comments on the paper by h.p. lee: design of a geneva mechanism with curved slots using parametric...

Letter to the Editor

Comments on the paper by H.P. Lee: Design of a Geneva mechanism with curved slots usingparametric polynomials, published in Mechanism and Machine Theory, 33(3), 321±329, 1998

The paper by Lee [1] recently published in Mechanism and Machine Theory is the last of aseries of contributions [2,3] aimed at improving the classical Geneva mechanism by replacingthe straight slots on the wheel with curved slots. It seems to me that the disadvantages of suchan ``improvement'' far outweigh the envisioned advantages, as I will try to explain hereafter.

1. A special con®guration

The key problem with the curved-slot Geneva mechanism is that Ð unlike the classicalGeneva mechanism Ð it has a change-point con®guration. At a change-point con®guration Ðalso termed uncertainty-con®guration, see [4] and [5] Ð a mechanism acquires a transitoryextra mobility. A notable example of a mechanism with a change-point con®guration is theparallelogram linkage. Normally it has one degree of freedom, but when it goes ¯at it acquiresone additional degree of freedom. In real applications, the transitory extra mobility of theparallelogram linkage is always attended to and carefully neutralized. In old-time locomotives,for instance, every pair of driving axles was coupled by two parallelogram linkages with cranksat right angles, so that no more than one parallelogram went ¯at at the same time.The curved-slot Geneva mechanism reaches a change-point con®guration when the roller

center becomes aligned with the axes of crank and wheel. In this con®guration, represented inFig. 1b, the curved-slot Geneva mechanism has two degrees of freedom, as can be easilyrecognized by considering that the relative instantaneous center of crank and wheel is notdetermined: it can be any point of the line through the axes of crank and wheel. Accordingly,the speed ratio of crank and wheel at the change-point con®guration can take any value.To grasp the implication of this occurrence, let us consider a curved-slot Geneva mechanism

at rest at the change-point con®guration (Fig. 1b). If the crank is turned in a given direction(say counterclockwise) by a given angle, the corresponding rotation of the wheel can takeeither of two values (see Fig. 1c and 1c ') depending on the direction of the external torqueapplied on the wheel when it was at the change-point con®guration.Proponents of curved-slot Geneva mechanisms keep overlooking this drawback. Confusing

the concept of change-point con®guration with the concept of dead-point con®guration, theycompare their invention to a slider±crank mechanism that is at rest with the slider at a dead

Mechanism and Machine Theory 35 (2000) 887±893

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point [6]. First they observe that no force applied to the slider can in¯uence the direction ofany subsequent rotation of the crank. Then they manage to see an analogy with the fact thatÐ for a curved-slot Geneva mechanism at rest at a change-point con®guration Ð the rotationof the crank cannot decide the evolution of the wheel rotation. Based on this alleged analogy,and on the honorable history of the slider±crank mechanism, they wrongly conclude that thecurved-slot Geneva mechanism cannot help being trouble-free too [6].Nevertheless, this comparison is deceptive and not pertinent. To begin with, a force is

applied to the slider of the slider±crank mechanism, whereas a rotation is given to the crank ofthe curved-slot Geneva mechanism. Secondly, only the direction of the possible motion is atstake for the slider±crank mechanism: regardless of this direction, the path followed by themechanism during its possible motion is known a priori. For the curved-slot Genevamechanism, instead, the direction of the crank rotation is given, the mechanism motion iscertain, and the indeterminacy is on whether the mechanism con®guration will evolve along thedesired path or, rather, along a completely di�erent path. Once the mechanism takes eitherpath, the direction of motion unambiguously stems from the given direction of the crankrotation.It is worth stressing this point even further. Let us suppose that the crank of a curved-slot

Geneva mechanism, following a counterclockwise rotation, makes the roller engage the slot onthe wheel (Fig. 1a) and ®nally stops at the change-point con®guration (Fig. 1b). If the cranksubsequently resumes its counterclockwise rotation, the roller can disengage the slot in either oftwo ways: by traveling along the slot branch it has already spanned during engagement(Fig. 1c ') or by escaping through the other branch (Fig. 1c). It should be noted that the formerway of disengagement is undesired because it involves a set of mechanism con®gurations thatare extraneous to the expected operation of the mechanism (represented, in Fig. 1, by theframed sequence of con®gurations). Once more, it is not a matter of direction of motion alonga given path (as for the slider±crank mechanism), but of path selection.

Fig. 1. Di�erent con®gurations of a curved-slot Geneva mechanism.

