On the classification of spherical four-bar linkages

Mechanism and Machine Theory Vol. 19, No. 3, pp. 283-287, 1984 Printed in Great Britain.

0094-114X/84 $3.00 + .00 © 1984 Pergamon Press Ltd.

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ON THE CLASSIFICATION OF SPHERICAL FOUR-BAR, LINKAGES C. H. CHIANG Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan, Republic of China

(Received 26 February 1982, /n revisedform 13 April 1983) Abstract--Spherical four-bar linkages can be classified according to the rotatability, i.e. the capability of performing complete revolutions, of one link relative to the other three finks. For a spherical four-bar linkage, two link lengths may be replaced at a time by their respective supplementary arc lengths. This reduction process is carried out until the sum of all four link lengths is a minimum. Grashof's rule may then be applied to test the rotatability of the linkage. The shortest link of a Grashof linkage is capable of making complete revolutions. A Grashof linkage may be a crank-rocker, a drag-link or a double-rocker. A non-Grashof linkage is always a double-rocker. Examples are given.

1. I N T R O D U C T I O N

Consider a spherical four-bar linkage AoABBo as shown in Fig. 1. OAo and OBo are two fixed axes of the two rotating links A0A and BoB. Suppose the link AoA is extended upwards along a great circle until the point A*, the opposite point of Ao, is reached. The half circle AoAA * represents then the whole link body which rotates about the axis A0A *. The link AoA may therefore be replaced by its supplementary arc AA * without affecting the motion of the original four-bar. In doing so, the fixed link AoB0 should also be replaced either by its supplementary arc BOA*, or even by the arc 360 ° - BOA,. Thus we may have spherical four-bar linkages with link lengths in the range from 0 ° to 360 °. It is therefore necessary to classify, or identify a given sperical four-bar. The terms "classification" and "type determination" are synonymous. The problem of type determination of a given spherical four-bar hnkage has been the subject of study of a number of investigators during the past few decades [1-7]. Because of their contributions we are now in a position of better understanding of the classification problem. However, it seems to be desirable at this stage to clarify this problem and to establish concise criteria for the type determination. First of all it has to be decided into what kind of catagories are we going to classify spherical four-bar linkages. If we consider the crank-rocker, the draglink and the Grashof double-rocker as one class, and the non-Grashof double-rockers as another class,

then it does make sense to classify a spherical fourbar linkage according to its rotatability. By rotatability it means the possibility of performing complete revolutions of one of the four links relative to the other three links. However, at present we shall exclude the case in which both opposite links are equal and hence there are two links making complete revolutions relative to the remaining two links. It reminds us that, as we deal with the classification problem of plane four-bar linkages, there is no need to refer to the law of cosine of triangles, nor have we to resort to the technique of vector algebra. It seems to be promising, in dealing with the classification problem of spherical four-bar linkages, also to use only simple geometrical concepts. 2. S U P P L E M E N T A R Y S P H E R I C A L F O U R - B A R LINKAGES

Consider now the spherical four-bar linkage

AoABBo as shown in Fig. 2. Each link is extended to become a full ring. Each pair of adjacent rings intersect in two points. There are altogether 8 intersection points Ao, A, B, Bo, A*, A*, B* and B*, dividing each ring into 4 segments. The points of the pairs Ao, A*; A, A*;... are opposite points on the sphere. These segments constitute, besides the original four-bar AoABBo, 15 additional but kinematically equivalent four-bar linkages, namely [3, 4]:

AoABbo, A*ABBo, AoA*BBo, AoAB*Bo, AoABB*,

A~A*BBo, A~AB'Bo, A~ABB~, AoA*B*Bo, AoA*BB~, AoAB*BL A~A*B*Bo, A~A*BB~, A~AB*B~, AoA*B*Bo', A*A*B*B~.

Fig. 1. Spherical four-bar linkage.

