Single-disk cam mechanisms with oscillating double roller follower

Mecham,mlandMacluneTheoryVol 19, No 4/5, pp ,1~3-415, 1984

0094--114X]84 $ 3 0 0 + 00 Pergamon Pros Ltd

Printed m the U S A

SINGLE-DISK CAM MECHANISMS WITH OSCILLATING DOUBLE ROLLER FOLLOWER W. WUNDERLICHt Department of GeomeWy, Technical Umverslty of Vienna, Gusshausstrasse 27, A-1040 Wien, Ausma

(Recetvedfor pubhcauon 10 March 1983) Al~trad--Planar cam mechamsmsof the Idnd consideredhere are subject m two condtUons[Duca and Sunionescu,

Mechanism andMachine Theory 15, 213-220 (1980)]. By weakening one of the conditions, wluch is resmcuve, mine practicable forms are aduevable_ The design is based on the choice of a certain part of either the cam profile or of the transmission law Mathematical analysis leads to a functional equation, which is the same as that m the sinular problem with flat-faced follower pairs, solved m a previous paper [Wunderlich, J. Mechanisms 6, 1-20 (1971)] All formulas for synthesis are developed Analytical solutions, among them algebraic ones, provide pamcular examples of plane curves with a movable closed chord polygon.

1. INTRODUCTION

For a flat-faced follower--p being a straight line (passing through P ) - - t h e analytical determination of the cam profile c has been given in a previous paper[2]. In the same paper the problem of singledisk cams operating a flat-faced follower pair has been completely solved: The respective cam profiles c are charactenzable as "curves with an lsoptic$ ctrcle"[3]. The simplest example, indicated already by Goldberg[4], uses an elhpse c to operate the arms of a right-angle follower pair; see [2], p. 7 Circular follower profiles are of special interest for roller followers; here the follower arm has a circular roller p of radius e at its extremity A. F o r this type of cam mechamsm it is convenient to consider the relative path F of the roller center A" The working cam profde c is equl&stant (at the distance e) to that more important "pitch profile" F (Figs. 5 and 9). In [2] the author doubted the existence of single-disk cams with an oscillating pair of rigidly connected roller followers that function precisely. This wrong conjecture, caused by omittang a very specaal arrangement, was corrected in a recent note of Duca and Sirmonescu[1]. To delete an unnecessary restricUon and to clarify the character of adrmssible transmission laws, the subject will be treated anew Moreover, a close connecuon will be revealed between the actual problem and the settled problem of flat-faced follower pairs

Cam mechamsms serve to transform a rotaUon into an oscillating motion. Such a mechanism consists of three elements: The fixed frame F,o, the rotating cam ZI, and the oscfllaang follower Z 2. In general the follower is combined with a spring wluch presses it against the cam to secure steady contact. To avoid vibrations which may occur m high-speed mechanisms, the spring may be replaced by a second follower, rigidly connected with the first and operated by a second cam integral with the first cam. The idea of combining the two cams into one restricts the variety of possible transrmss~on laws and leads to rather difficult geometrical problems[2]. In the commonly used planar type of cam mechanisms the motions Z./Eo and Z2/Zo are rotations about parallel axes. Reducing everything to a design plane orthogonal to the axes, we have a disk profile c turning about a center O, and a follower profile p rotating about a pwot P and always touching c at a varying point T(Flg. 1). Denoting the rotation angles of ,Y1 and Z, by 0 and ~/, respectively, the behavlour of the mechanism is charactenzed by the rransmisswn /aw ~(0), a function with period 2n To construct the u n k n o w n cam profile c for a given follower profile p, and for prescribed transmission law ~ (4'), we consider the mechanism from the point o i w e w of an observer in the cam system ,YI: c appears then as the envelope of aft the positions o f p occupied in the course of the relative motion ,Y2/~- The c o m m o n normal of c and p at the contact point T passes through the instantaneous pole I of this motion. According to the well-known A r o n h o l d Kennedy Theorem, this relative pole I is s~tuated on the hne OP and dixqdes the segment OP m the raUo 0I: PI = d ~ : d 0 (F~g. 1).

