A fourfold generalization of Peaucellier's inversion cell

A fourfold generalization of Peaucellier's inversion cell

Citation for published version (APA):

Dijksman, E. A. (1996). A fourfold generalization of Peaucellier's inversion cell. Meccanica, 31(4), 407-420. https://doi.org/10.1007/BF00429929

DOI:

10.1007/BF00429929

Document status and date: Published: 01/01/1996 Document Version:

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Meccarzica 31: 407-420,1996.

@ 1996 Kluwer Academic Publishers. Printed in the Netherlands.

A Fourfold Generalization of Peaucellier’s Inversion Cell

EVERT A. DIJKSMAN

Eindhoven Universip of Technology, Faculty of Mechanical Engineering, Mechanism and Machine Theory, Precision Engineering, Den Dolech 2; Eindhoven, The Netherlands

(Received: 19 September 1994; accepted in revised form: 18 July 1995)

Abstract. New inversors are proposed which are generalizations of the well-known inversor of Peaucellier. It appears that a kite in the Peaucellier cell is replaceable by an arbitrary 4-bar linkage (abhk) whereas the direction and length of the straight line, produced by the inversor, can be manipulated through the particular choice of the relative polar coordinates (m, /3) of a vertex A of the triangular input link. Formulas are derived for practical inversors with a revolving input link. The ones selected are basically governed by the choice of two transmission angles, ,ui and ,us, by the length L of the acquired line, as well as by its direction represented by the angle 7r/2-,B comprised between the line L and the frame.

Sommario. Vengono proposti nuovi inversori quali generalizzazioni de1 ben noto inversore di Peaucellier. Si mostra the un quadrilatero isoscele nella cella di Peaucelher e sostituibile da un’arbitraria connessione a quattro barre, mentre la direzione e la lunghezza della retta, prodotta dall’inversore, possono essere manipolate con la particolare sceha delle relative coordinate polari (AD, ,B) di un vertice A de1 collegamento triangolare di input. Vengono derivate formule per inversori funzionali con un collegamento di input rotante. Quelli solezionati sono principalmente govemati dalla scelta di due angoli di trasmissione, pi e ~3, dalla lunghezza L della linea ottenuta, cosl come dalla sua direzione rappresentata dall’angolo 7r/2+3 compreso tra la linea L ed il riferimento.

Key words: Kinematics, Inversion-cell, Planar mechanisms,

1. Introduction

Inversion mechanisms are very old and represent the first direct application of geume&

properties that are to be deduced from Appolonio’s circle (ca. 260 B.C.) by way of the harmonic ratio.

The very first application of geometric inveniun was invented by Peaucellier [1] in 1864 and independently demonstrated by Lipkin at the International Exhibition of Vienna in 1873. Later, other inversors were devised, such as those of Hart [2], Sylvester and Kempe [3], Hessenberg [4], Perrolatz, and Artobolevskii [S, 61, Dijksman [7] and others. (See Figures l-7.) Basically, all these inversors contain a so-called inversion ceZ1, having the geometric property to invert planar curves into their inverse shape being inverted with respect to a fixed unit circle, and vice versa. Such an inversion converts a circle, for instance, into another one of different size, sometimes used to produce very large circles mechanically. This leads to applications in a robot, or in calculating devices as part of a machine or an instrument. A well-known application of the latter has been demonstrated in [8], showing the utilization of these linkages in optical systems such as autofocus enlargers, based on the mechanical property of the cell to realize the inverse value of a distance.

Perhaps the most attractive application will be the possibility to produce true straight lines.

This happens when the input circle joins the fixed origin 0 of the cell leading to an infinite

large output circle, i.e. a straight line. (Note that point p, tracing the input circle, never reaches the origin 0 in reality; otherwise, point Q, then tracing the output line, would reach infinity.)