Letter to the Editor /Mechanism and Machine Theory 35 (2000) 887±893888

Due to the clearance between roller and slot, the undesired way of disengagement can betriggered even if the curved-slot Geneva mechanism is not stopped at the change-pointcon®guration, provided that the crank rotates at su�ciently low speed while a torque is appliedto the wheel.It can be easily recognized that, at a change-point con®guration, the roles of crank and

wheel are interchangeable: the path is still uncertain even if the wheel Ð instead of the crankÐ is the driving link (both con®gurations represented in Fig. 1a and 1a ', characterized by thesame orientation of the wheel, can be reached from the change-point con®guration of Fig. 1b).

2. Number of starters

Another disadvantage of the curved-slot Geneva mechanism with respect to the classicalGeneva mechanism pertains to the number of starters.Like the slider±crank mechanism, the classical Geneva mechanism has one degree of freedom

at any con®guration (the relative instantaneous center for any pair of links can always bedetermined). Moreover, the revolute pair between crank and frame is free from dead points.Therefore it can be actuated to start up the mechanism.For a curved-slot Geneva mechanism, instead, provision has to be made for the possible

need to start the mechanism at the change-point con®guration, where two degrees of freedomare present. Consequently, two starters are needed to apply external torques on both the crankand the wheel, lest the mechanism should take the wrong path and badly jam soon afterward(the locking device that keeps the wheel from rotating when the roller is disengaged from thewheel cannot accommodate the undesired mechanism con®gurations).The second starter could be spared if two rollers simultaneously engaged with the wheel in

such a way that their centers do not cross the line through the axes of crank and wheel at thesame time. This modi®cation would be the correspondent of the second parallelogram linkagefor locomotives (see above). Nevertheless the resulting mechanism could not be termed aGeneva mechanism any longer. Rather, it would be an in embryo version of one of the severalexisting types of parallel-shaft indexing drives.

3. Overconstraining

Another disadvantage of the curved-slot Geneva mechanism with respect to the classicalGeneva mechanism, is the stricter manufacturing and assembly accuracy demanded by theformer.At the change-point con®guration, the mutual constraints between any pair of links of a

curved-slot Geneva mechanism are ill-arranged: the mechanism both acquires one additionaldegree of freedom and becomes overconstrained. With reference to Fig. 2, the roller of acurved-slot Geneva mechanism can travel both branches of a slot only if the followingcondition is satis®ed

a� q � c �1�

Letter to the Editor /Mechanism and Machine Theory 35 (2000) 887±893 889

where a=AP, q=BQ, c=AB, A is the crank axis, B is the wheel axis, P is the roller center,and Q is the point of the slot centerline that is superimposed on P when P is the closest to B.Conversely, the slot radial extension is not a critical dimension for a classical Geneva

mechanism.

4. Shock loads

The problems with the curved-slot Geneva mechanism do not disappear as the mechanismspeeds up and inertia torques avert the undesired way of disengagement. To analyze this point,let us brie¯y consider the insurgence of shock loads in Geneva mechanisms.Contrary to what is stated in [1], shock loads are not caused by acceleration discontinuity.

As a rule, shock loads are caused by velocity discontinuity. In Geneva mechanisms, velocitydiscontinuity is triggered by the changing of side of the roller-slot clearance.In high-speed, pre-loaded, backlash-free mechanisms, elimination of acceleration

discontinuity contributes to further smoothing of the mechanism running. However, if backlashis present, shock loads mask any bene®cial e�ects stemming from the intention of eliminatingacceleration discontinuity.Unfortunately, Geneva mechanisms need clearance between the roller and the slot, otherwise

the roller would rub the slot sides and even seize up. Since both the classical and curved-slotGeneva mechanism cannot be pre-loaded, they are doomed to shock loads due to backlash.Let us suppose that the static torque on the wheel of a Geneva mechanism is negligible

compared to the inertia torque. Accordingly, the change of side of the roller-slot clearancestarts as soon as the inertia torque on the wheel reverts, i.e., when the roller center crosses theline joining the axes of crank and wheel. In this con®guration, the free travel of the rollerrelative to the wheel is parallel or orthogonal to the slot centerline depending on whether theGeneva mechanism has curved or radial slots respectively. For a given amount of sideclearance, the free travel of the roller relative to the wheel is therefore longer for a curved-slotGeneva mechanism than for a classical Geneva mechanism.Consequently, shock loads are comparatively higher in a curved-slot Geneva mechanism.

Fig. 2. Kinematically-relevant parameters of a Geneva mechanism.


5. Pressure angle

All of the problems pointed out so far for the curved-slot Geneva mechanism could havebeen anticipated by noting that the pressure angle reaches the limit value of 908 when the rollercenter is aligned with the axes of crank and wheel [7]. This is true regardless of whether thecrank or the wheel is considered as the driving link.At the change-point con®guration, the curved-slot Geneva mechanism behaves as a pair of

parallel-shaft spur gears with a pressure angle of 908. Needless to say, these extreme workingconditions are extraneous to any sound mechanism (for example, multi-roller parallel-shaftindexing drives have cams with portions of their pro®les relieved where the pressure anglewould reach abnormally-high values).

