Gear-wheel driven Geneva wheels

Citation for published version (APA):Dijksman, E. A. (1976). Gear-wheel driven Geneva wheels. Romanian Journal of Technical Science : AppliedMechanics, 21(4), 581-606.

Document status and date:Published: 01/01/1976

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GEAR-WHEEL DRIVEN GENEVA WHEELS

by E. A. DIJKSMAN"

The subject treated in thJs paper concerns a geared Geneva wheel me;hanlsmthat was initially proposed by Kraus [2] In 1963. The mechanism seems to beImportant enough to give It another approach and to add some graphs that mightprove useful to the designer.Apart from the examples given In the paper, detailed instruCtiollS are presentedto design the mechanism for any given number of stations and any given raUo oftimes moving/cycle that Is typical for intermittent motion mechanisms of the kind.

1. INTRODUCTION

There are many ·ways to drive Geneva wheels. Normally, this isdone by a single crank bearing a pin with a roller that intermittentlyinterlocks with the Geneva wheel, as is shown in figure 1. The center ofthe pin then traces a circle that touches the central-lines of two successiveslots that are engraved in the wheel. However, the mechanism that so

Fig. 1. - External Ge­ne·va wheel mechanismGIrlven by a single

crank.

........­,,I

I

III

\\ ,

.... ....

generates intermittent motion shows a number of disadvantages. Forinstance

~) The in-. and output axes are not co-axial iit) Values of 3600 or even 1800 for the output-angle of the wheel

are not possible for this type of mechanism iiii) The ratio of times 'J, which is the ratio of times between the

motion period of the wheel, and the cyclic period, which is the time-lapse

" Liverpool polytechnic, England.

Rev. Roum. Sci. Techn. - Ml!c. Appl., Tome 21. N0 4, p. 581- 606. Bucarest, 1976

582 Eo A. DIJXSMIAN 2

for motion and dwell together, is dependent on the number of stations n.It is therefore not possible to choose v and n independently from one­another. This limits the mechanisms' usefullness to the designer. For an,external Geneva wheel, for instance, the ratio v equals the value repre­sented by the equation:

1 1'JeT == - --,

2 n

For all 'internal Geneva wheel the ratio is represented by

1 1'1;.=-+-'

2 n

'iv) Caused by the re,~ular circle motion of the pin, an angular jerkappears at the start and at 'the end of the motion of the wheel.

When the driVing pa,rt of the mechanism is replaced by a gear­wheel mechanism, however the in- and output axis may be made coaxial.,The dependency of 'I and n may be relieved and the angular jerk of theGeneva wheel may be reduced to a lasser value.

The curve traced by the driving pin is then a cycloidal curve insteadof a circle.

Here, the pin, that produces the curve, is attached to a gear-wheelthat rolls about another wheel which is fixed. The fixed wheel comprisesa pitch circle whieh is the fixed powde ('It,) of the motion. Similarly,the moving wheel is comprhied by the moving polode ('Itm ).

If the moving polode rolls inside the fixed polode, a hypo-cycloidmotion is at hand. If it rolls over and about the fixed polode an epi­cycloid motion occurs. Finally, if the moving polode embraces the fixedone, a so-called peJ'i-e-ycloid motion lllay be generated. In figure 2 all. '

periC)lCloidmotion

Fill. 2.-

the mentioned possibilities arc shown: by gradually changing the size ofeither the fixed or the moving polode, it is demonstrated in the figurehow the occurring possibilities may blend into each other.

3 GEAR-WHEEL DRIVEN GENEVA WHEELS 583·

Not counting the particular motions in which one of the polodesis a straight-line, only the hypo-cycloid, the epi-cycloid and the peri­cycloid motion remain to be considered.

A (coupler-) point attached to the moving polode in these cases,will trace a hypo-cycloid, an epi-cycloid or a pericycloid respectively.Also, if the center of the pin lies inside the moving polode, the pointtraces a contracted cycloid, otherwise it is a prot"acted cycloid. Only ifthe curves traced by these centers are "suitable", the pin may be usedto drive a Geneva wheel (See for instance the curves shown in the figu­res 3 and 4).

