Jerk-free Geneva wheel driving - Pure - Aanmeldencoupler curve when the mechanism is moved (see Fig....

Jerk-free Geneva wheel driving

Citation for published version (APA):Dijksman, E. A. (1966). Jerk-free Geneva wheel driving. Journal of Mechanisms, 1(3/4), 235-283.https://doi.org/10.1016/0022-2569(66)90028-0

DOI:10.1016/0022-2569(66)90028-0

Document status and date:Published: 01/01/1966

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J. Mechanisms Vol. 1, pp. 235-283. Pergamon Press 1966. Printed in Great Britain

JERK-FREE GENEVA WHEEL DRIVING

Dr. E. A. DIJKSMAN

Technological University, Eindhoven, The Netherlands

(Received 23 March 1966)

Abstraet--A jerk-free motion of an internal Geneva wheel is obtained by using a special driving mechanism. For this mechanism a four-bar mechanism is chosen from which the coupler point drives the wheel. By special choice of the four-bar linkage the coupler curve can be made symmetrical and containing two Bali's points. For each number of stations the dimensions of the mechanism are tabled for optimal transmission angles and next to it also in those cases where the coupler point lies in one line with the two moving turning- points.

Drawings of the intermittent mechanism are simplified in cases where the circling-point curve degenerates.

1. INTRODUCTION

NORMALLY a Geneva wheel is driven by a crank with a crankpin. At the beginning as well as at the end of the intermittent motion at least two grooves of the Geneva wheel are then tangential to the circle traced by the crankpin. At those positions the normal acceleration of the rotating crankpin will be transmitted to the wheel with an impact. Hence, the wheel has an infinitely greatjerkt at such positions. This leads to an abrupt change in force on the wheel to the effect of undesirable vibrations in the mechanism.

With an internal Geneva wheel this drawback can be removed by using a four-bar linkage as a driving mechanism instead of a simple crank.

Such a four-bar linkage, indicated henceforth by (AoABBo), is a closed kinematic chain consisting of 4 links, one of them being thefixed link (AoBo). The opposite link, the coupler (AB), is the base of a coupler-triangle, the vertex of which, the coupler-point K, describes the coupler curve when the mechanism is moved (see Fig. 1).

I f the driving pin is now situated in a coupler point it is possible to trace an inflection point at the moment of entering or leaving the Geneva wheel. The tangents to the coupler curve in the two symmetrically situated inflection points of the curve must then coincide with the corresponding grooves of the wheel.

To avoid undesirable turns in the coupler curve, it is still better to replace the inflection points by what is known as undulation points or Bali's points [4].

At a Bali's point the coupler curve has a four-point contact with its tangent, through which the coupler curve, at least for a great part, runs on one side of this tangent. As a consequence, the motion of the turning table can begin and end with a zero angular jerk.

t By jerk is understood the time-derivative of the acceleration.

235

236 I-. A. DIJKSMAN

\ ,, \

.~ ,"/' ." . -

,i~... - - Ao ,,I "-~.~B °

F I ( ; . I

If the acceleration period may be made equal to the deceleration period, it is convenient to make the coupler curve symmetrical. The problem is then to determine the dimensions of the four-bar linkage in such a way that the symmetrical coupler curve has a Ball's point which is not on the axis of symmetry and in which the tangent to the coupler curve makes a given angle ~ with the axis of symmetry (see Fig. 2).

This problem has already been acknowledged by Hunt and his colleagues [1], who found a way to achieve a general solution of the problem. The present work deals with a further study on the subject, and is a condensation of the author's thesis [2] published in Dutch. The thesis contains a large number of graphs, from which the dimensions of the mechanism can be taken.

To drive the Geneva wheel, the input link has to make complete revolutions with respect to the fixed link. So only four-bar linkages complying with the condition of Grashofcan be

Jerk-free Geneva wheel driving 237

j. /

!

, , - f - /

I I

\ J \ /

\ j I

/

FIG. 2.

Z

I

/

considered. According to whether AoBo, AoA or AB is the smallest, we speak o f a drag-link a crank-and-rocker or a double-rocker mechanism. The coupler o f the last mechanism must be driven directly from the fixed link; if no gear wheels are used this is only possible by bringing the driving axle o f the rotor in the moving pivot A. This is an extra complication,

238 E. A. DIJKSMAN

which however, can be avoided by using one of the cognate mechanisms corresponding to the proposition of Roberts instead of the double rocker mechanism. We then get a crank- and-rocker mechanism with exactly the same coupler curve as the one we started from.

Up to the time of writing, no usable solutions of the problem have been found for the drag-rink mechanism. We can, therefore, as far as the solutions are concerned, confine ourselves exclusively to the crank-and-rocker mechanisms.

We know that symmetrical coupler curves are generated if BA =BB o =BK. The axis of symmetry thereby goes through the fixed turning-point Bo and makes an angle of ((rc/2)- .~.KAB) with the fixed link AoBo. It is also perpendicular to AoC o, a side of the fixed link triangle AoBoC o, which is similar to the coupler-triangle ABK. Turning the coupler-triangle 180 ° about its coupler causes the axis of symmetry also to turn 180 ° about the fixed link. If the crankpoint arrives at the opposite point with respect to the fixed link, the coupler point coincides with its opposite position with respect to the axis of symmetry. Hence, the two positions of the crank-and-rocker mechanism with the crank-point A in line with AoBo, have their corresponding coupler points at the intersections of the coupler curve with the axis of symmetry.

2. KINEMATIC FUNDAMENTAL THEORY 1"3]

The locus of those points of the moving coupler plane at which, at the moment of observation, the radius p of curvature in the described curves has a stationary value, is called the circling-point curvet and is indicated by ku. Thus, in accordance with this definition, ku is the locus of those coupler points for which (dp/dt) =0. Its equation in the polar co-ordinates r and q~ reads:

1 l 1 - - + - - ( 1 )

r Isinq~ mcosq~

The abscissa axis of the corresponding Cartesian co-ordinates coincides with the pole tangent p, and the ordinate with the pole normal n (see Fig. 3). The positive sense of y is taken in the direction from P to the inflection pole W. The positive sense of x changes into the positive sense of y after rotating the abscissa axis counter-clockwise about P through 7:/2 radians.

