A modified design for lambda-type double-cranks to approximate a straight line

A modified design for lambda-type double-cranks to

approximate a straight line

Citation for published version (APA):

Dijksman, E. A. (1988). A modified design for lambda-type double-cranks to approximate a straight line. Journal

of mechanisms, transmissions and automation in design : transactions of the ASME, 110(dec), 446-451.

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Published: 01/01/1988

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Reprinted from December 1988, Vol. 110, Journal of Mechanisms, Transmission, and Automation in Design

E. A. Dijksman

A Modified Design for À-Type

Double-Cranks to Approximate a

Straight Line

Faculty of Mechanical Engineering, Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, The Netherlands

Double-cranks containing a Chebyshev-dyad are investigated for their use as a straight-fine mechanism. The three availab/e design degrees of freedom have been used for the optimization of the minimum transmission angle, for the minimization of the maximum deviation and for the length L of the straight-stretch in the coupier curve. The resulting double-cranks are shown to have deviations that are about half as large as for those in which the coup/er point lies on the extension of the coup/er. For any maximum deviation, the length L of the straight-stretch also appears to be about 1.5 times as long as the one obtained when Ball's point lies at the base of the design. A graph showing the maximum deviation as a function of L allows the designer to pick his choice mechanism. A table also wil! be of help 10 find accurate dimensions of the mechanism that yields a given deviation or a given length L of the straight part in the coup/er curve.

Introduction

A four-bar linkage (Fig. 1) containing a "Chebyshev-dyad"

(Bo - BAK), having the form of a "crooked" -À, always

pro-duces a symmetrical branch of the coupier curve. (For non-Grashoflinkages, there is onJy one branch.) The proposition is based on Roberts' law, when applied on asymmetrical four-bar producing a symmetrical «coup/er-branch." Apart from the non-Grashof linkages, four-bars containing a Chebyshev-dy ad are either crank-and-rockers or double-crank mechanisms. Although double-cranks in comparison to crank-and-rockers do not approximate a straight line very efficient-ly, we still want to know how far they may be used for that purpose. In other words, we are trying to seek the best double-crank straight-line mechanisms. In order to get a rough idea of the possibilities, we may first look into the solutions based on

BaWs point. (At such a point, we have four coinciding

ac-curacy points.) A further expedient simplification would be to take P.l = P.2' in order to balance the two extreme values of the transmission angle appearing in the stretched positions of the input crank. If, additionally, the coupier point lies on the ex-tension of the coupier

h

=

0), only one solution ensues. But, an infinite number of solutions would be obtained, if for

-y?éO, the accuracy points are spread out in such a way that the maximum deviations from the straight line are equalized. Dijksman (1986) shows results that are obtained when the coup Ier angle -y

=

O. As expected, they appear to be better than the one based on Ball's point. That is to say, longer stretches of the straight line are to be approximated with smaller deviations.

In this paper, the angle -y is varied in order to improve the quality of the straight-Hne approximation. Results show that,

Contributed by the Mechanisms Committee for publication in the JOURNAL OF MECHANISMS, TRANSMISSlONS, MD AUTOMATION IN DESION. Manuscript received at ASME Headquarters, July 23, 1988.

446/VoI.110, DECEMBER 1988

for -y approximately equal to 30·, even Jonger straight-Hne segments appear when maximum deviations are held at the same value.

--.!"or ~oubJe-cranks containing a Chebyshev or À-dyad,

BoB=BA =BK. Thus the À is in fact a crooked one when -y ~KAB?éO. Generally, coupier branches traced by the coupier point K of this type take the form of a heart (Fig. 4). Any line normal to the axis of symmetry, therefore, intersects the branch at not more than four points. Rence, six intersec-Hons, that appear with crank-and-rocker mechanisms, are not possible in our case. Thus, since there are not more than four accuracy points, the approximation of a straight line by double-crank mechanisms is not as good as that obtained with crank-and-rocker mechanisms.

