On the type-synthesis of four-bar linkages for three accuracy-positions of the coupler-plane : an alternative way to solve C.R. Barker's problem

On the type-synthesis of four-bar linkages for three

accuracy-positions of the coupler-plane : an alternative way to solve

C.R. Barker's problem

Citation for published version (APA):

Dijksman, E. A. (1988). On the type-synthesis of four-bar linkages for three accuracy-positions of the coupler-plane : an alternative way to solve C.R. Barker's problem. (TH Eindhoven. Afd. Werktuigbouwkunde, Vakgroep Produktietechnologie : WPB; Vol. WPA0621). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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On the type-synthesis of four-bar linkages for three accuracy-positions of the

coupler-plane.

subtitle: an alternative way to solve C.R. Barker's problem. Dr. E.A. Dijksman

Report Nr. 0621 WPA

ON THE TYPE-SYNTHESIS OF FOUR-BAR LINKAGES FOR THREE ACCURACY-POSITIONS OF THE COUPLER-PLANE

subtitle: AN ALTERNATIVE WAY TO SOLVE C.R.BARKER'S PROBLEM

by Dr.Evert A.Dijksman

Mechanism and Machine Theory Faculty of Mechanical

Engineering Eindhoven University ot

Technology Eindhoven,The Netherlands

Summary, In order to map Grashof's border case for three positions, rational rotation curves (Rationale Polortkurven in german) are observed. These curves have a double point MG coinciding with an in- or an exoirole's center of stretchable four-bars. They further contain a distinct number of ground-pivot centers,meeting the posi-tion requirements.

As any of these curves is determined by her double point and her two (mutually perpendicular) double-point tangents (t

1,t2),the three de-grees of freedom,still available to the designer,may be used to vary the position ot this (movable) coordinate-system t1MGt2 with respect

to the (given) pole-triangle.

On the resulting maps tor ground-pivot centers, 'border curves' may

partition each map into three areas for which either a+b=c+d, or a+c=b+d, or a+d:b+c is valid.

If stretchable four-bars are to be found,meeting also 3 accuracy-positions for the coup~er plane in case Ao= P12 ,and Bo equals a freely chosen rnndom point, either two real solutions occur,or otherwise two,that are each others' complex conjugate solutiono and so unreal.

To prove this,we first choose a random point B and then determine the points

o

B01

=

BO ' B02 and B03 using the identiti~s:

.t1

P

23P~3BO ~

A

p

~3P

13BOZ and

A

P 23P 13Bo

~

Ll

p

~3P

13B03 •

The center of the circle joining B01 ,B₀₂ and B03 then coincides at the

point B1 ,we are looking for. The link-lengths c

=

B1BO and d

=

P12BO are then known.

1- (

As further, AO = P

12 ,point 41 has to join the straight-line P13P23• See Burmesters' Theory).

Point A1 also completes the four-bar under either one of the conditions: a + b = c + d t or, a - b

=

d - c ,or a - b

=

c - d , leading to a conic

section,as locus for A1 t of which the foci are the points A and B1•

1 0

The two then either real or unreal intersections of P12P23 and the appropriate oonic section then meet the requirements of motion-generation. Thus,

Statement 1

If one of the ground pivots coincides with either one of the poles P1ztP13

and P

23 teach time two solutions of stretchable four-bars may exist,meeting three accuracy positions of the coupler-plane.

In a similar way, we may prove the Rtatement: Statement 2

If one of the ground pivots say AO joins a side say,the side P1zP13 -of the pole triangle P₁₂P

a3P}1 ,each time two :::olutions of stretchable

four-bars occur if a conic section,having

P~3

=

A1 and BO for fooi,intersects this side at real pivots AO.(For other sides similar statements arise.)

Statement 3

If the two ground pivots join difterring sides of the po~e-triangle,say the

1

respective sides P1ZP13 for Ao and P1~23 for BO ,then P23P13 = the coupler b, and an infinite number of stretchable four-borG exist,meeting the three

ac-1

curacy positions for the coupler-plane.(Thus,A 1

=

P

2

-The two figures A and B ,here enclosed,show thnt Statement

4

IIf AC,as well as BO ,coincide with two differing poles out of the poles

I P12, P13 and P23 ,an infinite number of stretch:,ble four-bars exist,meeting

l

the three accuracy positions of the coupler-plane ..

Solutions D.ccording to the statements 3 .::md l. lIlay moet an additional.

re-quirement as a practical consequence of the infinite number of solutions that exist in these cases.

