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ANALYSIS of the

FOUR- BAR LINKAGE to the Synthesis , of Mechanisms I t8 Application

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ANALYSIS

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of the

FOUR- BAR LINI(AGE Its Application to the Synthesis of Mechanisms John A. Hrones

George L. Nelson

PROFESSOR OF

ASSISTANT PROFESSOR OF MECHANICAL ENGINEERING

MECHANICAL ENGINEERING

TilE MASSACHUSETTS INSTITUTE OF

TECHNOLOGY

Published Jointly by THE TEC~NOLOGY PRESS of

THE MASSACHUSETTS INSTITUTE of . TECHNOLOGY and JOHN WILEY & SONS~ INC., NEW YORK CHAPMAN & HALL, LIMITED , LONDON

from Design of Machinery 4ed by Robert L. Norton McGraw-Hill Copyright 2008

( viiiJ

Classes of Linkage Operation The basic four-bar

linkage is shown in figure 1.

The fixed member

are the cranks 1 is the line of centers. . Pinned to the The moving ends of the cranks are joined by the fourth bar , the extremities of

The F our- Bar Linkage

and

B.

A.

coIWecting rod

Determi~ate linkage motion results when the number of independent

input angular motions is two less than the number of links, All links are assumed to be rigid members and are pin-connected to one another. Freedom of relative angular motion exists between any two members at The minimum number of links which will permit relative motion between links is four. In the majority of applications one of the links (the line of centers) is stationary while a second link (the driving crank) is driven from an outside motion source. The motion of the remaining two links isa function of the geometry of the linkage and the motion the pin joint.

of the driving crank and the line of centers.

The driving crank 1 " and the follower crank

in rotation about their

G.

fixed axes located on the .line of centers

move The

in general moves jn combined translation and rotation. connecting rod and the follower The nature of the motion of the conhecting rod crank

fora given input motion to the

relative to the line of centers

All;

driving crank 1 is determined by the three basic link length ratios

, B and G. The shortest BIl; Gl1 which will hereaJterbe designated by link will always be designated as unity, The remaining links will be in order around the linkage. It is convenient to , B ahd labeled separate the geometrically

possible motions into three main categories

of operation.

Crank and Rocke

Four Bar Linkage Fig. 2 Fig. 1

A four-bar linkage isschema.tically shown in figure 1. It consists of the four links having pin-to-pin lengths of 1 , B and G. The geometry

All , Bl1

of the linkage is determined by the three ratios

and

GI1.

Up ,to this point the four-bar linkage has been represented as consisting of four lines, Actually, each

member is a solid body which from purely

theoretical aspects can be considered as being of indefinite extent (figure 6). Manufacturing and design considerations place very real limitations on the size of the members. Within these limits a wide variety of motions are available. In this volume the points indicated in figure 6 on the connecting rod included.. within a rectangular boundary extending a distance right equal to the driving crank length in directions parallel to and at angles to the centerline of the connecting rod have been investigated and, their trajectories and velocities presented, Thus , for each series of linkage

ratios the behavior of the points indicated in figure 6 has been studied and the results published for class

(a)

type linkage operation.

Crank and Rocker Fig. 3

from Design of Machinery 4ed by Robert L. Norton McGraw-Hill Copyright 2008 ~c-'" "

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---

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Class

(a)

One crank is capable of rotation through a complete revolution while the second crank can only oscillate.

Class

(b)

Both cranks are capable

Class

(c)

Both cranks oscillate ,

. '. ,.

.. ..

,......... , "

:.. .

of rotating through 3600

but neither can rotate through a

complete revolution. ' 1: '

~he above

t c

of the variolls

classifications arise from a consideration of the motions links relative to the fixed link. As any link can be fixed

, arbitrarily the classification of a given linkage is dependent upon the choice of a fixed

member. In figure

2,

link

is fixed. Crank 1 can make a

can only oscillate. The linkage operating as a class (a) unit commonly known as a crank-and-rocker linkage, Similarly, if link A is fixed (figure 3), Class (a) operation results, complete revolution while the crank

both cranks are capable of full 3600 rotation operation results, ' This linkage is often referred to as a

If link 1 is fixed (figure 4),

and

(b)

class

is fixed (figure 5), the two cranks A and can only oscillate , hence class (c) operation takes place. Though the above clas.sifications are helpful in design , it is important to realize that the relative. motion of any link to the remaining members of the linkage is the same , regardless of which member is fixed. drag link mechanism, If link

determining the class of operation when link ratios are known are listed below. Criteria for

Class (a)

(crank and rocker mechanism), crank mustbe the shortest link (1).

(1) Drive (2)

C.c (A

(3)

C;:. (IA

Class (b)

1). BI*

(drag link mechanism), must be the shortest link (1).