Table I. Inversion ceI1 of n _{h fd} Peaucellier (1864) 7 8 2 Hart (1874) 5 5 3 Sylvester-Kempe (1877) 5 5 4 Artobolevskii-Dijksman (1955-1993) 7 8 5(=4+land =3+2) Dijksman (1994) 7 8 6

Needless to say an inversion cell is quite different from a pantograph; the latter just transforms a curve into a similar one. Clearly, a pantograph never transforms a circle into a straight line.

Thus, inversion cells are a source for inventing true straight-line mechanisms.

As the input curve may be arbitrary and planar, the cell must have 2 degrees of freedom in motion. The cell further contains a fixed turning joint 0, an input point P as well as an output point Q, respectively tracing their input and output curve. The inverse of the input curve may be rotated about 0 over a fixed and given <QOP. Then, the transformation of the input curve represents the product of an inversion and a fixed rotation about the origin 0. Under these circumstances, both the scalar (dot) product /Z = (OP . OQ) as well as the vector (cross) product j = (OP x OQ) h ave to remain constant. Geometrically, the combined transformation may be presented in a vectorial manner, but also in a manner based on complex numbers instead of vectors. (The proof of the equivalency is shown in Appendix 1.)

The vectoriaZ manner transforms p into q with the formula: q = LOP - (P x .Mp. P)+

where p E OP, q E OQ, and the value k represents a given scalar, whereas

j

equals a given vector running normal to the plane of the mechanism.

The other manner, using complex numbers, though representing the same transformation, leads to the simple relation:

in which c is a given but complex constant, and p* represents the conjugate complex number of p, obtained from p by converting i = fl into 4 Then, GQOP = arccos[-&] = arcsin _r $$ = constant (with k* + jj* = cc*), whereas also the magnitude -- OP.OQ = Jm = &? remains the same. Thus cells, supporting the complex constant c = k - im, may be used as inversion cells. For the oldest cells such as those of Peaucellier and Hart, the points 0, P, and Q remain aligned. For others, such as that of Sylvester and Kempe, 4Q 0 P remains different from zero.

Apart from the angular distinction, we intend to classify inversion cells based on their number of links (n), their number of turning-joints (ti) and their crucial number fd offree design parameters. (See Table 1.)

After that, we intend to introduce and to design the most general inversion cell containing 7 links, 8 turning joints and even 6 free design parameters.

A Fourfold Generalization of Peaucellier ‘s Znversion Cell ₄₀₉

Figure I. Peaucellier’s inversion cell. (1864)

Figure 2. Hart’s contra-parallelogram cell. (1874)

2. Freedoms in Design

Peaucellier’s inversion ceZZ consists of a kite, a rhombus and a frame center 0 that coincides with a turning-joint of the kite (see Figure 1). So there are (6+ 1) links, the frame link included but not drawn. Since two sides determine the kite (or deltoid), Peaucellier’s cell has only two independent design parameters (fd = 2).

With Hart’s antiparaZZeZogram cell, n equals (4 + l), tj = 5 and fd = 2 -t- 1, the additional parameter following from the choice of 0 at one of the, eventually prolonged, sides of the anti- parallelogram (Figure 2). The so-called quadrupfane inversion ceZ1 of Sylvester and Kempe also consists of an anti-parallelogram, but with 3 sides forming similar triangular links, their apices representing the crucial points 0, p and Q (Figure 3).

In comparison to Hart’s cell, the fixed center 0 then represents an arbitrary turning-joint, not necessarily joining an actual side of the anti-parallelogram, whence jd = 3 + 1 for the quadruplane cell. Artobolevskii described a cell shown in Figure 52 of his book [6]. He derived the cell from the Hessenberg one [4] interconnecting two of these cells, both having coinciding points 0, P, and Q (Figure 7). Hessenberg’s cell was derived from the quadruplane inversion cell by the introduction of two sliders (Figure 4).

