Kinematic synthesis of geared linkages

Jnl. MechanismsVolume 5, pp.59-87/Pergamon Press1970/Printed in Great Britain

Kinematic Synthesis of Geared Linkages G. N. Sandor ° and R. E. Kaufman; with A. G. Erdman § T. J. Foster§ J. P. Sadlerl] C. Z. Smith fl and T. N. Kershawll Received 31 July 1969 Abstract Planar geared linkages readily lend themselves to function, path and motion generation. Function generation includes any problems in which rotations or sliding motion of input and output elements (either links or gears) must be correlated. In some cases, the designer may want to produce a formal functional relationship between the input and output. In these cases, the input and output rotations can be used as the linear analogs of the independent and dependent variables. In other cases, the designer has no particular functional relationship in mind, but merely wants to produce specific relationships between the input and output at certain "precision points". Path generation involves moving a tracer point along a specified path. A point attached to any "floating" link (such as a coupler) may be used as the tracer point. Motion generation requires that an entire body be guided through a prescribed motion sequence. The body to be guided can be attached to any floating link. Zusammenfassung--Kinematische Synthese von Zahnradkurbelgetrieben ; G. N. Sandor und R. E. Kaufman mit A. G. Erdman, T. J. Foster, J. P. Sadler, C. Z. Smith, T. N. Kershaw. Ebene Zahnradkurbelgetriebe eignen sich als Funktionsget~iebe und zur Bahn und Bewegungserzeugung. Funktionserzeugung enth~ilt Probleme in welchen Drehungen oder Gleitbewegungen der Antriebs-oder Abtriebs-Elemente (entweder Glieder oder Zahnriider) einander zugeh6rig sind. In einigen F~illgn hat der Konstrukteur den Wunsch ein formelles Funktions-Verhiiltniss zwischen dem Antrieb und Abtrieb zu erhalten. In diesen F~illen k6nnen die Drehungen des Antriebes und Abtriebes als die linearen Analoge der unabh~ngigen und afhiingigen Ver~nderlichen verwendet werden. In anderen Fiillen hat der Konstrukteur keine vorgesehene funktionale Verh~ltnisse, sondern m6chte nur spezifische Verh~ltnisse zwischen dem Antrieb und Abtrieb in gewissen "Pr~zisionspunkten" erhalten. Bahnerzeugung hat das Ziel einen bestimmten Punkt in einer vorgegebenen Bahn zu bewegen. Ein Punkt befestigt zu einer Koppelebene kann als Erzeugungspunkt benutzt werden. Bewegungserzeugung verlangt manchesmal, dass ein ganzer K6rper durch vorgeschriebene Lagen gef6hrt wird. Der gef~hrte K6rper kann dann an die Koppelebene befestigt werden. * DipI.M.E.; D.Eng. ALCOA Foundation Professor of Mechanical Design; Chairman, Division of Machines and Structures, Rcnssclaer Polytechnic Inst., Troy, N.Y.; Fellow, ASME; Mere, N.Y. Acad. Sc. **Assistant Professor, Massachusetts Inst. of Tech., Cambridge, Mass.; B.S.M.E.; M.F.A.; M S.M.E.; Ph.D.; Associate Member, ASME. § I t . U.S.A.R.; B.S.M.E., M.S.M.E. Graduate Students, Associate Member, A.S.M.E.; Rensselaer Polytechnic Ins., Troy, N.Y. 12181, U.S.A. [[ Graduate Student ASME, Rensselaer Polytechnic Inst., Troy, N.Y. 12181, U.S.A.

59

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Introduction A FREQUENTLY encountered industrial problem involves the generation of intermittent or non-uniform motions. Simple linkages, such as the four-bar drag link mechanism, are usually the least complicated and most satisfactory devices for such tasks. However, when the desired motion is too complex to produce with a four-bar linkage or a slider-crank mechanism, a geared linkage can often be designed to economically fulfill the design requirements. For example, in a packaging machine, it may be necessary to connect an input shaft and an output shaft so that the output shaft oscillates with a prescribed dwell period and timing while the input shaft rotates continuously. A simple geared five-bar mechanism, such as that shown in Fig. I, can be readily designed to produce such a dwell period [1].

/

\ --F--

\

Figure 1. Geared five-bar dwell mechanism. In the case shown, the gear ratios have been chosen to produce a hypocycloid of four cusps (astroid). The curvature between cusps is approximately an arc of a circle, and the length of the coupler is equal to the radius o f that arc. The follower crank has been arranged so that in its extreme position its crankpin falls at the center o f the arc. If desired, the coupler could be attached to an output slider rather than to an output crank. Still different motion characteristics could bc obtained through the use o f a three or five cusp hypocycloid.

61

This problem of relating the rotations of the output (~) to the rotations of the input (4) can be considered one of"function generation". A desired functional relationship, # =f(~), relates the rotations of the input and output cranks. It is the mechanism designer's task to synthesize a linkage capable of generating the desired function. As another example of function generation, suppose that a valve actuator linkage must be designed to connect an electric motor to a large globe valve. Because of the low starting torque of the motor, it is necessary that at the start and end of the valve motion the motor should run unloaded. In other words, when the actuator is in an extreme position, the motor shaft must turn without moving the output shaft. A general statement of this type problem would be that the angular positions of the input and output shafts must be correlated and, in addition, the angular velocities of the two shafts must be correlated at certain prescribed "precision points". This is an example of "higher order function generation". Geared linkages can be readily designed to fulfill this type of requirement. A third type of problem which can be solved using geared linkages is "path generation". For instance, in an electric motor armature winding machine, the wire bobbin must be passed through a prescribed path in space. The synthesis problem is to design a meehan;.sm to produce the desired path. Endless varieties of geared linkages can be synthesized using the approach illustrated in this paper. Single or multiple-degree-of-freedom linkages can be designed. Either gear rotations or link rotations can be used as mechanism input or output variables. The simplest mechanisms are single loop, single-degree-of-freedom mechanisms, although the same design approach is adaptable to multi-loop, multi-degree-of-freedom linkages. Degrees of Freedom Mobility of a planar geared linkage can be studied through the use of the equation

F=3(n- l)-2j- G

(1)

where Fis the number of degrees-of-freedom of the linkage, n is the number of links, j is the number of revolute joints, and G is the number of constraints imposed by the gears. If the gears are attached rigidly to p links, then G=p[2. For instance, the mechanism of Fig. 2

$

A\

\$

Figure 2. Geared five-bar function generator linkage. GearA is stationary, gears B and C are idlers and gear D is rigidly attached to the output link.