As it, has already been pointed out by G. Bellerman [lJ in 1867, howe­ver there are always two gear-wheel pairs that will produce the samecurve. Therefore, the two gear-wheel mechanisms that are linked inthis way are curve-cognates of one another.. So, it may be proved that protracted kypocycloids and contractedhypocycloids are identical. In the same way one may prove that a pro­tracted epicycloid and a contracted pericycloid' are identical, and reversed.

This fact is of some importance for the designer of these mechanisms:once the curve is chosen, he still has the choice of the mechanism. Naturally,the two curve-cognates do not occupy the same space as can easily beverified. So, if a compact mechanism· is wanted, it is possible to meetsuch a demand.

2. PR.OOF OF THE TWO·FOLD GENERATION (See figures 3 and 4)

Suppose the source mechanism produces a protracted hypocycloid.(See figure 3). The curve may then be traced by any point C that liesoutside the moving polode. Thus, during the rolling motion, the centerM of this polode and the point 0, once it is chosen, are at a constantdistance from each other. The same applies for the centers M and Mo.Therefore, M and Mo may be joined by a single bar MMo.

Next, we adjoill the dyad MoM'C such' ~hat a linkage parallelogram,j1[oMCM' is created. ...

We then investigate the motion of the adJoined bar M'C with respectto the frame. We shall prove then that the motion of this bar may be

. directed or reproduced by another pair of wheels. In this case we shallprove that the bar M'C can be attached to a circle that .rolls insideanother circle, which is the fixed polode for that motion.

To do this, we first determine the location of the velocity-pole P'of the observed bar. This is easily done, using the Kennedy-Aronholdtheorem about the three relative poles that join a straightline.

According to this theorem we find that the pole P' of the barobserved has to join the Kennedy-line PC as well as the Kennedy-lineMoM'. Thus, P' = (MoM', PC). Here; P resembles the pole of the barMC which is attached to the moving polode of the source mechanism.Further, we see that

P'M M'M PMo M'M Ro t R'0= 0'---= o·-=constan = 0PM R

584

I langaft. " tho Cyrv.

,............\( \.' .. .

"

Fig. 3. - Identical hypocyctoids traced by cognate gear-wheel mechanisms.

/

Fig. 4 . - Identical curves traced by cognate gear-wheel mechanisms .

585

586 Eo A. DlJK'SMlAN 6

Throughout the motion, therefore, the locus of poles P' in the frameresembles a circle about M o with radius R~. Thus, the fixed polode (7'C;)of the motion observed is a circle about M o that joins the pole P'.Similarly, we find that

P'M' = P'Mo - M'lIio = )J1'lIio (~ -1) = constant = J(,'

Hence, the locus of poles P' in the moving plane of the observed b~u'

M'G is a circle too. Therefore, the moving polode (7'C~.) which is thefixed polode of the inverse motion, is a circle about M with radius R'.Conclusion: the motion of the bar M'G may be generated by the l'olliugmotion of a circle inside another one.

Since the point G of the bars M'G and MG is now a common pointthat takes part in two different motions, the curve traced by that pointis to be generated by two different wheel-pairs. These pairs are represent­ed by the source mechanism and by the curve-cognate of the sourcemechanism. .

Both mechanisms produce hypocycloid motions, since for the twomechanisms the rolling motion occurs inside the fixed wheel. 1\10reo\'8l',since C lies inside the moving polode of the curve-cognate, the point Ctraces a contracted hypocycloid if it is attached to that mechanism.If C is attached to the source mechanism the point traces a protractedhypocycloid. Thus, the contracted hypocycloid is identical to the protrac­ted hypocycloid.

3. RELATED DIMENSiONS OF THE CURVE·COGNATES

•It is further seen, that

R' R---=1---R~ R o

(1)

Thus, ~f by definition

we so' have found that

k = RojR and k' = R~jR' (2)

(3)1 1-+-=1k k'

By further introducing the angular velocities wand w' of the respectiv(~

moving polodes of the source mechanism and of the curve-cogn'1.te,.we find that

w _!!-~ _ 0.21£__ MMo= PMo - PM = 1 _ !lo_ = 1 _ k."w;- GP - lIIP - MP MP R

Thuswjk = -w'jk' (4)

Through"fig~e 4 it may be similarly.shown that a protracted epicycloidis identical to a contracted peri-cyclOId. .

