Mechani,~m and Mac,'~me Theory 1973. VOI. 8, pp 257-269 Pergamon =~ress Prlnteo Jn Great Britain

Synthesis of Spatial Slider-Crank Mechanism for given Slider Stroke and Crank Length

P. Antonescu '~:

and

C. Udriste~

Received 5 June 1972

Abstract With the end positions of the stroke and the radius of the crank specified, the locus of the fixed end of the crank in the plane of crank rotation can be graphically or analytically determined.

1. Introduction A SPATIAL slider-crank mechanism of the RSST type can be represented in pictorial projection (Fig. 1) or by double orthogonal projection (Fig. 2). The axis (D) of the fixed revolute joint (,4o) is shown as a straight line perpendicular to the frontal plane and the fixed guide (A) as a straight line parallel to the horizontal plane.

In both projections the crank (AA,, = r), being in a frontal plane, appears in actual size. Guide (.~) and the angle (a) between (A) and frontal planes appear in actual size only in the plan view.

Any position of the crank determines the location of the spherical joint (,4) on the crank circle (CA). The corresponding position of the slider (spherical joint B, Figs. 1 and 2) can be graphically[l , 9] or analytically [5] determined. It is represented by the second spherical joint B (Figs. l and 2).

Stroke s of the slider is defined as the distance between the extreme positions B~, B~ [1, 8]. When the slider is in either of these positions (Fig. 3), axis (D), crank (AA0), and couplor (AB) are contained in plane Pt or Pll [5.6].

When stroke s and length l of the coupler are known, fixed joint A0 must be on a conic (ellipse or hyperbola) defined as the geometric locus of points radially equidistant from two circles [3,4].

With stroke s and crank length r known, joint Ao will occupy positions in the frontal plane defining curve F.~,, named the curve of fixed centers. Graphical and analytic methods for determining this curve are discussed in the remainder of this paper.

For practical reasons the limitation of solutions can be obtained by choosing the eccentricity, coefficient of productivity, etc.

~Associate Professor dr. eng. Dept. of Theory of Mechanisms and Machines. Polytechnic Institute. Bucharest, Roumania.

-Lecturer dr., Dept. of Mathematics IV. Polytechnic Institute. Bucharest, Roumania. Editor's Note: American readers should recognize that the drawings in this paper use "first-aagle" projec-

tion. The "fold" between views is labelled the X-axis so that the front elevation {above) becomes an XZ plane and the plan view {below) is an XY plane. Any point {such as.l) in space is shown as a in the plan view and as a' in the front view.

257

MMT Vol. g No 2- H

258

\ (i i?

Figure 1

x \

i

,/

\

iZ

Figure 2

2. Graphic Determination of F~,, During one rotation of crank AAo of the spatial slider-crank mechanism the slider B

moves sequentially to each of the limiting points (Bt. Bit) and returns to its initial starting position. The corresponding motion of the coupler A B describes a ruled surface (conoid) having crank circle C~ and segment BiB. as director curves (Fig. 4).

For a fixed segment BtBn and crank radius r, any change in the length of the coupler requires that the crank rotate a different center. Thus, as shown in Fig. 5, lengths / and I' correspond to C~ and C~.

Figure 3

×

. . ' d ~ ! \ , . I ~,

( ~ I !

(D) 2

(CA

c ~ l ~ " i " if"

/

Figure 4 (D) 3

(C~) ' Ao 5

all @ 6 9, 7 ',1! rr

{D~ z

2 (CA) ' 3

~Y

259

Figure 5

260

Planes P~ and Pn contain axis D and are, therefore, perpendicular to the frontal plane. Projections of these planes on the frontal plane are straight lines which also represent the frontal projection of the coupler when end B is located at B~ or Bw

If the length of coupler A B is k, then for any h, the locus of end A is a sphere of radius R = h. When end B is at B~ a sphere centered at B~ with radius R intersects the plane of the crank at circle Ct. Similarly, with B at Bu a second sphere intersects the plane of the crank at circle CH. Centers of these circles are b~ and b~, the frontal projec- tions of Bt and Bn (Fig. 6).

