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Chapter 4 Planar Linkage Mechanisms (??????)

- 4.1 Characteristics(??) of Planar Linkage

Mechanisms

- Linkage mechanisms are lower-pair mechanisms. The

main practical advantage of lower pairs over

higher pairs is - The contact pressure(??) is lower.
- Better ability to trap(??) lubricant(???)

between enveloping(??) surfaces. - (3) The lower pair elements are easy to

manufacture. - As a result, the linkage is preferred(???)

for low wear and heavy load situations.

- A planar four-bar(?) mechanism is the simplest

planar linkage mechanism with one degree of

freedom. Four-bar mechanisms are extremely(???)

versatile(???) and useful devices(??). For the

sake(??) of simplicity(??), designers should

always first try to solve their problem with this

device.

4.2 The Types of Four-bar Mechanisms (???????)

- 1.Revolute four-bar mechanism(??????)

If all lower pairs in a four-bar mechanism are

revolute pairs, as shown in left, the mechanism

is called a revolute four-bar mechanism(??????),

which is the basic form of the four-bar

mechanism.

- In a revolute four-bar mechanism, the links

connected to the frame are called side

links(???). Usually, one of the side links is an

input link, and the other side link is an output

link. The floating(???) link couples(??) the

input to the output. The floating link is

therefore called the coupler(??).

- If two links connected by a revolute can rotate

360o relative to each other, the revolute is

called a fully rotating revolute(?????)

otherwise, a partially rotating revolute(???).

The revolutes A and B are fully rotating

revolutes, while the revolutes C and D are

partially(???) rotating revolutes.

- If a side link can rotate continuously(???)

through 360o relative to the frame, it is called

a crank(??) otherwise, a rocker(??).

According to the types of the two side links, the

types of the revolute four-bar mechanisms can be

divided into

- Crank-rocker mechanism(??????)
- one side link AB can rotate continuously

through 360o relative to the frame while the

other side link DC just rocks(??). Therefore, AB

is a crank while DC is a rocker. This mechanism

is called a crank-rocker mechanism.

Applications of crank-rocker mechanism

Applications of crank-rocker mechanism

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The input link may be the crank or the rocker.

- In the foot-operated sewing(??) machine, the

oscillation(??) of the driving rocker is

transformed into the continuous rotation of the

driven crank.

(b) Double-crank mechanism (?????) both the

side links AD and BC can make complete

revolutions relative to the frame AB. Thus, both

AD and BC are cranks. This mechanism is called a

double-crank mechanism

- If one crank rotates at a constant speed, the

other crank will rotate in the same direction at

a varying(???) speed.

- Applications of double-crank mechanism

(c) Double-rocker mechanism(?????)

- Both the side links DA and CB can only rock(??)

through a limited(???) angle relative to the

frame. Therefore, both DA and CB are rockers.

This mechanism is called a double-rocker

mechanism.

- The crane(?????) is a famous(???) use of the

double-rocker mechanism. In order to avoid

raising or lowering the load while moving it, the

centre E of the wheel on the coupler should

trace(??) a horizontal(??) line.

- ?????????,????????????????????????ABCD?????????

- ?????,?????????????????,???????????

- The three kinds of mechanism can transform each

other by following wayFor the same kinematic

chain, different kinds of linkage mechanisms will

be generated(??) by holding different links fixed

as the frame. Such kinds of variations(??) are

called inversions(??).

It is of importance to note that inversion of a

mechanism in no way changes the type of revolute

and the relative motion between its links.

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2. Replacing a revolute pair with a sliding pair

- If the revolute pair D in a crank-rocker

mechanism is replaced by a sliding pair, the

revolute four-bar mechanism turns into a

slider-crank mechanism(??????).

- Applications of slider-crank mechanism Internal

combustion engine

- Applications of slider-crank mechanism punch

machine

- If the extended(??) path of the centre of

revolute C goes through the centre A of the

crankshaft, the mechanism is then called an

in-line(??) slider-crank mechanism , otherwise,

an eccentric (or offset)(??) slider-crank

mechanism . The distance from the crankshaft A to

the path of the centre of the revolute C is

called the offset (??), denoted(??) as e

- Both mechanisms in left are in-line slider-crank

mechanism , while the other two in right are

eccentric (or offset) slider-crank mechanism .

Rotating guide-bar mechanism

Translating sliding-rod mechanism

Crank and oscillating block mechanism(??????).