6. Another proof

Proponents of the curved-slot Geneva mechanism ®nd the shape of the slot centerline as thetrajectory Ð relative to the wheel Ð of the roller center when both crank and wheel are madeto rotate with prescribed laws of motion. They do not seem to care whether a curved slot, onceperfectly machined on the actual wheel of a Geneva mechanism, is really able to reproduce thesame relative motion between crank and wheel that was invoked for slot generation. A possibleanswer to this reasonable question is reported hereafter.By referring again to Fig. 2, let angles a and b parametrize the orientations of crank and

wheel relative to the frame. In particular, b is the angle formed by segment BQ (®xed to thewheel) with segment BA (®xed to the frame). As mentioned earlier, Q is the point of the slotcenterline that is superimposed on the roller center, P, when P falls on segment AB.The shape of the slot centerline is considered as known with respect to a reference frame

®xed to the wheel. It can be expressed by polar coordinates (b, y ) as a function of a curvilinearcoordinate s with origin at Q

b � b�s� �2�

y � y�s� �3�In Fig. 2, angles a, b, and y are all supposed as positive.The closure equations of the mechanism can be easily written by considering triangle ABP

2a c cos a� b2 ÿ a2 ÿ c2 � 0 �4�

a sin aÿ b sin�b� y� � 0 �5�By di�erentiating Eqs. (4) and (5), the ensuing relations can be obtained

ÿa c sin a da� bdb

dsds � 0 �6�


a cos a daÿ sin�b� y�dbds

dsÿ b cos�b� y��

db� dyds

ds

�� 0 �7�

Since s is a curvilinear coordinate, the following condition is satis®ed�db

ds

�2

��b

dyds

�2

� 1 �8�

Moreover, the assumption b>0 is now introduced.The transmission of motion between crank and wheel is possible if a relationship between

in®nitesimal rotations da and db can be found that does not explicitly involve ds. If db/ds$0,Eq. (6) provides the value for ds that can be inserted into Eq. (7) in order to obtain thesought-for relationship.For the classical Geneva mechanism, the parametric equations of the radial slot are

b, (s )=q+s and y(s )=0. Accordingly, db/ds is always di�erent from zero and the transmissionof motion between crank and wheel is always e�ective.The case of the curved-slot Geneva mechanism is now analyzed. Since the roller has to travel

both branches of the slot and point P gets its closest to point B when it crosses segment AB, Qmust be the point on the slot centerline that is nearest to B. Therefore, when P issuperimposed on Q (a=0) and angles b and y vanish, the derivative db/ds also vanishes (breaches its minimum, i.e., q ). Thus Eq. (6) becomes a useless identity and Eq. (8) provides

dyds� 1

q�9�

(in writing Eq. (9) Ð and with no loss of generality Ð inequality dy/ds>0 has been supposedas holding).Finally, Eq. (7) provides

a daÿ q dbÿ ds � 0 �10�Eq. (10) clearly shows that Ð for a=0 Ð the curved-slot Geneva mechanism has two

degrees of freedom: the in®nitesimal rotations da and db of crank and wheel are mutuallyindependent, since Eq. (10) can always be satis®ed by a suitable ds.Concluding, curved slots do not seem able to play any role in the evolution of the classical

Geneva mechanism. Such an evolution, started decades ago along a di�erent route, has alreadyresulted in o�-the-shelf or custom-made reliable devices, i.e., pre-loaded, backlash-free indexingdrives for parallel shafts.

References

[1] H.P. Lee, Mechanism and Machine Theory 33 (3) (1998) 321.[2] R.G. Fenton, Y. Zhang, J. Xu, ASME Journal of Mechanical Design 113 (1) (1991) 40.

[3] Y. Zhang, R.G. Fenton, J. Xu, ASME Journal of Mechanical Design 116 (2) (1994) 647.[4] K.H. Hunt, Kinematic Geometry of Mechanisms, Clarendon Press, Oxford, 1978.[5] B. Paul, Kinematics and Dynamics of Planar Machinery, Prentice-Hall, Englewood Cli�s, NJ, 1979.


[6] R.G. Fenton, ASME Journal of Mechanical Design 117 (4) (1995) 663.[7] C. Innocenti, ASME Journal of Mechanical Design 117 (4) (1995) 662.

Carlo Innocenti*Department of Mechanical Engineering, University of Bologna, Viale Risorgimento, 2-40136,

Bologna, ItalyE-mail address: [email protected]

6 May 1998

* Tel.: +39-51-2093450; fax: +39-51-2093446.


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