IfBtb~ = a and B.blt = c, radii Rt and Ru can be determined from

Rt = X / ~ - a Z ; R. = X/Xo.-c z. ~2.1)

Assuming that h >~ c ~ a , then Rt I> Rw Fixed joint Ao is located distance r from end A of the coupler. Distance of A0 from

centers b~ and bit is, respectively

p, = lR~+'rl; po = IR,,-+'rl. (2.2)

Thus the locus of Ao can be graphically determined by the intersection of circles C~ and Co. with radii

12.3)

3. Analytic Considerations (a) The equat ions o f the circles (C0, (C.)

Let O x y z be the Cartesian coordinate system in space and Bt(o, a, o), Btt(b, c, O) be the extreme positions of the slider. The projections of Bt, BII on Ox are Or(o, o, o) and On(b, o, o), respectively, where (Fig. 7):

b = s c o s a : c = a + s sina, (3.1)

If h is the variable length of the coupler then the spheres with the centers at Bx, Bu,

2

7

Figure 6

261

having the same R = ?,,determine in xOz the circles (CO, (Cll) with the equations:

( c ) [ X 2 + Z z = h 2 - a '. ( C n ) I ( x ~ - _ b ) ' + z 2 = h 2 - c a ' [ 3 ' = o t y =

(3.2)

which have the radii (2.1) respectively. As in Section 2 we can suppose a ~< c and hence h ~> c, since the other case can be

reduced to the former. At xOz, the sys tem

x z + z 2 = h2 _ a 2

( x - b ) 2 + z 2 = h2--ca (3.3)

gives us information about the reciprocal posit ions of the circles (CO, (C,). Subtracting there follows

and

ca - a 2 - b 2 ( 3 . 4 )

x - b = 2b

Z 2 = h 2 - - ca -- ( c2 -- ~.~a2 -- b2~2.j = X2 - c ' 2 (3.5)

I f z is a real number , then (C~), (Cn) intersect. I f z is a complex number , then (Ci), (C,) are external or internal. We suppose ca < a2-e M. Then x < b and we have: for c ~ h < c' the circles are

external; for X = c ' the circles are externally tangent; for x > c' the circles intersect on the straight line (3.4).

I f c ~ = a -~ + b 2, then for any X ;~ c. z is a real number. The relations (3.4), (3.5)

become

x = b (3.4')

z 2 = x z _ c a (3.5 ')

and hence the circles ( G L (Ca) intersect on the straight line (3.4'). I f c z > a ~ + b 2. then x > b and we get: for c ~< X < c' the circles are internal: for

X = c ' the circles are internally tangent; for h > c' the circles intersect on the straight line (3.4).

(b) Remarks on the angle a A complete and adequate geometrical synthesis demands evaluation of the angle a

be tween xOz and (A). F o r that reason we give the equation which determines the angle a with respect to a, b, c.

F r o m Fig. 7 it follows that a satisfies

o r

sin ~ = c --___E (3.6) s

S . ¢ 2 _ a 2 _ b2

sin ~ = ~c-t 2sc (3.?)

262

/ ,' . \ \

I ". o,., ., c~ \

Figure 7

The equation (3.6) shows that a ~ [0. ~-/2] when a ~< c. The equation (3.7) shows that a has a limit value a0, when c" = a" -~ b", which permits us to write

c= - - a" - - b 'z (3.7') sin a = sin ao + 2 s c

From the latter relationship we conclude that the relations

c z -~ a " ~ b ~ 13.8)

are respect ively equivalent to

c~ ~ c~o. 13.8')

Thus it is possible to use the angle a for the determination of the reciprocal positions of the circles (CO, (Cn).

4 . A n a i ~ i c D e t e r m i n a t i o n o f 1"4,, The geometrical locus F~,, consists of points radially equidistant from (Ct), (Ctt).