(1), Line of centers (2)

C.c (A

(3)

C;:. (IA

Class (c)

+1).

BBI

1). " + I),

Same as class

(a).

Two

(double rocker mechanism). All cases where the connecting rod is the shortest link (1).

(1) (2) All linkages in ~hich (2) and

(3) for classes

(a)

and

(b)

Rockers are

not satisfied;

Fig,

The above conditions have been graphically expressed in figure 7. is the abscissa; and values of ratio A are plotted as 450 lines, For a given value of A , if the point specified by the coordinates Band falls inside the oblique rectangular spa~e bounded by the lines of constant A , the ' linkage will operate in ' class (a) or (b): class (a) if the shortest link is made a crank; class (b) if the shortest liI~lk is made the line of centers, If Jhe point determined by the coordinates B). (A IA- BI = absolute value of The ratio

(ixJ

is the ordinate; ratio

Fig. 6

from Design of Machinery 4ed by Robert L. Norton McGraw-Hill Copyright 2008

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-

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Ex)

Band class

C falls outside the rectangular area bounded by the lines of constant (c) operation is indicated, Crank and rocker linkages (class (a)

operation) included in heavily outlined area are covered in this book.

Example: '1;

Data:

Find:

Drive crank = 4" ; Connecting Rod = 10" Follower Crank = 8" ; Line of Centers = 12"

(1) Class of operation. (2) Variation in link ratios possible without changing class

10 of operation.

Solution

(see figure 8): - = 2, 5;

= - = 2;

C=

Drive crank is the shortest link, C = 3) falls within oblique rectangular area-

Refer to figure 7. Point

Therefore

class (a)

= 2, = 2. =

(B

= 2

=2.

bounded by lines

operation is indicated

= 2 C may be varied from 1.5 to 3. may be varied from 1.5 to 4. 2, C = may be varied from 2 to 4 , C = 3

bs.

I\~ bs.

'1.

Fig; 7

FOUR~BAR LINKAGE CLASSIFICATION CHART Linkage whose ratios Band

Link,

area bounded by lines of constant ratio

determine a point which lies in the oblique rectangular operates as a Crank and Rocker or as a Drag

;~,;.it1i'&"~;,:J!f;;t.

Fig,

iii

III

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from Design of Machinery 4ed by Robert L. Norton McGraw-Hill Copyright 2008

Mathematical Relationships (See figure 9. ) The follower crank ,angular

Driver-Follower Crank.

position is given by 4.

given in reference 1.

al-

1ft =

IUustrative Exam les, (See reference 2. Figure 10 shows the path DwellPeriod from Straight-Line Path. 1. of a point on the connecting rod of a linkage which is approximately straight b. A rotation of 550 of the drive crank produces this and between points straight portion, This is indicated by the eleven dashes which constitute and the path lying between If the point having this displacement path drives a member with a

sin 8 al

C+cos8 cos 8

K2 +2C a2

K2

= cos

2BL

= 1 +

B2

C2

= 1 +

C2

2C

M'i

1. 'Y

cos 8

cos 8

K2 +2C

S2

=, an ...J 4B2V

cos

2C cosO

K2

2BL

+ cos 8

(1)

Differentiating equation 1 with respect to time yields the following

dt V S2

equation for the velocity of the follower crank:

sin 8

dt

(C

cos 8 + 1)

2 +

A second differentiation yields an expression for the angularaccelera-

tion of the follower crank: " 1ft

dt2

l(Ccos 8 +

d28

1) +

202

2C2

VS2 sin2 8

sin2 8

S2

cos 8

M2) C

sin 8

2(C 1 -

1 -

27P

These figures are obtained directly from the trajectory by counting the number of dashes in each phase of the motion and multiplying by 50 the total angular motion If the pivot of the rotating link is located at . At a uniform drive crank speed approximately one-half of the link is 200 of each cycle is used for the forward stroke , one~third for the return stroke and one-sixth of each cycle for the dwell. All alternative method of obtaining a dwell is to drive a " scotch yoke a 2200 forward stroke , and an 850 return stroke.

move in the

2 +

(2B2 -

of

ab

a member slotted parallel to the straight portion

sin 8

dt2 V

an angular motion will result. In terms of drive crank angle there will be a 550 dwell l;,

radial slot constrained to rotate about the fixed point

M4

sin

d1ft

The relationships for displacemen~, velocity, and acceleration of a point located anywhere on the connecting" r()d are considerably more cu~bersome than the above equations and are Motion of Points on Connecting Rod.

L2

cos 8 + I

direction

ef

perpendicular to

ab,

but constrained to

In this case the ratio of

forward return stroke is not adjustable" and is equal to 51/21 or ratio of the length of stroke to drive crank length is 1.37.