62 has five bars and a train of four gears. The gears are fixed at two points, to the frame and to the output crank. Thus n = 5, p = 2, j = 5 and G = 1, so the system has a single degree of freedom. All of the illustrative examples worked out in this paper have a single degree of freedom.

Complex Synthesis Equations The method of complex numbers [2] is particularly well suited to the synthesis of geared linkages because links and gear ratios between links are readily represented and manipulated mathematically. When a limited number of precision conditions is imposed on a linkage, the method provides synthesis equations which are linear in the unknown link vectors describing the mechanism in its starting configuration. In function generation with a singleloop mechanism, a linear solution can be obtained for one fewer precision conditions than the number of bars in the loop. For example, in the case of a geared six-bar (such as is shown in Fig. 12), a linear solution can be obtained for up to five first-order precision points or for, say, one first-order and two second-order precision points or, as still another possible combination, for one second-order and one third-order precision point. (By second-order, we mean that both position and velocity are specified with respect to the reference variable. In third-order approximation position, velocity, and acceleration are specified.) One can prescribe more than five precision conditions for this mechanism, but the solution is made more difficult because some of the coefficients of the link vectors must then be treated as unknowns. Nonlinear compatibility equations must then be solved. Computer techniques for handling nonlinear problems will not be discussed in this paper. For finite synthesis, vector loop equations and displacement equations written in terms of complex numbers form the system of synthesis equations. In higher-order synthesis involving prescribed derivatives, velocities, accelerations, and higher accelerations, the loop equations [3] are differentiated with respect to a reference variable or time. In both finite and infinitesimal (higher-order) synthesis, the system of synthesis equations is linear in the unknown link vectors. The coefficients contain the prescribed performance parameters and the gearing velocity ratios. These can be arbitrarily assigned convenient values by the mechanism designer. By varying these arbitrary choices the designer can obtain an infinite spectrum of solutions from which to select a suitable mechanism. All of these solutions will satisfy the given precision conditions. Selection of the best available solution can be based on such optimization criteria as most favorable transmission angles, best gear ratios, or ratio between longest and shortest link lengths closest to unity.

Geared Five-Bar Example Suppose one wishes to synthesize a geared linkage ofthe type shown in Fig. 2 to generate a function y =f(x) over some given range. Let the rotation of the input crank (~b) be the linear analog of x and the rotation of the output link (~k) be the linear analog o f y . One can represent this mechanism by a dosed vector pentagon (Fig. 3). In the reference position of the mechanism, vectors Z l l , Z12, Z13, Z~4 , and Z~ 5 define the orientation and length of links 1, 2, 3, 4 and 5. In a general displaced position of the mechanism, say the j-th position, at which requirements on the motion have been prescribed, the mechanism is defined by the vectors Z/l, Z j2, Z j3, Z i4, and Zjs. If these vectors are written as complex numbers, Zjk =Zik,, + iZjk,, then the rotation of link # k through an angle c<1 from its reference (1-st) to its j-th position can be written Zjk = e~'JZ 14.

(2)

63

Thus, ei=j can be considered a "rotation operator". In terms of the reference vectors for the mechanism of Fig. 3, Z/t= Z j2

Z t t, =

Z j3 =

ea~JZt 2, ei~Z t a,

(3)

Z i4 = et~'JZ t 4,

and Zjs = ei~'JZts.

~B

_z

Figure 3. Vector schematic of geared five-bar mechanism. Note that gear A is fixed to vector 1 which represents the frame, and gear D is fixed to vector 5 which represents the output link. Gears B and C are idler gears and are free to rotate.

A mathematical relation must be developed to express the relative, and finally, the absolute rotations of the various links. In other words, for a specified rotation of the input crank (Link 2), can an expression for the rotation of some other link be found in terms of this input rotation and the gear ratios ? Obviously, there must be some such relation since the mechanism has but one degree of freedom. D e t e r m i n i n g t h e Effect of Gear Ratios on t h e Rotation of t h e Links In reference to Fig. 4, the following general relation will prove very helpful in solving this rotation problem: + Tkt ~j(k+l)=~jlg (~bjk-$j(k_t))Tk+ 1 . (4)

T refers to the number of teeth on some k-th link. For the linkage in Fig. 3, let ~ B and 9: C be the absolute rotations of gears B and C for any given input rotation ~b. 1' and # are likewise absolute rotations of links 3 and 4, respectively. ~ is the rotation of link 5, the output. From Fig. 5:

or

64

~j(k+l)

Figure 4. A general geared pair showing notation for the link rotations.

&

Figure 5. Input side of geared five-bar.

Applying the general relation (see Fig. 6): gC=y+(V- ~cB)~T~.

(6)

Finally, applying the general relation (Fig. 7):

¢/=u+(u- ~C)r~.

(7)

65

¢C

Figure 6. Idler gears and floating link Z13 in geared five-bar.

Figure 7. Output side of geared five-bar. Now,

"¢c=~+(~- CB~T~

Tc Tc ¢=~+(~-

(8)

Tc'

*:c)r~

[ T. T. rs.VlTc = g+/#-t~ +7--- _-~---¢.l-\

L

TD and

Tc Tc

T.