These points appeared at the intersection of 4 pairs of circles (C ,), (C2) with rays [7]:

(C~')pl- = r + R i ; (C~-)pl- = l r - R l l (C.,_-)p.,- = r ~ - R n : (C._,-!p.,_- = I r - R . l

(4.1}

where r = constant. For R . < R~ < r we obtain a pair of symmetr ica l points with respect to OtO~l given

263 by (C1 +) 71 (C._,-). For R . < r < R~ we get two pairs of symmetrical points with respect to O10. given by (C1-) 71 (C2 ÷) and (C~-) f3 (C2 ~ ).

For r < R . < R~ and b ~ 2r we obtain three pairs of symmetrical points with respect to O~O, g ivenby (C, +) 71 (C.~*), (C~-) 71 (C.~*), (C~-) f3 (C . - ) .

Fo r r < R , < R~ and b > 2r we get 4 pairs of symmetrical points with respect to OtOH given by (C1 ÷) 7~ (C2"), (C1-) 71 (C2+), (C,- ) f3 (Cz-) , ( C , ~ ~ (Co-).

(a) Case a = 0 I f a = 0. then a = c and b = s. The curve F.4o is the conic (ellipse or hyperbola)

( x - b l 2 ) " ,."° - - = 1 (4.2) r 2 r ~ - - b21'4

represented in Fig. 8. The spatial mechanisms obtained by choosing the fixed joint Ao on FA,. are of sl ider-

rocker type [5]. In the extreme positions, the conditions OiAt = OziA H are verified.

(b) Case ~ E (0. rrl2) Let a ~ (0. rr/2), then a < c and b = s cos a. The circles (C0, (C2) of rays (2.3) are

(CI) x"-+z 2 = ( V ' X 2 - a 2 _ r ) 2 (4.3)

(Co.) ( x - b ) 2 + z 2= ( ~ ~ r ) " - . (4.4)

/(~ I

\ [b<__ 2r i

z

i ,

/ .

/ ' L

02

~ , / / , " /

I / i b < 2 r :

/,'

i / I • J .. /' i0;~2 r

\

Figure 8

264

By subt rac t ion there follows

x = b-+ c " - a a - b 2 r X j ~ = . V / X . , _ c 2 1 2b ~-b(=

and then

~4.5~

z=. ,_ /h.,__c=, r Z + 2 r ~ _ [ c a - a " - b ' - ' r /-- . ] -- 2b +. ~ (= ~¢ ~.a _ a.-, .7_ V ~.e _ c2 2.

~,4.6)

The relat ions (4.5) and (4.6), where we imposed ~ I> c and the condi t ion o f being of z, represent the parametr ic equat ions of the curve F4..

This curve is a symmet r i ca l quart ic with respec t to x-axis and hence for a given h ~> c we obtain, general ly , four pairs o f points o f Fa:,.

T h e curve o f centers (F,,) has a vert ical a s y m p t o t e

C a -- a'~ _ b.~ x = b + !4.7)

2b

This straight line is the radical axis o f the circles (CO. (Cu). F o r the general case a # ao the implicit equat ion o f F~o is compl ica ted , so that for

the concre te problems of synthesis one will p roceed to the graphical de te rmina t ion o f Fao with the help o f the pairs o f circles (4. l) and utilizing for a bet ter or ienta t ion the results o f the case ~t = a0 and the results o f Sect ion 5.

F o r a = c~0 the equat ions (4.5) and (4.6) b e c o m e

x = b - ' - b I : h~¢/-~-~-' - a 2 = V'X a - c 2 ) t4.5' )

; r 2 . . . . . . t4 .6 ' ) Z = m ~'k- ' c2--r".=.2r~C'X " c~---~(-,- X/X2 a, ,:N/-~'-_c,)"

The el iminat ion o f x f rom (4.5') and (4.6 ') lead us to the implicit equat ion o f F~,,:

[ ( b - 2 r ) X - b r ] [ ( b + 2 r ) X - b r ] ( X - r ) t X - ; - r ) - 4 1 a X " Z ~" = 0 . 14.8)

where X = x - b. Z = z represent a t ranslat ion along the Ox-axis. T h e curve IA,, cuts the O X = Ox-axis at 4 points:

br br X~ = r ; X ' - ' = - - r : X a = b - - 2 r ; X ~ = b - 2 r 14.9)

when b ~ 2r, and at three points: r

X~ = r: Xz = - r: Xa = -q'- 1 4.9' )

when b = 2r. T h e OZ-ax i s is an a sympto t e for the curve I ' , . (Fig. 9).

