb Cfl = rp2 = 2 Cf3 (4.4) (VI i. e. link 3 rotates with an angular velocity of W;j = -,2 Complete the link mechanism 01AB02 through a mechanism 01A'B'02 similar to the former but advancing relative to it by 180" (Fig. 10c). The two links will keep a constant advance of 90° relative to each other. As a consequence they can be replaced by a rigid cross member. The bearing support at point 0 has by t his become superfluous. By connecting points A' and B instead of 0 1 and O2 to the frame and dropping the members 1, 4, 2 and 5' the Oldham coupling a::: :::ho,m in Fig. 2a will be arrived at (Fig. 10d). ·LYALYSIS OF COUPLIXG JIECHAXI:;Jh 75 The Oldham coupling is the simplest means for securing a true uniform transmission of rotation bet-ween parallel shafts. But because of construc-tional feature" it can be used only in such cases 'when the distances of the t,vo shafts are not too large. If the distance of the two shaft;; i:; somewhat large, the device shown in Fig. 9 "ill be used in practice. K 13, and K 13', for instance may be cardan joints. Fig:. 10 b) By rotating the ~ystem 2-3" in Fig. 3 about the axis z through 180°, a uniform transmi"sion of rotation bet'ween intersecting shafts lC' ::,ecured (Fig. 11). Such a layout i~ symmetrical in respect to a plane (x, y) i. e. it is symmetrical as an image seen in a mirror. c) Finally. rotate system 2-3" through 1300 about an optional vector v = cos fJ J + sin (3 k lying in the plane y, z. As a result the "kew shafts shown in Fig. 12 are obtained. Between the shaft angle 2 0.' of 1 and 2, the angle o. and the angle of inclination I' of the two planes determined by thc "hafts 1-3' and 2-3". respectively, the following relationship holds true: 0) '1 , cos- a - cos- Cl cos i' =--= .. , Sln- a (4.5) Hence follow:3 that a coupling of skew shafts gin>s a true uniform trans-mission of rotation if the t,,,-o coupling elements mounted on the two end:3 of the shaft 3 are rotated relatively to the opposite-hand layout through an an!?le y about the axis x. 76 J. HERLYG In the cases described in b) and c) the couplings K 13, and K13 rt are· usually cardan joints. An objectionable feature of the double card an joint devices is that though the angular velocities of the shafts 1 and 2 are equally uniform, the-Fig. 11 b. Fig. 12 angular velocity of 3 will keep yarying, which mav gin' risc to considerabk illertia forces. Variability of the angular velocity 'of ,.hafts 3 can be eliminated hy application of synchronous driYeE'. AXALYSIS OF COUPLLYG JfECHASISJfS 77 5. Synchronous drives True uniform transmission of rotation between shaft 1 and 2 has been obtained by application of two couplings and a third shaft 3, the latter rotating with variable angular velocity. In order to eliminate the variability in speed also at the shaft 3, a direct coupling 8uitable to secure true uniform transmission of rotation between two intersecting shafts has to be de3igned. As has been seen in paragraph 4b for intersecting shafts, a true uniform transmission of rotation can be obtained through a device symmetrical to a plane, i. e. representing an opposite-hand view of the counter-part. The same requiremcnt can also be fulfilled by an only. symmetrically design-t~d joint. Fig. 13 Such a solution is shown in Fig. 13, where t'wo shaft ends developcd symmetrically are connected directly to each other. The contact of the shaft cnds is at a single spot. While the shafts are rotating, the contact spot as well as the point of intersection of the center lines are located, always in a plane bisecting the shaft angle and perpendicular to the common plane of the two "hafts. This deyice ensures keeping up a perfect symmetry by a complete re\-olution, i. e. the transmission of rotation is true uniform. \Vith this solution the contact of the shaft ends is at a point of the circumference. As a con;;;equence, a large surface pressure and considerable bending forces arise. Bending forces can be taken care of through a distrib-ution of the transmission of moments over several points of the circum-ference, and the contact at a point can be avoidcd by means of halfcylindcrs making contact on plane surfaces and rotating in their seats (Fig. 14). By another solution moments arc transmitted through halls placed along the circumference (Fig. 15). The balls are located at the points of inter-section of semi-circular races made in the driving and driven shafts. A ball-and-socket joint located at the point of intersection of the shaft cent er lines prevents axial displaccments. This is important as any axial displacement 78 J. REBUt; of the shafts upsets the symmetry and brakes off the uniformity of the trans-mission of rotation. The races must be made so that their point of intersection al·wavE lies in tht' plane of symmetry and their angle of intersection is fairly largt'_ Should the angle of intersection of the races be too "mall, they, might come Fir;. 14 into covering at a certain angular displacement crI' and therbalI would lose its definite location in the plane of Eymmetry. Therefore it is most advisable to design the races in such a way that they intersect each other at some con-stant angle, for any angle of rotation CPl' This requirement may be met by Fig. 15 races designed as logarithmic spirals. In order to facilitate processing loga-rithmic spirals can succesfully be substituted by osculatory circles. Such couplings securing a true uniform transmission of rotation through direct connection of intersecting shafts are termed synchronous drives. By using synchronous drives instead of the couplings shown in Fig. 9, 11 and 12, shafts parallel, intersecting or skew can be connected in such a way that not only shaft 2, but also 3 rotates with the same angular velocity as driving shaft 1. Summary Kinematic and dynamic analyse5 of several possibilities for connecting shafts arc described in the present paper. Con~truction methods for the determination of moments and forces acting on shafts are expounded. A5ALYSIS OF COUPLLYG jJECILUISMS 79 General principles for couplings suitable to ensure uniform trausmISSIon of rotation for shafts of any chosen location are examined and examples of their applications are shown. Finally, several examples of direct coupling devices suitable for ensuring a true uniform transmission of rotation for intersecting shafts, viz. of synchronous drives are given. References 1. BRICARD, R.: Le«;ons de Cinematique. Paris, 1927. T. n. 2. DIETZ, H. : Die Ubertragung Yon }iomenten in Kreuzgelenken. Z. Y. D. 1. 82, 825-28. (1938). 3. GREEN, W. G.: Theory of }Iachines. London and Glasgow. 1955. 4. GROSSMANX, K. H.: Die 3fomenten in Kreuzgelenk. Schweizerische Bauzeitung. 113, 27. (1939) . . 'i. HERI?;G, J.: Cardano Gelenkwellenkupplungen. Acta Technica. XXI. (1958). 6. KUTZBACH, K.: Quer- und Winkelbewegliche Wellenkupplullgen. Kraftfahrtteehnische Forschungsarbeiten. VDI-Verlag. Berlin, 1937. No 6. p. p. 1-25. I. JI.hlCOB J\t H. Kap)laHble nepe)latUI aBTo;;w6Hm'L (Cardan Joints of lIotorcars) MaIllrH~ 1951. MOCKBa. BblTI. 2. 8. R.-I.UH, K.: Praktische Getriebelehre. Berlin, 1939. Bd. n. 9. RZEPPA, A. H.: Gniversal Joint Drives. 3fachine Design. 1955. Apr. p. p. 162-170. J. HERING, Budapest XI., Budafoki ut 4-6, Hungary 

ANALYSIS OF COUPLING MECHANISMS

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ANALYSIS OF COUPLING MECHANISMS

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Periodica Polytechnica (Budapest University of Technology and Economics)