TD

Tc)JT,,

To

TD

d/=#(1-Tc'~ i'Tc+Ts'~--['Tx+T.~ +

(9)

So, it can be seen that there is a direct relationship between ~, ~, #, y, and the gear ratios. As expected, the rotations ~ B and g C of the idlers do not appear in this expression. Since ~band ~ are prescribed by the function to be generated, if values of 7 are assumed then corresponding values of ~ are automaticaUy defined.

66 For convenience, let

to/'

/

s=(Ta+ TB~ \To.l"

(10)

Therefore,

¢=I~Q- ~R+dpS,

(11)

v=~0+~R-¢S).

(12)

Determining the Number of Precision Positions for Which the Mechanism Can Be Synthesized For the mechanism in thej-th position, the vector equation of closure can be written as follows: Zl 2e~J + Zl 3eiyJ+ Z t , e i~J+ Zt 5e ~*J= Zt t = 1.

(13)

Note that Z~ 1 is assigned the value 1 for convenience. This is permissible because only angular relationships are of interest. Recall that the ~b's and ~'s are prescribed by the function and the V's are chosen arbitrarily. Thus, the #'s are incidentally known by g=(1/Q)(~+yR-dpS). The following table can now be constructed :

Table I. Precision positions

Real equations

Real unknowns

1

2

Z12, Z13, Zt4, Z1s

2

4

3

Number of real unknowns

Arbitrary choices

8

6

,, ?'2

9

5

6

,, 7'3

10

4

4

8

,, 3'4

11

3

5

10

,, ~'5

12

2

6

12

,, Y6

13

1

7

14

,, ~'7

14

0

Through four precision positions, the designer can pick the y's arbitrarily. For five, six or seven positions nonlinear compatibility relationships must be solved for the y's. Thus, it can be seen that the limiting number of precision positions, beyond which nonlinear compatibility equations become necessary in the solution, is 4. For predetermined gear ratios and scale factors, the mechanism may be synthesized for 7 positions. However, this problem is considerably more complicated. Only the synthesis of the first 4 positions is given in this example. If the gear ratios and scale factors are also regarded as unknowns, the number of attainable precision points can be further increased.

67

Synthesis Equations for Geared Five-Bar For four finitely separated first-order precision points the synthesis equations for this mechanism consist of the following: 1

1

1

1

Z12

1

ei,~z

eiT,_

ei~Z

ei~Z

Z13

1

ei¢~ ~

ei;.3

elU ~

eiCJ 3

Z14

1

ei~ 4

ei7 4

e i,~4

e i,#,~

215

1

(14) .

As all the quantities in the coefficient matrix are either prescribed or arbitrarily assumed, one can easily solve this linear system of complex equations for the four unknowns, Z12, Z13, Zt4, and Z15. Varying the arbitrary values Yz, "/3, and Y4 and varying the choice of gear ratios and scale factors allows one to obtain an infinite spectrum of solutions. Optimization criteria can then be used to select the best of these solutions.

Determining the Spacing of the Precision Points and Specifying Ranges Suppose that the function to be generated is y = tan(x), 0 < x < 45 °. For four accuracy points with Chebyshev spacing [4], values of x and y can be found geometrically by the construction shown in Fig. 8. Thus : x t = 22.5 - 22.5 cos(22.5) = 1.712711 x2 = 22.5 - 22.5 cos(67.5) = 13.889623 x 3 = 22.5 + 22.5 cos(67.5) = 31.110377 x~ = 22.5 + 22.5 cos(22.5) = 43.287289

(15)

Y1 = tan(xl) = 0.029901 Y2 = tan(x2) = 0.247283 Y3 = tan(x3) = 0.603486 y¢ = tan(x4) = 0.941934.

(16)

Let R ÷ = range of input crank = 90 °. Let R• = range of output = 90 °.

7 /

r0 XI

X2

X=

X445°

Figure 8. A geometric construction for Chebyshev spacing of four precision points.

68 Knowing the values of x and y for the four positions, values of q5i and Oi must now be found. These values of q~j and Oj correspond to the rotation of the input and output, respectively, from the first precision position to the j-th precision position ( j = 2 , 3, 4). e j and ~ks. are computed as follows:

q~i=xiAT~ R~,= ~ ( 9 0 ) = j = YJA-YlR, = ~ ( 9 0 ) =

2(x , - xx) 90(y j - y 1)"

(17)

The results of these computations are: q~2= 24.353824°

¢2 = 19.564380°

~b3= 58.795332°

¢3 = 51.622650 °

~b4=83.149156 °

~k4= 82.082970 ° .

(18)

Utilizing the Computer-Synthesized Output Figures 9, 10, and 11 and Table 2 show some typical computer synthesized linkages for generating the function y = tan(x). Note that the gears are not drawn in. For any particular linkage which may be synthesized, it is possible to calculate the actual size of the gears which are needed so that the mechanism will in fact move through the prescribed precision positions. The pitch radii of a pair of mating gears are proportional to the tooth numbers in the following manner: rl ~

r2

TI °

T2

Taking the mechanism shown in Fig. 9, for example, the length of Link 2 must equal the sum of the pitch radii of gears A and B. (Length Link 2) = r,t + rs = I. 18

T a - 3 - r a , ra=3rn, 4rn=1.18 TB I re rB=0.27 r A= 0.91.

1 1

Figure 9. Synthesized geared five-bar generating the tangent function. Example (1) of Table 2 is shown in its first (solid), second (short dash), third (uneven dash), and fourth (long dash) precision position. Gears are not shown.

69

GL-._ "',,I

,

---.

Figure 10. Example (2) of Table 2 is shown in its four precision positions (same notation as Figure 9). Gears are not shown.

Figure 11. Example (3) of Table 2 is shown in its four precision positions (same notation as Fig. 9). Gears are not shown.