(c) Case a = 7r/2 Ifc~ = rr/2, then a < c and b = 0. The curve F. , is r educed to the circle

.v2~.= 2 = ~ V k : : - - ,~ /2 - - r )~ = / ~ v ' k 2 - - C : : ~ r~::

concentr ic with I C~I. ~C.I and radially equidistant from them I Fig. 10~.

14.10)

265

x" x

(C:)o(C;>

r=l s ,2 c -2

, ~ °

(C,)n(C~;

O,

)n(C 2 )

Io

Figure 9

Jz

Figure 10

266

From the equation (4.10), for the parameter X there results only the magnitude:

1 A = ~r V'( aa ~ c" + 4re)-'-- 4a"( a. (4. I I )

O n e obta ins a s ingle m e c h a n i s m (Fig. 10) w h a t e v e r pos i t ions o f the fixed joint .4,, on (CA,) = (F~,,) are.

x - X

Z ¸

s , 2 c , 2 ¢ ~20 °

c7 °cci-> {Fa, e )

~z

Figure 11

.(-;( (C~')n(C~/

i

s = 2 ~y

c , 2 " "

Iz

Figure 12

267

_e J ~ ¢=

I -

. ,<

; J . ¢J

~ . C d ~

Z

t . - e -

Z

v i,, ~

÷ I I

'--- i %

li II

- k ; - t - i~ . ~

i A -~-

~1 h li

,~ ,2 ,2

m C

li II IJ I! II

i .+ + I I : ~ ~ ~ ,~

I -"k ! I ~ + -t- I

A

C c

C

c c c

268

5. The Values of h for Which I'ao Intersects the Ox-axis Let us now re tu rn to the gene ra l case a ~ ~0 2. F o r a b e t t e r o r i e n t a t i o n in the g raph ic

r e p r e s e n t a t i o n of F.~o the va lues o f X for which F.,o i n t e r sec t s the Ox-ax i s a re needed . T h e s e va lues can be o b t a i n e d f rom the c ond i t i ons of t a n g e n c y o f pa i rs o f the c i rc les (C~-), (C2+): ( C , ' ) , (C.,-) : ( C , - ) . (C.,*); (C~-), (C~-).

S u p p o s e tha t b < 2r. T h e n T a b l e 1 g ives all the p o s s i b l e cases . A s e x a m p l e s , for r = l . s = 2. c = 2 the re have been c o n s t r u c t e d the c u r v e s F~,

c o r r e s p o n d i n g to the th ree d i s t inc t c a s e s (Figs . 9. l l and 12):

(a) a = 30 ° = oLo; a = 1; b = ~ ; ; ( b ) ~ = 2 0 ° < c = o ; a = 1 . 3 1 ; b = 1.87; (c) • = 45 = > c~0: a = 0,59; b = 1.41.

References [1] ANTONESCU P., Contribution to the graphical synthesis of spatial slider-crank mechanism, J. Mech-

anism 3, 3 (1968). [2] ANTONESCU P.. Contributii la sinteza grafic~t a mecanismelor spatiale articulate, Te~a de doctorat.

Bucure~ti (1969). [3] ANTONESCU P., Beitnige zur kinematischen Synthese riiumlicher Schubkurbeltneb¢, Rev. M~c.

Appl. 14, 2, Bucarest (1969). [41 A N T O N E S C U P., Beitr~ge zur dynmraschen Synthes¢ raumlicher Schubkurbctrieb¢, Proceedings o f

the I1 International Congress on the TMM, 2, Zakopane, Polska (1969). [5] A N T O N E S C U P., Calculul cursei elementului condus la mecanismele spa[iale cu 4 elemente, St. ~i

cerc. de Mec. apl. 30, 1, Bucuresti (1971). [6] KOJEVNIKOV N. S., K voprosu o kinematike i sinteze prostranstvenn~ krivo~}ipno koromislovi'h

mehanizmov. Tr. sere. po T M M 4. 14 (1947). [7] ANTONESCU P. and UDRISTE C., Contributions to the kinematics synthesis of the four-link spatial

mechanisms. Volume H of the third word congress for T M M ( 1971). [8] DUFFY J. and GILMARTIN J. M., Limit positions of four-link spatial mechanisms, J. Mechanism 4,

3 (1969). [9] LICHTENHELDT W., Zur konstruktion der raumlichen Schubkurbel, Sonderabdruch aus Konstruk-

tion 11.2 (1959).