If X w is the intersection of a path normal PX with the inflection circle, the angle ~o is determined by the equations: ~0 = ~ XwPx where 0 < ~0 < 7r.

r is the distance from the origin P to the point of path X. If such a point is lying in the upper semi-plane, then r is considered positive, in the other case negative.

The length m is given by the equation:

36 m - (d6/ds--) ' (2)

where 6 is the diameter of the inflection circle and s the length of arc along the fixed polode. 6 is always positive.

The length l is given by the equation

3 1 l 2 1 (3) I = / ~ + 6 = R Ro

t Or cubic of stationary curvature.


/

i.c./'/ / /' / /

/ // ,/

asymptote of ku J ~ / [ :// ./~ ,

k,,,,\, /asv, , ~ , ~ / ~.'-'~\'~ / dire, lion of,~a /

oe \ Y' \ O : P

\

\

\

,sn

/

/ !"\\

/ Y - - / \ ~ , ; - \ -

, ku

1~o. 3.

/

/'

/

/

where R and R o represent the ordinates of the centres of curvature of, respectively, the moving and the fixed polode in the pole. It is known that ku has a double-point in the origin P with radii ½1 and ½m, the former coinciding with the pole normal, the latter with the pole tangent.

240 E.A. DIJKSMAN

In accordance with the definition, the turning-points A and B of a four-bar linkage (AoABB o) lie o11 the circling-point curve. Therefore, the circling-point curve can be considered to be the locus of possible points of choice A and B of the four-bar linkage.

The locus of those centres of curvature, of which the corresponding points of the path lie on the circling-point curve, is called the instantaneous centering-point curve and is indicated by k,. The centering-point curve k a is the circling-point curve of the inverse motion.

In accordance with the definition the centres of pivot on frame, A o and Bo, of a four-bar linkage (AoABBo) lie on the centering point curve, so curve ka may also be seen as the locus of possible points of choice A o and Bo of the four-bar linkage.

In the polar co-ordinates r 0 and ~p, k~ is given by the equation

1 1 I + (4)

ro l osin(p tacos,p"

where l 0 is defined by the equation

or by

3 2 1 I I - , ( 5 )

/, Ro R Ro o

1 1 1 t6)

3. BALL'S POINT OR THE UNDULATION POINT U [4]

Bali's point is defined as that point of the path where the path tangent has a four-point contact with the path. This point lies on the inflection circle as well as on the circling-point curve. These curves generally intersect at 2 × 3 =6 points, viz. the two isotropic points, the triple counting pole and a sixth point, Ball's point. I f the pole is a triple intersection, a real Ball's point exists. In some special cases the inflection circle coincides with part of k. and then any point of the inflection circle may be considered to be a Bali's point.

Generally Ball's point is identified by intersecting the inflection circle with the line through the pole in the asymptotic direction of the centering-point curve. This direction is fixed by the direction coefficient - m / I o (see Fig. 3).

4. DEGENERATIONS OF k,, AND A.

Parts of this chapter which are printed in small type give rise to earlier derived driving mechanisms or to mechanisms unsuitable to our purpose.

4.1 The general eycloidal position of the Jour-bar linkage I f db/ds=O (or m-1 =0), it follows from (1) that either ~o =-~Tr, which is the equation of

the pole normal, or that r = /s in 9, which is the equation of a circle through P with its centre on the pole normal.

In this case the k, curve is degenerated in the pole normal and a circle through P(0, 0) and L(0, I) with its centre on the pole normal 17. In the same way we find the k, curve to be


degenerated in the pole normal and a circle through P(0, 0) and Lo(0, 1o) with its centre on ?/.

The relation between the ordinates of the points L(0, l), Lo(0, 1o) and the inflection pole W(0, 6) is expressed by

1 1 1

1 Io

(see Fig. 9). Hence, the three circles are interdependent. designed as follows:

(a) choose X anywhere in the moving plane; (b) connect X with P, L and W; (c) draw P Y parallel to XW and find Y on XL; (d) draw YL parallel to PX and find L o on n.

This interdependence can be

Proof. From P'W PX PL

LoP LoY LoL

we find tS/(- 1o) = 1/(l- 1o), which is another form of equation (6).

~a

F1G. 4. Pericycloidal position: Ro < R < ~Ro, -- J < l< 0, -- ½J < lo < 0.

242 E . A . DIJKSMAN

F1o. 5.

' 'i c"

\ \

J c;ku--C.C.

Perieycloidal position: R=¼R0, 1~--~, 1 0 = - ~ , c.c.=cuspidal circle=inflection circle of the inverse motion.

i.C.

P //,'~

clkB =C.C. \ /

! ku

FIG. 6. Cardioidal position (=kinematical inversion of the cardan position): R = 2R0, l = - co, 10 = - c~.


Fie. 7. Evolvent position: R = ~ , l=3~, 1o=--]-~.

¢

P ka

Fta. 8. Elliptical or eardan position: R=½Ro, l=~, lo=--oo.

/

/ C. I a

\ ,

\

E. A. DIJKSMAN

ta

-0

u Iw /

"-'~ y

!P P

\

\ \

to

,// " t 6

ia

1~o. 9. Hypocycloidal position: ½Ro


i I P

kU

FI~. 11. Hypocycloidal position: :~Ro

246 E . A . DIJKSMAN

ku:ka-

P--ka

~/ fJf~ /]]

/ A

al \

\ \

K=UI

" "-\!.C.= c-ku

/// /

Bo=W=Q -13

Ao~..~ ~'i "i

FIG. 12.

is infinitely large. In that case only the pole tangent forms part of k,. As also in this case the line at infinity belongs to ka, the inflection circle is a branch of k~. So any point of the inflection circle is then a Bali's point and may be considered to be a coupler point. The design is shown in Fig. 12. As the quadrilateral does not satisfy the condition of Grashof, this case can be left out of consideration.

(b) A as well as B on the circle that is a branch o f ku (see Fig. 13). (Henceforth a circle that is a branch of ku or of k , will be indicated by c - - k , and c - k a respectively.)

If c - - k , and the inflection circle are lying on opposite sides of p, then BBo < BU, and, hence, if U=K, BBo%BK. Only when c--ku and the inflection circle are on the same side of p, that is to say, when I/6>0, is it possible to find a point B, for which BBo=BU. Hence in the cycle of Fig. 4 to 11 the degenerated configuration of Fig. 6 is a limiting situation for the solution of our problem. This is also the case with the degenerated configuration of Fig. 10, where l/lo=½.