The next condition to be satisfied is (Length Link 3) = r a + r c = 0.85. The r a of the first calculation will not satisfy this condition and still give the desired ratio between gears B and C. This is e x p ~ t e d since this feature was not built into the program. However, this is no problem sinc,~ we are free to m o u n t another gear B' on the same shaft as B. Then, r s, is calculated by r a, + r c = 0.85, and T a , / T c = ½. ra,=0.28 rc=0.57. Similarly, rc,=0.59 rv=l.18.

7O Table

2

(1) Geared 5-bar Figure

(2) Geared 5-bar

9

Function

10

y = tan.r

Range

y = tanx

0<~x-q<45"

0~
R~b =90"

Rq~ ~90"

R¥ ~90 °

R ~ =90"

Ta:Tn:Tc:To 3:1:2:4

Ta:Ts:Tc:To 3:1:2:4

+1.000, +0.000i

+1.000, +0.000i

Zt 2

+0.402, -- 1.115i

+ 1.335, +0.027i

Zt3

-0.709, +0.475i

--0.888, +0.846i

Z14

+ 1.714, --0.4688i

+ 1.919, --0.61 li

Zl5

--0.408, + 1.109i

-- 1.366, --0.263i

Scale factors

Gear ratios Link vectors Zll

Zt6

Arbitrary link Rotations

Figure

.

.

.

72=20 . 73 = 0"

72= O* ~,3= O"

74 =

y4----~6 0 "

0°

(3) Geared 5-bar

(4) Geared 6-bar

11

15

y = tanx

Function

.

y~x

3

Range

0~
0~
Scale factors

R~b =90"

A$=90 °

Rot----90"

A~60

Gear ratios

o

Ta:Tn:Tc:Tn 3:1:2:4

TI:T3; T4:T6 2:6; 2:6

Link vectors ZI

+ 1.000, +0.000i

+ 1.000 +0.000i

Z2

+0.333, -- 1.126i

--1.553, +2.403i

Z3

-- 1.225, --0.482i

+0.742 -- 1.779i

Z4

2.029, +0.254i

+2.475 +0.256i

Zs

--0.317, + 1.355i

--2.313, -- 1.404i +1.653 +0.529i

Z6 Arbitrary link Rotations

7z= O" )'3=20"

72 =

)'4 = 4 0 "

74=12" 75=16"

4*

73--'-- 8 °

71 Table 2 (cont.) (5) Geared 6-bar

(6) Geared 6-bar

16

17 & 18

Function

y =xt.5

y = 1/x

Ran~

0~
1 ~
Scale factors

Acb=90"

Aqb=90"

A~=90*

A~ =90"

Tt:T3; T4:T6 .5:1;1:1

Tt:T3; T4:T6 1:1;3:10

Link vectors Zl

+1.000, +0.000i

+1.000, +0.000i

Z2

+0.807, --2.299i

+2.298, +0.433i

Z3

-- 1.245, + 1.778i

--0.489, +0.848i

Z4

+0.301, +0.922i

--0.494, +0.765i

Z5

-- 1.422, -- 1.148i

+0.717, --0.901i

Z6

+ 2.559, +0.747i

-- 1.032, -- 1.144i

y2= 10" 73 = 20" ~)z= 1 '~3= 1

Y2= 106" Tt = 0" "~z=--l.6 ~t=--l.6

Fi~tr¢

Gear ratios

Arbitrary link rotations and derivatives

Although gears are the basis of the preceding derivation, belts, tape wheels, chain drives, or other such devices could be utilized to obtain the same velocity ratios between the links. Geared Six-Bar

As another example of the application of this general approach to the synthesis of geared function generating mechanisms, we will synthesize the geared six-barlinkage shown in Fig. 12 for point, order and combined point-order approximations. Figure 12 shows the mechanism in the j-th position with the links represented by complex numbers of the form Zjk=Z#,+iZjk ,. TI, T3, T4 and T 6 are the "tooth numbers" for the various gears. The link rotations are measured from the firstprecision position, and the relationships between the firstand j-th positions are as follows:

Zjl~

Zll,

Zyz = ei~Z12,

Zja =

e~Zt3,

Zj 4 =

eiBJZ14,

Z j s = eiT~Zl s,

and Zjs = e ~ Z l e .

(19)

72 The link vectors can be related through vector equations of closure. From Fig. 12, the first-order equation of closure is

Z j2 "~Z j3 ,-~Zj~,..~Z j5 "+'Zj6 -~-Zjl e~*~Ztz + easZ~ 3 + eiaJZt,t + ei~Zl s + ei¢sz 16 = Z t i •

or

(20)

Velocity synthesis can be attained by taking the derivative of this expression with respect to ~bi. The following second-order equation of closure results (superior dot means derivative):

iei~Zl 2 + i~jeV'~Zl 3 + it~jeiP~Z~ 4 + i¢/jei'°Zl

s

+ i(#j e ~ Z l 6 = 0

ei*'Z 12 + ~je~'~Zl 3 +/~leiP~Zl ¢ + 4/jei'tsZt s -F I[IyeiCtSZl6 = 0

or

(21 )

where the i has been cancelled out. If angular accelerations are of interest, then a second derivative of equation (21) must be taken:

iei#~Z 12 dr ( -- ~ dl-i~tj)ei=Jz13

i/b)e'P'zl, + ( - ~ + i~j)e/#'Zl 6 = 0.

(22)

Due to the two pairs of gears, there are two relationships involving angular rotations of the various links. Therefore, two of these angular rotations can be eliminated in terms of other rotations. The first gear constraint gives:

OtJ = ~ J fl- ~33~ J = (1-k ~33)~ J or

(23) where

GEARED SIX-BAR MECHANISM

=j

uj =OUTPUT

Figure 12. A geared six-bar linkage is shown in itsj-th position. Angles labeled are the displacement angles measured from the starting position of the mechanism.