2. EXISTENCECONDITIONS Let us consider a planar single-disk cam mechamsm with pitch profile ~ rotating about the center

tEmentus Professor of Geometry. CAn lsopac curve is the locus of points from which two tangents to a given curve make a constant angle. 409

~

c

Fig 1. Cam with osollating follower

410

W WLrNDEILLICH

q

where 2 and # < 2 are natural numbers without common factor. The corresponding condlUon (2) in [1], demanding an integer value of n(p = l), imposes a superfluous restriction, which is n~schievous for the actual purpose (compare Figs. 5 and 9) However, there is another restriction to be mentioned, it will turn out m Section 3 that the characteristic number n = 2/# must have an odd denominator p, otherwise the pitch profile E would cross itself. We shall see that under circumstances that satisfy the three stated conditions the construction leads to suitable results, although only small numbers ,l and # are useful for practical purposes.

h

,

\

,hi "hi

//

Fig 2. AdJaCentpomhons of a double follower

3. CAM DESIGN

O. The arms PA and PB of the double follower are supposed to have equal length PA = PB = 1; the case of unequal follower arms seems to be rather hopeless. The mvariant angle 4: APB = 2~e will be measured in that sector which does not contain the cam center O. The distance from O to the follower pivot P is denoted by L (Fig. 2). Starting from an arbitrary position A~P~B~ of the follower, and observing all from the cam system ~j, we identify after a certain tame an adjacent position A2P2B2 with A2 = B t Proceeding m this way, we obtain an equilateral polygon AtP~A2P2... of side length/, whose vertices P, are situated on a circle h with center O and radius L, whereas the vertices A, are alternatively distributed over two concentric cardes h~ and h2 (Fig. 2). These points form a chord polygon AtA2A3 .. of E with constant side length s = 21. sin `'. Each of the chords Av4,+~ = A~B~is seen from the center O under the same angle ft. In general this angle is variable, and it depends on the value x/B whether the polygon is closed or not: For n i t ratiohal, the polygon closes after a finite number of steps; othervnse the polygon never closes. Consequently, in general (i.e. for fl # cons0 the pitch profile ? will contain, at any instant when n/IJ takes an ~rraUonal value, two infinite and everywhere dense point sets { A , A , A , . . } and {A2A4A,... } on certain concentnc cireles h~, h2, and this circumstance excludes pracucable earns[2]. The situation, however, m quite different m the case # = coast, overlooked in [2], but investigated m [1]" Observing the mechanism from the follower system ,r:, we can say that all points O from wberat the (fixed) segment A B subtends a constant angle fl must lie on a circle passing through A and B. Due to OP -- L = coast, this circle has its center at P, and we obtain the necessary condition

Let us measure the rotation of the follower A P B by the angle ~, which the bmector of the angle 4: APB = 2`" makes w~th the base line OP. Considering the mechamsm from the cam system 2~t, we see that two adjacent positions A P B and A ' P ' B ' vath A ' = B are m~rror images with respect to the axis OB, provided condition (2.1) is satisfied (Fig. 3). Hence the passage from A P B to A ' P ' B ' induces the change of ¢ to - ¢. Due to 4: A O B = 4: A ' O B ' = `" the next position A"P"B" w~th A" = B' is related with A P B by rotaUon about O through the angle 2,, Continuing the procedure of adding adjacent positions, we amve at a remarkable hnkage which is closed, if--according with condition (2 2)---the ratio g/= is a rational number n = 2//.z This overconstramed linkage consists of a chain of 22 isosceles mangles joined by 22 spokes w~th a fixed central hinge at O If all joints are movable, the linkage adrmts a one-parametric deformauon_ Smce all odd-numbered spokes form invariant angles 2a, they constitute a rigid system, and the same holds for all evennumbered spokes. Hence the linkage is composed of 22 mangles and two 2-gons; the number of (binary) joints is 22 + 1 Figure 4 shows the linkage for the case n = 3 (a = 6o°). From the fact that the deviation angle ¢ of the follower changes its sign d u n n g the phase A P B - - , A ' P ' B ' , it follows (for reasons of supposed continuity) that there exists at least one mean position A~P~B~ wath ~k~= 0. Starting from such a zero position, the chmn of adjacent potations AjP,B, with A, = B,_ l generates a regular polygon AIA2.. • A2a, provided /l is odd (condition 3), for # even, the