In comparison to four coinciding accuracy points (Ball's point), the noncoinciding ones are known to result in Jonger segments of the branch approximating the straight line. Therefore, it is possible to improve a first approach that is based on Ball's point. The method we recommend doesn't ex-actly use the accuracy points, but calculates the dimensions of the mechanism from the coordinates (-L!2,h_{min )} of the coupier point when it occupies the same level as in the mid-position. The maximum deviation, which is half the band-width of the wavy curve, may then be minimized through variation of the height-coordinate hmin of the lower level. The

minimization of (hmax - hmin ), therefore, may be carried out

for any chosen length L of the straight segment in the coupier branch.

2 Determination of the Coupier Angle l' and "'min from

L/b

and hmin/b

One of the major properties of a Chebyshev-dyad is that

-1.:KBoA

=

7r12 - 'Y remains a constant during the motion. (See

Transactions of the ASME

/r

or

/( (x,;')

and for double-cranks

giving

a=cos p.m;n/2+sin Jl.min/2

d = cos Jl.m;n /2 - sin Jl.min /2

(10) (11)

ad=cos Jl.min (12)

When the coup Ier point is on the same level hmin as in the mid-position its coordinates are x= -L/2 and h _{hmin . The} equations (2) and (4) then turn into

d+a cos rf>L

2 cos Jl.L/2 (13)

A.,-A" and Fig. 1 Coup Ier motion split up into a rotation aboul Bo and into a car· dan molion

Fig. 1 in which <!KAB= <t:.BKA='Y.) This property enables the designer to divide the coupier motion into two parts, a pure rotation about the fixed center Bo, followed by the weIl-known elliptic or cardan motion. In the figure, three positions are shown; the positions 1, 2, and 3, of which 2 and 3 are in-stantaneous coincident positions, corresponds to the mid- or symmetry position of the mechanism. Now, if the rotation about Bo brings the dyad from position 1 into a fictitious posi-tion I', when rotated through the angle KlBoKl' the elliptic motion finally turns the coup Ier from this fictitious position into the midposition. From this, a number of formulas may be derived. Defining a coordinate system, such that the origin 0= Bo and the h-axis runs along the axis of symmetry, the coordinates of our coupler-point K(x,h) then give the

equa-tion arctanx/h <f.=ABoK2 î:ABoAl (1)

Thus

(2) or

x -a sin tf>

.J

x

2 _{+ hl 2 cos} (3)

in which a=AoA; d=BoAo; b=BoB 1; rf>

=

1:

AAoA 2; Jl/2=

1:

BBoA.

From the law of Sines, applied to CJ(BoA, we further have

(14)

rf>L and JlL are related through equation (9).

The coordinates of the coupIer point in the midposition are, respectively: x=O and h=h min . Hence, equation (4) transforms into

h_min = 2 sin [(p.rnin /2) 'Y

1

Then, defining Po and I' through the equations

Po = _{arc sin(hmin /2)}

so that

1'= arc sin

+.J[(LZ/4)+h~inl

Jl.min 2(1'0 + 'Y) Jl.L = 2(p+'Y) we find that hmin - . - cos(P+'Y) =d+ smp cos 2(p+'Y) d and similarly h_min - . - - cos(po + 'Y) smvo Subtraction gives hmin .cos'Y·(cotv - cotpo)

d + cos 2(1'0 + 'Y)

d

cos 2(p+'Y)-cos 2(po+'Y)

d (15) (16) (17) (18) (19) (20) (21) (22) (4) or

In fact, the equations (2)-(4) represent a parametric form of the coupler-curve equation, provided the crank angle rf> and the transmission angle Jl. are related to one another. The latter may be derived from the law of Cosines in MoAoA

a2 + dl + 2ad cos rf> 4 cos2 _{Jl./2 (5)}

From the stretched positions of the input crank AoA, we further derive

a+d 2 cos Jl.z/2 a-d=2 sin Jl.l/2 Therefore

Hence

ad(cos rf> - I) = cos Jl. - cos Jl.2 ad(cos rf> + 1) = cos Jl. + cos Jl.l

cos cos tf>;; --'---'---...:'----'-cos Jl.l + cos Jl.z (6) (7) (8)