73

!

=j

e Link 6 is Temporarily Fixed

Figure 13. Diagram for second gear constraint equation in geared six-bar.

To develop the second gear constraint equation, first imagine link 6 fixed as shown in Fig. 13. Then, with link 6 fixed:

P;=(vj-¢j)+~(~j-¢j)=(1+~)(v1-¢j) (24)

or

where

/~J= "(~'1- ¢i) \

TJ

Now with link 6 free to rotate,

#i =#J+ ¢'i = n~j- .¢j+ ¢i or

/~j=nTj+(l - n ) ¢ i

(25)

Then

:j=n~j+(l -.)¢,j and

Bj=,6j+(I -.)~j. The following substitutions can now be made in the loop closure equations (equations 20-22). 0tj-- m~i

/~j=nTj+(1 -n)dl 1

di=m

/~j=m)l+(1 - n ) ~ i

~j=0

]b=,6j+(1 -,)ff~.

(26)

Suppose now that one wishes to synthesize a geared six-bar mechanism of this type for five first-order precision points. Four displacement angles are specified for ~j and ~kj,

74 measured from the starting position. The five equations which describe the mechanism in its starting position and in its four displaced positions are:

b

-

1

1

l

1

t

ei4~:

eim4~2 ei[n;',_+(t -n)~z]

el','2

ei~,,.

eiO3

elm4~3

el.:3

el,#3

e14~4

eiratb4 ei[n't4+(t-n)qt4]

e174

ei4~4

ei4~s

elra4~5 ei[,w~ + ( t -

eleS

ea4,s

ei[n~.3 + ( 1 - n),/Jj]

nh0~]

-I

Z~2j

Zlt [

Z13

Ztt

Zx4

=

Zll

Z,5

Ztl [

Z16

211

(27)

Here, Z l l may be set equal to unity: Zil=l+0i. The gear ratios enter into m and n through equations (23) and (24) so m and n should be selected so as to give convenient gear ratios. The link rotations, 7j, can be arbitrarily varied so as to obtain a wide spectrum of solutions. Even with all of these free choices, the system of equations is linear in the unknown link vectors Z12 , Z13 , Z14, Z~5, and Z t r . The equations can be solved by any of the conventional techniques for solving simultaneous equations (keeping in mind that these equations have complex elements). For instance, matrix inversion, Cramer's rule, or Gaussian elimination can be used to solve for the unknown Z vectors. Suppose one wishes to design this mechanism to generate the function y ~ X3

in the interval 0 < x < 1. The spacing of the precision points can be accomplished by choosing Chebyshev spacing. A simple formula provides the determination of the spacing to be used so that the error in the synthesis of the prescribed function will be at a first approximation of a minimum within the indicated range. The equation [4]: Ak = a + h cos ~ , ( 2 k1)re 2n

k = 1, 9, ....

,n

(28)

is used to determine the required spacing. In this equation the terms are identified as follows, with reference made to Fig. 14: a: mid-point of the interval h: one-half the length of the interval n: the number of precision points Ak: a specific precision point, k = 1, 2 . . . . n.

75

o-Z5 A4 h

+h

o

Figure 14. Geometric construction for Chebyshev spacing of five precision points. For this problem: n= 5 a=0.5 k=0.5 .'.Ak=0.5+0.5 cos

2k-1 10

k = l , 2. . . . 5

At =0.5 +0.5 cos ~ =0.9755300 10 A 2 = 0.5 + 0.5 cos -_2.'"-_ 0.793849 10 A3=0.5+0.5 cos n--=0.500000 2 71r A4=0.5 +0.5 cos "_v.7.=0.2061050 '" 10 Olr As = 0.5 + 0.5 COS~ = 0.0244700. 10 The input and output angles for the mechanism are determined by two simple equations. They are: • ~bj=xL'xlA~b Xf -- Xs

g/j=Y!YY~A(,, Y f - - Ys

where: ~b~: input angle from position 1 to positionj ~kj: output angle from position 1 to positionj y~, x j: j-th precision pt. x~,, Ys: limits of the ranges (higher)

x~, Ys: limits of the ranges (lower).

(29)

76 In this problem the ranges of x and y result in the following simplified equations for the determination of the angles.

Ax= I,

Ay= 1,

q~y= (xi-xl)At~,

t) j = ( y y - y t ) A ~ .

By varying the values of Aq~ and A¢ the designer can obtain an infinite variety of solutions with all the other mechanism specifications held constant. Figure 15 shows example 4 of Table 2 in its first precision position. For this example Aq~= 90°, A~, = 60°, while the assigned values of V2, 73, 74, and 7s appear in Table 2.

Figure 15. Synthesized geared six-bar linkage (Table 2, example 4) for five first-order precision points. The linkage is shown in its first precision position.

Higher-OrderSynthesis Now suppose that instead of satisfying five first-order constraints the designer wants to synthesize a geared six-bar mechanism of this type to have three precision points, two of which are of second-order and one of first-order. For th~s case, the system of synthesis equations is: 1

1

1

1

1

Z12 Z13

ei~2

elmer2

eiA2

el~Z

e~2

ei03

eim~

e i A3

eiTa

elg,~

ei4,2

mei,.,~2 A2el^,

~2e~,~

!//2el#2

ei÷~

meim~

93etTJ ~3ei~'3

A3e'^~

!

ZI 1

Zll / Z~4 = ZO~[ Zts Zl6

I

0 J

(30)

77 where A z = [ny2 + (1 - n)tP2], A a = [ny3 + (1 - n)~3], A2 =In')2 + ( 1 - n ) ~ 2 ] , and /~3 = [n~a + (1 -- n)~3].