B=A'

1 = L,

stated as condition (1) in [1] As an ~mmediate consequence we have fl = =. Moreover, to exclude infinite chord polygons we have to assume a rational value of lt/fl = it/`', explicitly ~/`" = n = 2/la

I ~I~--~ / / / / ~

(2.1)

> 1,

(2.2)

/

.~t~/

,~'~_[' ~- " ~ = ~ B ' = A "

".

8

0

Fig 3 Chebyshev dyads forming a plagxograph of Sylvester

411

Single-disk cam mechamsms with oscallating double roller follower CZ ..

'43

•. '

X

"'%-.%--Cj

C;, !

A"

/ %,

n=3

A~

a=60"

plagiograph of Sylvester, composed of two Chebyshev dyads (Fig 3) Consequently the next portaon (3 = c~ = c'~ of the profile ~ is directly congruent wRh cL (Fig. 4). Analogously we have c4 "" c2, c5 "" c3 ~- cl, etc Thus, we state that the working profile c of the cam also adnuts the rotation w~th angle 2a about O, and likewise consists of alternating arcs of two kinds. Figure 5 shows a possible form of a single-disk cam mechanism with an oscillating double roller follower, based on the scheme of Fig. 4 (n = 3, • = 60°). The arrangement is somewhat unconventional, as the follower pivot P hes within the cam disk. 4. ANALYTICALSOLUTION

Ftg 40verconstramed hnkage

polygon would close already after 2 steps. All vertices A, belong to the required pitch profile 3 of the cam; for # even, they would be double points In the case of an integer number n(2 = n, # = 1) the regular chord polygon A~A2... A~ is convex. To construct a suitable profile curve ?, we may prescribe arbitrarily (but witban reasonable limits) the arc ct from At to A2 = Bt. The rest of the profile ~ is then completely determined" Using an auxiliary hnkage of the kind represented in Fig. 4, and leading its joint A along the given arc ?m, the other joints A', A", .. will descnbe the massing portions c2, c3,-.-, ~2~ of the required profile ~. In the case of a fraction n = 2/# (# > 1 and odd), the regular chord polygon A t A : . . . A 2 a is star-shaped (Fzg. 9). Now we may arbitrarily choose only a partial arc ?t from At to one of the two n e i g h b o u n n g vertices, but the way to find the rest of the profile ~ is the same as before. The angular distance from At to its neighbours is 6 = ±nil

The decisive point-to-point mapping X: A ~ A ' which transforms the imtlal are ~ Into a new part ~ = 72 of the profile ~, and generally any known arc ~, into another part ?~ = ~,÷~, is mechanically affected by means of a simple device consisting of the mangle APA' and the crank OP turmng about O (Fig. 3) With respect to OP = PA = PA' = I this mechanism of freedom 2 may be called a generahzed Chebyshev dyad, its original form appears with ,~ APA' = 2a = n(n = 2, Fig. 8) The mapping X, which is also simple to execute graphically, is an algegbraic two-to-two transformation of order 4: When the point A is led along a straaght line g (not passing through O), then the xmage point A' descnbes a (symmetric) coupler quartic g' of the shder-crank mechamsm g-AA'P-O In Fig. 4 the zmtial arc ~'j was chosen as a part of a ctrcle (center C~), consequently, the derived arc ~'~ = c2 is part of a (symmetric) computer sextic generated by the four-bar C~-AA'P-O. Repeating the transformation Z, we obtain the mapping X2 A ~ A " which is, due to OA = OA" and ,~ AOA" = 2% a rotation about the center O through the angle 2,,. It Is effected by means of a particular