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- If the crank BC is longer than the frame BA, the

guide-bar AE can rotate continuously. It is a

rotating guide-bar mechanism

If the crank BC is shorter than the frame BA, the

guide-bar AE can only oscillate(??). The linkage

mechanism is called an oscillating guide-bar

mechanism(??????).

- The quick-return(??) mechanism in a shaper(????)

is one of the applications of the oscillating

guide-bar mechanism

- The hydraulic(??) cylinder(??) is one of

applications of the crank and oscillating block

mechanism. The hydraulic cylinder is used widely

in practice. The self-tipping(??) vehicle(??) is

an example

- Hand-operated well(?) pump(?) mechanism is one of

applications of the translating sliding-rod

mechanism.

3. Replacing 2 revolute pairs with 2 sliding pair

Crank and translating guide-bar mechanism

- there are two sliding pairs. The output

displacement X of the translating(???) guide-bar

3 is the sine(??) function(??) ofthe input angle

? of the crank AB, i.e. XRsin(?). Thus, this

crank and translating guide-bar mechanism is

often used as a sinusoid generator(?????).

- Applications of crank and translating guide-bar

mechanism

If the link 3 is fixed as the frame, then we get

a double sliding block mechanism(?????).

Crank and translating guide-bar mechanism

- The right mechanism is called an elliptic

trammel(???). This name comes from the fact that

any point on link AB traces(??) out an ellipse(??

- If the link 1 is fixed as the frame, one obtains

a mechanism known as the double rotating block

mechanism (?????) or Oldham coupling(???).

- Double rotating block mechanism(??????????) or

Oldham coupling is used to connect two rotating

shafts(?) with parallel(??) but non-collinear(??)

axes.

- If the link 4 is fixed as the frame, one obtains

a sinusoid generator(?????).

- output displacement Y of the slider 3 is the

tangent(??) function of the input angle ? of the

oscillating(??) guide-bar AC, i.e. YLtan(?).

Thus, this oscillating guide-bar mechanism can be

used as a tangent generator

4. Enlarging(??) a revolute pair

- The length LAB of the crank AB is determined

according to the kinematic requirements, while

the radii(??) of the revolutes are determined by

the transmitted power(??). Note Enlarging a

revolute pair in no way changes the motion

relationship between any links

5. Interchanging guide-bar and sliding block

- Any link in a sliding pair can be drawn as a

guide-bar, and the other link as a sliding block.

The centre line of any sliding pair can be

translated without changing any relative motion

4-3 Characteristics(??) Analysis of Four-bar

Linkages

- 1 Condition for having a crank(Grashof

Criterion??)

In a revolute four-bar mechanism, the input

motion is usually obtained through a side link

driven by an electric motor directly or

indirectlythrough belt mechanism or gears.

Therefore a designer must ensure(??) that one

side link is a crank, which can be used as the

driving link.

- Suppose we wish to design a crank-rocker

mechanism ABCD, in which the side link AB is an

input crank, while the side link DC is a

follower(???) rocker.

If the RRR Assur group can be assembled(??) onto

the basic mechanism by the two outer revolutes B

and D, the lengths of the three sides in ?BCD

must obey (??) the triangle inequality(???)

The distance f is a variable(???) value during

the motion of the mechanism.

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Suppose dgta

(4-2) (4-3)

- Thus from the inequalities (4-3), we can see that

the crank in a crank-rocker mechanism must be the

shortest link.

Again, from the inequalities (4-2), we

can conclude that the sum of the shortest and the

longest links must be less than the sum of the

remaining(??) two links. This is called Grashof

criterion(??) or the Condition for having a crank.

- The Grashof criterion can be expressed as LMAX

LMIN lt Lb Lc. A linkage mechanism which

satisfies the Grashof criterion is sometimes

called a Grashof linkage mechanism.

- If LMAX LMIN gtLb Lc, the linkage mechanism is a

non-Grashof linkage mechanism, in which no link

can rotate through 360o relative to any other

link and all inversions(??) are double-rocker

mechanisms

- In a non-Grashof linkage mechanism, no link can

rotate through 360o relative to any other link.

However, in a Grashof double-rocker mechanism,

the coupler can rotate 360o with respect to other

links.

- A well-known example of the Grashof double-rocker

mechanism is the swing(??) mechanism of a swing

fan. First, ?21 can be determined according

?51. Then ?1 can be found in the single DOF

mechanism 1-2-3-4 according to ?21.