The procedure followed by Artobolevskii did not result into an arbitrary four-bar linkage, but in one having perpendicuzar diagonals. This property originated from the fact that each of the two interconnected Hessenberg’s cells did have the same angles, E and 6. (If the two

&WQ = con&an?’ Figure 3. Inversion cell of Sylvester and Kempe, (1877)

Figure #. Hessenberg’s inversion cell. (1924)

A Fout$old Generalization of Peaucellier 5 Inversion Cell 411

Figure 6. 1st inversion cell of Artobolevskii-Dijksman. (1955) (1993)

Hessenberg’s cells would merely have had the same value for E t 6, Artobolevskii would have discovered the most generalized cell, lying at the base of this paper.)

Dijksman [7] found a shorter way to design Artobolevskii’s cell, starting from an arbitrary four-bar linkage with perpendicular diagonals. The two triangular links, contained in the cell, were shown to be similar. Artobolevskii’s cell appeared to have 5 independent design parameters, namely, 3 for the conditioned four-bar linkage and 2 for the additional choice of an apex. Thus, fd = 3 •t 2 = 5.

More cells appear in this category, starting from a completely arbitrary four-bar linkage and an additional turning-joint at a diagonal of the four-bar linkage, giving fd = 4 + 1 = 5. (See Figure 55 of [6] or Figure 3 of [7].) Hence, the inversion cells of this type equally possess 5 free design parameters (see Figure 6).

As already indicated, a further generalization may be obtained by the adjoinment of a final independent design parameter, leading to an inversion cell for which fd = 5 + 1 = 6. The cell is derived through the interconnection of two dis.simiZar quadruplane inversion cells at four corresponding apices being replaced by common turning-joints [ 11, 121. After omitting a dyad with one of these apices as its joint, an inversion cell of a more general nature appeared. The liberty to do so arose after the discovery of the possibility of interconnecting two entirely different anti-parallelogram chains with more than two (in this case, 4) turning joints.

One of the aims of the present paper will be the direct design of this most general inversion cell, thereby avoiding the cumbersome use of the quadruplane inversion cell. The design starts with the arbitrary choice of a four-bar linkage and with the choice of the joint of the dyad that is adjoined to the linkage. The number of independent design parameters of the cell then equals fd = 4+2 = 6; hence, one more than available with Artobolevskii’s cell.

The property of geometric inversion of the obtained cell has to be proven. In order to obtain a practical set of true straight-line linkages, we are then going to deprive the new inversor from three of their available, 6 + 1 = 7, independent design parameters. This is induced by our wish for the inversor to have a completely revolving input link and a reasonable transfer of motion. The arbitrary choice of the four-bar linkage contained in the inversor will then be a practical tool for that aim. The arbitrary choice of the joint of the dyad is needed for the length and the direction of the straight line with respect to the frame.

Figure 7. 2nd inversion cell of Artobolevskii-Dijksman. (1964) (1993)

3. Generalized Inversion Cell

Peaucellier’s inversion cell consists of six bars in which one recognizes a deltoid linkage as well as a rhombus. This cell has the disadvantage of collapsing, since both the deltoid and the rhombus may become aligned. Particular devices have to be designed in case the cell has to be driven by a revolving input link. A way to avoid collapsing or an uncertain motion-transfer, caused by branching problems, can be found through generalization of the cell. A generalized inversion cell, possibly containing a completely revolving input link, is to be obtained through the assignments (see Figure 8):

a. Start the design with an arbitrary choice for the four-bar linkage OS’D’R’ with corre- sponding sides abhk.

b. Choose an arbitrary point A and form the rigid triangle ADS’ based on the chosen 45”DA = /3 and the chosen distance DA (with D = 0).

c. Make A DABi N ADSID’.

d. Determine the image B(= Q) of Bi with respect to the perpendicular dropped upon the line BiD from point A.

e. Determine point C with the conditions: <CBA = T - <R/D’S’, _-- and 4’DR’ = IT - p, or use the relation CD/AD = -{(b2 - u2)/a}{k/(h2 - k2)} (see below).

f. Finally, form the four-bar linkage ABCD, the linkage-dyad DR’D’ and the rigid triangle CDR’.