Introducang polar coordinates r, 0 in the cam system Et, it may be convenient for the observer in Zt to count the polar angles in opposite sense (Fig. 3). The important mapp,ng Z: A(r, O ) ~ A ' ( r ' , 0') Is then determined--applying the cosine theorem to triangle A O A ' - - b y

r2+r'2-2rr'cos~=412sin2`',

0'=0+~.

(4.1)

The rotation Z2. A(r, O)--,A"(r", 0") is described by

r"=r,

0 " = 0 +2,,.

(4.2)

Prowded we know the polar equation r = r(O) of the chosen arc ~1 of the pitch profile 6 of the cam (Section 3), we find the equaUon r'---r'(O') of the corresponding arc ~'t = 62 by solving the quadratic eqn (4.1), thus, obtaining r ' = r cos Gt + ( x / / ~ - r 2) sin ct.

(4.3)

Wzth respect to condflaon (2.2) and the consequences revealed in Section 3, the function r(0) zs supposed to be defined m an interval of length 6 = 7r/i = ~/#, for instance in 0 _< 0 < 6. The associated function r'(O') is then obtained in the interval `" < 0' _< ,, + 6 = (# + 1)a/#. For reasons of continuity eqn (4.1) has to be sattsfied watt r = r(0) and r ' = r(6). All of the addmonal portions c3, c4. . . . of the profile ~ might be found by successively con-

A

.. "

•

n=3

AC,

"--'-'--'---

-'-'--" L_: . A~

-y .....

C"-"I"

--

Ae a ~ 6 0 °

Fig 5 Single-disk cam mechamsm wRh osollatlng double roller follower

412

W. W t y ~ m ~ c n

lmuing the procedure, but we know already that they are congruent either ~ath ?~ or w~th ?:. This fact is in accordance w~th eqns (4.2), although the latter are not the only possible consequences of eqn (4.1), see [3], eqn (3.5). Unifying now all the partial functaons defined in different intervals of length 6, we obtmn a periodic function describing the profile ~ as a whole. In order to find the transmtsswn law ~k(~) associated with the g~ven profile function r(O), we take from Fig. 3: cos (~b - 0) = r(O)/21,

~, + • = 2(~b - 0) (4.4)

By solwng these equations we obtain ~b and ¢ as functions of 0, hence a parametric representation of the reqmred law. Moreover, we see from Fig. 3 that the rotation angles ~b and ~b', belonging to adjacent follower positions A P B and A ' P ' B ' , are related by

8

a

-d

10" .

.

o . . . . .

.

. 6o*,

i

," ,' ,," ," ,"

,~o.

F~g 6 Transtmssion law of the cam mechamsm of Fig. 5

To construct a cam for a given transtmssion law ~b(q~) which satisfies the condition (4.9), the polar function r(O) of the pitch profile ~ may be derwed from eqn (4A). We obtain 1

0 = ~b - ~ (~, + a),

~+,, r = 21- c o s - - - - ~ ,

(4 10)

where ~b serves as the independent variable. @ + @" = 20' = 20 + 2¢t = 2c~ - ~ + ~t

(45)

With attenUon to the corresponding oscillation angles ~b and ~ , ' = - ¢ , we state a linear transformation

4" =,~ -~' +~,

~,'= - ~ .

(4.6)

RepeaUng this step, we attain ~b"---~b'-~b'+a =4 +2%

#"=~,.