Thus real points B can only be found if [10l >=6. It is also evident that if c--ku and c - - k , are on the same side of p, no real points A on c--ku can be found that satisfy the condition BA=BBo. Thus there

~o

o

~,~.

I 0

! /

J J

\ , n Q~

j %

S~

~ b~'v

/

~Q

248 E . A . D1JKSMAN

/ / / / 7

"" / A t'

axis of symmetry

\ \

\

I

/

~B

\ /

I

I

\lc°upLer curve J

~o

FIG. 14.

(c) B on c - ku and A on n, as another branch of k, (see Fig. 15). The construction then is as follows: Dependent on the desired angle 2~, we start from a ratio l/lo, taken from Fig. 16, between the diameters of c - k u and c - k a. Just as in case (b) we h a v e / > 0 and I/ol > 6 . (A negative value of lo means that c - k u and the inflection circle are on opposite sides ofp. ) With the aid of equation (6) we now draw the circles c - k , , c - k a and the inflection circle. Moreover, the points P, B and B o lie on one straight line in such a way that B

lies on c - k , and Bo on e - k a . On the assumption that BBo = B K we have


km

\ \ "\

coupter curve

\

\

". circte of Apd.tonius

/

P ÷X

FIG. 15.

Hence, besides c -k , a second locus can be indicated for point B, viz. the locus of those points for which the ratio of the distances to two fixed points U and P is given. This locus is the circle of Apollonius (see Fig. 15).

250 E. A. D I J K S M A N

i

I

/ I I . . . . ! i

i ,

/

F :

+ !

. . . . . i V-

I ,

L

L

r

' I

I

" T

I

. . . . . _ _ L

,.6

4

r

] J

i

?

k

I

i

I I r 1

! ~ - + . . . . .

] ,

g


As BA =BBo the turning-point A is found at one of the two intersections of the pole normal with a circle with centre B and radius BB o. One of those intersections exactly coincides with the Bali's point, which can not be taken as turning-point A. The fixed point A o lies also on the pole normal and can be found by the equation PA 2 =AA o . AAw. This follows from the Euler-Savary equation. The point Aw coincides with W, U and K. The corresponding coupler curve appears to have a path suitable for our purpose.

We are also able to determine the dimensions of the mechanism. We do this by using a number of equations, the derivation of which will, for the sake of brevity, be omitted. With

Moreover,

and

while

Finally

b

and

q = 1 -- 2(I/lo), (7)

tan 2 ~ = 2qa(q 2 + 1)(2q 2 + q + 1) 2 (3q2 + 1) 2 (8)

a 2q2: q + l "~½ b = 3q-~ '~\2q2- q + 1]

d 2 : 2 q 4 + 2 q a + 4 q 2 + q + l ) ~

b = 3 q ~ k , q + l '

(9)

tan Yl = ~ ( 2 q ( q 2 + 1)) ~r.

(10)

2 (2q(2q 2 - q + 1)(2q 4 + 2qa + 4q2 + q + 1)~½ =(q + 1)(2q 2 + q + 1) \ J

(11)

(12)

2q b = 2 q 2 + q + l k ( q + l ) ( q 2 + l ) ] (13)

For a number of angles 2T, the results are given in Table I.

4.2 The cardan position of the four-bar linkage

When in addition to d6/ds =0 (or m-1 =0) also Ro =2R (or lo-1 =0), the circling-point curve ku is degenerated into the inflection circle and the pole normal. Then the centering- point curve/ca is degenerated in the pole tangent, in the pole normal and in the line at infinity (see Fig. 8). The cardan position may be presented as a special case of the general cycloidal position, as treated in section 4.1. Choosing the coupler point K also in the inflection pole W of the cardan position, we then have only one angle 2~, for which the problem can be solved. In the cardan position however, the inflection circle forms part of ku.

TAB

LE 1

2r

0 °

15 °

a/b

0 0

.07

29

d/b

2 1

.92

54

?1

0 °

--2

4"5

9 °

Pm

in

0 °

4"6

3 °

(BoE

ol)b

0

0'8

76

7

(Eo

Ui)

/b

0 0

-15

37

~

½

o .4

068

~b

1

0"8

62

8

30

° 4

5 °

60

° 7

2 °

90

° 10

5 °

12

0 °

13

5 °

1

50

° 1

65

° 1

80

°

0.1

62

9

0"2

49

2

0.3

26

2

0.3

79

3

0.4

42

7

0.4

78

8

0-4

97

8

0'4

96

6

0"4

69

8

0.4

02

5

0

1.8

27

5

1"72

41

1.6

18

7

1"5

33

4

1"4

03

3

1.2

91

5

1-1

74

4

1.0

48

2

0.9

05

5

0-7

26

3

0

--2

4.2

1 ~

--

21

.94

° --

18

.88

° --

16

.08

° --

11

.42

° _

7.1

3 °

__

2.3

8 °

-I-3

.03

° +

9-6

2 °

+

19

.03

°

90

°

11

.25

° 1

8.7

7 °

26

.98

°

33

.99

° 4

5.2

7 °

47

.95

° 3

9.5

5 °

3

2.0

2 °

25

.17

° 1

8.6

3 °

0 o

0.9

47

3

0.9

80

1

1.0

03

6

1'0

21

0

1.0

48

8

1.0

74

8

1.1

03

9

1'1

35

1

1-1

64

6

1'1

73

9

0

0-2

40

1

0.3

01

7

0-3

50

4

0.3

83

7

0.4

27

5

0.4

60

0!

0.4

88

6

0.5

11

4~

0

.52

18

0

-49

66

0

0-3

54

9

0.3

09

5

0-2

65

4

0.2

29

0

0.1

69

4

0.1

11

3

+0

.03

98

--

0.0

56

01

--

0-2

02

9

--0

.50

18

--

oo

0.8

21

5

0-8

11

5

0.8

13

0

0.8

23

4!

0.8

54

6

0.8

97

41

0

.96

11

1

.05

73

1

.21

42

1

.52

93


So besides W, any point of the inflection circle is a Bali's point. Hence the whole inflection circle is at our disposal for choosing coupler point K, which means that, in the cardan position, any angle 2~ can be realised by means of a four-bar linkage.