Since

-- m = 2.mi. COS(~AB~D) - m = 2.m.(b/a). cos(&‘D’D) - DA.DD’/u

A Four$old Generalization of Peaucellier ‘s Inversion Cell

413

Figure 8. Generalized inversor.

we have

--

(u/m).(DQ.DD’)

= 2b.m. cos(&D’D)

- (m)2

= 2b{b - u. cos(uDS’D’)} - {u2 j- b2 - 2ub. cos(qDS’D’)}

c b2-u2.

Hence

DA.(b2 - u2)/u = m.m

= -m.(h2

- k2)/k.

--

Which proves, indeed, that for this linkage

OP’.OQ

= constant, whereas also

4JOP’ =

r - ,0 = constant. (Note that by agreement 0 G

D,

Q G B and

P’

z

D’.)

Therefore, the cell so designed may be used as an inversion cell. Thus, if

P’

traces part of an input circle (about

PA)

containing the fixed center 0, point Q will trace part of a

true

straight-line. The line traced encloses an angle of (7r/2 - ,L3) radians with the fixed link

_--

PiO.

In the vq particular case for which

R’S’ 1 D’D, SD = R’D, &‘/DA = 0

and

DA =

DS,

Peaucellier’s inversion cell will be regained. Apparently

four

conditions are necessary to obtain Peaucellier’s cell from the general case. We conclude that the new inversion cell represents afoqGoZd generalization of Peaucellier’s cell.

!

R r

~ J_

%,

/

/

/

/

/

/

/

Figure 9. Straight line segment, being traced by Q, runs parallel to the frame D6D. Generalized inversor.

4. Choice of Crank and Mechanism

It is recommended to drive the cell by way of the rigid triangle S J D A . Since c = d (i.e.

P ~ D ' = P ~ D ) , the four-bar linkage D S ' D ' P ~ (or abcd, for short) may be designed as a crank- rocker mechanism with acceptable transmission angles between coupler and rocker. Not only

that, but also < D R ' D ' as well as <~CBA = 7r - <~R!DrS r should not reach neighbouring

values of zero or 7r.

To obtain these requirements, a particular optimization has been carried out in Appendix 2 with the recommended result as demonstrated in Figures 8-10. For these inversors,

afb = ½; ep,

=

83-(1 + -v/3);

c = d ;

h l a =

g3; ht~ = ½V3;

k = b, D C = 6 . D A ,

with

! !

A Four$old Generalization of Peaucellier ‘s Inversion Cell 415

C

Figure 10. Pin-joint Q, tracing a segment of a straight-line, running normal to the fixed bar DAD. Inversor driven by crank DA.

Figure 1 I. Determination of ,C.

The point A (or, alternatively, the tracing point Q) may still be chosen arbitrarily. Therefore, this freedom-in-design results in a 2-parameter manifold of solutions with compZetely revohing input links driving the mechanisms.

The direction of the straight-line being traced by point Q may be varied through variation

416 Evert A, Dijksman

&we 12. Determination of L.

(see Figure 9). Whereas, if p = 0, the line traced runs even normal to that bar (Ph E DL, see

also Figure 10). Note, that for /3 = 0 one may regain Artobolevskii’s inversor of Figure 7.

Finally, the length L of the straight-line happens to be proportional to the chosen distance

DA and is shown to be independent

of m&e ,B. Application of the Law of Sines to triangle

!Z’DQmtn yields:

(see Figure 11) with

cosa? = $.

Similarly,

-

--

lx? lnax

= (Al? -I- AD) sinLfs; ‘) ,

with

cosu = @y-$.

(see Figure 12).

For the length L of the straight line we obtain the expression:

(2)

(3)

A Four&old Generalization of Peaucellier k Inversion Cell

417

or

L = y[(b + a)

sin7 - (b - a) sin%],

whence, for inversors with input links

DA,!?’

making complete revolutions with respect to the

bar

Al?,

the length

L

of the straight line has the general form

In case b/a = 2 and d/b = i(l t d3), we obtain the simple expression:

L = AD(3

sinv - sinz).