It is understood that, in tracing a certain four-bar loop from one joint to the next joint, a shorter route rather than a longer route is taken. Although a link length say, AB, is kinematically equivalent to a link length of 360 ° - AB in motion transmission, only the 283

C. H. CHL~N~

284

i

Fig. 2. Supplementary spherical four-bar linkages. shorter part of the full ring shall be considered in dealing with classification problems. Thus, for example, if the four-bar loop AoA *BB*Ao is t o be traced, we trace first from A0 downwards and then upwards to get to A *, instead of from A0 passing through A and A* to A *. The same principle applies to the routes A *B, BB'~ and B'~Ao. In other words, the link lengths A *B*AB, BBoB*B'~and B*A '~BoAoshall not be considered. Hence none of the above listed 16 four-bars contains a link length longer than 180°. Even if a given spherical four-bar contains a link length longer than 180°, it can always be reduced to an equivalent four-bar which contains only link lengths less than or equal to 180° It seems that the 16 spherical four-bars were "cognate" linkages. In plane mechanisms two four-bar linkages are said to be cognate if the two couplers have a permanently common point. In other words, two cognate linkages produce the same coupler curve at this particular common coupler point. This is, however, not the case with the spherical identical linkages, because the 16 spherical four-bars represent only four relatively moving bodies. The coupler is always the ring ABA *B*, and nothing else. Any point on the coupler will of course trace the same coupler curve by any one of the 16 linkages. In fact these 16 linkages are all identical and represent just one single linkage instead of 16 different linkages. The word "cognate" may have been taken over from plane mechanisms to indicate these linkages, but a true spherical cognate such like the Roberts' cognates in plane mechanisms does not exist because parallelism does not exist in spherical geometry, although for certain spacial mechanisms there are actually Roberts' cognates [8, 9]. It seems to be adequate to term the linkages in Fig. 2 as supplementary spherical four-bar linkages.

Among the 16 linkages, 8 of them are reflected ones of the other 8 linkages. Thus for example, A '~AB*Bo is a reflection of AoA * BB'~ and A*A *B'B* is a reflection of AoABBo. As far as the link lengths are concerned, a reflected linkage possesses the same link lengths and a symmetrical configuration as those of the linkage before reflection. Thus, only 8 linkages of different link lengths shall be considered [3, 10]. These linkages, or, more precisely, 8 four-bar loops are tabulated in Table 1. Here the symbols 1-4 are used to designate the four relatively moving bodies, as labeled in Fig. 2. From this table it can be seen that adjacent links remain adjacent, and opposite links remain opposite. It is also evident that any four-bar loop can be transformed into another four-bar loop by replacing two or four of its link lengths by their respective supplementary arc lengths. Therefore we may conclude that: (1) If a given spherical four-bar linkage has an even number of link lengths which are >90 °, it is always possible to reduce it to a four-bar loop in which all link lengths are < 90 °. (2) I f a given spherical four-bar linkage has an odd number of link lengths which are > 90 °, it can always be reduced to a four-bar loop in which three link lengths are < 90 °, and one link length which is > 90 °. Therefore as long as there exists a single link length which is = 90 °, it is always possible to reduce all link lengths to within ~<90 °. Consequently we can draw the following important conclusion: Any given spherical four-bar linkage can always be reduced to a four-bar loop in which the sum of any two link lengths is not greater than 180°. 3. REVIEW OF G R A S H O F ' S RULE

In a general spherical four-bar linkage, let the lengths of the two rotating links, the coupler and the fixed link be denoted respectively by a, b, c, a n d f a s shown in Fig. 1. Suppose a is the crank. We set forth now a prerequisite that the sum of any two link lengths is not greater than 180°, which is always achievable according to the last conclusion stated in the preceding Section. Let us review Grashof's rule for the rotatability of a plane four-bar linkage, or more precisely, of a plane crank-rocker. As will be shown later, with the above mentioned prerequisite for a spherical four-bar linkage, there is no difference whether the linkage in question is planar or spherical,

Table 1. Supplementary spherical four-bar linkages ]~u...]~z. ;,oop Link 1

moAo 1s o ° - . ~

AB 180°- A 1~

leo°-

*o* ]B ~ ( ~ ) .C~"~' ~(.~) ~'~(~) ]~ ~ ( ~ )

180°-lmo~ 180°..~,o leO°-]~o

]~ Bo

AB 1eO°-AoA

~ * m ~o(Z,)

Link 4

Link 2

.%A

180° -

leo°..Aoa 180°..ao.a

A

A~ A ]B

180e - A ]

A]B

~e0°-~ ~, 160%3 ~o B~o

On the classification of spherical four-bar linkages in the derivation of the conditions to be fulfilled by a crank-rocker. There are four critical positions through which the crank a must pass in order to ensure its rotatability. These positions are shown in Fig. 3(a)-(d), in which the crank a is in "line" with its adjacent links c and f respectively. The configuration Fig. 3(a) demands that

a+c<~b+f

(1)

and Fig. 3(b) demands that

a+f<~b+c.