4.2.1 Design of the driving mechanism. Restricting ourselves to finite lengths of the links, the turning-points A and B can not be chosen on the inflection circle. Point P is an exception to this rule, for in this point the radius of curvature of the path may take any value between 0 and oo. By not choosing A or B in P we get a folding linkage, in which all links are superimposed, and the minimum transmission angle It~n=O. By choosing A in P we get the mechanism of Fig. 12, which has already been discussed in part (a) of section 4.1.1.

I f B and P coincide, the construction is as follows (see Fig. 17):

(a) Draw the inflection circle i.e., the pole tangent p and pole normal PW.

(b) Determine angle 71 in Fig. 18 at a given angle 2~. (For instance if 2~=90 ° then 71 = 16"47°.)

(c) Take ~g WPK=2[Tx[. Choose one of the two possible positions of the free side of this angle. Let K be the point of intersection of this side with the inflection circle.

(d) Draw a circle with centre P = B and radius BK=B-Bo =flA.

(e) Let A be the point of intersection of this circle with the pole normal (n) outside the inflection circle.

(f) Let Bo be that point of intersection of the circle mentioned under (d) with p that lies on the same side of n as K if 71 >0, and on the opposite side of n if 71

e,, I : J c

j -

/

r r l ~

/ / ' /

t \

J ! /

/ / i

I I \ -

\ /

\ \ J

C r

/ /' i ~ ~ /

! I /// ] / : -- I \

~ J " C ',,

n

\ , , \

J /

/

\ \

\

/ , , \

," \

/ i \

'il

.J J

/ J

/ /i /


0 21r

T ~ o

256 E.A. DIJKSMAN

EoU 1 b = cos2yl -sin2Ylt_.~_~nz (18)

It may be seen that always

#mi. ----> 2 arcsin ¼(5 ½ - 1) ~ 36 ° ,

which is a reasonable and permissible angle.

It will be clear that only when yl = 0 does the coupler point K coincide with the inflection pole I4I. This will be the case when 2z=2arc tan2=126 .87 °. Only then is the solution according to the general cycloidal position identical with the one according to the cardan position. For significant values of 2T the results according to the cardan position are given in Table 2.

4.3 Degeneration of the ku curve, when l-l =0 (or R=2Ro) If l -1 =0, it follows from equation (1) that either 40 =0, which is the equation of p, or

r =mcos~p, which is the equation of a circle through P with its centre on p. Hence, the ku curve degenerates into p and a circle through P(0, 0) and M(m, 0) with its centre on p. The k a curve keeps its general shape. The circle that is a branch of k,, and the inflection circle intersect at the pole and at the only Bali's point.

/ /

' \

\ \

/ / • h¢.

/, " , \ \ \

! ,,\ _~ ~ p t , , < ' ~ ' t " , \ ! a,.',.~ ":e;'7

. -. /; ( xOlelo IF~,l , , ,'

\ \ '\

FIG. 19.

Jcrk-fr~ Geneva wheel driving 257

q . ~ . . ~ ~ , . ~ . • ° 0'1 ~ °

~- -" ~ . ~ ~ ~ ~ 7. ~.

• o ~ " .

O0 r ~ o ~ ~ '~

._p:ku

2 5 8 E . A . D I J K S M A N

4.3.1 The Jour-bar linkage in tile position, where R =2Ro. In this degenerated position of the four-bar linkage there are three possibilities for placing A and B on k,,.

(a) B on p and A on c - k , . From the theorem of Hartmann it follows that Bo =P. A~, in addition, BA = B B o =BK, point B can only be taken in the centre of c - k , (see Fig. 191.

Point A can be chosen freely on c - k . , while the centre of curvature Ao corresponding to A is given by the equation PA z = A A w . A A o. Since the coupler point Ki s a Balrs point, this point turns out to lie exactly on the axis o f symmetry of the coupler curve. This means that in any case 2r = 180 °. Hence the mechanism can be applied to drive a two station Geneva wheel only [1].

i r

I

Eo c~gp~er curvl~ i

| / , axis of s y m m e t r ~

7 [

I ' \ , ; "\ L ' "-. \\ i 1 ~ - - . - . . -C- , , U j L.

W

P=ku _

¥ FIG. 2 0 . V,

/ i

J

,~ J A 2

tqcL 2 I.


(b) A on p and B on c-ku. In this case Ao = P (see Fig. 20). Consequently, points B, Bo and Ao are in a straight line. In Fig. 21 the coupler corresponding to this position is indicated by A1BI. The position in which the coupler point has arrived at the opposite point with respect to the axis of symmetry, corresponds to the position in which turning-point A has arrived at the opposite point A2 with respect to the fixed link. It can now be seen from the figure that the two positions of the coupler can be obtained from one another by rotating it about the rotation centre P12 =B0. Moreover, ABoAIBI =---ABoA2B2. So ~ B2A2Bo = ~ BIBoA2. This indicates that in the second position the coupler runs parallel to the fixed link, in which case ku and k, are both degenerated: this is the ease which we have already discussed in part (b) of section 4.1.1.

(c) Both A and B on c--ku (see Fig. 22). As 4c QAP= ¢: BMP and as from the theorem of Bobillier it follows that ~c QPA= ~cBPM, :. APQAIII'fAPBM. Hence, ,;cPQA= ~PBM=90o. This means that the collineation axis PQ.LAB.

In Fig. 23 the coupler triangle corresponding to this position is indicated by AtB1KI (P1QI being assumed to be perpendicular to AtBI). By rotating the crank AoA, the coupler triangle arrives at a second position A2B2K2 if A2 lies opposite to At with respect to the fixed link. The corresponding coupler points K1 and K2 are symmetrical with respect to the axis of symmetry of the coupler curve, as can be seen from the fact that the two coupler triangles can reach each other's position by rotating them about the rotation centre P12=Bo, and from the fact that the axis of symmetry goes through Bo.

A third position of the coupler triangle, which is in an opposite position to the second with respect to the fixed link, is shown in the same figure. It will be clear that any geometrical property attached to four- bar linkage No. 3 can also be applied to four-bar linkage No. 2.

Now the first and the third four-bar linkages have the crank and the fixed link in common, while the remaining four sides form a rhombus. It will be shown by means of Fig. 24 that the collineation axes ql and q3 of the linkages 1 and 3 are parallel, even in any other position of the common crank.