(31)

Once again, Zlt can be set equal to unity. In the above equations, ~bi, ~b2, ¢k3, ~kt, ~k2, ¢3, ~1, ~2, ~3 are given prescribed values and Zt2, Zt3, Zt4, ZI 5, and Zt6 are the unknowns which are to be solved for. Here m, n, ~2, ~3, ~2, and 7a can be given arbitrary values. Then, the equations are linear in the unknown Z vectors which describe the mechanism in its reference (Ist) position. The computer program calculates the precision points and the prescribed values of Oh, ~, and ~ in the following manner. Suppose that the function is y = x t'5, 0 < x < 1. Then the accuracy points may be chosen as the roots of the Chebyshev polynomial, x = a + h cos

( 2 k - 1)~ - - , 2n

k = 1, 2, 3

(32)

where a=0.5, h=0.5, n=3. In the computer solution, the corresponding y values and ) values, along with the x values, are printed out. ( y = x t's, j)= 1.5xO.S). Calculation of ~b values:

~bj_ x j - xt A~

(33)

Ax

where ~bj is the change in ~b from the 1st precision position. Ax= 1 Set A~b= 90 °, so ~j = ~ ( x j -

xt) = 90(x j - xl),

~b2= 90(x2 - xl), and ~b3= 90(xa - x l ) . Calculation of ~b values:

~--~s ~j Yj--Yt A~b-A~-

Ay

where ~'J is the change in ~b from the 1st precision position. Ay=l Set A~b= 90 °, then ~ j = 90(yj--yl), ~2 = 90(y2--yl), and ~/3 = 90(y3 --Yl)-

(34)

78 Calculation of ~ values: = ~s + ~(yY - Ys) = @s+ 90(y-- ys). d~O dy dy dx ~-~ = 90~-~ = 90~xx~-"~ and

(35) dx Ax 1 d~b A~b 90

Thus dO dy

Values for m, n, '/2, Y3, 92, and 73 are read in as input data for the computer program. Table 2 (Example 5) lists the input data and a synthesized linkage which resulted from a computer solution of equations (30). The first, second, and third precision positions of this example are shown in Fig. 16.

1

p/,

2#

Figure 16. Synthesized geared six-bar linkage (Table 2, example 5) for one first-order (position 1) and two second-order precision points. The three precision positions are shown.

79

As a final example of higher-order synthesis using this geared six-bar linkage, suppose two precision points are required. One, let us say, must be accurate to third-order. Then the other precision point can be accurate to second-order and the designer will still be able to attain a linear form of solution. The system of synthesis equations would then be:

Ztt I

I

I

1

I

I

Zt21

1

m

At

7t

61

"~t3 I

0

Z141 =

0

I

Z~I

Ztt

I

_Zt6]

L o -I

i

el~2

-m 2 (-h~+i~t)

e~m~

e~Aa

(-~+i~t)

e02

(-6~+i~t)

eleZ I

(36) where

At =[n~t + ( 1 - n ) ~ z ] ,

/~t= [/'r~!d~(l--/l)~l], At = [me + (I - n)¢2], and A , = [n;~ + (1 - n ) t d .

(37)

As before, ZIt can be set equal to 1 and values for m, n, 7z, 71, 1)z, and 71 can be arbitrarily chosen. Values for Oz, ~t, ~;t, ~2, and ~z are prescribed from the input data. The equations are linear in the starting Z vectors. Figures 17 and 18 show an example of a synthesized linkage in its two precision positions with the arbitrarily chosen values listed in Table 2 (Example 6).

First Position Figure 17. Figures 17 and 18. Synthesized geared six-bar linkage (Table 2, example 6) for one second-order, and one third-order precision point. The first precision position appears in Fig. 17, while the second precision position appears in Fig. 18.

8O

Second Position Figure 18. Analysis of the Geared Six-Bar When a mechanism is synthesized, it will exactly generate the desired function only at the precision points. The purpose of an analysis program is to determine whether the linkage can move between the precision points without binding, and secondly, to determine the errors in the generated function as compared with the "ideal" function throughout the range of motion. Also, as the mechanism moves, the quality of its transmission will vary. Each type of mechanism has one or more "transmission angles" which serve as critical indices of the quality of transmission. For the geared six-bar considered in this paper, the significant transmission angle is the angle 7' shown in Fig. 19. A value of ~,' near 90 ° means that the mechanism has good force transmission properties. Ify' = 0 or 180° the mechanism is locked. The transmission angle is the angle between a line drawn through point A and gear mesh-point P and a line through point B and gear mesh-point P.

A

4

3

Figure 19. Transmission angle ~' of geared six-bar. 90 ° is optimum.

81 Figure 20(a) shows the three-force member 4, with the forces acting on it. Figure 20(b) is the freebody diagram of link 6. Acting on this link is the gear force F~6, which is equal and opposite to F64, and force F56 (through two-force member 5), which is equal and opposite to F54. Therefore, the resultant force FR transmitted to link 6 is equal to F3¢ and acts on a line through points A and P. The transmission angle is the angle between the line of this force and a line from the rotation point B to the point P where the force acts.

(a)

(b)

Figure 20. Force diagram on links 4 and 6 of geared six-bar, used in study of transmission characteristics.

The poor transmissibility, for the case when ~' =0, is illustrated from the velocity point of view in Fig. 21. The velocity of point P is perpendicular to line A B and the velocity of A¢ relative to P is also in this direction. Therefore A4 can only move in a direction perpendicular to line AB. But A 3 (point A on link 3) wants to move on a eycloidal path which is almost perpendicular to this direction, so the linkage is locked.

Pat of

A/

Path of A4

I~ O

F i g u r e 21. Geared six-bar in immobile position : transmission angle ~' is zero.

To check the mechanism for errors between precision points, consider it as if it were broken into two parts as shown in Fig. 22.