Taking P.l Jl.z

=

P.min for equalized minimum transmission angles, we find that

cos P.min cos tf> = cos Jl. (9)

cos 'Y sin(v + 'Y + Po + 'Y)

sinp cos(vo + 'Y) - sin(po + 'Y) (23) giving

tanz'Y- (2cot(p+po)+ . . cosPo + sin Po ) tan 'Y smp.sm(v+po) cospo sinpo

+

-1 0 sinv'sin(p+ po) (24) or 1 tan 'Y= 2 (p+ +2q)- T.J[(P+ +2q)2-4(p_ -1)) (25) where cospo - sinpo p - = --:----':---:----=--:-sinp.sin(p + 1'0) and cosPo + sinpo sinp.sin(p+ po) (26) In case 'Y

=

0, equation (24) gives p _

=

1; hence equation (25) needs only the minus sign before the square root. When 'Y=O,

we further find th at cot Vo = tan2 v

+

tanv

+

I; see also,

Dijksman (1986) for this case.

Generally , however, we may determine the coup Ier angle 'Y from the quadratic equation (25) provided hminlb and Llb are known quantities. The remaining dimensions of the double crank are then calculated from the equations (18), (10), and (11).

The transmission angle PL for the extreme positions of the coupier point may be calculated further from (19), whereas the corresponding crank angle ~L follows from equation (9), i.e., cos cpL cos PL -cos -1 Pmin or its equivalent

cp L ::I:

[~o

+

arc sin ( : sin

~o)

]

(27)

where

orinverted

t = _-_s...:;o~_1_+_..j_s..:.Ö_+_6_s....::0~+_1 2 So

(Ball's point) (35) where So tan Vo tan[(Pmin12) - 'YBaIl]' (As So = 0, when t = 1, the plus sign before the square root has been taken.)

Instantaneous

kinematics lead to the same result, Fig. 2. Then, Ball's point coincides with the mid- or symmetry posi-tion of the coupier point, when the crank -pin A is aligned with the fixed link. Ball's point also joins the inflection circle and the cubic of stationary curvature of this position, the latter degenerating into the pole-tangent BoB and into the circle about BI with radius b= 1. From the geometry of the figure and from Euler-Savary's equation, we then find that

Ll2

CPo = arc tan

-h_min (28) Bo Wsin Pmin12

_ _ PA2

w=AAw-(a+d)= AAl -(a+d)

AAo

The geometry of these positions further reveal that

Y;L

=PLI2::1:~O (29) which are the angles of the secondary crank BoB correspon-ding to these postions.

3 Ball's Point as a Particular Case

This case applies when LI2 0; thus when v = vo, equation (23) then leads to

cos 'YB or

2 sin Pmin/2 cos Pmin/2 cos Pmin/2 sin Pmin/2

tan 'YB

=

tan Pmin/2 (1+

Taking tan Pmin12=t, we may write that

or I I ( I \ \ \ 1 (

_lt)

tan 'YBaIl

"2

t2

+

t

+

1 (Ball's point)

I I \

\

\

"

1 t t(1+t) (Ball's point) (30) (31) (32) (33) (34) d (a+d)

a

Therefore

12 cos Pmin/2-sin Pmin12

cot Pmin • .

cos Pmin/2

+

sm Pmin/2 readily leading to equation (34).

Therefore, for double-cranks producing Ball's point, the following may be summarized

Pmin 90· ,'YB = 45· Pmln 57.065° , 'YB = 0° Pmin 45° , 'YB -22.5° Pmln 41.0708°, 'YB -30· Pmln 30· , 'YB - 50.1039·

Pmin

O·

'YB -90·

4 Determination of h

max and I1h

for

the Optimized

Linkage

At this stage, all dimensions of the mechanism are assumed to be known. The height h of the coupier point, at a random position, may generally be calculated from the equations (2) and (4).

h= (d+a cos ~)(cos 'Y tanpl2-sin 'Y) (36)

The maximum value for h is attained when O=dhld~= (d+a cos cp) -::--::;-~

dp .