Statement

5

I For a random choice of the ground-pivots ,A

O ,md BO' no stretchable four-bars will si.multaneously meet the three accurncy po[)itions of the coupler--plane.

Though not everything is worked out in detail,fip;ures A and B show some

details of statement 4. In theso figures, I fur Lller UGcd the l)roposition that statement

6

II

·

:&'our-bars of which all sideu are tungents to folding four-bars,and vice versa. 11 circle,are Btretchable or (See also the fig?:res A1 to A8 of the Appendix of ~m enclosed EUT-report

in the dutch language,needed for a curve having a &ero minimum transmission-angle, ~min= O,and,therefore,concern folding four-bars only. The curves, contained in the figures 17,22,23,24 and ~6,for wh~ch ~ .

=

O,combine the

mJ.n

pre'sence of a Ball-nodal point in the 4-bar coupler curve t with the

stretcha-b~lity of the four-bar. Examples of ctemonstrution,are in fact the mentioned figures A1 to ABe)

Clearly,the combination of stretchability and motion generation, represents quite another project,though the necessity to investigate the border case

.

of stretchability is a common factor, with the cornmon goal to reduce the area of practical solutions.

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In addition to my evaluation of Prof.Barkers' research proposal,named:

l~aEping the location of the 1st ~round pivot for defect-free 3 position

Synthesis of Elanar 4-bar linkages", I have done some in-depth investigation of my own,in order to see if I am able to come up with a better plan,than the one proposed by Prof.Barker.

As an expert on Burmester Theory,I am probably in a better position to find such an alternative,which indeed exists. ,

t..

. The new plan,which I am going to suggest is a prom.wsing one,snd really seems the thing to do.

In comparison to the proposed plan,the alterna.tive one replaces Barkers'

(a,~)-plane by a plane fitted with a grid of double points of rotation curves • . [4]

According to an investigat10n of a dutch professor,named Prof.Dr.G.R.Veldkamp, (whom's paper I enclose),these curves are very signifioAnt with regard to the reoognition of the

.!W.

of linkage. Each one of the 8 types,that exist with 4-bar linkages,may indeed be recognised on the basis of these rotation ourves. To demonstrate this type~anallSistI just draw a rotation curve for (Grasho!'s) border-oasetto wit,a rotation curve,having a double-point MG. Then two bilities exist,i! a small deviation from this case is contemplated. One possi-lity is the increment-change into a unicursal curve(drawn in green),representin a total-rocker; the other possibility shows minute-transformation into a two-branched one (drawn in red) ,having an ~ (even) part and a separate odd-branoh with a real point at infinity. The two-branched (red) curve represents

the Grashof-linkages of whioh the smalles bar may rotate 3600 with respect to all others. According to Veldkamp, each of the coloured curves contains 3 different types of the four-bar. So each of these curves hav~ to be segmented in order to recogniae them from one another.

- 2

-~

From this,! think it wouldVeasy to arrive at a segmentation of the

rotation curve,representing the border case.(the one having a doub~e point.) The mapping of the first Ground-Pivot ,f.ouGht nfter by Prof .. Barker, boils

down then to the mapping of the rotation curves,representing the required loci. I fUrther state,that each one of the rational rotation curvee,depende entirely on the choice of their doub~e-point with respect to the pole-triang~etwhich

ie fully determined by the three (given) accuracy positions of the coupler-plane.

What remains is a design of these curves,starting from Ma and the pole triangle. The design,I propose to follow is in fact based on Burmester Theory for 3

positions of the coupler plane and further on the properties of the mentioned rotation curve. The ~atter bears severa~ names,to w't:3rd order circular curve,

•

focal curve (used in the design for Kempe's focal mechanisms),po~e curve

(as she contains the po~es _{P12,P13,P14,P15,P16'P1?, ••• etc.)or rotation curve.} The first treatment of the curve stems from Burmester [1] , then Lohse

[3]

used a singular branoh of this (Po~ort-Kurve) curve for his design-method of' mtahan1r:sma,wherells,f'1nnlly,VeldkHmp

[41

pl'ovttd H number of

Inllth~mut1onl

and kinematic featUres of this curve.

It is not difficu~t to prove that the

4

sides' of any stretchable four-bar all touch a circle. Say,the center of this circle is named(MGrashOf or ) Ma in case the coupler AB reaches the position 1. Point Ma then proves to be a doubl,e point of Lohse's(Polortk~rve) rotation curve, being the locus of points seeing o~posit~ sides of the four-bar AoA1B1Bo under the same angle or under angles differing 180°.