7 GEAR-WHEEL DRIVEN GENEVA WHEELS 587

The derived formulas too, are valid for such a case. However,one has to keep in mind that the gear-ratio of the wheels that generatean epicycloid motion, always obtains a negative value. Thus, k < 0 forthe epicycloid' motion. .For the pericycloid motion however, 0 < k < 1. Also for the hypocycloidmotion we note that k > 1. We further note that

M'G MloM R o - R (k - 1) R (k)2 R--=--~-----= .-= - ~

MO MoM' - R~~R' (k' -1) R' k' R'(5a)

This relation shows how the location of the common point 0 is transform­ed from its location with respect to the source mechaniRm into thatwith respect to the corresponding curve-cognate.From the figures 3 and 4, we further see that

and

where M~ = Mo.

R' M'O k--=--=-MO R k'

M~O MoO R--=---.--,R~ R o MO

4. DIMENSIONS OF THE MECHANiSM

"

(5.b)

(5c)

As pointed out earlier, the in- and output aix of the m~chanism

may be made coaxial, which means that the input-crank MoM ,and theGeneva wheel must rotate about the same fixed center Mo. Besides, theslots of the Geneva wheel are radial. So, in order to obtain smoot;b. out­put motions, that is to say motions without a jump in the angular 'velo­city of the wheel, the driving pin of the wheel may only enter or leavea .slot if the tangent to the curve is directed on the center Mo (Seeagain the figures 3 and 4). This fact relates some dimensions of the mecha­nism. In order to investigate this relationship, we shall introduce thedesign-position of the mechanism, which is the position where the drivingpin just enters or just leaves the Geneva wheel. In the figures 3 and 4for instance, such design positions are drawn.

As in the design-position the tangent to the curve at 0 is directedon the center Mo, the point 0 of such a position has to join a locusthat resembles the circle with diameter .MoP (Point P being the velocity­pole of 7tm with respect to 7t1 in the design-position). Generally, therefore,the mechanism considered allows for two degrees of freedom in design,viz. the choice of the point 0 on the mentioned locus in addition to thechosen ratio RjRo of the polodes. Thus, in comparison to the Genevawheel that is driven by a single crank, ,,:e have obtained an additionaldegree of freedom in design. This fact actually gives us the freedomto choose the ratio of times Vo and the number of stations 1t, independentlyfrom one another.

",

588 8

, We BOW define y as the angle enclosing two wheel-diameters touchinga lobe of the curve on both sides. We further confine ourselves toangles y for which

_ 27t' -4 1':Y --n - ,a, '"n

where n resembles a positive integer.

I

~

(6)

Fig. 5. - Protracted hypocycloid generatinggear-wheel mechanism.

In addition, we define the angle " as the angle needed for theinput-axis to turn from the. position in which the driving pin justenters the wheel to the position in which the pin is at the point of leavingthe wheel.

For the hypocycloid motion this angle at may be calculated asfollows: Since in the design-position, at 0, the curve-tangent joinsMOl the tangent MoO must be perpendicular to the path normal PO.And so

(See figure 5) (7)

According to the rule of Sines for !i. MoMO, we additionally have

.JiO.sin..!... ~ = MoO· sin..!... (at + y)2 2

in which~ ~. R = ~at.Ro2 2

Thus

(8)

(9)

Further

1 1Ro -R= MoM = MoC,cos-(at + y) - MC·cos-~.

2 2

From eq. (7), (8) and (9) we so derive that

MO sin (at + y). Ro = -; sin..!.. k at

2

Simil.arly, if we combine eqs. (7), (9) and (10) we find

Jr.-I - Sin2 ..!.. (at + y)MO 2--- = -----:-----

Ro cos~kat2

589

(10)

(11)

(12)

Thus, equating the right-hand sides of the last two equations, we get

sin(at + y) = 2 {k- l- sin2 ~ (at + y) }.tan ~ kat.

whence we find, after some calculation

(2k- 1 -1) sin ~ ak-Sin{at(l- ~ k) + y} = 0 (13)

This formula is derived only for the hypo-cycloid motion for which k > 2.The derived formula determines the value at if the ~ar-ratio k and thenumber of stations 'II, are known. However, since illdthe design positionthe coupler-point C joins the circle with diameter P~, only those values

for at are permissible for which at + y :E;; ~ .Thus any value for at that.. 2 2is'· derived through eq. (13) has to meet the condition at :E;; 7t - y.·In case the derived at- values do not meet this condition no mechanismcorresponds to the given gear-ratio and given number of stations.