- - - -a sm ~(cos 'Y·tan pl2- sin 'Y)

dep

From equation (9), we derive

dp cos -sin ~

~ =_....:...:.::::::....p_.:- (37) Substitution of this value in the foregoing equation leads to

(d+a cos cp)cos 'Y

2

cos Pmin sin P

(38)

Fig. 2 Osculating posItIon, uslng BaU's point doesn't glve the best having divided by the common factor sin~. which is zero only

stralght·llne mechanlsm, though 1I isn't so 'ar olf in the midposition for which h h_{min and not} h_{max .}

or

Table 1 Crooked·À·lype double·cranks having equlllbrated deviations from the straight Une that ara mlnimized

0.5000 t)~5200 0.5400 (I. ~ó()o 0.58,)0 0.""00 O.62i)() tJ"ó400 0 ... 00 ,) ... 800 0.7000 0.7200 0.7400 0.7"(JO 0.7800 0.80(;(1 0.8200 l~. 8400 0.8600 0.8800 0.9(1)0 0.9:200 0.94<1>':-0.'1"<10 0.98<:'0 1.0000 1.0200 1.0400 1. 0 .. 00 1.0800 1.1000 1.1200 1.1400 1.1600 1.1800 1.2QfJ9 1.2200 1.2400 1.2600 1.2800 1 ~ 3000 1 ~ 32(Jü 1."400 1.3 .. 00 1 .. 3800 1.4000 1.4200 1.4400 1.4600 1.4800 1.5000 1 .. 5200 1.5400 1.5600 1.~8t)O 1.6000 1.620l) 1.6400 1 ... 00 1.6800 1.7000 1.7200 I . 7400 1;7600 1.780(1 1 ~ 8000 1.8200 1.8400 1 .. 8600 1 ~ 88ÜO 1.90(1) 1.9;;00 1.9400 1 ~ 9600 1.9BOU 2.0(01) lllu(mln) 39.8117 ,39.766 .. 39.7158 39.6592 39.6082 39.5510 39.491 .. 39.4442 39.3743 39.3178 39.2500 39.11322 39.1136 39.0454 38.9751 38.9025 38.8194 38.7431 38.6677 38.5855 38.4992 38.4186 38.3284 38.241 .. 38.148 .. 38.05 .. 5 37.9 .. 34 37.8 .. 73 37.7680 37.6 .. 89 37.5669 37.4622 37.3555 37.2512 37.137'1 37.0398 36.9171 36 .. 8029 36.6808 36.568::> 36.4465 36.3244 3 ... 2015 3 ... 0751 35.9481 35.8190 'lS .. ó8eCI 35.5531 35.4169 3!:i.2828 35.1427 35~0025 34.8 .. 04 34.7153 34.5702 34.4224 34.2723 34.121 .. 33.9 .. 78 33.8117 3.3 .. 6552 33.4951 33~3350 33.1728 :;~3. (;ü77 32.8416 32.6732 '32.5ü37 ::;2.3315 32.15'72 "!1.981(1 ~l. >3037 '31.6241 31.4423 :~1 .. 2589 .31.07.38 gamma -31.0940 -":51.0742 -31.0609 -31.0543 -3 I. o:n4 -31. 0198 -31.00 .. 3 -30.966.3 -30.9043 -30.9332 -30.9190 -30.9009 -30.8802 -30.8546 -30.8289 -30.8035 -30.7934 -30.7667 -3C'.7343 -30.7104 -30.6900 -30.0551 -30.6339 -30.6024 -30.5781 -30.5485 -30.5165 -30.4860 -30.4572 -30.4242 -30.3925 -30.3 .. 16 -30.33(13 -30. 29l)7 -30 .. 2636 -30.2210 -30. 18ó5 -30.1490 -30.1106 -30.0726 -30. <)"372 -29.9978 -29.9560 -29.91 .. 5 -29.8740 -29.8313 -29.7881 -29.7478 -29.7022 -29.6561) -29 ... 131 -29.5662 -29.518e -29.4727 -29 .. 4225 -29 .. 3731 -29 .. 3238 -29 .. 271~ -29.2207 -29.1099 -29.1158 -29 .. 