- If LMAX LMIN Lb Lc, the centre lines of the

four links can become collinear(??). At these

positions, the output behavior

may become indeterminate(????). These positions

are called change-points. Such linkage

mechanisms are called change-point mechanisms.

The configuration AB2C2D is called a

parallel-crank mechanism while the configuration

AB2C2?D is called an antiparallel-crank mechanism.

- the change-points are handled by providing the

duplicate(???) linkage 90? out of phase(??). As a

consequence(??), each linkage carries the other

through its change-points so that the output

remains(??) determinate(???) at all positions.

Double-crank

Double-rocker

Table 4-1 Type criteria for the revolute four-bar

mechanisms

Frame LmaxLmin ltLbLc LmaxLmin gtLbLc LmaxLmin LbLc

Grashof Non-Grashof Change-point

Shortest link Double-crank

Opposite to the shortest link Double-rocker Double-rocker

Adjacent to the shortest link Crank-rocker

- From the above, we know that the Grashof

criterion LMAX LMIN lt Lb Lc is only a necessary

condition, not sufficient(???) condition for

having a crank. To determine the type of a

revolute four-bar mechanism, we must check not

only whether the necessary condition is satisfied

but also which link is the frame

- In an offset slider-crank mechanism, the sum of

the length a of the crank AB and the offset e

must be less than the length b of the coupler BC,

if the crank AB is to rotate 360o relative to the

frame.

2. Quick Return Characteristics(????)

- ?C1DC2 is called the angular stroke(??) of the

rocker, denoted as ?max. ?C1AC2 is called the

crank acute angle between the two limiting

positions(????), denoted as ?.

- If the crank rotates counter-clockwise(???) at

constant speed, it will take a longer time for

the rocker in its counter-clockwise stroke than

its clockwise stroke. The ratio(??) of the faster

average(???) angular velocity ?f to the slower

one ?S is called the coefficient of travel speed

variation(????????), denoted as k.

k

where tf and tS are the time durations for

the faster stroke and the slower stroke,

respectively.

From the above, we can see that k is also the

time ratio of the slower stroke to the faster

stroke.

- the counter-clockwise stroke of the follower

rocker should be the working stroke(????), and

the clockwise stroke should be the return

stroke(??). If the clockwise stroke is needed to

be a working stroke, then the rotation direction

of the crank should be reversed(??).

- A crank-rocker mechanism with special dimensions

may not have quick return characteristics.if

a2d2b2c2, then ?0 and k1. This crank-rocker

mechanism has no quick-return characteristics

- In the offset slider-crank mechanism, the

distance C1C2 is the stroke H of the slider.

?C2AC1 is the angle ?. If the driving crank AB

rotates counter-clockwise with constant angular

velocity, the slider will take a longer time in

its rightward(??) stroke than in its leftward(??)

stroke. The coefficient k of the travel speed

variation, or the time ratio, is

(180o?)/(180o-?).

- Since, an in-line(??) slider-crank mechanism has

no quick-return characteristics because of ?0o

In an oscillating guide-bar mechanism, two

limiting positions CD1 and CD2 of the follower

guide-bar CD occur when the driving crank AB is

perpendicular(???) to the oscillating guide-bar

CD. Note The limiting positions of the

follower guide-bar CD do not occur when the

driving crank AB is horizontal(???).

- ?D1CD2 is the angular stroke ?max of the

follower. The acute angle between AB1 and AB2 is

?. For this linkage mechanism, ? happens(??) to

be equal to ?max. The coefficient of

travel speed variation K, or the time ratio, is

K(180o?)/(180o-?).

3 . Pressure Angle(???) ? and Transmission

Angle(???) ?

- The acute angle(??) between the directions of the

force F and the velocity of the point

receiving(??) the force on the follower is

defined as the pressure angle(???) ? of the

mechanism at that position.

- Only the tangential(???) component(??) F t can

create the output torque(??) on the driven link

DC. The radial(???) component F r only increases

pivot(??) friction(??) and does not contribute to

the output torque. For this reason, it is

desirable that ? is not too great or ? is not too

small.

- The complement(??) of the pressure angle ? is

called transmission angle(???) ?. The

transmission angle ? is also the acute angle(??)

between the coupler(??) and the follower. If

?BCDlt90o, then ??BCD. If ?BCD gt90o, then ?180o-

?BCD.