Proof. From AP1AIBIII[AA3P3B3 it follows that PIB1 .P3B3=AIB1 .A3B3, so with A Q1B1B 0 1tl ABoB3Q3 we find that

QIB1. Q3B3 =BIBo.B3Bo:A1B1 .AaB3=P1B1 .P3B3. This may be written as

so with

we have

(P3B3/ Q3B3) =(QIB1/P1BI),

¢~ P3B3Q3 = 9:QtB1P1

AP3Q3B3II[ A QtPxBa.

Hence ~ B3P3Q3= ¢: BI QtP1, so ql is parallel to q3.

Since the coUineation axis Px Q1 of linkage 1 is perpendicular to the coupler A1B1, the collineation axis P3Q3 of linkage 3 will be perpendicular to the rocker B3Bo. The same property holds for linkage 2.

From the theorem of Bobillier we can then deduce that the crank AzAo is perpendicular to the pole tangent P2. So circling point A2 lies on n2. The equation of ku2 shows that if one circling point lies on n2, n2 must be a branch of ku2. Another branch of ku2 is a circle through P2 with its centre on n2. So B2 lies on c-ku2 as long as position 2 is not a folded position. This leads to the position already investigated under part (e) of section 4.1.1, so that no new viewpoints arise from this consideration.

4.4 Degeneration of the k a curve, when lo t =0 (or R o = 2R)

In this case only the ka curve degenerates into p and into a circle through P(0, 0) and M(m, 0) with its centre on p. Bali's point is found at the intersection of the inflection circle with a line through P in the asymptotic direction of ka. This line coincides with the pole tangent p, so only the pole P can be a Bali's point. Since P is also a cusp, it is not suitable as a coupler point; so in this position there is no solution of the problem.

4.5 Degeneration of the k u and ku curve, when P-~oo

Both the ku and ka curves are cubic. In general both curves pass through the two isotropie points and through one asymptotic point. If the double point of ku and ka, i.e. the pole P, becomes infinite then the line at infinity will intersect each cubic curve in not less than 5 points. This is only possible when the line at infinity is in itself a branch of ku and ka. Then the pole tangent is an asymptotic line of the remaining conic sections, since the two curves touch the pole tangent at P®. Hence the two conic sections arc hyperbolas.

The corresponding position of the four-bar linkage will be reached where the crank AoA runs parallel to the rocker BoB, as occurs if P=Poo (see Fig. 25).

t Is similar to.

M,c

/ / / / // / / [ I J I

O

!.

FKL

22,

/ f

/

rs B

(,' .-4~:-~ cou

pler

cu

rve

"M

axis

o~

sj

mm

et

r~ _

p=k~

/ / ..

....

. ~

B

w

/

/i r~

.> > z


/ j /

~E

!o_ ~-

oj I i i

IN

f T .

/' /

/;

/' /

// , ""

// /" ~1

I I L

\\

\ "

. B

~

-\

b c

qll~

q3

> > Z

Yl~

,.

24

.

ortho

gona

[ hyp

erbo

t; ku

\ ka

or

thog

onat

hype

rbota

'~i = \/

/>z

\

FIG. 25.

264

\ "\

\ \

x%

F. A. DIJKSMAN

K= U / /

ku

/

f 5 z <

A

~Y" ~ \,

~ . ~ - ~ i.c.p=ku BO •

-\ '\

Flo. 26. L \ \

If Q is the collineation point in this position and S the point of intersection of the coupler AB with the pole tangent p, it follows from the theorem of Bobillier that QA =BS.

It is further known that any line intersects a hyperbola in two points; the middle of the distance between these points coincides with the middle of the distance between the intersection points of the line with the asymptotes of the hyperbola. Hence the second asymptote of the hyperbola passes through the collineation point Q. Since the second asymptote runs parallel to the pole normal, it is perpendicular to p.

A similar reasoning holds for the hyperbola that is a branch of k,. Hence both hyperbolas are or- thogonal.

The pole tangent p is a branch of the inflection circle, because to each point of p is added a centre of curvature coinciding with P=P~, as can be seen from the theorem of Hartmann. So in general there is no finite Bali's point that could be a finite intersection of ku and the inflection circle.

This is still true for the case in which the hyperbola that is a branch ofka, degenerates into its asymptotes p and AoBo. But, if the hyperbola that is a branch of ku degenerates into its asymptotes p and AB, then any point of p becomes a Bali's point (see Figs. 26 and 27). In Figs. 26 and 27 the points A, B0 and K lie on a chosen circle about B, of which K lies also on p since it must be a Ball's point. In Fig. 28 the considered position of the four-bar linkage is indicated by (AoA1BIBo) and will be referred to as position,Wl for short. In position 2 the erankpoint A has reached point A2, the opposite of A1 with respect to the fixed link AoBo. Hence

AA2B2Bo=_~A1BIBo, so -'g A2B2Bo= .,): AIBIBo=900.

Sinceit follows from position 1 that d2=b2+(b--a) 2, AzAo coincides with A2B2, and therefore position 2 corresponds with the cardan position; this case was discussed in section 4.2. We conclude that starting from position 1, we can only expect to find solutions such as have already been calculated with the cardan position as the starting position.


/

4"

h.

Ao

b

. /

_ J a=sof / / coup le r cu rve l s y m m e t r y /

.," Eo K:U~

ku

or hogon hyerbot

ka i. -p=k u FIG. 27.

• .~ ~ ' /

J

J Qf

266 E.A. DIJKSMAN

5. INVESTIGATION OF THE MAIN PROBLEM

In what precedes, all possibilities of degeneration of k,, have been investigated in connec- t ion with the product ion o f V-shaped, symmetrical coupler curves by four-bar linkages.

When k , is not o f degenerate form, we get one more degree of freedom in design. This additional degree o f freedom may then be used for different purposes. It allows us. for example, to choose the coupler point K on the produced part of the coupler for any given angle 2~. (The choice corresponds to the condit ion 7~ =0°.) The corresponding dimensions are calculated and given in Table 3. Besides, Figs. 29, 30, 31 and 32 show how the Geneva wheel moves with respect to time for 2v =45 °, 60 °, 90 ° and 120'-'.