Die Synthese der Raumschubkurbelgetrlebe mit gegebenem Schubhub und gegebener Kurbelltnge

P. Antonescu und C. Udriste

Kurzfassung- Die vorliegende Arbeit verfolgt die Bestimmung des geometrischen Ortes des Festgelenks A) for verschiedene Koppellangen / = h (Bild. 5) in einer gegebenen Ebene xOz, mit gegebenem Schubhub s = BxBu und gegebener Kurbellange r = AoA (Bild. 3).

AIs Kurve ['A. nimmt man die "Mittelpunktskurve", die sowobl graphisch als auch analytisch bestimmt wurde.

FOr eine gegebene Koppellange / = h schneiden die Kugeln mit Radius R = k (Bild. 6) die Ebene xOz in den Kreisen (C[), (C.) mit den Radien (2.1).

Die Position des Festgelenkes At ergibt sich aus den Schnittpunkten der Kreise (C,), (C2) mit den Radien (2.3).

Die Gleichung der Kurve F.4,, ergibt sich aus den Gleichungen (2.3) der Kreise (C[), (Cn), deren gegenseitige Lage mit Hilfe der Beziehungen (3.5) untersucht wird.

Mit Hilfe des Winkels e, (Bild. 7) gegeben durch (3.6) oder durch (3.7') als Funktion von a+ werden verschiedene M6glichkeiten untersucht.

269

Die Radien der vier Kreise, deren Schnittpunkte die kurve I'a,, bestimmen, sind mit Hilfe der Beziehungen (4.1) definiert.

FOr o~ -- 0 ist FA,, eine Ellipse oder Hyperbel (Bild. 8) und hat als Gleichung (4.2). FOr a E (0, ~r/2) werden die parametrischen Gleichungen (4.5), (4.6) tier

Kurve FAo mit Hilfe der Gleichungen (4.3), (4.4) bestimmt. In diesem Fall ist es eine Kurve 4-ten Grades, die die Ox-Achse als Symmetrieachse hat, und deren Asymptote durch (4.7) gegebenen ist.

Im allgemeinen Falle a ~ ~, ist die Gleichung der kurve ['A,, schwer zu bestimmen.

FOr konkrete syntetische Probleme bestimmt man FA,, graphisch mit Hilfe der Kreispaare (4.1) und verwendet, um eine genauere Losung zu erzielen, die Ergebnisse des Falles (x = ~ , wie auch die Werte aus der Wertetabelle 5.1, for welche Fa,, die Achse-Ox schneidet.

FOr (~ = ~ sind die parametrischen Gleichungen von ra,, durch (4.5), (4.6) gegeben, aus denen die Gleichung (4.8) durch eine Verschiebung entlang der Ox-Achse folgt.

Die vier Schnittpunkte der Kurve ['ao mit der Ox-Achse for den Fall b ~ 2r werden in den Beziehungen (4.9) gegeben, w&hrend (4.9') for den Fall b = 2r gL~ltig ist.

FOr e = ~r/2 reduziert sich ['.40 zum Kreis (4.10) und wir erhalten eine einzige LSsung (4.11 ) f~r k; daraus ergibt sich ein einziges Getriebe (Bild. 10).

Die Kurven Fao for r = 1, s = 2, c = 2 (Bild. 12), die den Werten ~ = (xo, c{ < ~o, c{ > cxo entsprechen, sind in Bildern 9, 11 und 12 dargestellt.

Es ist zu bemerken, dab der geschlossene Teil der Kurve der Raumschubkurbel entspricht, w&hrend der offene Teil for die Raumschubschwinge bestimmt ist.