(4.7)

This means that ¥(~b) is a periodic function with period 2et = 2n/n = 2/an/2 Talong into account the natural period 2n, it follows that any hnear combination u . 2a + v - 2rt with integers u and v is also a pened of ~u(~b). Choosing for u, o a solution pair of the Diophantine equaUon/zu + 2o -- 1, we find the fundamental period 2J = 2n/2 Hence it holds

5. CONNECTION WITH THE PROBLEM OF FLAT-FACED FOLLOWER PAIRS

The soluUon expounded in the precechng seettons depends on a suitable choice of a part of the pitch profile ~ or of the corresponding part of the transrmssion law ~,(~). Hence the complete cam profile is "patched up" of components of different kind: Compare the example of Figs. 4 and 5, where the profile is composed of circular arcs cl, F3, F5 and of parts c2, c4, 66 of coupler sextics. From the mathematical point of view an analytically closed solution would be more welcome. Thas means that the function r'(O") should be the analytic continuation of r(O) Hence r(O) should satisfy the functional equation r 2 + r "2 --

¢ ( $ ) = ~,($ + 26)

with

Introducang now ~, = / ~ in eqn (4 6) and remembenng that/z ~s odd (Sectton 2), we find the following pair of values also corresponding in the transmission law ~ = ~b-~b + h ,

2rr" cos tz = 412 sm 2 cz with r"

~ = x/g = ~/#. (4.8)

if-- -~,.

(4.9)

The geometrical interpretation of tins hnear mapping (~, ¢,)--,($, ~ ) means that the Cartesian graph of the law ~(~b) admits a certain affinity. Its effect is illustrated in Fig. 6 by applying it to the rectangle with the vertices (0, 0), (d, 0), (t$, e), (0, e); the image under (4 9) Is the parallelogram with the vertices (t~, 0), (2J, 0), ( 2 6 - t, - e), ( J - e, - t). Consequently, the transmission law ~ (q~) can be prescribed only in a relatively small interval of length 6 = x/2; the rest is completed by application of eqn (4 9) and repetition wath the period 26. In choosing the imtial part it is necessary to take care to avoid discontinuities and corners of the graph.

=

r(O + or).

(5.0 Tlus equaaon is of just the same type as eqn (4.1) is t h e key equation for the sLmilar problem of single-disk cam mechamsms with a flat-faced follower pair. Indeed there exists a close connection between the two problems: Erectang in Fig. 3 the perpendiculars p ± OA at A,p' ± OA ' at A ' etc., we obtain an equiangular polygon inscribed m a fixed circle with radms 21. Since Rs angles have the common constant value n - ~, the sides p, p', represent adjacent positions of a rigidly connected pair of flat-facedfoUowers operated by a single cam disk, whose profile c* is the envelope of the varying positions of the polygon sldes. Hence c* is a curve with the isoptic circle (0,2/); this characterisac property of c* was mentioned m Section 1. Conversely we state: Our pitch profile ~ is a pedal curve of c*, I e., the locus of the bases (A, A', .) of the perpendiculars dropped from the center O onto the tangents (/7, p ' , . .) of c*. Instead of adopting the analytical solution (4.9) from [2] and duly adapting it for the present purpose, in [2] w h i c h

Single-disk cam meehamsms ~nth osollatmg double roller follower

413

a.,4'

we prefer to develop the solutaon ofeqn (5.1) directly. First we introduce an auxihary Cartesian frame (O; X, Y) which rotates about the origin O in such a manner that the symmetral X = Y always coincides with the bisector of the angle A O B (Fig. 7). Hence the coordinate axes form equal angles 4: X O A = ~g B O Y = er with the rays OA and OB; their constant value is

rI

Y

O ,4

~r = ( n / 4 ) - ( ~ / 2 ) .

(5 2)

Fig 7. Auxlhary Cartesmn frame. The radial distances r = OA and r" --- OB are linear combmataons of the coordinates X and Y of the pwot P, namely

with

r = 2(aX + b Y), a = cos o,

r ' = 2(bX + a Y) b = sin a

(5.3)

Exadently the quantltxes X and Y are periodic functions of 0, with the same penod 2,, as r(O) and r ' = r(O + ~t). Moreover they satsify the relations x ( o + ,,) = r ( o ) ,

r ( o + ~) = x ( o ) .