TABLE 3. SOLUTIONS WITH B INSIDE THE INFLECTION CIRCLE AND )'1 ~ 0

2r

a/b

120'

0.4988

90"

0.4630

72

0.4187

d/b 1"1514 1"3018

Pl 68.80 56-14

P2

BoEo/b

EoUj/b

0 62"78

1.4015

48 94 ~

60 45' 30"

0'3787 0 " 3 1 5 2 0-2339

I 4749 1"5773 1"6952

a4 10 37'73 30'58

38.09' 49-59 58.86 66.47 78.25 93 -87

1.0688 0'8715 0-7597 0 6846 0 5861 0-4756

0.4870 0.4257 0.3856 0.3564 0.3154 0.2649

58.75 '~ 55.77 ~ 53-62' 50.82 47 90

h/h 0.7607 0.8746

(pTl)/b 0.2877 0.1058

(pT2)/b 1.1346 1.4825

1.4T

7756-07

7.51

63"98 Cm

0.9133

0~0546-

0.8432

11 '05 I

20-87 11.81

0"9219 0"9046

0'0330 0-0154

0"6260 0'3598

13-35 16'17

6"45

0.837t~

0.0055

O. 1468

18.90

3 77

It is also possible to pursue an optimal #ml, for each angle 2z. The dimensions complying with this op t imum are represented in Table 4. With the aid o f the data taken f rom this table, drawings have been made o f the driving mechanisms for n =3 , 4, 5, 6, 8 and 12 stations (see Figs. 34 to 39). Moreover , graphs o f the resultant mot ion of the Geneva wheel are shown in Figs. 40 to 43 for n = 8, 6, 4 and 3 respectively.

The calculations behind the general case are based on the following construction (see

Fig. 44):

(a) Draw the pole tangent p, the pole normal n, and the inflection circle.

Lock

ed

__p_

e~_

__

2 S 0 °

.129

,18 °

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ank

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the

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. 29

. C

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ven

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a f

ou

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15

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.82

° d/

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1.5

773

h/b

=0

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46

P

min

_ 3

7.7

3 °

yl-

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0 °

BoE

o/b=

0"58

61

2r=

45

°

EoU

1/b-

-0"3

154

Fo

r th

e m

ea

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g o

f th

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mb

ols

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Fig

. 33

.

Lock

ed

peri

od -4-

-22 so"

Lock

ed

Locked

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l pe

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I -126.31

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\ \

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lar

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the

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vel

ocity

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the

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en

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of t

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term

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ry

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>

7.

7~

> Z

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. 30

. C

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ism

dri

ven

by a

fo

ur-

bar

hn

kage

, a'

b

0'3"

787;

0=

53

"62

°;

d/b=

1"4

749;

h/

b=O

,921

9; lt

mm

=4

4"1

0:;

71

=0'

00~

: B

oEo/

h--0

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6:

2r

60

: E

oUl/h

=-0

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4, F

or

the

mea

nin

g o

f th

e sy

mb

ols

see

Fig

. 33

.

|Kkod


-7"~.~,t

..iS*

FIG. 31. Curves showing the motion of a 4-station internal Geneva mechanism driven by a four-bar linkage.

a/b=0.4630 0=58.75 ° d/b = 1 "3018 h/b =0.8746

/Zmin =49"59° ~'l =0 "00°

BoEo/b=0.8715 2~=90 °

EoUdb =0-4257

For the meaning of the symbols see Fig. 33.

270 E. A . D I J K S M A N

i I

_ _ + _ _

I

y

- i

I :+50" ] -

- 1

" + 1

i

I

m

a/b -- 0 . 4 9 8 8 dlb 1 .1514 //min £ 3 8 " 0 9 '

BoEo/b 1 "0688 EoUI/b = 0 " 4 8 7 0 0 = 62,78'"

bib == 0 . 7 6 0 7 } ' 1 = 0 ' 0 0 '

2 r - 120 '

locked per~,~g~d

FtG. 32.

._]_

F

L

'ii I

1

/

..... I---- +---÷--- ;- i ' ] i

i

i/ ; i : i

j

I 11"/~22 e . . . . ~ -

I

! ' 1 '

-4 - +

i !-*

- ~ ,~ T ~ i + - - - - £ - - % o -

? Lo : angul~r ~Loc,~ of th~ ~rooved wheQ~

~ - i ~°~ :~;2 :;Y2'4', 2::2; ..... , / -P . . . . . . ~ T - - T - - r - - : . . . . r - - -Y-~

~ ; . 2 ..2,2~2 ~.,,o-~U. =-~-U_5

I i * t ' q - " : i , : ~ + ; t - 1 % - 1 T--~

f2_ L_ _J4L_2_ i] i _ mov~ pz~jod . . . . . . . ~locke~ p~r~.d

For the m e a n i n g o f the s y m b o l s see F~g. 33


,T,

I

'o

j ~

\

\

\

\

d

/ J 8o

FIG. 33.

bo

.,-d

TA

BL

E 4

. S

OL

UT

ION

S W

ITH

B

INS

IDE

TH

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LE

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ION

CIR

CL

E A

ND

fll

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/2 =

/-/r

ain

2T

a/b

d/b

)'l

flm

in

(BoE

o)/b

(Eo U

, )/b

0 h/b

fpT~

)/b

(pT2

yb

~B

15"

0.1

25

1

1.4

08

6

+3

7 "

27 °

79

"84

0.0

79

5

0'0

73

4

48

.58

~

0.3

50

7

0.0

00

3

0'0

10

4

0.5

ff

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3ff

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33

7!