W

Note that the dimensions suggested

for the inversor are taken so that, for all values of ,B and

DA,

the

three

occurring

minimum transmission-anglesp,,,i”

are 30’. (see Appendix 2.)

5. Conclusions

Interconnecting two dissimilar quadruplane

_-

inversion cells of Sylvester and Kempe, provided

_-

they possess

the same constant value for (Op. OQ) and also an equal constant value for the

enclosed

CIPOQ,

a generalized inversion cell is to be obtained after omitting two links not

containing the joints 0,

P

or Q.

A direct design of the cell, circumventing the cumbersome

derivation mentioned above, has

been obtained. The generalized cell consists of

an arbitrary

four-bar linkage and an adjoined

dyad, the latter also being dependent

on the

arbitrary

choice of a single joint

A.

The (4 + 2) = 6 independent design parameters

may be used to regain Peaucellier’s cell,

Perrolatz’s cell and Artobolevskii’s cells, as well as all others with 7 links and 8 turning joints.

In this paper, three out of seven independent

design parameters

have been used to derive a

new inversor with a completely revolving input link, simultaneously supporting a reasonable

transfer of motion. Attention has been given to transmission angles, in order to reckon with

possible friction in the joints.

Further, a model has been made to demonstrate

the feasibility of the idea.

Appendix 1 Proof of the equivalency of inversion-formulae

If the dot-product LZ =

(p . q)

as well as the vector product j =

(p

x

q)

are given constants, it

follows from their ratio that tan(p,

q)

= /c-’

[jl

= constant.

Thus, ]p]]q] = /CCOS-1

_{k7 4 = dV2 i- Li 4H re P resents a constant too. It is possible to}

express

q

explicitly in

p, j

and k; namely:

or

giving

Another way of expressing the transformation will be through the formula based on complex numbers:

!I= ;,

P

c being a complex constant, p* being the complex conjugated number of the complex input number p, whereas q resembles the complex output number. Clearly, arg q = arg c - arg p*, giving arg q - arg p = arg c. Thus, 4JOP = arg c which is a constant.

As also q* = 5, we find that ip/.[qi = dm = @. Indeed, [qi = e. As in the plane, the scalar and the vector products turn out [’ ] to be the real and imaginary parts of the complex product pq*, we have that

c* =pq* =

(paq) t id(j-j)

= k+Q./(j-j)

and

c=p*q=k-id(j.j).

Hence, cc* = k2 + (j .

j),

interrelating the constants.

Appendix

2 Derivation of the dimensions of an inversion mechanism

Taking SD as the crank a, being the shortest input link of the four-bar linkage &cd and naming 45”D’Dh the minimum transmission angle ~1 appearing in the overlapping position of the four-bar linkage, for which the &“DD~ = 0, we may write

(d - u)~ = b2 + c2 - 2bc. cos pl and

(d i- CL)~ = b2 + c2 - 2bc. cosp2,

in which ~2 = &“D’D~ in the stretched position of the four-bar linkage, for which CIS’DD; = T.

Knowing that c = d, we obtain the relations u/b = ;(COS/Q - cosp2)

and

d/b = ;.[l - (a/b)2][cospI - (u/b)]-!

The four-bar linkage DS’D’R’ = abhk having the dyad linkage h - k linked to the initial four-bar linkage ubcd, possesses a smallest link u making complete revolutions with respect to the bars b, h and k. Thus,

(b - a)2 = h2 + k2 - 2hk. cosp3, (b + u)~ = h2 + k2 - 2hk. cosp4, (k - u)~ = h2 + b2 - 2bh. cosp5, (k + a)2 = h2 + b2 - 2bh. cosp6,

’ Zwikker, C: The advanced Geometry of Plane Curves and Their Applications, Dover Publications (1963), New York 14, p. 19.

A FoutjQold GeneraEzation of PeaucelZier 5 inversion Cell 419 where p3 = <DR’D’ if <DS’D’ = 0, ,LL~ = 6DRfD’ if 6DS’D’ = T, p5 = GR’D’S’ if qR’D,S = 0, ph = 6R’D’S’ if 4R’DS’ = T.