0639 -29.0079 -28.9518 -28.8965 -28. S::,[J'j -26.7612 -28.7215 -:'8.6623 -28.6026 -28.5421 -28.4795 -28.416-/ -28.35::.5 -28.2690 -28.2231 1. 5545 1.5536 1.5528 1.5521 1.5511 1.5503 1.5494 1.5480 1.5473 I. 54 .. 1 I. 5451 1.5440 1.5429 1. 5417 1.540:5 1.539·3 1.:5363 1.5370 1.535 .. 1.5343 1.5331 1.5316 1.5303 1.5289 1.5275 1.52 .. 1 1.5246 1.5231 1. :5217 1.5201 1.5186 1.5171 1. 515:5 1. 51:SS I. 5124 1.5106 1.5090 1.5073 1.~OS6 1. 5'):39 1.5022 1.5005 1.4987 1.4970 1.4952 1.4934 1.4916 1.4898 1.4880 1.48 .. 1 1.4843 1.4624 1. 4805 1. 4787 1.4767 1. 4748 1.4729 1.4709 1.4 .. 90 1. 4 .. 71 1.4 .. 51 1 ~ 4632 1. 4612 1.4592 1.4572 1.4552 1.4532 I . 4512 1.4491 1. 4471 1.4451 1. 4431 1. 44!() 1.4390 1.4370 1.4349 h(minl/b 1.5543 1.5534 1. 552:5 !. 5517 1.5507 1.5498 1. 5488 1. 5474 1.5460 1.5453 1.5442 1.5431 1. 5418 1.5405 1.:5392 1.5378 1.53 .... 1.5352 1.533 .. 1.5322 1.5307 1.5290 1.5275 1.5259 1.5243 1.5225 1.5208 1.519() 1. 5172 1.5153 1.5134 1.5115 1.509 .. 1.5075 1.5056 1.5034 1.501:1 1.4991 1.4969 1.4946 1.4924 1.4900 1.41376 1.4852 1.4827 1.4802 1.4777 1. 4751 I . 4725 1.4098 1.4071 1.4043 1.4 .. 15 1.4587 1.4557 1 .. 4528 1. 4498 1. 4467 1. 4436 1.4405 1;4373 1.4341 1.4308 1. 4274 1.4241 1.4206 1.4171 1.4131:> 1. 4100 1.40 .. 3 1.4026 1.3989 1.3950 1.3912 1.3673 1.3833 .ah/b 0.00023 0.00027 0.00031 0.0003 .. 0.00041 0.00047 0 .. 00054 0.000 .. 1 0.00069 0.00077 0.00087 0.00097 0.00108 0.00120 0.00133 0.00147 0.00102 0.00179 0.00196 0.00215 0.00234 0.0025" 0.00278 0.00302 0.00328 0.00355 0.00384 0.00414 0.00440 0.00480 0.00:515 0.0(':553 0.00593 0.00 .. 34 0.00 .. 78 0.Q0724 0.00772 0.00823 0.00875 0.00931 0.00989 0.01049 0.01112 0.01178 0.01240 0.01,317 0.01392 0.01469 0.01549 0.01033 0.01719 0.01809 0.01903 0.02000 0.02100 0.02204 0.02311 0.02423 0.02538 0.02;'57 0.02780 0.0'2907 0.03039 0.03174 OJ)3314 0.03458 ,0.<:<3 .. 07 0.03760 Ç'.03918 0.04061 0.04248 0.04421 0.04598 0.04781 0.04909 (1.05161 b i L 0.00046 0.00051 0.00057 0.000 .. 4 0.00071 0.00079 0.00087 0.00095 0.00104 0.00114 0.00124 0.00135 0.0014;' 0.00158 0.00171 0.00184 0.00198 0.00213 0.00228 0.00244 0.00261 0.00278 0.002941 0.0031:5 0.00334 0.003:55 0.00376 0.00399 0.00421 0.00444 0.004 .. 9 0.00494 0.00520 0.00547 0.00575 0,001003 0.00633 0.00 .... 3 0.0069:5 0.00727 0.00700 0.00795 0.00830 0.008 .... 0.00903 0.00941 0.00980 0.01020 0.01061 0.01103 0.01146 0.01190 0.01236 0.01282 0.01329 0.01317 0.01427 0.01477 0.01529 0.01582 0.01635 0.01 .. 90 0.01746 0.01803 0.016 .. 2 0.01921 0.01982 0.02044 0.02107 0.02171 0.022,36 0.02303 0.02370 0.02439 0.<)2509 0.02:581

