A note on symmetrical self-oscilating coupler curves

Mechanism and Machine Theory Vol. 17, No. 3, pp. 229-232, 1982 Printed in Great Britain.

0094-114X182/030229-04503.0010 © 1982 Pergamon Press Ltd.

A NOTE ON SYMMETRICAL SELF-OSCULATING COUPLER CURVES K. H. HUNTt Department of Mechanical Engineering, Monash University, Clayton, Victoria 3168, Australia and J. E. KIMBRELL~: PWR Systems Division, Westinghouse Electric Corporation, P.O. Box 355, Pittsburgh, PA 15230, U.S.A. (Received 16 April 1981) Abstract--Some special symmetrical coupler curves, from hinged planar 4-bar linkages, are presented which not only touch themselves but which osculate with themselves where three double points coalesce. Possible applications in point-guidance are touched on, 1. INTRODUCTION

For more than a century the properties of the path traced by a coupler-point of a hinged planar 4-bar linkage have been well known, and many coupler curves have been studied. Burmester[l] and Miiller (see a series of papers translated in[2]) were amongst the earlier investigators. Subsequently planar curvature theory was applied to coupler curves with a view to their use in linkagemechanisms (to give dwell-motion, for example). A broad view of such work can be gained from [3] and the references listed therein, Hain[4] (Section 12.03) describes how to devise a double-dwell linkage using two discrete regions of identical curvature on a coupler curve. In the present paper we concern ourselves with two superimposed regions of identical curvature in a symmetrical coupler curve; accordingly a common osculating circle exists. Our searches have not revealed any earlier mention of this form of coupler curve.

2. COALESCENCE O F BOUBLE POINTS

It is well known (see [3], Section 7.4, for instance) that a real double point of a non-degenerate planar 4-bar coupler curve can occur only on the circle of singular foci, and moreover that a coupler curve cannot cross the circle of singular foci except at a double point. Also the maximum number of real double points is three (unless the coupler curve degenerates by virtue of the 4-bar linkage becoming flattenable). A non-degenerate coupler curve, whether it has one or two closed circuits, could not return and again pass through a regular double point (so creating a triple point-- or three coalesced double points), because the as yet unjoined-up branches of the coupler curve would then be on opposite sides of the circle of singular foci; the coupler curve is prevented from closing its circuit(s) because it cannot cross this circle again, the full quota of three real double points having been exhausted. However, MOiler[2] considers the possibility of the

coupler curve touching the circle of singular loci. The point where the curve touches it (namely intersects it at two infinitesimally separated points) can be none other than two consecutive double points to which the coupler curve must return once more (at some other linkageconfiguration) and touch both itself and the circle of singular foci at a tacnode. A simple example of a tacnode is shown in Fig. 7.6 of[3]. Miiller[2] shows that a tenthorder transition curve on the coupler lamina contains all coupler-points whose paths pass through tacnodes. A coupler curve with a tacnode can have a third real double point elsewhere on the circle of singular foci, and linkages to achieve this combination may be synthesized fairly readily. In this paper we concern ourselves with the third double point actually at the tacnode, i.e. three coalesced double points corresponding with no more than two discrete configurations of the linkage. At such a point of coalescence the coupler curve has three consecutive points in common with the circle of singular foci and then returns at another configuration to these same three consecutive points. Accordingly the coupler curve osculates not only with the circle of singular loci but also with itself. Figure I shows a coupler

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*Professor of Mechanism. *Senior Engineer.

Fig. I, A near-self-osculating coupler curve from a non-Grashof 4-hinge linkage. The circle of singular foci, f, is shown. 229

K H. Ht;Nr and J. E. KIMBREI[

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curve in which the three double points are distinct but not far removed from coalescence. Clearly the condition for self-osculation of a coupler curve is that the centre of curvature of the path of a coupler-point C coincides with the centre of the circle of singular foci f at the instant when C lies on [. This is a highle restrictive condition, We confine attention in this Note to symmetrical self-osculating coupler curves, suspecting, but not exhaustively proving, that no unsymmetrical self-osculating coupler curves exist. Pursuit of this further possibility we prefer to leave to others. For meaningful self-osculation the radius of the circle of singular foci must be finite, self-osculation along a straight line not being attainable. Also a 4-hinge linkage tracing a symmetrical coupler curve must have the coupler-triangle ABC isosceles.