(5.4)

These relahons correspond to the passage from A P B to A ' P ' B ' , and may be formally checked by means of the mversious ar - br" ar" - br X = 2 sin--------~' Y = 2 sin-------~'

(5.5)

Inserting now the expressions (5.3) in eqn (5.1), tlus functional equation reduces to X2+y2=l

2 with

Y(O)=X(O+~),

(5.4)

a simpler form of the condation 01) = L Due to eqns (5.4) the variable difference X 2 _ y2 = A

(5.7)

2n, has the fundamental period 2if and satisfies the conthtion ,4(0-6)=-/1(0)

(5.8)

6. A L G E B R A I C E X A M P L E S

A----rn2cos20

y 2 = ~(l 2 - d )

(5.9)

with

a=cos

-

;)

,

b=sin

(:

-

(0)] "

(5.10)

with

0
The resulting profile 6, represented by its polar eqn (5 10) or in Cartesian coordinates by

Making use of eqns (5.3), we finally attain the required solunon of the functional eqn (5.1). r = x / ~ [ a x / ~ + ,4 (0) 4- b / x ~ - / t

(5.11)

The simplest choice of a suitable auxiliary function d(0), satisfying condition (5.11), is

F o r an arbitrary function ,4 (0) we find X 2 = ~1( I 2 + A),

t~=~t/2=,,/p.

Consequently, d (0) can be chosen only m an interval of length cL To secure real values of the square roots in formula (5.10), the restncuon IA(0)I <12 is necessary. The associated transmission law ~(¢k) is to calculate as indicated m eqns (4.4). Another way to solve the problem has been used in [5]: Fastening to the follower A P B a third arm PQ of length l and which is positioned along the bisector of the angle APB, its endpoint Q will descnbe a certain path q in the cam system ,~. Since Q repeatedly passes through the center O, this path q has the shape of a rosette Choosing one of the congruent leaves of q (witlun rather narrow hrmts), and repeating ~t, the motion Z2/Zt is completely deterrraned by leading P along the orcle (O, I) and simultaneously Q along the rosette q. Hence the pitch profile 6 of the cam is kinematicaUy generated as the common path of the points A and B. The corresponding eqn (6.2) m [5], describing the profile 6, is easy to derive, but not so advantageous as eqn (5.10) above.

is a penodic function of 0 satisfying the relatton ,t (0 + ~ ) = - '4 (0).

with

x=r.cosO,

y=r.sinO,

(6.2)

Is algebraic, but m general not rational. In the case n = 2, for instance, we have ). = 2, ~t = 6 = ~t/2, a = 1, b = 0. The pitch profile ~, represented by

r 2 = 2(12 + m 2 cos 20) Argtung as in Section 4, we state that the auxdiary function d ((9), possessing the commensurable periods 2a and

or

(X 2 4. y2)2 = 2(12 + m2)x 2 + 2(! 2 _ m2)y2,

(6.3)

414

W. WUNDERLICH

is a bicircular quartic known as "Booth's lemniscate" (Fig. 8). Corresponding to a remark in Sectaon 5, it is the central pedal of Goldberg's ellipse[4]. Ttus curve E and also the equidistant cam profile c are ovals, if m: < 1:/3. The follower now has stretched arms, since A P B = 2= = it. Considering the closed chain of adjacent follower positions around the cam, we detect an apparently unknown property of Booth's lemniscate" It contains a movable chord rhombus A A "A"A " with constant side length s = 21 (Fig 8)_ The order N of a curve E belonging to the family (6.1) is the number of intersection points (real and ~mag~nary) wath any straight line, and may be found by considering the line equation A x + By = C, after inserting there the quantmes (6.2), (5.10) and (6.1), and then passing, by means of Euler's formula, to the algebraic parameter t = e '° The degree N of the resulting equation m t has (w~th excephon of the case n = 2) the value

N=

4(2 + 2) 2(2+2)

for 2 odd, for2even.