- ? and ? change during motion. The maximum value

of ? should be less than the allowable(???)

pressure angle ?40o, or the minimum value of ?

should be larger than?50o . Thus we should

find the extreme values of ? and ?.

- ?BCD reaches its extreme(??) when the driving

crank and the frame link are collinear. ?min will

occur in either of the two positions. It is

common practice to calculate both values and then

pick(??) the worst case, i.e., ?min

min?min?, ?min??.

- Suppose lABa, lBCb, lCDc and lADd

- For the same kinematic chain, the positions and

the values of ? and ? will change, if a different

link is chosen as the driver. ? and ? must be

drawn on the driven link!!

- If the crank is an input link and the slider is

an output, then the acute angle(??) between the

coupler BC and the slider path is ? at that

position. ? 90o- ?.

- The extreme(???) values of ? and ?, ?max and ?min

, occur when the crank AB is perpendicular(???)

to the slider path, i.e., ?max90o- ?min

sin-1(ae)/b

4 Toggle(??) Positions and Dead-points

- In a crank-rocker mechanism, the rocker DC

reaches its two limiting positions DC1 and DC2,

when the crank AB and the coupler BC become

Overlapping(??) collinear (AB1C1D) and extended

collinear (AB2C2D).

- if the rocker DC is a driver, then at its

limiting positions, the force applied to the

follower AB passes through the fixed pivot(??) A

of the follower. Therefore, the output torque is

zero regardless(??) of the amount of the input

torque applied. In this sense, the limiting

positions are called dead points(??).

- However, if the link AB is a driver, then near

the limiting positions of the rocker DC, a small

torque(??) applied to the link AB can generate a

huge(???) torque on the follower rocker DC. In

this sense(??), the limiting positions are called

toggle positions(????).

toggle positions dead

points

- In any four-bar mechanism (except the

change-point mechanisms), the dead point will not

occur if the crank is a driver. The dead points

will occur if the rocker or the slider is a

driver. The dead points occur when the driver

reaches either of its two limiting positions

toggle positions

dead points

- Note the limiting position of the rocker DC is

different from that where ?min may occurs

- Obviously, the limiting positions of the slider

are different from that where ?min occurs.

???????

- ?????

If a rocker or a slider is the driver, a

flywheel(??) on the driven crank will be required

to carry the mechanism through the dead point.

- A flywheel on the driven crank will be required

to carry the mechanism through the dead point.

- ????????

- the dead points are overcome by providing the

duplicate(???) linkage 90? out of phase(??).

- In some circumstances(??), the dead point is very

useful. An example of the application of a dead

point is the clamping device(??) on machine

tools(??). The mechanism is at the dead point

under the force from the clamped work piece(??).

??????

Shown is a landing(??) mechanism in airplane.

When the wheel is at its lowest position, links

BC and CD are collinear.

- Therefore the mechanism is at a dead point. A

small torque(??) on the link CD is enough to

prevent(??) the link DC from rotating.

??

3-4 Dimensional Synthesis (??) Planar Linkage

Mechanisms

- Synthesis Qualitative synthesis and Quantitative

synthesis (?????????). - Type Synthesis is a form of qualitative. It

refers to the definition of the proper type of

mechanism best suited to the problem(???). - Dimensional Synthesis of a linkage is the

determination of the proportions(lengths) of the

links necessary to accomplish the desired motions

(????).

- Dimensions affecting(??) the motion of the

mechanism are called kinematics

dimensions(?????). LAB between the centres of

the two holes in the coupler is the only

kinematics dimension.

- Dimensional synthesis(??) of a mechanism is the

determination of the kinematic dimensions

necessary to achieve(??) the required motion.

- Usually, different problems will use different

methods. - Graphical method
- Analytical method
- Experimental method

Three types of synthesis tasks(1)

1. Body Guidance(????) A linkage mechanism

is to be design to guide a line segment on the

coupler passing through some specified positions.

Such a synthesis problem is called body guidance.

Example 1 ?????????

Example 2 ???????

Three types of synthesis tasks(2)

2.Path Generation (?????) A linkage mechanism

is to be design to guide a line segment on the

coupler passing through some specified positions.

Such a synthesis problem is called body guidance.

Crane

Blender

Three types of synthesis tasks(3)

3. Function Generation(?????). A linkage

mechanism is to be design to guide a line segment

on the coupler passing through some specified

positions. Such a synthesis problem is called

body guidance.