82

Figure 22. Displacement analysis of geared six-bar: matching of D t and D 2 leads to determination of output angle. Starting at the first precision position, Ztt, Zt2, Zt3, Zt4 , Zts , and Zt6 are known. For an input rotation ¢ from this position, the vector D t can be determined Dt = - Z1 + Z2 + Z3

(38)

where Zt = Ztt = constant for all input rotations ¢, Z2 =efCZt2, and Z a =eimCZt3 . Now link 6 is held fixed and angle 6 is varied (using Newton's iteration method) until ID I=IDxl, where D2=-Z6-Zs-Z

4

(39)

where Z 5 = et6Zt 5, and Z , = e~"6Zt,. Now, A~ = a r g D 1 - a r g D 2 is the increment in ~. Starting at this new position, the input variable ¢ is incremented, the next A~ is found in a similar manner, and the process continues throughout the range. By this procedure, the error (@ld,=l--~b==tua~ can be determined throughout the motion of the mechanism.

Path and Motion Generation by Geared Linkages

Geared linkages can be effectively utilized in specialized path and motion generatioz applications. In path generation a single point fixed to a moving coupler link is guided through certain specified "precison points" lying on the path to be generated. These points may be either finitely or infinitesimally separated. In motion generation, the body to he guided is rigidly attached to a moving coupler link. One point of the body is then guided through its desired location in consecutive "precision positions" of the body and, at each of these precision positions, the rotation of the body is controlled so that every other point of the body also falls in its corresponding desired location. Thus, motion generation imposes more constraints on the design of a linkage than does path generation. Certain applications may require that the rotations of the input or output cranks of the linkage be coordinated with the motion of the body or tracing point so as to give prescribed timing. When a limited number of precision positions is specified, a geared linkage can be

83

designed to correlate the motion of the moving body, the rotation of the input crank, and the rotation of the output crank. As the rotation operators are all specified under these circumstances, the only remaining unknowns are the link vectors; thus the system of synthesis equations will be linear. However, the number of precision positions is limited. To increase the number of precision conditions, the designer can relax the prescription of the rotations of the output crank, maintaining prescribed timing of the path or motion generation with respect to the rotations of the input link. He can then prescribe additional precision conditions, but at the expense of needing to solve a nonlinear system of equations. Still more precision conditions can be prescribed if the rotations of the input crank are also treated as unknowns, in which case control of timing is sacrificed. Finally, in the case of path generation, the rotations of the body carrying the tracer point can be treated as unknowns. The maximum possible number of prescribed path points can then be attained. Suppose the designer wishes to utilize a geared five-bar linkage of the type shown in Fig. 23. In path generation, he specifies the desired path by means of vectors rj which

01

to be generated 'ath to be traced

iY

r

_zj

;2 ZI ~,/ Figure 23.

• ~uj

Geared five-bar path or motion generator guides plane x through sequence of prescribed coplanar positions.

locate the consecutive positions of the tracer point ,4. In motion generation, one specifies the positions of the moving body g by means of vectors r i which locate the positions of point A of the body and by means of the angles yj which specify the rotations of the body from its first precision position to its j-th precision position.

84

Rotations of the coupler link 4 are related to the rotations of links 3 and 2 by the expressions ,

=

72

+ (YJ

=

-

(40)

+

where r,/

The position vectors rj can be expressed as rj = Z 0 + Z j2 -~ Z j 6 = Z0 ÷ el#JZ 12 + e~Y~Zl6.

(41)

In addition, a closure equation can be written for the mechanism as Z j2 "1-Z j3 ÷ Z j4 ÷ Z j5 = Z 1

or

(42) e~#~Z12 + e~Y~Zt 3 + e ~ Z ~, + e~°lz t s = Z t.

Here, the vector Z 1 must be treated as an unknown, so it cannot be arbitrarily specified. We can now construct Table 3 and Table 4. As shown in these tables, a linear solution with prescribed timing of all rotations (input, coupler, and output) can be obtained for either path or motion generation when three precision positions are specified. The equations for path and motion generation are the same. One can choose a vector, say Z o and solve for the remaining vectors. The equations required are: 0

1

0

0

0

1

0

e ~'2

0

0

0

e I~2

Z12

r 2 - Zo

0

ei*~

0

0

0

e I;'3

Zt3 ]

r3--Zo

I

I

1

1

0

Zl,

0

-- 1

e to2

etn

e t~2

e i~2

0

Z15

0

-- I

e I¢3

e i73

e 1#3

e i#~

0

_Z16]

0

-I

Zl

I

rt-Z°

(43)

Here, the vectors ry specify the given positions of point A. In the case of motion generation, the angles 71 specify the orientation of the moving body in the precision positions. The angles/zj are specified through equation (40) once the ratio 17is chosen. Values of ~bj and ~bj are chosen to give the desired timing. (Thus, function generation is an incidental byproduct of the path or motion synthesis.) The unknowns are the link vectors Zt, Zt2 , Z~3 , Zt,, Zls, and Z16. Conclusion

A wide variety of applications can be handled using geared linkages. A few possible configurations for such linkages have been given in this paper, but specific applications will probably suggest other designs. However, the synthesis technique illustrated in this paper is general, and can readily be adapted to any specific configuration. Although only linear solutions were illustrated by examples, the procedures for the nonlinear systems involved in use of incxeased numbers of precision conditions can be readily handled by methods illustrated elsewhere [2, 12, 13].