Between the curve-cognate and the source mechanism there existthe relations

cu = dat'/dt and cu' = dat/dt

So, with eq. (3) and (4) we arrive at the relation

at' =0:(1 - k) or k'at' = - kat.

SUbstituting, therefore, the expressions for *at and k from

at = at'(1 - k') and k = k'/(k' - 1)

into (13), we find that the left-hand side of eq. (13) remains inva­riable. Thus, ell.. (13) is valid also for the curve-cognate mechanism.

590 E.A.DIJKSMAN 10

Hence, eq. (13) is valid for all the hypo-cycloid driven Geneva wheelmechanisms. That is to say: eq. (13) holds true for k > 1.

For negative values of k such as for the epicycloid driven Genevawheel mechanisms, we similarly arrive at eq. (13).

Since, in addition, the equation remains unchanged if we transformthe driving mechanism into its curve-cognate, eq. (13) must be validalso for the peri-cycloid driven Geneva wheel mechanisms. Thus, finally,eq. (13) holds true for any value of k.

5. THE RATIO OF TIMES Vo

According to its definition the ratio of times '10, answers the equation

'10 = ~f2rc,

whence, according to eq. (9),

'10 = krxf2rc

Thus, according to eq. (15), we See that

'olD = - '10

(16)

(17)For the designer, this means that, apart from the sign, the two

curve-cognates always spend the same time for the motion period inrelation to the time needed for the full cycle.Eliminating rx from the eq. (13) and (16) we obtain eq.

(2k- 1 - 1) sin '"'10 - sin {1tVo(2k-1 - 1) + 2:} = 0 (18)

which is still valid for all values of k = RofE, providing that eq. (16)holds.

The equation shows that unlike as for the single crank drivenGeneva wheel, the ratio '10 is not dependent on the number of stationsalone, but may be varied instead by choosing other values for k. Forthe designer, this is very practical.

6. THE GEAR·RATIO AND THE ACTUAL NUMBER OF KNOTS

Generally, however, not all the real values of k are allowed.They are restricted to rational numbers only. In order to give the reader:more insight in this respect, we shall define a new number m by makingm equal to the number of lobes that could be placed between two succes­sive lobes of the curve. Thus, m equals the number of unreal lobes thatjust fit between two successive, real ones, that actually appear in thecurve.

If m is a rational positive number, instead of a positive integer,~uch as ~fm2' we say that m1 unreal lobes just fit between the firstand the (m2 + l)th lobe that are really there.

GEAR-WHEEL DRIVEN GEINEVA WHEEILS 591

So, we define m bym = (max. possible number of lobes that would fit for any number of .. ,.cycles minus the actual number of lobes that appe.ar for that number . ',,:, :.: ."of cycles) divided by the actual number of lobes that appear for those. \cycles.Thus, if the curve needs n l cycles to repeat itself, the maximum numberof lobes that would fit for n,. cycles equals nl·n. Therefore, the actualnumber* of lobes s that appear in the curve equals

nl,ns=--.m+1

For the protracted hypocycloid we so find that **

~=(~)= m + 1 (k > 2).k s n

For the contracted hypocyloid we then have

-.!..... = 1 - m + 1 (1 < le' < 2).k' n

Similarly, we find for the epi-cycloid the relationship

1 m + 1-- = -~-- (le < 0).k n

And for the peri-cycloid

_1 __ 1+m+1 (0 < k < 1).k' n

These equations agree with the fact that either

21tR = ±(m + 1)yRo,

or

(19)

(20a)

(20b)

(20c)

(20d)

(21a)

21tR' = {27t - (m -:- 1)y} R~. (21b)

So, if we choose the values m and n, in addition to the kind of curvewe are going to apply, the gear-ratio is fixed. This can be done usingthe equation (20) that corresponds to our choice of curve or mechanism.