[l].

A vast variety of algebrmc cam profiles is obtainable by combining several terms of the kind (6.1) m a tngonometric polynomial

(65)

)

Statable choice of coefficients aj, bj will allow approximation to any wanted transrmssion law, and that without &scontmuitles of derivatives. Similar but less simple algebrmc solutmns are obtained by means of the other method, indicated at the end of Section 5, by choosing an algebraic rosette q (e g. a suitable hypotrochoid, see Fig. 6 in [5]).

n=2(==90")

,y

~

l=3, m=l

0 Fig 9 Cam mechanism and transmzssion law.

(6 4)

Thus the p~tch profile E in Fig. 9 which illustrates the case n = 4/3 is of order 12, here we have 2 = 4, - = 135 °, u = - 2 2 . 5 °, a =0.924, b = - 0 . 3 8 3 Since the double follower is right-angled and the pivot P is outside of the cam disk, the arrangement looks more familmr than in those devices which are obtained under the restrictave condition of integer values n m

A =~,(ajcosjgO+b:sinjgO), j odd

n=413

1=3, m=2

B=A'

13 ......

.....

Fig. 8. Booth's lemmscate with movable chord r h o m b u s

In 1939 Fischer[6] raised the problem concerning the existence of plane curves (apart from the circle) m which an equilateral chord polygon with a certain side length s always closes, wherever it begins. Obviously the solution of our cam problem offers a family of special solutions of the geometric problem. The chain of adjacent follower segments AB of length s = 21 sinai is a movable (and deforming) chord polygon of the pitch profile ~; see the rhombus in Fig 8, the hexagon in Figs. 4 and 5, and the star-shaped octogon in Fig. 9. Although specaal, as distinguished by the fact that all chords subtend the same constant a n g l e , from the center O, our solutaons are insofar remarkable, as they contain an infinity of algebraic examples, whereas Flscher's solutions, if analytical at all, are transcendent. By the way, for the purely geometric question the restrictions # odd and I A I < I 2 may be dropped.

7. CONCLUSION

Planar single-disk cam mechanisms wRh a oscillating double roller follower--whose possibility was doubted in [2], but proved in [I]---exist under three condiaons. 1 The common length I of the, follower arras P A and P B must be equal to the distance between the rotataon center O of the cam and the pivot P of the follower. 2. The angle ~ A P B - - 2 g of the follower arms must be commensurable with the full angle 2n. 3. The rational fractmn n/= = n = 2 / p > 1 must have an odd denominator #. (For practical purposes only small integers 2 and p are useful.) The cam disk shows a certain penodlcity (cychc symmetry): it admits the rotation about the center O through the angle 2~ = 2~r/2 ----2~/la. The polar equa.

Single-disk cam mechamsms with oscillating double roller follower tlon r = r(O) of the pitch profile ~ (equidistant to the worhng cam profile c) satisfies the functional equation r2(O) q- r2(O + ol) -- 2r(O)r(O + or) cos • = 412 sin: at. (7.1) Analytical soluiaons of this equation are represented by formula (5.10). They depend on the arbitrary choice of an auxiliary periodic function A (0) obeying the law A(0 + 6 ) = - - d ( 0 ) . Appropriate functions d(0), e g (6.1) or (6.5), provide algebraic solutions; these curves are also of purely geometric interest, as they possess movable closed chord polygon (Section 6). There exzsts a close connection between the present problem and the sinular problem of fiat-faced follower pairs operated by a single-disk cam (Section 5). The transmission law O(~b), relating the oscillatory motion of the follower w~th the rotation of the cam, has the penod 26 and satisfies the conditions ~=$-~+6,

ff=-$

(72)