Taking b = k simplifies matters considerably, yielding ~5 = ~3 and p6 = ~4. We thus obtain the equations

u/h =

&(cosp3

- cosp4)

and

b/h = ;.[l - (u/h)2][cosp3 - (u/h)]-‘. Since,

u/b = (u/h).(h/b) = (u/h).2[cosp3 - (u/h)][l - (u/h)2]-1 we find that

@/h12t2 - W)] - +/h) cosp3 + (u/b) = o.

A maximum value for ~3 will be obtained if the discriminant of this quadratic equation vanishes, i.e., if

from which we obtain

u/b = 1 - d(l - COS2p3) = 1 - sinp3 and

u/h = {2 - (u/b)}-’ .cosp3 = (1 + sinp3)-‘.cosps.

For practical reasons, 30’ < ,LQ < 90’. As further the crank-length u should not be too small on the grounds of the desired length of the straight line as it is traced by point I?, we have to take p3 = 30°.

(Note that the length L of the straight line is generally given by formula (7) of section 4, and is directly proportional to the distance DA.) Whence,

u/b = ;; u/h = &/3 and

As for practical reasons 30’ < ~1 c 90°, we may take pl = 30°. Then, d/b z i( 1 j- ,/3). As further k = b, we derive that

- -

DC/DA = (b/a)(b2 - a2)(b2 - h2)-’ = (2 - sinp3)( 1 - sinp3)-‘. sin-’ ~3.

Yielding

DC = 6.DA for ~3 = 30’.

Of course, other values for ~1 and ~3 result in other dimensions of the inversor. References I. 2. 3. 4. 5. 6. I. 8. 9. 10. 11. 12.

Peaucellier, A., ‘Note sur une question de geometric de compas’, Nowelles Annalesde Mathkmatique, S&II, Tome III, (1864) 344.

Hart, H., ‘On certain conversions of motion’, Messengerof Mathematics, IV (1874) 82. Sylvester and Kempe, A.B., ‘How to draw a straight-line’, Nature, XVI, Part II (1877) 86-89.

Hessenberg, G., Gelenkmechanismen zur Kreisvenvandtschaft, Titbinger naturwissenschaftliche Abhandlun- gen, Heft 6, Ttibungen 1924.

Artobolevskii, I.I., An InversorMechanism, DAN, USSR 104, Nr. 6, 1955.

Artobolevskii, 1.1. Mechanisms for the Generation of Plane Curves, $ 7 and 3 10, Pergamon Press, Paris 1964.

Dijksman, E.A., ‘A way to generalize Peaucellier’s inversor’, In: Proc. 6th Int. Symp. on Linkages and CAD-Methods (Theory and Practice of Mechanisms), Vol. I, Paper 10, SYROM ‘93, Bucurqti, Romsnia, June 1993 pp. 73-82.

Hartenberg, R.S. and Denavit, J., Kinematic Synthesis of Linkages, MacGraw Hill, Inc. 1964, pp. 185. Dijksman, E.A., ‘A strong relationship between new and old inversion mechanisms’, Journal OfEngineering

for Industry, Transactions of the ASME, Series B, 93 (Feb. 1971) 334-339.

Dijksman, E.A., Motion GeometryofMechanisms, Chapter 8 Inversors, Cambridge University Press, London 1976. Cinemdtica de Mecanismos, Cap.8, Inversores, Editorial Limusa S.A., Mexico 1981.

Dijksman, E.A., ‘%o non-similar contra-parallelogram linkages interconnected at the vertices of a para- llelogram’, In: I Congreso Iberoamericano de Ingenieria Mecdnica, Madrid, 21-24 Sept. 1993, Vol. 3, pp. 401-406.

Dijksman, E.A., ‘A general inversion cell, obtained from two random contra-parallelogram linkages inter- connected at the vertices of a parallelogram’, Mechanism and Machine Theory, 29 (6) (1994) 793-801.