7. PRACTICAL INDICATIONS

Clearly, we may confine ourselves to the protracted hypocycloidand to the epi-cycloid driven Geneva wheel emchanisms. If necessary,we can always apply the cognate transformation and use the curve-cog-

* Numbers R 1 and s a~e positive integers only** For each revoluticn of It,,, a knot Is produced. Thus s(2ltR) =R 1 • (:rn~o)'

592 Eo A. DI.lDMAN 12

nates instead of the one mentioned. The dimensions· for the curve­cognate mechanisms are then easily derived from the source mechanismsthrough the cognate transition formulas already given in this paper.So, for briefness' sake we shall only refer to equation (20) if it is writ­ten in the form

k = ± 'It

m+ 1

It we substitute this value into eq. (18) we arrive at the relation

(20)

(m + 1 ). . { ( m + 1 ) 27t }± 2--

n- -1 sm 7tVo - sm 7tVo ± 2 -n-- -1 +---;- = O.

(22)

For each integer '1'1" and rational number m it is then posible to ca.lcu­late the ratio "'0' From graphs that are made that way, we choose thepractical values '11 0 and n and then determine the number m from whichwe calculate the gear-ratio, uaing eq. (20).

We then determine the values for « and y, according to the equa­tions (16) and (6) respectively.

The rema.ining dimensions, such as MoC/Eo and MC/H are finallycalculated through relations

MC 1_0_ =cos- (a. + y), (k< 0 or k> 2) (7)

Ro 2

MC k--=--

R 2sin (Cl + y)

. 1 kSID- CIt

2

(k< 0 or k> 2), (23)

the last one being derived through equations (2) and (11). Thegraphs we have been referring to just now are demonstrated herewithunder the numbers 1 and 2. The dots in these graphs represent mecha­nisms having only positive integer values for m, illCluding the numberzero. (Later, we will $ow that also rational numbers of m are allowedas soon as we allow lobes to be unused.)

If the lobes that appear in the curve are all used to drive the wheel'" = "'0 = k«/2rc otherwise "'". "'0 = k«/2rc. But even if we use themall, the designer of this kind of intermittent motion mechanism is stillleft with a large number of values '" that are equal or less than one.

Examples are given in figures 6 to 15.

8. THE MAXIMUM NUMBER OF SLOTS go ON THE WHEEL THAT COULD BE USED

The number of slots or grooves 90 that have to be made in thewheel, does not necessarily has to be identical to the number of stations1'1, of the· mechanism.

n ,,4

m = Ik =-l9=90=4ex = 21·6°Y ,,0·12Me" 1.263 RoIIy;: O·562 Ro

Fig. 6. - Epicycloidal-drivenGeneva wheel.

593

....--------. -'-,,/ .-- ............

// 1b. -".\

/ \( \\ ;\ /'" \..1 /',- V /'

'-'~ '---._."-

n .. 4m .. 0k ... -4g .. 90" 2ot .. 14° '/.I';Y .. 0·16 ......-.-.~.-.Io4C.. 1·0053 Ro./ /'. .........,

104oC= 0·6129 Flo ./ I M \. \

! . - )./ \I .-' '~.'V'/' "".}(" X

.. "-.--. .--./' )\ / \ i~ /\ " / /

" .". ./',_.~<._--"

Fig, 7, - Epicycloidai-drivenGeneva wheel.

594

n =, ;m = 2k = - ~

9 = 90 = I •at = 25-65Y = 0·095 IMe. 1·538 Flo ;MoC ()'5325 Flo

Fig. 9. - Hypocycloidal-drivenslot with Instantaneous dwell in

the output-lIIotion.

. n =". m =0

k ="9=9o=2IX =,90"V ,. 1Me= MoMMoC: 0

Fig. 8. - Epicycloidal-drivenGeneva wheel.

n :II 12m.: 3k , 3

9 , Ilo' 12IlL • as'!25'

v ,0737,,",C,0598Ro""'oc. 0512 Ro

Fig. 10. - Hypocycloidal~'

driven Geneva wheel.