1-3

94

7

+2

9-1

6 ~

70

.97

'

0'1

91

5

0"1

58

6

51

.66

°

45

~,

0-3

21

0

1'3

77

2

+2

0-8

3

63

-7Y

0.3

38

7

0.2

45

3

54

.07

°

0.6

36

1

0-8

28

4

0.0

02

9

0.0

11

1

0.0

85

8

1.98

°

8.0

0

0.2

69

0

4"0

8 °

8 "8

8

60

°

0"3

86

2

1.3

60

4

+1

2.4

2 °

58

.30

~

0"5

17

3

0'3

22

8

55

'8T

0'9

20

5

0-0

28

7

0-5

45

4

6.2

8 °

12

-66

72

~

0"4

23

6

1'3

49

2

+

5-7

6 °

55

.13

~

0.6

77

0

0-3

73

9

56

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~

0-9

28

1

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1

0.7

98

1

7.8

5 °

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0.4

58

2

1.3

37

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--3

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31

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3

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71

9

9.6

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69

2

1.3

34

0

--

11.7

7 °

51

.24

°

1'1

46

6

0"4

56

1

58

-70

°

0"7

27

1

0.1

79

8

1'4

53

4

10

-46

°

24

55

.7

12

0 °

0"4

64

3

1"3

35

7

19

-22

°

51

.66

°

1.3

55

1

0-4

65

7

59

.33

°

135

~

0"4

41

9

1'3

43

3

26

.30

°

53

"57

"

1.5

50

9

0.4

56

4

59

.92

~

0-5

71

0

0.4

00

7

0'2

87

1

0"4

54

1

15

0 °

0"3

96

2

1'3

57

5

32

.98

°

57

-45

~

1'7

28

7

0'4

22

5

60

.48

°

0.2

33

3

0.7

45

0

165

°

1-3

79

8

39

.24

°

64

-68

°

1.8

83

2

0'3

44

2

60

.95

~

0"0

87

8

1.4

15

0

175

°

0'1

94

3

1.4

00

7

-43

.15

~

74

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"

1'9

67

7

0'2

25

0

60

.98

~

0'0

17

7

3,0

96

0

1.7

11

o

1.9

74

3

2.3

18

3

3.o

14

o

4.7

13

o

1o

.69

- lO

.24

~

9.0

3 °

6

.72

°

3.9

1 °

t49

.o7

2

7.9

2

1o

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5

.37

3

.45

rd~

~r

z


A,

A

\

/

/ / 1

a Ao

U1

' \ \

J \ / . . /

f _

r- o

\ \

t'Lmm: 51"66°

y1=-19.22"

FIG. 34.

\ =90 °

Ao

- " [ ~ 1 1

) ~min= 52.B °

[

¥1--'3"97 ° FIG. 35.

/

274 E. A. DIJKSMAN

,F

/

/ j t

i, o p, mm: 55'13

a Ao d " ' ~ - ~

F]o. 36.

Y1 ~ 5 . 7 6 °

' P.m~n = 5 8 ' 3 0 "

2~ d

/ L

2Z~60o /

Vi=12.42 °

FI

Jerk-flee Geneva wheel driving 275

. /

~ rn~n`63`75° U~

. , ~ l l l , ~ t°' ; ~ A ' E o

A / /

t .'- ~J~A° ,' , y~f:i2~13o

FIG. 38.

E ~ ~, :- ..%°. . ~ ~ i.,t mi n :, 70.97 ~, 1 + ' ~ j . x

." i f

, i

" Yi-- 29.'16 *

F~¢;. 39.

toct

ed

rock

ed

perio

d =[

m

ovin

g pe

riod

~.

perio

d

I÷

1 o

I I

I I

I i

\ /

! !

t "~

)

~O=W

ot ~z

~,

~ s0

o E)

~...

! so

" ',

I 2-

-" "T

''

,,o"~

-.~a~

,z~.

(.do

: an

guta

r ve

loci

ty o

f th

e dr

iven

cra

nk

of

the

inte

rmed

+ary

m

echa

nism

FIG

. 40

. C

urv

es

%('l

O\V

lng t

he

mo

rio

n

of

an

,~-,

ccfi

on i

nte

rnal

Gen

eva

mec

han

ism

d

riv

en

by

,l

fou

r-b

ar

lin

kag

e.

a/b=

0.32

10

0 =

+5

4-0

7

d/b

:- 1

.37

72

h/

b =0

"82

84

Pm

in -

- 6

3"7

5

71 =

+

20"8

3

BoE

o/b

0'3

38

7

2r

45

'EoU

I/b

0"2

45

3

1~i

I;2

1-:o

1" th

e m

ean

ing

o

f th

e s.

',m

bo

N s

ee F

ig.

33

7T

> ~u

Z

rock

ed

peri

od

mov

ing

peri

od

rock

ed

I_pe

rlod

I--

"I;:

30 °

i I t t t I 11

24.1

8o

1 I I [

I !

I

-lO 3

°

I I

--..

..

~/t

Oo

\

\ \

.50

o /~,,,

, [ra"

l \

\

I

. O.S

__

_

/ /

o/ \

t

I t

i I / \

-0'5

J J

\

F[o

. 41

. C

urv

es

sho

win

g t

he

mo

tio

n o

f a

6-s

tati

on

in

tern

al G

en

ev

a m

ec

ha

nis

m

dri

ven

by

a f

ou

r-b

ar l

ink

age.

a/

b=0.

3862

0

=5

5.8

2 °

d/

b =

1.3

604

h/b

=0

.92

05

/J

min

= 5

8" 3

0 °

71 =

12-

42 °

BoE

o/b=

0.51

73

ltl

=P

2

EoU

t/b=

0.32

28

2z

=6

0 °

50 °

\/

/

/

J J

/

\ f

q):'t

aot

103

° ~

_ IU

=

~ 12

/.1.1

8o

.30

° I tO

:

an

ou

tar

veto

citv

o

f th

e ar

oove

d w

heet

.

(do

: an

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r ve

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CO : angular va~oclty ot ~h." groo*ad wheet

angutar veloclty o| ~he driven crank

of ~h~ in ~rm~d~arv mechar,lsm

FIG. 42, Cur~es s h o w i n g the 111o[1oll O1" ~l 4 - s t a t i on intc~-n~ll (Jcnc\~a m e c h a n i s m tit ixcn b\ ,~ f o u r - b a r l inkage .

a/h ~ 0.4582 0 57.9~

d/b 1"337q h/h 0"8506

lqn,n 52"1~ ,r ~ 07

BoEo/h = 0.93 ! 6 ,'tl / t ,

EoUl/b 0.4293 2r 90

[ o v the m e a n i n g of tile ~ m b o l s ~cc Fig. ~3.

Jerk-free Geneva wheel driving

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b \ \

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l l l L i i l i Curves show;rig the mot ;oh at a

) . s t a t i o n i n te rna t Genevs mechanism

driven by a tour .bar t;nkage,

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moving period

F~G. 43. For the meaning of the symbols see Fig. 33.