This means that the Cartesmn graph of ~b(~) admits a certain affine mapping onto itself. For the synthesis of the cam mechanism with a given follower angle 2ct either of two posslbihties ,s indicated

EINSCHEIBEN-NOCKENTRIEBE

(a) A section of the pitch profile ~, subtending the angle 4, from the center O, may be suitably chosen. The rest of the profile is then completely deterrmned and simply derivable (Section 3). The corresponding transnusslon law is obtained by means of eqns (4 4). (b) Prescribing the translmssion law @(4,) in an interval of length t~, the rest is completely determined by the relations (7.2) The pitch profile 6 of the associated cam is found by means of eqns (4.10). A remarkable overconstrained linkage, connected with the cam mechamsm, is discussed in Section 3.

REFERENCES 1. C- D Duca and I Slmionescu, The exact synthesisof singledisk cams with two oscdlatmg rigidly connected roller followers Mechan~m and Mactune Theory 15,213-220 (1980) 2 W. Wunderlich, Contnbutions to the geometry of cam mechamsms with oscillating followers. J. Mechantsms 6, 1-20 (1971) 3. W Wundedlch, Kurven nut isoptischem Kres dequat math 6, 71-81 (1971). 4. M. Goldberg, Rotors m polygons and polyhedra Math. Comput 14, 229-239 (1960) 5 W. Wunderhch, Ebene Kurven nut e,nem beweghcben geschlossenen Sehnenpolygen Archtv Math. 38, 18-25 (1982) 6 H. J. Fischer, Kurven, in denen ein Zug yon Sehnen glelcher Lange sich unabh~in~g vom Ausgangspunkt schheflt Deutsche Math. 4, 228-237 (1939)

MIT DOPPELTEM

ROLLEN-SCHWINGHEBEL

W.WUNDERLICH

Kurafassun~ - Ebene Nockentriebe in [2] bezweifelt,

Jedoch

415

der genannten A r t - deren Moglichkeit in [1] nachgewlesen w u r d e - exlstieren unter

d r e t Bedlngungen: 1. Die gemeinaame Lange 1 der belden Schwlnghebelarme PA und PB ~uB Bit dem Abstand des Nockenzentrums 0 vom Hebellager P ubereznstlmmen, 2. Der Winkel ( A P B = 2a muB mit den vollen W1nkel 2. kommensurabel sein. 3. Der rationale Bruch ./Q = n = A/u • I muB einen ungerader rennet u haben. Die zugehdrlge Nockenscheibe gestattet ale automorphe Drehung um 0 durch den Wlnkel 26 = 2 ~ / ~ = 2a/u. Die Polargleichung r= r(e) des Leitprofils ~ befriedigt die Funktlonalgleichung (7.I). Analytlsche Losungen, darstellbar durch (5.10), hangen yon elner wlllkurlichen Funktlon 6(0) ab, welche die Bedlngung 6(8÷6) = -A(8) erfullt. Passende Annah~en yon 6(e), z.B. (6.1) und (6.5), liefern sogar algebraische Losun~en. die auch von rein geometrisehem Interesse sind. Das Ubertragungsgesetz ~($) des Getrlebes weist ebenfalls die Perlode 26 auf und unterllegt der Bedlngung (7.2), welche besagt, dab oas karteelsche Diagramm eine gewlsse Affinitat vertragt. FUr die Synthese des Nockentriebs bleten slch zwei entgegengesetzte Wage an, die auf der Wahl sines (relativ klelnen) Abschnitts entweder das Leltprofils oder abet des Ubertragungsgesetzes beruhen. Beatehende Zusammenhange mit dem verwandten Problem elnes gerad£1anklgen Schwlnghebelpaares [2], mit einem ubergeschlossenen Gelenkwerk (Fig. 4 und 8) cowls mit elner alten geometrisehen Frage [6J warden h e r a u s g e s t e l l t .