..........

fi95

':;-~'-2----I

1m = 5 'Ik =-2

,'J~90=~2 Ijill = 30

'

" =0·167 IMe =0·866 Rc

!MoC= 0·866 Ro.

..//

I

I(

\'"'"'---

9 - Co Jla~

.'......

\

\

1

//

/. . ~

/.J'..... •. ...........~._--,._--- .

Fig. 11. - F.plcycloidlll-dri\'clIGeneva wheel.

596

n-~8-'-1

m .5 I

Fig;. 12. ...,. .gpicycloill.~I,drl ...eD• . Geneva wheel. . '

Fig. 13. - EpicyclDldal-driven .Geneva wheel.

k • - 39=\10=18or. = 24°'11'

II .0·21Me = 0·589Moe. 0,927 Ro

(

\

\

"-

\

I/­

I

'/

n =18m ,,8k =-29=l3o=9IX ,,'29°22'y' " 0·163Me =0·168 R"Moe:: 0·91 Ro

Fig. 14, - Eplcycloldal-drlvenGeneva wheel.

n ,,18,m .11k .-19"llo·18.CIt .3fs3'Y "Q.l0SMe ,,1,3 Ro MMb 0·851 R" • . .-'~

/'/ .'"I \( \\' j

/\ /

''''. /"'-.. /''--.._---.

597

Fig. 15. - Eplcycloldal-drlvenGeneva whi'el,

598 Eo A. DlJKISM!AN 18

, Clearly, the number of slots needed equals either n, ~, ~. ~, ... ,234:

or 1. Which number it actually is, is decided by the next reasoning:If the driving pin leaves a Slot, it enters the next one just (m + 2)y

or (m + 2). 27t radians further on the wheel. So, on the wheel, that11.

is to say for 21t radians, there are at least _11.__ slots. If _11.__ is am+2 m+2

't' . tn,POSI lve ill eger, go = -- .m+2

If it is not, we have to multiply it with the smallest possiblepositive integer so as to make it one.Thus

go = n/(greatest common divisor of 11. and n~ + 2) (24)

(For applications: see the given examples in figures 6 to 15).

REDUCTION OF ."

In order to reduce the value of "', we may diminish the numberof slots. The lower values of v, obtained in this way, are sometimesvery practical, since they represent the circumstances in which onlya small portion of time is needed for the actual motion of thewheel. Naturally, if there are fewer slots, the locking time of thewheel will be enlarged and more time is available for completion ofproducts for instance that are moving around with the wheel.

How to find the number of slots in those cases will be explainedthrough the next reasoning:

If the driving pin leaves a Slot, it may find the next one

(m + 2) 21t rad. further on the wheel. However, if no slot is available at11.

that position, it may still find another one (m+1) 27t rad. then further11.

on the wheel. Again, if no .slot is present at that position, we may find

the next one (m + 1) 27t radians further on the wheel, and so on.n

Therefore, the slots that are used subsequently are either21t 27t 27t

(m+2) - rad., (2m+3) - rad., or (3m+4:) - rad., etc.... set apart on11. 11. 11.

the wheel. As before, we find that the number of slots g, that are engravedin the wheel, has to meet the equation:

9=11./9.,(11., m+2), or g=n/g.,(n, 2m+3), or 9=11./9."(11., 3m+4) etc. (25)

where g., resembles the greatest common divisor of two positive integers,one of them being 11., which is the number of stations. Which integerthe other one has to be depends on the number of lobes that are unused

19. GEAR-WHEEL DlUVEN GEINEVA WlHE!LS 599

in the mechanism. For instance, if 3 lobes are unused 9 =?~/ge:tl (n, 4m+5)•.As an example, we observe the case for which n = 8; m =0 and 'k =8.(See figure 16.) If all lobes are used then to drive the wheel intermittent­ly, we find that go = 8/2 = 4. Thus, the maximum number of slots goequals 4, as is demonstrated in the figure. In addition v .vo=ka./2rc .=0.65 (vo then resembles the maximum ratio if the maximum number ofslots (go) is applied). If we want to diminish the number of slots, it isnecesary to skip at least two lobes. Then, 9 = n/gctl (n, 3m + 4) ;= 8/,1=2

slots. Such a possibility is demonstrated ill figure 17. Then v =..!... Vo =3

= 0.22 since now the cyclic time is three times as large in comparisonto the cyclic time needed if go slots were in use. .