For more information about such a proof,I may refer to a dutch booklet of mine (here en~lo6ed),which is entirely devoted to Burmester's Theory and its applioation. I refer in particular to the figures 18,19,20 and 21 in addition to the dutoh text of the pages 12 to 14 of this boo~et~

These figures are direotly applicable if opposite poles are replaced by 'opposite turning-joints'

of

the four-bar. (Opposite turning-joints are those joining the same diagonal of the four-bar.)

The existence of the double point of the focal curve,turns the curve of

-I(. genfs 1 into a'rational' curve,having the shape of the greek character !p.

As a consequence,this cp-curve then has the same (geometric) properties as those of the instantaneous circling point curve for which MG is replaced by the instantaneous velocity pole P. For the design of the latter,see

in particular figure

3.4

of page 96 of my book : 'Motion Geometry of Mechnnisms Cambridge University Press,New York 19?6u

3

-AS

the mentioned ~-curve represents a common locus for all 4 turning-joints of the four-bars,we are looking for,this ~-curve ~ play an important rOle in the ~l2e-sxethesis of these linkages. So does the double point MG of this curve.

Instead ot the unpredictable (~,~)-planetused by Prof.Barker, I,therefore, strongly recommend to use the fixed plane,dotted by points Ma ,we are going to choose.

As for 3-position synthesis,the pole triangle P12P23P31 is completely deter-mined,we so ohoose MG viewing its position with respect to this triangle. Generally,two design-degrees of freedom are taken away as soon as we choose Ma in the plane containing the pole triangle. They represent the condition

that MG joins a rational rotation curve as well as the condition that such a joining occurs at the double point of the curve.

The coordinates,preferably the triangular - or better still,the normal coor~inat~s _{(d1,d2,d3) with respect to the fundamental triangle} P1~23P31' of MG ,then are the design-coordinates,we are going to vary in order to aohieve our goal in type-synthesis.

Once MG is ohosen,one may find a singular curve as a locus for the

4

turning-joints of the 4-bar.Clearly,this curve happens to be the rational ~-curve,we

spoke off.

It is important to show that this ~-curve is completely determined by the position of Ma with respect to the pole triangle. We will also show,that an additional choice of a Eermissible turning-joint at this curve,completely determines the remaining 3 turning-joints at this curve. Indeed,totally,only 3 design-degrees of freedom determine a stretchable four-bar of which the coupler plane has to meet 3 accuracy positions.

cartesian-In our approach,they are the twovcooru1nates (~ ,Y_M) or otherwise the normal-coordinates of MG with respect to the

fun~amen~al

triangle P12P23P31 • in ~ddition to the singular design-parameter (such as the radius of the . .

~-6/1lL ~

.~. incircle) that determines the first-ground-pivot A at the ~-curve.

o

In

order to find the ~-curve from MG and the pole triangle,we use Burmester's 3-position synthesis: For th1a,we choose a center Co at MG ,find her isogonal transformed location C

F with respect to the pole-triangle and then determine

CFts . image-point,c_1f with respect to the side P₁₂P₁₃ of the po~e triangle. We further take D, at MG in a reverse procedure, reflect D1 to the fundamental - or image point DF through the mirror P₁₂P13 and then determine DFts iso~onal

4

---..

...

The locus of points seeing C

1Co and D1Do under the same angle or otherwise under angles differing 1800, then identifies the ~-curve,we are looking for.

The poles P12 and P13 should join this curve,but ~ the pole P23 1

and her _{image F2,.}

The ~;9.tot:i.c. direotion of the ~-curve equals the direction of the

lL~e of gravity (median) MGZ of triangle C1MGDo ,whereas the two,at right angles running,double-point tange~s to the curve,are the two

angle-!,isee,tors e.t MG of this triangle. These angle-bisectors are needed in a oonfiguration showing that the star-points P~2 ,P~3 t c~ and D: indeed

join a straight.line,as is necessary for the ~-curve.See again figure 3.4 of page

96

of the mentioned book Motion Geometry of mechanisms.

According to the ~-curve's similarity with an instantaneous circling point curve,star-points such as P~3 may be found as follows:

~

for

this,we draw the normal to M_GP

13 at P13 and intersect the normal with the two double point tangents of the ~-curve. The two intersections as well as MG and P~3 then constitute a rectan5le.