2,

4. The

Figure 11 shows the path Dwell Period from Circular-Line Path. whose basic ratios bar linkage on tbF connecting rod of a fourof a point andb an approximate circular arc exists are 4; 2, 5;3. 5. IBetween points 2.

pinned to the midpoint of the connecting rod drives a bell crank rotating about the same fixed

over a 900 drive crank angle displacement, Link

axis as the drive crank,

Link proportions are selected such that

is the

The bell crank has a dwell period of ab. center of curvature ' of the arc one-fourth the total cycle and a total angle of travel of 340 with approximately equal times for advance and return. The "time ratio of forward to return stroke and the angle of oscillation

OIC': "2

f/I

can be adjusted by choosing other locations for the bell crank with corresponding changesin the length of the bell crank arm so that remains the center of curvature of the arc

6r~9)

ac

ab.

fixed axis

For instance , it is possible

using this linkage to locate the bell crank axis at the fixed axis of the

follower crank, Care must be taken to avoid a dead center position between ac and the" bell crank arm to which it is p illned. the iink The same fundamental linkage can be employed to produce a straight

Fig,

(xi)

from the end of the line reciprocating motion with the same dwell period at Design of Machinery 4ed by Robert L. Norton McGraw-Hill Copyright 2008

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(xii) ,

for the bell crank ~ slider constrained to move in a

stroke, Substitute fixed" straight slot passing through point c,

Adjust the direction of this

ratio of forward to return stroke. For , the slider will dwell at example , using the line cd as the axis of the slot , and return to point c in 1200 point cfor 900 , advance to point d in 1500

slot to obtain the desired time

Four-bar linkages are often used as computers.

Computer Linkages.

time required for each part of the cycle is obtained by counting the dashes between the points of tangency b , c , d , and e and is given in the table below, The linkage has the basic ratios 2

rotation of the drive crank,

3,

whose trajectory exhibits an interoscillations can differ in amplitude and time, section, The two output is of the sa~e Figure 14 illustrates the special case where each oscillation point a. The amplitude : The pivot of the output member is located at

driven by a point on the connecting rod

Because of the infinite number of output-input

coordinate location (+1

In this event the primary linkage approximates the

desired relationship

function4'=~(~l5

finding over a range of (Hrom 0 to 550 , when used as indicated below, In crank angle was this mechanism the procedure was as follows. The drive assumed to be the variable (), The value of 4'corresponding to values of calculated angular () from 0 to 750 at 50 intervals was calculated. The Repeated positions 4' were then accurately laid out on transparent paper.

superposition of this layout on various charts resulted in finding a trajectory cp (of the overlay) fell on which the lines representing the angular position on successive dash terminals, The point on this particular linkage giving pivoted

sheet.

this desired result can be used to drive a radially slotted member a~ the intersection of the lines on the overlay The selected linkage has the basic ratios 2; 3; 2,

sentation of

cp

5, The drive point

location + 1.5

, + 1. The slotted

the range of () from 0 to

135

2d return

errors

within the tolerance limits required, (See reference 3.

Figure 12 shows a linkage which closely satisfies the

Crank

Degrees

Dashes

1st forward 1st return 2d forward

necessary,

while additional linkages apply corrections to bring the maximum

Within

Path

Stroke

accuracy is required more than one four-bar linkage is often

member is pivoted ata.

1). Drit'e

functions. Where a high degree of

the conne~ting rod has the coordinate

The drive point is at the

relationships available a

wide variety of functions can be represented over limited ranges of the

variables appearing in the desired

, and 2,

5.

Symmetrical-Motion

Pat~s. In a number

of applications it is

desirable to obtain a path which is reference line. Linkages where the connecting rod and

symmetrical with respect to some

follower crank are

have points on the connecting rod whosetrajeccircle of radius A tories meet this condition, The locus of such points is a the follower crank.

of equal length (A =

B)

on the connecting rod with its center at the moving end of '

Figure 15 shows the paths of twelve points on the connecting rod points are equally bar linkage with the basic ratios 2; 2; 2, 5. The symmetrical , spaced on the dashed circle. The trajectory of each point is about the straight line passing through the follower crank fixed axis and The twelve trajectories illustrate a the zero position of the trajectory. symmetrical-motion paths and show the great variety of a four-

typical set of

curves available for use where symmetry of forward and return stroke is essential.

550 a good repre-

is obtained.

. J

is the logarithm Figure 13 shows a linkage in which the output position the previous problem an of the input position over a limited range. As in overlay was constructed and the charts searche d for satisfactory matching of the overlay over the

desired range. In

the mechanism shown , the

radially slotted output member pivoted at a

has angular displacements

proportional to the logarithm of the drive crank

displacement in the range

position 1 the accuracy is good, Two linkages of this type feeding a differential unit could be used as a multiplier, location of , The basic linkage ratios are 2, , 2, 5, and 1.5. The coordinate of positions 1 to 10. Except at

, + 1. the point on the connecting rod is + 1.5

from Design of Machinery 4ed by Robert L. Norton McGraw-Hill Copyright 2008

A slotted crank whose frequency of

14, It oscillation is twice that of the drive crank is shown in figure 4,

Double Oscillating Crank,

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A.