If more lobes are skipped, the slot number decreases even further.For instance, if we skip 6 lobes, then g = n/gcrJ (n, 7m + 8) . 8/8 =

1 slot in such a case. Thi~ is dem~nstrated in figure 18. Herei v = -.!.... 7Vo = 0.094.

Remarks: Contrary to what is demonstrated with the last figures,it is not in all cases possible to create redundant lobes. For instance,if we consider the case where n = 8, m = 1,' as shown in figure 19, 9 == go = 8. It is then not possible to reduce the number of Slots, unlesswe allow the subsequent dwell periods to be unequal (3). Therefore,we are not looking for cases where this mayor may not be done for agiven curve, but we are looking for curves instead that may containredUhdant lobes.

If we assume redundant lobes, generally, the v-value that corre­sponds to the mechanism may be obtained from the equation:

v=2rc(1 + uaJ

(26)

in which Uk resembles the number of unused or redundant lobes thatlie between two successively used ones. Therefore,

Vov = -----(1 + Uk)

(27)

in which Vo = ~/2rc resembles the maximum ratio of times moving/cycling.This number is actually used as the y-coordinate in the graphs 1 and 2.So far, we have only displayed examples in which m represents an integer.However, as soon as we allow lobes to be unused this is no longer necessary.

For instance, if each time 1 lobe is unused as it is shown in theexamples demonstrated in the figures 20 and 21, only 2m has to be aninteger. This follows from the fact that for two successively used lobes(2m + 1) lobes could be thought to lie in-between. Therefore, if eachtime 1 lobe is unused, the quantity m may obtain the values ?n = 0,

...!.. .1.~, 2, 2..!.. ... , etc.2 2 2

600

Fig. 16. - Hypocycloidal-dri­ven Geneva wheel.

Fig. 17. - Hypocycloldal-drl­ven Geneva wheel.

n .8m .0k .89 .2.III .2(5'II .0·22,l4C.0.5365MoCo 0.7982 RO

Fig. 18. - Hypocycloidal-dri­ven Geneva wheel.

n .8m .0k .8II .1Cl .215'II .0.014Me. Q.S365M"l:. 0·7982

Cl01

Fig. 19. - H)'poc)·cloidal-drl­veil Geneva wheel.

602 E. A. DLTKSMAN 22

a~o

r

;1 .~. s: C:'s: II; s: c: I: c: r

---- I II I i I I .} } I I

I il il il II • II il

/ / II ill/ '/ I II IIA~'

~ V--- c... :

i.-o'i"'" I-"" ~

I--"'0 II ratlll of tl_ ./IlOlIl"lil/qlCleI'B .. I'lI.Imber of RatiOll.·

c),.' ~"II""'~

0.10 UIt"·l)eln 11'1(. .. DIn {1i''ilZIf'.lI.~}

I . i I I J J I I I I I Io 12345 Ii 7 i I lOti 1213I4I51!!if7I819202IU

m .. nurntMfr of ICibIlI th!It fit In I!0twMn tlOIO 1II.R:C8Hiw !lM!'l

Graph 1. - HypocycloidJI-ca-lwn Gene~a whc·~J mechanisms.

Figures 20 and 21 show examples in which m = 3/'2 and m =1/2respectively. Since in both cases, the number of slots g answers the equa­tion g = n/gca ('1'1, 2m + 3), one finds that

g = 1 if n = 6, m = 3/2, and similarly,g = 2 if n = 8, m =!if/1~

So the ~umber of slots is very much reduced for these examples.

1 2345".I~ftGG~el8f7I819~~22D24~Bv~~m~n~lIlIo~ I!XI 0lIillft lh!n til In llvi¥lRn. 1loo __

Graph 2. - Epicycloidal-drivcn Geneva whce mechanisms.

o

; ..If, • raiiO of Ii.... .....,;ng/~

~ltn ........... of fltotiona..... k •• Mh-

(2k"-llfltnll'i. .IMn {1l'!J2k"-Il.~}

~21~ i:"241\ 1\

",A i\ 1\. l"'"

~ t'-.r'" r-. r-.16 'j\' i"~ r0- T"" r-o. ~

I..... 1'0.. .... I"'"

12 ~ I""'-~ ~ roo!\ I" r--:-.. 1"1- -n

n."'" I'.. "",,- I" .... 1""-1- n.