Using equations (9)-(12), th en gives

sin2_{JL tan 'Y(1}

₊

_{cos JL)sin JL =}

_rJ2

_{+ cos JL = (1-sin JLmin)}

+ cos JL

Therefore, a recurrent relation, based on equation (39), such as

(1 +COS JLs)·cos(JLs-'Y) cos 'Y. sin JLmin

b.JLs = _sm . (2 _{JLs 'Y + sm} ) ' ( _{JLs - 'Y} )

1800

(41)

(1 +cos JL).COS(JL-'Y) = cos 'Y. sin JLmin (39) For any given values of JLmin and 'Y, therefore, this equation holds when dh/dt!> = O. We assume that this occurs when JL

=

p..

One may prove that for 'Y

=

'YB.lI' that is to say for a curve con-taining Ball's point

may efficiently start with the initial value JLs JLmin' after which JLs+\ =JLs+b.JLs, JLs+2 JLs+l + b.JLs+l , etc. Two or three steps are then usually sufficient to find the end value

p.

that solves equation (39). The value

p.

obtained, then gives the posi-tion for which

(1 + cos JLmin)·CoS(JLmin -'YBaU) = cos 'YB.U·sin JLmin (40) (42)

Fig. 3 For all double·cranks: /L1 ::: /L2 = /Lmin

obtained through elimination of (d+a cos tP) from the equa-tions (36) and (38).

With the help of equation (15), we find the maximum devia-tion

I

-t:.h

2

I sin jlosin2 _{[(jll2) - 'Y1}

2

(hmax - hmin ) d cos 'Y

sin [(Pmin 12) - 'Y1 (43)

It is this value that has to be minimized by variation of hminlb. The minimization has to be carried out for constant values of

Lib. (Dne may verify that for Llb=O, indeed ,),,='YBall;

jl

=

Pmin and t:.h

=

0).

The mechanism with a minimal value for t:.h may then be found from the known values of Llb and hminlb. Results of the optimized double-cranks are tabulated as weil as plotted in Fig. 3.

5 Resulting Deviations Compared With Those

Obtained Through Ball's Point

The resulting deviations and subsequent linkages with those based on Ball's point are compared only with those having the same LI2-and Pmin-values. Given these values, it then re-mains to calculate t:.h for the coup Ier curve concerned, con-taining BaIl's point. Therefore, the corresponding generating mechanism is completely defined by and the coupier angle 'YBalI that follows from 'YBan

=

arc tan ( + t+ I -l/t}l2, where t=tan (Pmin12). Further

hmax =2 Sin«Pmin12)-'YB) (44)

whereas hmin follows from the equation 4501 Vol. 110, DECEMBER 1988

I

I

r

I

\

\

/ / I

\

\

\

"-/ ' / ' ~~==~~~~~~~,:~

"

"'-"

"-

'-

'"'''---~ mt?,O ~mJn ~.J~(J'.31/

J"

= -J!8,22-"; ó&r-It-B/c'

-1!-

=

O,O'?51J

Fig. 4 Double·crank tor which the (hmax -hmin)-value has been minimlzed. Comparlson with deviation when Ball's point Is used.

(45) wh ere PB = PBan may be calculated initially through the expres-sions derived from equations (3), (4), and (9)

L/2

cos

tPB

cos PBoCOS-1pmin or through the equation

LI2 -

cr.I(l-

COS2_PB

.cos

_{-2pmin )}

.(tan(PBI2)cos 'YB sin 'YB) = 0 where

tPB

has been eliminated.