(2)

Fig. 3(c) demands either

c-a>~f-b,

i f f > b,

(3)

c-a>~b-f,

ifb>f

(4)

f-a>~c-b,

ifc >b,

(5)

f-a>~b-c,

if b > c .

(6)

or

Fig. 3(d) demands either

or

The inequality (3) leads again to (2), and the inequality (5) leads again to (1). The inequalities (4) and (6) are identical, leading to a + b ~
(7)

Altogether we have, for the rotatability of the crank a, only three independent conditions (1), (2) and (7). Setting them together gives:

a+b<~c+f] a+c<~b+f I. a +f<.b + c J

Co)

(8)

285

Let us examine the extent of validity of these coCiditions for spherical four-bar linkage. With b <~ 180° a n d f ~< 180° in Fig. 3(a), condition (1) is valid as long as a + c ~<180°. Similarly with c ~<180° and b ~< 180° in Fig. 3(b), condition (2) is valid as long as a +f~< 180° . Furthermore, with a~<180 °, c ~< 180° and f ~<180° in Figs. 3(c) and 3(d), conditions (4) and (6) are valid as long as b ~< 180°. It is therefore evident that the prerequisite set forth before for a spherical four-bar loop is quite sufficient to ensure the validity of these conditions. On the whole, the conditions (8) are sufficient, for any linkage which fulfills them ensures the existence of the four triangles shown in Fig. 3, hence the rotatability of the crank a. These conditions are also necessary, for if any one of the three inequalities is violated, the linkage becomes a double-rocker. The " = " signs in (8) indicate the limiting cases in which the triangles in Fig. 3 degenerate into a single great circle. The conditions (8) claim that, the sum of the length a of the crank and that of any other link should be less than the sum of the lengths of the two remaining links. Consequently a is the shortest link length. In fact, Grashof's rule claiming that "the sum of the lengths of the shortest and longest links should be less than the sum of the lengths of the other two links" is quite a clever statement, for, no matter which one of b, c and f is the longest link, once one of the three inequalities is satisfied for the longest link, it automatically includes the other two inequalities. 4. CRITERIA OF CLASSIFICATION

According to the foregoing there are therefore two classes of spherical four-bar linkages, and the following criteria may be established: (I) Class 1 linkages are those which satisfy Grashof's rule. They may also be termed as Grashof linkages. For this class of linkages, the shortest link is capable of making complete revolutions relative to the other three links, and (i) a crank-rocker exists if the link adjacent to the shortest link is fixed, (ii) a drag-link exists if the shortest link is fixed, and (iii) a double-rocker exists if the link opposite to the shortest link is fixed. (2) Class 2 linkages are those which do not satisfy Grashof's rule. They may also be termed as nonGrashof linkages. For this class of linkages, none of the links is capable of making complete revolutions relative to the other links, and a double-rocker exists whatever a link is chosen as the frame. The linkage exhibits (i) internal rocking angles if the fixed hnk is the longest link (Fig. 4), (ii) external rocking angles

(e)

Fig. 3. Fdur critical positions of a crank-rocker.

Fig. 4. Non-Grashof double-rocker with internal rocking ang]~.