" ........ ....- n•n.I'-- -t- n.n.

n.n.n.

,

o

o

o

Fig. 20. - Eplcycloldal-drl­ven slot •

.n a 8m cik oJg=2~1Io

« =46.12°·11 =0·342Me =()'596 ReoM"C=().700 Reo

Fig. 21. - Hypocycloldal-dri­"en Geneva wheel.

604 E. A. DI.TKSl\IM.N 24

In order to find out if a mechanism could be composed at aUfor given values of n and m, we first of all observe t;he ratio nj(m + 1).Only if the ratio is larger than 2 we may assemble a hypocycloid mecha­nism as well as an epicycloid mechanism. In case the ratio is less than2, only an epicycloid mechanism (or his cognate) could be assembled.

To decide if a hypocycloid mechanism could be used in case '11,/

(m + 1) > 2, we further have to look at graph no. 1, and find out if forthe given data of '11, and m, a value for Vo or ex could be found (Inthis connection see also the remarks made just after formula (13)).

In case, for instance, '11, = 6, m = 3/2, eq. (13) gives rise only tovalues ex for which ex > 7t - r. So there are no permissible values forVo or ex in this case as could also be seen from graph no. 1. Thus in thiscase only an epicycloid mechanism may be applied as is carried out infigure 20.

In case '11, = 8, m = 1/2 graph No.1 indeed provides us with apermissible value for Vo or IX. Hence, a hypocycloid as well as an epicy­cloid may be used in this case. Figure 21 desmonstrates how it worksout for the hypocycloid. Once the values for k and ex are derived, theremaining dimensions may then be calculated using the equations (2), (7),(11) and (27).

If each time two lobes are unused, as is shown with the examplesdemonstrated in figures 22, 23, 24 and 25, 9 = n/gcrl ('11" 3m + 4),

and furthermore, the quantity m may obtain the values m = o,..!... 2...3 3

1 21, la, Is, 2, ... etc.

The examples displayed in figures 22 to 25 show cases withonly a few number of slots. Technically easy to manufacture are thosemechanisms with only 1 slot. Figures 22 and 23 show examples uf them.

The author wishes to acknowledge the support of the Science Re­search Council for this work under Grant No. B/RG/I0l6.5, and alsotheir support through Grant No. BS/SR/4359 having enabled the authorto make extensive use of Analogue Computer EAL. 580 installed in theMechanism Laboratory at Liverpool Polytechnic.

Finally, the author also wishes to thank the students A.H.G. Kersten,C.P. van Luyk and R.A. Zoethout for their active cooperation.

Received July 26, 11174

REF ERENCES

1 G. HELLERMAN, Eplcyklolden und Hypocyklolden, Berlin (1867) (Doppelte Erzeugung dercykllschen Kunien).

2 R. A. l<RAus, Malleserkreuz Schallwerke mit glelchachslgem An-und Ablrleb, Konslruktion,15 (1963), 10, S. 392-399.

3 E. A. DUItSMAN, Mallezer-krulsmechanlsmen. Polylechnlsch TijdschrUl Editie A, 23 (1968),p. 273-280. .

Fig. 22. - Eplcycloldal drl1

- vensot.

n '.7• 1

II _;

;.1 '120fll.··.85.4°)I .0·28'Me. o·sn RoMoCa 0.36.8 Ro

(.-..

\.)

.-/'Each tirM 2 lobesan.. s~ipped

Each tlrrHt 2 lobMQf~ ,skipped

Fig '>3. _ . - HSpocycloidal-drl-ven slot.

FIg. 24. - Hypocycloidal-dri­'len Geneva wheel.

Fig. 25. - Hypocycloldal-dri­ven Geneva wheel.

n .30m • §b ••1Q

II • 6Cl • '·24"1 • 0·154Me. (1.1824MaC- (I.!11a29 R"