85

Table 3. Path generation with geared 5-bar

Precision points

Complex equations

Equivalent real equations

Unknown reals to be solved for

Unknown reals to be chosen arbitrarily

Linear solution--prescribed link rotations and timing I

2

4

4

(Contained in any two vectors) 2

4

8

8

(Contained in any four vectors) 3

6

12

12 (Contained in any six vectors)

I1 (Remaining 5 vectors and ratio 17) 7 (Remaining 3 vectors and r/) 3 (Remaining vector and r/)

Non-linear solution--prescribed timing of input rotations Sj 16 (Contained in vectors Z0, Zi, Zl2, Zt3, Z14, Z15, Zl6, ~'2, ~'3)

5 (72, Y3, 3'4, ~'4, r/)

4

8

16

5

10

20

20 (Above plus ~/4, ~/5, Y2, 73,)

3 (Y4, Y~, r/)

6

12

24

24 (Above plus ~v6, Y4, ~'5, ;'6)

(~)

1

Non-linear solution--no prescribed timing 7

14

28

8

16

32

32 (Above plus ~8, 7s, ~4, d)5)

9

18

36

36 (Above plus ~'9, Y9, ~b6, ~7)

10

20

40

22

44

4 3

(,l, ~8, ~9) 2

40

(Above plus ¢/zo,Ylo, q~8,$9) I1

5

28 (Above plus t//7, Y7, ~2, ¢~3)

(~, ~ o )

44

(Above plus ~11, Ylz, ~Io, ~btt) 12

24

48

48

(Above plus W12, :'12, ~12, ,/) Thus, a geared 5-bar can be synthesized for a maximum of twelve path points

1

(,I)

86

Table 4. Motion generation with geared 5-bar

Precision positions

Complex equations

Equivalent real equations

Unknown reals to be solved for

Unknown reals to be chosen arbitrarily

Linear solution--prescribed link rotations and timing 1

2

4

4

(Contained in any two vectors) 2

4

8

8

(Contained in any four vectors) 3

6

12

t2 (Contained in any six vectors)

11 (Remaining 5 vectors and ratio r/) 7 (Remaining 3 vectors and t/) 3 (Remaining vector and u)

Non-linear solution--prescribed timing input rotaions ~ 4

8

16

16 (Contained in vector Z0, Z1, 212, 213, 214, 215, 216, {¢2, ///3

2 (v/4. q)

Non-linear solution--no prescribed timing 5 6

10 12

20 24

20 (Above plus ~4, ~'.~, ~bz, ~b3)

(~, q~4, 4~5)

24 (Above plus C/s, ~b4, q~S, q~6)

! (t/)

3

Thus, a geared 5-bar can be synthesized for a maximum of 6 precision positions of a moving body

Acknowledgements--The authors would like to express their appreciation to the National Science Foundation for the support of their research at Rens~lacr Polytechnic Institute through Grant No. G-1231 sponsored by the Engineering Mechanics Program, Enginecring Division of N.S.F. They would also like to thank Mrs. Frances K. Willson for her skill, meticulous care and patience in typing the manuscript. The present paper was presented at the Applied Mechanism Conference, held July 31 and August 1, 1969, and sponsored by Oklahoma State University. This paper is released for publication by the directors of the conference, Dr. A. H. Soni and Dr. I . ~ Harrisberger.

References [1] CHI~ONlS14. P. Machine Devices and Instrumentation, p. 11. McGraw-Hill, New York (1966). [2] Sm',raORG. N. A general complex-number method of plane kinematic synthesis with applications, doctoral di~ertation, Columbia University, N.Y.; UnivcrsiW Microfilms, Library of Congre~ card No. Mic.59-2596, Ann Arbor, Mich., (1959) 305 pp. [3] SANI:~RG. N. On the loop equations in kinematics. Trans. of the Seventh Conference on Mechanisms. Purdue Univexsity, pp. 49-56 (1962). [4] H A K ~ o R. S. and DENAvrrJ. Kinematic Synthesis of Linkages. pp. 140-147. McGraw-Hill, New York (1964). [5] AlcrOBOLEVSKIIlI., BLOK/-IS. Sh. and DOBROVOLSlCdIV. V. Synthesis of Mechanisms. pp. 253-374. State Publishing House, Moscow and Leningrad, USSR, (1944) (Russian). [6] Fe.~trDENS'tm~F. Structural error analysis in plane kinematic synthesis. 3". Engng lnd., Trans. ASME, Series E, 81, 15-22 (1959).

87 [7] ROTXB., FREtrO~,~r~ F. and SnNVORG. N. Synthesis of four-link path-generating mechanisms with optimum transmissiou characteristics. Trans. of the Seventh Conference on Mechanisms, pp. 46--48. Purdue University, (1962). [8] McL~.N^N C. W. On linkage synthesis with minimum error. J. Mech. 3, 101-105 (1968). [9] F ~ ~ F. and S^~ooR G. N. Synthesis of path-generating mechanisms by means of a programreed digital computer. J. Engng Ind., Trans. ASME 81B, 159-168 (1959). [10] ~ B A C H P. W. and T~Ag D. Optimization of four-bar linkages satisfying four generalized coplanar positions. ASMEPaper No. 68-Mech-30, Mechanisms Conference (1968). [11] Fe,~tnD~STV~F. Four-bar function generators. Machine Design 119-123 (1958). [12] I C ~ t ~ z ~ R. E. and SANDORG. N. The Bicycloidal crank--a new four-link mechanism. ASMEPaper No. 68-Mech-14, presented at the 10th ASME Mechanisms Conference, Georgia Institute of Technology, Oct. 10, 1968, published in the I. Engng Ind. 91, 91-96 (1969). [13] SAI,a~OR G. N. and WILT D. R. Synthesis of a geared fore-bar mechanism. Second Internationai Congress on the Theory of Machines and Mechanisms, Zakopane, Poland, Sept. 24-27 (1969).

Appendix Table 1 shows that the geared five-bar linkage of Figs. 2 and 3, with arbitrary scale factors and gear ratios, can be synthesized for up to 7-point approximate function generation. If we now relax scale factors and gear ratios, the number of attainable precision points will increase to 12. However, this means gear ratios which, in general, will necessitate compound intermediate gears. If simple intermediate gears are desired, as shown in Figs. 2 and 3, then the number of teeth of the gears and the link lengths must satisfy the following equations:

TB+Tc Z13 Tc+ To Z14 TA+Te-Z12' Ta+ TB-Z12' which reduces the maximum attainable precision points to 10.