C. H. CHIANG

286 e

Fig. 5. Non-Grashof double-rocker with external rocking angles. B

•

one having a minimum sum of all four link lengths. The procedure is quite straightforward, but will best be explained by the following illustrative examples. Examples. 10 miscellaneous examples are listed here in a table form. The given link lengths are reduced or not reduced according to the principle mentioned before. The results found by applying the criteria are listed in the righthand column. Note that in the last two examples, the results could also be obtained without reducing the sum of link lengths to a minimum, although the reduction process is always recommended, in order to avoid any possible mistakes. Thus in the seventh example, a non-Grashof double-rocker would be mistaken for the drag-link without reduction. 5. COMPARISON OF THE PRESENT CRITERIA WITH ANOTHER SET OF CRITERIA

Fig. 6. Non-Grashof double-rocker with overlapping rocking angles. if the coupler is the longest link (Fig. 5), and (iii) overlapping rocking angles if one of the two links adjacent to the fixed link is the longest link (Fig. 6). It should be noted that, before applying the above criteria, the four-bar loop should always be reduced to the one which satisfies the above mentioned prerequisite, namely, the sum of any two link lengths is not greater than 180° . The reduction is carried out, according to Table 1, by replacing each time the two longest link lengths by their respective supplementary arc lengths until none of the sums of two link lengths is greater than 180° . It is easy to conceive that the final four-bar loop is unique, and that it is also the

Another set of criteria of classification of spherical four-bar linkages as given in[4], is fundamentally the same as the present criteria, but are in a different representation. These are: (I) A given spherical four-bar linkage with an even number (0, 2 or 4) of obtuse link lengths is rotatable, if, after replacing all obtuse link lengths by their respective supplementary arc lengths, the sum of the shortest and longest link lengths is smaller than the sum of the other two'link lengths, or smaller than the half of the sum of all link lengths. (II) A given spherical four-bar linkage with an odd number (I or 3) of obtuse link lengths, is rotatable, if, after replacing all obtuse link lengths by their respective supplementary arc lengths, the sum of the shortest link length and 90 ° is smaller than the half of the sum of all link lengths.

Table 2 f£xod

lJ~k

~otaUnK

link

Result

coupler r o t a t ~

link

|

l

•

b

Given: ;lvenx

800 ~0°

200 60°

600 6O°

7~ 0

~.lvenz

SO°

75°

25°

70 °

P-iven:

8~ °

75 °

65 °

70o

GJ.ven.

100 °

160 °

120 °

105 °

~0°

260° 2.~°

/5 ° 100°

Crank-Hocker

120°

~(~0° 110°

C:snk-Rocke:

lehoed to" ;lvonl ,.du,.od to, ;1yen: ;iven:

Y50

I(~°

J5 °

~o

1]0~o

1 ~5°

60°

70 °

80°

;oo

,'.o

1 ~5°

.50°

65°

e O°

"°

~.du©.d to,

285o

~o

J,o

1100o

;lven:

60 °

80 °

25°

110°

100°

40 °

900

,.+++.,o, ;1yen:

/oo

Crank-Rocker nmc-Link Cr&shor Double-Rocker NmI-G~ud~oF Double-Rocker v£th internal rockAn~ amlrloa

~n~-LJJ~ ]io~-G~Jhof Double-noeke~" ~i~ m ~ l a p p ~ roek£ng ~ l e ~ Cralhof Double-Ro~er

600 Czsmk-Rocker

On the classification of spherical four-bar linkages These two criteria may be easily verified on the basis of the present criteria. It is evident that the former part of criterion (I) is identical with our criterion (1). Let the shortest and longest link lengths be represented respectively by l~. and 1~,~, and the lengths of the other two links be lm and 1.. Grashof's rule states that

287

21,.,. + 180 ° ~< Ii + lx + 13 +/4, or

/m,. + 90° ~

11+1x+13+14 2 '

(12)

which is a statement of criterion (II). 6. C O N C L U S I O N

/rain "1- lmax ~ lm "k-/n, i.e.

2(&,. + &.O <. l.~. + t.= + lm + l., or

l.,i. + l.~x<~ l*.io + l . - + l,. + l . (9)

2

The inequality (9) represents the latter part of criterion (1). To show the validity of criterion (II), let us start from our criterion (2). Suppose the four-bar lbop has been reduced to its minimum sum of four link lengths: !1, 12, 1~, and 14. Without loss of generality we may assume that lz be the shortest one of all four link lengths, and that/2 be the only one which is obtuse. Grashof's rule demands that (10)

Ii + 12 ~< 13 + 14.