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3. FORM OF THE LINKAGE

Let the fixed pivots OA and Oa lie on the circle of singular foci f at equal distances q on either side of a diameter PC (Fig. 2). Join OaP and OBP and if necessary produce these lines, setting off (in opposite senses) any convenient distance r along both OaP and OBP to find the coupler-hinges A and B. The coupler-triangle ABC is isosceles and similar to triangle OAOBC. At this configuration P is the coupler's instantaneous centre, and C's path (shown in Fig. 2) osculates with itself and with circle f at a point on the fixed frame conciding with C. It is not difficult to follow accepted means of applying the Euler-Savary equation or performing a construction such as Aronhold's ([3], Section 5.5.1) to confirm that the radius of curvature of C's path is equal to the radius of circle f. If P were taken at some point not on the circle f, and/or if Oa and O~ were not symmetrically disposed about a diameter through P, then, as far as we have been able to ascertain, the Euler-Savary equation cannot be reconciled with the condition for self-osculation except for cognates of linkages like those of Figs. 2--6. Figure 7 gives two examples of cognates. 4. FORMS OF SELF-OSCULATING COUPLER CURVES

Let the coupler angle ~ ACB = 3'. The coupler of Fig. 2 has 3' < 90°; for Fig. 3 y > 90°. Both of these linkages are non-Grashof double-rockers ([3], Section 3.10), and

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Fig. 2. A unicursal symmetrical self-osculating coupler curve with r = 1.8q and <[ACB] = 3' = 60°.

Fig. 3. A unicursal symmetrical self-osculating coupler curve with r = 1.8q and '( ACB = 3' = 60°.

here the longest link-length is AB, which is easily seen to be 2(r 2 sin2 ½3'+ q2),t2, the shortest link-length being r. The sum of the shortest and longest,

r + 2(r 2 sin2 ½3'+ q2)m is then greater than the sum of the other two, r + 2q. However, for r > 2q, OAOR becomes the shortest link, and the Grashof condition, shortest plus longest less than the sum of the other two lengths, is satisfied when

AB + OaO. < 2r, which reduces to r cos 2 213,> 2q. 21

At the transition, r cos ~y=2q, the quadrilateral OaABOB can flatten and, as expected (see[3], Section 7.2 for instance), another double point materializes. Figure 4 (r = 0.8q) shows a non-Grashof double-rocker with 3, = 90°. Figure 5 (r = 4q) presents a transitionlinkage also with y = 90°. Figure 6 (r = 5q), also with 3' = 90°, shows a Grashof double-crank. The linkages of Fig. 7(a, b) are cognates of Figs. 4 and 6, respectively. Using as continuously-rotating input the turning pair Oa in the cognate double-crank of Fig. 7(b), the transmission characteristics of the linkage appear to be satisfactory. With the origin of coordinates at the self-osculating point (see Figs. 6 and 7, for instance) the equation of the coupler curve for 3' = 900 can be derived following the long-recognized approach described in[3] Section 7.3.

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A note on symmetrical self-osculatingcoupler curves

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Fig. 4. A unicursal symmetrical self-osculating coupler curve with r = 0.8q and 47A C B = 3' = 90 °.

The resulting equation {(x - y - 2q)(x 2 + y2) _ 4qZy}2 + {(x + y + 2q)(x: + y2) + 4q2y}Z = 4(r 2 + 2q2){(x2 + y2) + 2qy}2

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conforms to the usual pattern for equations of tricircular sextic coupler curves. Comparable equations, though somewhat more complicated, can be written for 3,~ 90°, but in the interests of brevity are not given here. There is a remarkable superficial likeness between the symmetrical self-osculating coupler curve, eqn (1) for instance, and the self-osculating "p-curve" (see for example[5] Section 45, two forms of which are there illustrated). The p-curve relates to curvature theory and gives those points on moving lamina whose paths have identical radius of curvature, p, at a given instant. Then, with the inflexion-circle diameter equal to rs, the p-curve can be expressed as (x ~ + y~)~ = p'{r~y - (x ~ + yS} ~.

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Fig. 6. A bicursal symmetrical self-osculatingcoupler curve with r = 5q and 47A C B =3' = 90°. The linkage is a Grashof doublecrank. Curves for various values of p : r s ratios in eqn (2) are not reconcilable with those of eqn (1) or the more general equation for the self-osculating coupler curve when 7 # 90°. Yet the p-curve is a tricircular sextic and can be regarded as the infinitesimal limit of its finitely-separated equivalent (described in[3] Section 6.2 and[5] Section 12 for instance) which is a coupler curve. However the above comparison between self-osculating coupler cur-

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Fig. 5. A symmetrical self-osculating coupler curve with r = 4q and 47A C B = 3, = 90° showing the additional double point arising from the linkage being at a transition between non-Grashof and Grashof.