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279

280 1. A D I JKS MAN

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(b) Choose polar co-ordinates rB and tpn of the turning-point B with respect to the pole and the pole tangent in such a way that

1-_ { 1 - sin q~B "X t']

if B is chosen inside the inflection circle, and

"-[-1 ' { 1+sin q). '~t'] rs/6-->½sinq~8 L -r~ l_-'~-~sin;e ) J '

if B lies outside the inflection circle. (In any case we may confine ourselves to the first quadrant for the choice of B.)

(c) Find B o from equation PB 2 =BBo. BBw.

(d) Draw circle b with its centre at B and radius BBo = b.

(e) Choose the coupler point in one of the intersections K1 and K z of circle b with the inflection circle.

(f) Find the Bali's point U in the chosen coupler point K.

(g) Draw a line through K perpendicular to the path normal PK. The line segments which this line cuts from the two axes p and n form the sides of a rectangle with K. as the fourth vertex.

(h) Determine point Bo in a similar way, starting from point B.

(i) Draw line K.B. which intersects p and n at M and L respectively. Then with P M = m and PL =l, k , is completely determined. (As is shown by Beyer [5], k, is determined by p, n and two points of ku, the points being U and B in the case under observation. This is done by determining the line M L as before, from which other points of k, can be derived by reversing the process.)

(j) Point A can then be located at one of the intersection points of k, and circle b. As k, is a circular curve, the isotropic points are to be counted as two intersection points. Since we forced Bali's point on b and k,, this point of intersection may not be taken as a turning-point. So from the six intersection points of circle b with the cubic k~, only three of them remain from which to choose turning-point A. Two of them may or may not be complex. If all three intersection points for the choice of A are real and b intersects the inflection circle in two different and real points K, six solutions exist in addition to the chosen polar coordinates r n and tp n of B.

For a real point A on b, the added point A° must lie on KoB.. Hence this leads to the construction of the real intersection(s) of the cubic bo with K,B., as can be easily effected by way of trial and error.

(k) Find the point Ao from point A using the equation PA 2 = A A o . A A w.

(1) Draw the coupler curve, check the position of the axis of symmetry and judge the length of the straight guidance.

282 I:. A. D I J K S M A N

Remarks. The limitation in tile choice of tile position of turning-point B has three causes:

1. the equivalence of the left- and right-hand half plane; 2. the very existence of real intersections K, and K 2 of circle h ~ ith the inflection circle:

and 3. the equivalence of solutions corresponding with the two different poqm)ns of the

coupler point in the two Ball's points of the coupler curve. The last point will require some further explanation: The tuo different mflecnon circlc~ corresponding to the two positions of the coupler ill which the coupler point coincides with a Ball's point, can be made of the same size by increasing or decreasing one of the two four- bar linkages. This makes it possible to indicate the positions of two points B producing equivalent solutions, in one moving plane (of drawing), in which the lnutual relation betu.een these points must be investigated.

Let the two positions of the coupler be represented by A ~B, and A,B,. A~ is knou n, the points U, and U2 are opposite points to one another with respect to the axis of symmetry of the coupler curve, the latter going through Bo. The perpendicular bisectors of the distancc~

UtU2 and B,B, will then intersect at P,2 =B0. So ,4, and A 2 are opposite points to one another with respect to the fixed link AoB o. To study the relationship between the t~ o positions of the four-bar linkage, it is more convenient to replace the second by its image with respect to the fixed link. The positions under consideration will then be A~B~ and ,43B ~, A 3 coinciding with A,. These positions are shown in Fig. 24. It is further proved in part (b) of section 4.3.1 that PIQ~ runs parallel to P3Q3. F'rom the theorem of Bobillier, it can then be seen that gp,P1Bx = ~zA~P,Q, = -.):.A3P3Q 3 = ~ P 3 P 3 B 3 . Hence the two related points B, lying in one moving plane as mentioned above, are on the same path normal. II i,

also proved that P~ B,. P3B3 == B I B o . B 3 B o.

N o w we h a v e P~ B~ =B,Bo. B,B,,.I a n d P3 B2= B3B,). B3B~,. 3 s o B 1B,, i. B3B, , = 131B o . B3Bo,

o r

(6, sintpn, - P, B,)((/3 sin(pR, - P3 B3 ) =-- BI Bo. B3 Bo = PI/~1 • P3 B3,

consequently with (PB, =¢Pn,, for the t~o four-bar linkages the follmvmg relation holds:

PIBI ( P 3B301/()3) ~ - ½ s i n tp m = ~sin (p~, fi, ,5,

In the moving plane of drawing the two couplers have the same inflection circle' 4o for the two related points B in this moving plane we have the equation

rn-½6 sin ~%= }6 sin CPB- r}~.

Hence, the midpoint between two related points B of the moving plane must coincide with the intersection of a path normal with a circle which touches the pole tangent in P and passes through the centre of the inflection circle (see Fig. 45). (In general this midpoint does not coincide with P.) The equations referring to this general case have been omitted. They can be found in the author's thesis [2].


REFERENCES

[I] K. H. HUNT, N. FINK and J. NAYAR, Linkage Geneva mechanisms. A design study inmechanismgeome- try. Proc. Mech. Inst. Engrs. 1'74, 643 (1960).

[2] E. A. DIJKSMAN, The Four-Bar Linkage as a Driving Mechanism (in Dutch). The Technical Publishers, H. Stam 0964).

[3] A. S. HALL, Kinematics andLinkage Design, pp. 97-105. Prentice-Hall (1961).

[4] SIR ROBERT BALL, Dublin Proc. (2) 1, 243 (1871).

[5] R. BEYER, The Kinematic Synthesis o f Mechanisms (translated from the German by H. Kuenzel), p. 208. Chapman & Hall 0963).

[6] J. HIRSCHHORN Kinematics and Dynamics o f Plane ~techanisms, pp. 349-356. McGraw-Hill 0962).

[7] K. RAUH and L. HAGEDORN, Praktische Getriebelehre I: Die Viergelenkkette (3rd edition), Vol. l, pp. 95-96. Springer-Verlag 0965).

Jerk-free Geneva wheel driving - Pure - Aanmeldencoupler curve when the mechanism is moved (see Fig....

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Transcript of Jerk-free Geneva wheel driving - Pure - Aanmeldencoupler curve when the mechanism is moved (see Fig....