Now according to criterion (II), 12 should also be reduced to an acute link length. Let 12 = 180 ° - !,. Then (10) becomes (11)

180 ° + (ll - 1,) ~
There are two possibilities, namely, either I~ < 1~ or Ix /> iv If Ix < 11, then the inequality (1 l) implies that/3 +14 > 180 °. But it has been assumed that none of l3 and/4 is obtuse, therefore such an Ix does not exist. The only possibility is that Ix t> l~, hence 1~ = Imi,, and the inequality (11) can be written as

F r o m the foregoing it is believed that the derivation of the present criteria is more straightforward than the other known methods. No reference has been made to the law of cosines in spherical trigonometry or to any complicated spherical geometry concepts, nor have we used the technique of vector algebra. The application of the present criteria is also quite simple, as has been shown by the examples. It is hoped that these criteria could be extended to spacial four-link mechanisms. REFERENCES

1. N. G. Bruewitsch, Kinematics of spherical four-link mechanisms. Vestnik inzenerov i technikov No. 8, 465-468 (1937). 2. V. V. Dobrovolskii, Theory of Spherical Mechanisms. Moscow (esp. pp. 154-155) (1947). 3. F. Freudenstein, On the determination of the type of spherical four-link mechanisms. Contemporary Problems in the Theory of Machines and Mechanisms, Academy of Science of the USSR, 193-196 (1965). 4. FI. Duditza and G. Dittrich, Die Bedingungen ffir die Umlauff'fihigkeit sph~irischer viergliedriger Kurbelgetriebe. Industrie-Anzeiger 91, 1687-1690 (1969). 5. M. Savage and A. S. Hall, Jr., Unique descriptions of all spherical four-bar linkages. Trans. ASME, J. Engng Ind. 92B, 559-563 (1970). 6. A. H. Soni, Discussion on [5], Trans. ASME, J. Engng Ind. 92B, 563-566 (1970). 7. M. J. Gilmartin and .I. Duffy, Type and mobility analysis of the spherical four-link mechanism. Conf. on Mechanisms 1972, l.Mech.E., 90-97 (1972). 8. A. H. Soni and L. Harrisberger, Roberts' cognates of space four-bar machanisms with two general constraints. Trans. ASME, J. Engng Ind. 91B, 123--128 (1969). 9. A. H. Soni and PI R. Pamidi, Roberts' cognate of space five-link RHHHH and HHRHH mechanisms. Trans. ASME, J. Engng Ind. 93B, 227-230 (1971). 10. A. H. Soni and L. Harrisberger, The design of the spherical drag-link mechanism. Trans. ASME, J. Engng Ind. 89B, 177-181 (1967).

U B ~ DIS KLASSIFZZIERUNG YON S P H ~ R I S C ~ VIERG~L~KGETRIEB~ C. H. C h i a n g Kurzfassun~ - 3phArische Viergelenkgetriebe Uml&uffHhigkeit versteht

kSnnen nach d e r U m l a u f f L h i g k e i t k l a s s i f i z i e r t

man d i e F g ~ i g k e i t e i n e s G l i e d e s ,

zu kUnnen. FUr e i n s p h ~ r i s c h e s Y i e r g e l e n k g e t r i e b e

werden. U n t e r

g e g e n U b e r den d r e i a n d e r e n G l i e d e m u m l a u f e n

k~nnen a u f e4nmnl Zwei G l i e d e r d u t c h i h r e J e w e i l i g e n

g r g ~ n z u n g s b ~ g e n e r s e ~ z t w e r d e n . D i e s e s R e d u z i e x , m g - v e r f a h r e n w i r d so a u s g e f U h r t , b i s d i e G e s a ~ t l ~ n g e a A l e r T i e r G l i e d e r e i n Minimum i s t . keit

Dann kann d e r Sa~z yon G r a s h o f angewandt w e r d e n , um d i e Uml&uffAhig-

des G e t r i e b e s zu p r U f e n . Das k l e i n s t e

Glied eines Grashofachen Getriebes ist

umlauff~Aig. Bin G r u °

h o f n e h e s G e t r i e b e kann e i n e K u r b e l s c h w i n g e , e i n e D o p p e l k u r b e l o d e r e i n e D o p p e l s c h w i n g e s e i n . Gr~hofsehes

Getriebe ist

immer e i n e D o p p e l s c h w i n g e . g i n i g e B e i s p i e l e

werden b e h a n d e l t .

Ein Nicht-