Hrones Ond G.

L.

Nelson

A= 2. B=

C= 1.

Fig.

LOGARITHMIC LINKA.

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from Design of Machinery 4ed by Robert L. Norton McGraw-Hill Copyright 2008

'= -

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Cam versus Linkage

struct a simple scale

The cam is in wide solution. In many instances ,

. =

the linkage for bad angularity however , the cam design limits the perform-

ance of the machine. Often , in such circumstances , a suitable linkage

removes the cam-imposed difficulties. Because of the

sizing a linkage to achieve a eliminate the linkage from the design in favor of the cam despite certain

limitations. It is

, excessive ' torque ratio , and

possible interferences. II!

Velocity Determination

. Consider the trajectory shown in figure distanceab measured along the path in an interval

the trajectory traverses a

required for the drive crank to

linkages . in such . problems. The cam has certain distinctive

can be easily designed to give a wide variety of motions. small and compact. , however, subjected to very high stresses at the points of contact. The cam contour either by a spring which

or by a, grooved slot in which the backlash causes trouble at high

If the drive crank

rotating at a uniform angular speed the average velocity this travel is the length ab divided by the time the drive crank requires to move through 50 (ab)"" (V"b)

where

= average velocity of point

V"b

point

(inches per second).

to point

measured along trajectory "" = angular speed of drive crank

of operation. In order to limit the

ab

able levels the degree of contour is beyond the abilities of most shops and in any event results in an extremely expensive operation.

of many machines are set by the limitations of results in excessive design and careful machining can raise such limitations.

and

= length between

Acceleration Determination The acceleration of a point moving in space relationships. = lim

dt ~t

Linkages very often offer distinct motion is determined by the relative length of the , small percentage variations in these lengths do not radically influence the Furthermore , the machining With precision in any shop. the link pivots is standard of ::i::O. OO2 inch are easily

acc.

dV

= hm V"b

Vb" -

~t

~t--O

~t--O

(be -

~t

Thus , an approximation of the acceleration at the vector

25 .

is found by evaluating 2/25).

facture required tolerances for high-speed operation are of the ::i::O. OOO2 inch. In addition

more favorable than they are at the cam Balancing of Linkages

Ifa must be paid to the dynamic

keep

the mass of the moving links as small as possible.

. alloys with a low-weight members with low inertia oscillating forces to the linkage supports , inertia forces should be balanced.

The subject of balancing Use of Models in Design Work Ti!e charts give motion characteristics of many linkages aIid therefore lead directly to the selection of the basic link ratios. It

(xix)

Fig. 16

from Design of Machinery 4ed by Robert L. Norton McGraw-Hill Copyright 2008

," (xx)

As the acceleration is determined by subtraction , where the acceleration is small the vectors

be

and

ab

are nearly equal and very

acceleration determination can ' be encountered. However , at points the trajectory relatively low. , errors in acceleration from as low as 10 per cent to as high as , greater. the error tends to be excessive , owing to the. equal velocity vectors , the use of

drive crank angle is

on

LIST

(1) Talbpurdet ,

, instead of using average

vectors for a 50 crank increment , values for a 150 -increment

(2) Nelson , George L.

desirable. Accuracy

Guy J.

Machine Design;

OF

REFERENCES

Mathematical Solution of Four-Bar

May, June and July, Synthesis of Four-Bar

duction of Desired Motio n

of the machine over

Technology; June , 1949. (3) Svoboda , Antonin Computing Mechanisms and Linkages; HubertM. James , McGraw-Hill; 1948.

detectable by the naked eye.

(4) Hrones , John A.

005 inch. For a given

::!::0. 01

Linkages for the Repro-

Paths; " Master of

Dept. of Mechanical

Traject~ries Link lengths of the machine used in producting the ,trajectories were of

held

Linkages;

edited by

Key Factors in

April , May and June , 1949 , and " An Analysis of the Dynamic Driven System; Transactions of the American Sodety of Mechanical Engineers Vol. 70 , 1948.

inch. The

Machine Design;

resulting from a possible accumulation of machining errors on the timing cam , variable time lag in the pen operating mechanism , etc.

tions are that the errors in usually are very much smaller.

The length of the drive crank of the

00 inches ' and

tolerances indicated above are referred to this basic size. are one-half the scale of the originals. For

crank the point moves the dot to the tail of the dash.

'Y

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