Naturally,the four-bar CoC1D1Do has an in- and excircle with sero radiuB at MG. To reach the 2nd or 3rd position of the coupler plane,the mentioned

four-bar has to be stretched first. So,the four-bar indeed is a folding one,simultaneously meeting the 3 position require~ents of the coupler-plane. As we can do this for anl point MG in the plane, an infinite square number of solutions exist for which a a b ~ c

=

d , representing kites,for which

a

=

input crank- or rocker-length, b

=

coupler-length, c

=

length of secundary crank or rocker and

d I:: frame length.

Kites simultaneously meet ~ border conditions of Grashof; in this case b + d a a + c

and b + C

=

a + d

(1)2 of them do exist. And we also know how to find them.

In case we do ~ want this Earticula~ type of linkage,but are looking for 4-bars instead,meeting only one border condition,we have to exploit

-our last design-parameter.

We employ this freedom in design,by choosing a radius of the inscribed or exscribed circle about MG of the stretchable four-bares). A~l tangenta to such a oircle are then considered until a pair of intersections (Ao,A,) with the ~curve are in accordance with their mutual eoordinancy in view of the pole-triangle.More solutions will appear. But only those may be combined to form a four-bar (AoA1B1Bo) when the angle bisectors at MG

5

-of the triangle MGAoB1 coincide with the double points' tangents

ot

the curve.,

The four-bars meeting the position-requirements,must have four

angle-bisectors all intersecting at the ~ _{point MG. These four-bars {Ao• 1B1Bo>} then meet Grashof's border condition as well 8S the 3 accuracy positions

for their coupler planes.

Which of the

4

possible border conditions are actually met,depend on the turning-joints' location at the rational rotation curve.

It is possible that the initial choice of MG is already decisive in this partition.

According to R.Beyer's Proposition Nr.11 (Ref.[?] >,the minimum length of any moving bar (a,b,or c) equals the diameter of the pole-trianglets

incirele.(Otherwise,the4..:~~~~€-l?fd6h~.n~inFo~

a closed quadrilateral.)

;rlrt~ThiS border-case of motion o~frs when MG joins the center of the pole

( triangle's incirqle. ' _

r . If MG

~oin.1on~:,;ot'91cne

pole triangle's excirclestH.Alt'o in 1936 described

'J~

l\t-curve - normally,a I+-u/;\r couplt:r CUl'VH -

(~(mtl'l\ctn iu~o

H t:lnj';ulur

1

point ( MG

=

Ao

=

~).

As 0. consequence, the dinmetoro of the 3

pole-triangle's excircles are unique rocker- (or crank-) lengths,probably extreme values under the ~2 possibilities.

In this eontext,we may further use the general proposition:

li

lt two adjacent bars of • stretchable four-bar are interchanged as if they were vectors, the'interchanged four-bar' is stretchable too. r This proposition probably turns the dotted MG-plane into partioned areas, \ dividing different cases of stretchability.

In the particular case for which a center A (or B ) of the rational

ro-o 0

tation curve joins the circumscribed circle of the pole triangle,an infinite turning-joint A~ (or B~ > occurs,resulting into inverted slider cranks or ditto-rockers,already considered by Lohse (See also the enclosed pages from his book on this subject.)

A similar thing happens,if A1 (or B₁) joins the cirole circumscribed about triangle

P12P~3P31'

Then,A: (or

B~

goes to infinity,resulting into crank-slider - ,or rocker-crank-slider meohanisms for which~internal rocking anele changes into an external one or vice versa.

At this point of the alternative proposal,I must point out,that not all the points of the rational rotation curve are permissible. .~

wlt,f

For instance,Ao taken at the intersection of the side P12P13 . the ~-curve,

To find the solutions directly,instead of trying them out one by one, would be solved by isogonal transformation of the (rational) rotation curve,f'ollowed by a reflection into the side P₁₂P

13Q The curve obtained this way,then has to be intersected with the initial rotation curve Q The points of intersection are the permissible points A1 of the rota-tion curve,we are looking for.

An

alternative way would be to transform the rotation curve isogonally with respect to triangle

P12P~3P31

and then to tntersect its image curve in view.of the mirror P1~13t with the initial curve. Their in-tersections are the permissible ground-pivots (Ao or B_o).

I am sure I have left some vital gaps that have to be bridged. Computer-Programs have to be set up to cover this pl'ocedure.

I am equally sure,however. that the equipment,available to Prof. Barker enables him to find all the answers of remaining problems that arise in this procedure.

Dear Dr. !'I.arsh. it is a long letter,l know, lmd maybe the consequence will be to take hold of another referee,who is able to recognize the pitfalls of both systems,as you must have done the other time,when I suggested a different approach of research.