(46) (47)

(48) This equation may be solved fOT PB' using a (derivabie) recurrent relation, based on the Newton-Raphson method. For the initial value needed, it is recommended to use the vaIue PL from the "optimized linkage," already obtained in the preceding section. Having found PB' it is then easy to calculate

(t:.h)IL _{(hmax - hmin)/ L using equations (44) and (45).}

The results are shown in the diagram, in order to compare them with the deviations obtained with the optimized linkages. In addition, Figs. 4-7, demonstrate double-cranks, each containing two different coupier curves, one with minimized deviations and the other containing Ball's point. The coupier points producing minimized deviations are located in the ex-treme positions of the straight segments in the coupleT curve. The ones producing Ball's points are already farther out.

Conclusion

The possibility of varying the coupier angle 'Y has been used to minimize the maximum deviations already equilibrated. FOT a fixed coupleT angle (such as 'Y = 0), the maximum deviations may only be equilibrated. Even then, the results are better than the on es based on Ball's point. Generally, the best results Transactions of the ASME

1/.6

-~;r ;/mio ~,",G5'6.l"

cr

= -élU; 1158°;

!iwt""

-1t.3, '1/81/

1 "

401845;

~II

=

o,O~~tJ6

Fig. 5 Double-crank lor which Ah·value has been minimized. Com· parison wlth deviation based on Ball's point (U).

! - ' - , " - - - - ; . . . ..c:!... _ - -0"" .... ;;r ,

-../ /'

"

/ ~

/

~

I

~

\

/

\

/

\

I

l./,z

\

I

I

\

I

\

I

\

/

\

/

\

/

\

/

~ /

"

_"-

_/ /

"-

../ ' - / '

"'-..---o -

I ; .-/min = 38, tJS1J5"

I

=-345#350 ; ~/=J5,G1JS'

~h

"" f),tJOJ55 ;

~&II

=

(JiJ/IS'! 0

Fig. 6 Double-crank wlth mlnlmlzed deviatIons of the straight IIne. Comparison wlth deviation when Ball's point Is used.

/ / /

"

--

_.--"

At "-

,

\ \ \ \ / / 1 I I I \ \ \ \ 1 I I I I I

Fig. 7 Double-crank for whlch Ah·value has been minimized. Com-parilon wlth devlatlon based on Ball's point U.

are obtained for coupier angles 'Y in the region of about - 30· . For the same deviation and coupier length, the length L of the straight segment in the coupier curve, appears to be about •. 5 times as long as for double-cranks having a Ball's point in their coupier curves.

For 0.5

<

L/b

<

2, the minimum transmission angle of the optimized double-cranks remains within the interval

39.8° <I'min <31 .• •.

Acknowledgments

A software program, based on the equations in this paper, has been written by Ir.A.T.J.M. Smals, Senior Research Of-fleer in the Faculty of Mechanical Engineering, Eindhoven University of Technology. The flgures in this paper were drawn by Mrs. M. Dijksman-Margarit.

References

Dijksman, E. A., 1972, "Approximate Straight·Line Mechanisms Through Four-Bar Linkages," Romanian Journalof Technica/ Sciences, Applied Mechanics, Vol. 17, No. 2, pp. 319-372.

Dijksman, E. A., 1976, Motion Geometry of Mechanisms, Cambridge University Press, London/New Y ork, pp. 114-116.

Dijksman, E. A., 1986, "Optimized Double-Cranks With a Nearly Straight Segment in Their Coupier Curve," Design Engineering Technical Conference, Columbus, OH, October 5-8, ASME Paper No. 86-DET·175, pp. 1-12.

Schlütter, 0., and Tolle, P., 1985, "Auswahl·kriterien für symmetrische Gelenkviereck ·geradfürungen," Stuttgart-Tagung (Nov. '85), V .D.I.-Berichte 576, pp. 181-202.

Tao, D. C., 1965, "A Four·Bar Linkage Adjustable for Variabie Straight-Line Motions," ASME Journalof Engineering for Industry, Series B, Vol. 87, No. 3, pp. 287-290.

Tesar, D., 1965, .. Analysis of Approximate Four-Bar Straight·Line Mechanisms," ASME Journa/ of Engineering for Industry, Series B, Vol. 87, No. 3, pp. 291-297.

Wunderlich, W., 1965, "Zur angenäherten Geradführung durch Sym-metrische Gelenkvierecke," ZAMM, Vol. 36, No. 3/4, pp. 103-110.