(b) Fig. 7. Cognate linkages, (a) of Fig. 4, (b) of Fig. 6.

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K.H. Hu•l and J. E. KIMBREU.

yes and the p-curve confirms the statement in[5] that the p-curve is not a coupler curve. Moreover the self-osculating symmetrical coupler curves constitute a twoparameter family (disregarding scale), since r, q, and y are at our disposal: there is only a one-parameter family of p-curves. 5. POSSIBLEAPPLICATIONSANDCONCLUSIONS If it should be required to guide a point C so that it approximates the same circular arc twice in succession, approaching it first from inside the circle and later from outside, then linkages like those of Fig. 7 could be used. Whichever may be chosen the input actuator can be the turning pair at Oa, the linkage of Fig. 7(a) requiring input-reversal. The sheer bulk of the double-crank linkage of Fig. 7(b) is likely to be a disadvantage to be offset against the potential of having a continuously rotating input. Unfortunately there is no reasonable prospect of synthesizing a mid-travel double-dwell linkage with two further links added to either of these arrangements. The grounds for this pessimism relate directly to the arguments presented in[3] Sections 8.3 and 8.4. In short the driving point C sweeps over too large an area; whatever arrangement is chosen the linkage requires dismantling and re-assembly in order to achieve the desired input-output requirement. Moreover, as explained in [3] Section 7.10, special measures have to be adopted to make full-cycle use of a coupler curve from

any double-crank linkage, and this fact vitiates against the arrangement of Fig. 7(b). Whilst these arguments have here been pursued for 3' = 90° similar conclusions are reached for the other coupler-shapes. Notwithstanding the poor prospects of sensible mechanical application we consider this form of coupler curve to be interesting. We forbore the temptation to expand this contribution to more than a "Note", but accept that some reader may feel inclined to enlarge the study of self-osculating coupler curves emphasizing their geometrical properties and perhaps finding further forms of them. Acknowledgements--Our thanks are due to Mr. P. M. Herman (Department of Computer Science, Monash University) for assistance in programmes to plot the coupler curves both by simulating the linkages and directly from the curves' equations, and to Mrs. L. J. Ryan (Department of Mechanical Engineering) for her typing assistance. REFERENCES

I. L. Burmester, Lehrbuch der Kinematik. Felix, Leipzig (1888). 2. R. Miiller, Papers on geometrical theory of motion ... (18891903) (Trans. by D. Tesar from originalGerman). Kansas State University Bulletin Vol. 46, No. 6, Special Report No. 21 (1962). 3. K. H. Hunt, Kinematic Geometry of Mechanisms. Clarendon Press, Oxford (1978). 4. K. Hain, Applied Kinematics (Trans. from original German). McGraw-Hill, New York (1967). 5. R. Beyer, Kinematische Getriebesynthese. Springer, Berlin (1953). Trans. H. Kuenzel, as The Kinematic Synthesis of. Mechanisms. Chapman and Hall, London (1963).

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B ~ E R K U N G E N ~BER SYMMETRISCHE SELBSTOSKULIERENDE KOPPELKURVEN K. H. Hunt und J. E. Kimbrell Ku2z2assun~ - Obwohl die Koppelkurven des ebenen Gelen~vlereckes Gbe~ eln Jahrhundert eingehend studle~t worden sind, scheint bls jetzt die MGgllchkeit der Selbstos~lation Gbersehez wooden zu seln. Es werden die Bedingungen unte~sucht, unter demen bel s~rmmetrischen K o p p e l ~ v e n Selbstkontakte zweite~ O~dnung auft~eten. Wahrscheinllch besitzen unsymmetrlsche Koppelkumven diese Eigensohaft nicht. In den Abbildungen wezden die gefumdenen Kuzvenarten gezelgt. Anwendungen fGr Punktftthrung sind mSgllch, abet der Antr£eb grSBere~ Mechanlsmen dutch einen Antriebspunkt, der eine2 selbstoskulierenden Kurve folg~, erscheint p~aktisoh umbzauchba~. Trotzdem ist die Eigenschaft der Selbstoskulatlon, vom Standpunkt de2 Geometrle aus gesehen, interessant.