Of course,i! you seek out Prof.Dr.P.Lohse (8901,KissingtFliederstra~e 15. Bundesrepublik Deutschlarld) he will most probubly favour the approach in which I suggested to use'his' Polort-KurveQ

But

maybe Prof Barker too,may favou.r this approach? References:

[1) Burmester,L.: "Die Brennpunktmechanismenll_{• Z.Math.Phys.i(1893)p.193-223}

-~J

Beyer,R.:"Kinematic Synthesis of Mechan:isms",Chapman and Hall 1963,

If

or in german:qKinematische Getriebesynthese,Springer Verlag 1953,pp.13-23 (Sat .. 11)

~ Lohse,P.: "Getriebesynthese."( Bewegungsablaufe ebener Koppelmechanismen) Springer Verlag,New York 1980,pp 50-53 and 61 to 65.

[4]

Veldkamp,G.R.: "Rotation Curvesll,Journal of Hechanisms ~(1967) ,pp.147-'156

[5~ DijksmantE.A.: "Het Ontwerpen van stangenmechanismen",Polytechnisch Tijdschri!t-Monografie Nr.33 (1968)

[6~ D1jksman,E.A.: ''Motion GeometrY' of Mechanismsu,Cambridge University Press,

New York, 1976,Chapter 3.

APPENDIX

li

The way to map the ground-pivot loci for stretchable four-bars.havins coupler planes meeting three accurac,-posi~10ns. A rational rotation curve (Polortkurve in the german language) is tully d etermined by the 2 poles P₁₂ ,P13V ' of this(curve tne rando£ly chosen double point )

NG ~ a perpendicular set of double-point tangents (t₁,t₂) at M_G

-To demonstrate this,we may regard the curve as an instantaneous ~ircling

EOint curve for its characteristics. If the rco-curve would have been the ICP-curve,the double-point MG would act as the instantaneous center of rotation (velocity-pole p). Using the design for such a curve,we draw the perpendicular to M_GP₁₂ at P

12 and intersect the perpendicular with t1 and t₂• The rectangletformed by these 2 intersections and by the

:H :H Jf

points MG and P₁₂ then results into the point P₁₂ on the diagonal M_GP₁₂

of this rectangle. Similarly, P~3 is determined from P

13 _ This results into the 'star-line' P~2P~3 at which all starpoints appear originating from the points of the rc -curve. Clearly,a reverse procedure that starts _o from the starpoints so leads to the original curve rc ,we wanted to

deter-o

mine. The curve rc_o drawn this way, is dependent on the choice of MG with· respect to the (given) pole triangle as well as on the direction of one of the double point tangents (t₁ or t₂).

The isogonal transformed curve (;c-)i of rc 's image curve with respect

- 0 0

to the side P₁₂P

13 of the pole triangle,then has to be intersected with rc _{0 0 0} ~iving solution-points A (or B ) meeting the 3-position requirements as well as the fact that they indeed join rc simultaneously_

o

Having a solution-center Ao,the 'opposite-turning-joint' B1 For this,we just intersect the line,connecting MG with Ao's with respect to t1 ' at the point B1 of rc_o•

may be constructed. image point

The respectively to Ao and B1 coordinated pOints,A₁ and Bo ' then constitute a stretchable four-bar,meeting the 3 accuracy positions for its coupler plane.

To find more solution-centers A ,we just have to rotate our t1MGt2 -tangents'

.0

-system about M_G- Each position of this system leads to a distinct number of solution-centers A •

o

At least one solution-curve a is obtained by o

successive rotation of our tangent-system about M_G -After

90

times a 1 degree increment-angle of rotation,

the earlier found solution-centers A appearGagain; o

o

so a scanning of

90

is sufficient to cover all the possibilities. Thus, each point MG leads to a curTe a o as a consequence ot the increment-rotations of t1 about M

2

-continued Appendix about 'The way to map the ground-piTot loci,for

~tretchable four-bars,h&Ying coupler-planes meeting three

accuracl-~sitions'_

As we haye seen ,the curve a represents the locus of solution-centers o

Ao for a given point MG but for different directions of t₁

-A bundle of curTes &0 then results from a locus for points

Ma-

For this locus we prefer to take a circle about M ,the incircle's

cen-o ter of the pole triangle _

The radius of this circle then indicates the map-number containing such a bundle of a -curves. Another radius, gives another map_

o

Each curTe &0 on such a map should bear the angular polar-ooordinate

of MG at the (chosen) circle about Mo.

This way,the mapping ot all stretchable four-barstmeating the 3-posi-tion requirements become possible.

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