1. Body Guidance(????)

- A revolute four-bar mechanism ABCD is to be

designed to guide a line segment(?) AB on the

coupler passing through three specified positions

E1F1, E2F2 , and EiFi.

- The fixed pivots(??) have been determined

If the points E and F are selected as the moving

revolute centres B and C respectively, the fixed

pivots will be A and D

1) ?????,????? ????, ????? lBC,??? ??l??

2) ??????????????? ???

4) ??????

3) ?????????????D,C?

lABAB ?l m lCDCD ?l m lADAD ?l m

(2) The fixed pivots have not been determined

- Points B1 and C1 can be chosen arbitrarily(???)

on the first position of the coupler.

The shape of the quadrilateral(???) BCFE should

remain the same in all positions.

- Constructing quadrilaterals B2C2F2E2 ?B1C1F1E1

and B3C3F3E3 ?B1C1F1E1, we get points B1, B2, B3

and C1, C2, C3.

Since the locus(??) of the point B relative to

the frame is a circle the centre of which is the

- fixed pivot A, a circle is constructed passing

through the three points B1, B2 and B3. The

centre of the circle is the fixed pivot A.

- Similarly, bisect(??) C1C2 and C2C3. The

intersection(??) of the two bisectors(???) is the

fixed pivot D.

- The accuracy(??) of the graphical methods by

hands is insufficient(???). However, the accuracy

of the graphical methods is good enough if

AutoCAD is used.

- It can be seen that the synthesized mechanism

cannot move the coupler through all three

specified positions in a continuous(???) motion

cycle

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- For this reason, the mechanism must be checked

after synthesis to see whether the assembly

mode(????) of the Assur group remains the same in

a continuous motion cycle. This check is called

consistency(???) of the assembly mode of a Assur

group. - Furthermore, the synthesized mechanism should be

checked for the Grash of criterion and ?max and

?min when required.

If the three points C1, C2, C3 locate on a

straight line, the the rocker become a sliding

block.

Analytical Synthesis

- A revolute four-bar linkage ABCD is to be

designed to guide a line segment MN on the

coupler BC through three positions M1N1, M2N2,

and MiNi

????

????

???????????M???????????Mi(xMi, yMi)

???????????2i ? ???? ?????Oxy,?????????????????

??????

???????

???????

???????

???1i???

???5?????xA?yA?a?k??? ??5??????????

N lt5 ?,???????

????????N3?????xA?yA

??????????

X0?X1?X2

????????

????

??????????

2 Function generation

- Synthesis problem that involves(??)

coordinating(??) the rotational and/or

translational orientations(??) of the input and

output is called function generation(????).

(1)??????????????

Supposea/a1, b/am, c/an, d/al?

???????????????,??????

????5?????,????????? ????????????N5?,???????? ??

N gt5 ?,??????,??????? ??N lt5 ?,

?????????N05-N,???????

(2 )Design of Quick Return Mechanisms

- Crank-rocker mechanism
- Suppose that the length c of the follower

rocker, the angular stroke ?max, and the

coefficient(??) k of the travel speed variation

have been specified. A crank-rocker mechanism is

to be designed

- When the driving crank AB runs at a constant

speed, the coefficient k (180o?) / (180o-?) ,

Therefore,

- By a well-known geometrical theorem(????), for

any point Ai on the arc C1P of a circle, ?C1AiC2

is constant. If ?PC1C2 90o, then PC2is the

diameter of the circle.

- Choose a fixed pivot D and draw the two limiting

positions, DC1 and DC2, of the follower rocker

with the known values of c and ?max. - Calculate ? according to the specified value of

k. - ?(k-1)/(k1)180o

- Through C1 construct a line perpendicular(???) to

C1C2. - Through C2 construct a line so that ?PC2C1 90o

-?. - Draw a circle with the midpoint of C2P as the

centre and the length of the line C2P as the

diameter.

- Choose a suitable point on the arc C1P as the

fixed pivot A. Measure the distances AC1and AC2.

The actual lengths, a and b, of the crank AB and

the coupler BC can be calculated. - a (AC2-AC1)/2 b(AC2AC1)/2

- Since any point on the arc C1P can be used as the

fixed pivot A if the length of the frame is

unknown, there is an infinity(???) of solutions.

Check the minimum transmission angle ?min after

synthesis. If it is not satisfied, then the

location of the fixed pivot A on the arc C1P

should be changed and the mechanism should be

redesigned. ?max can be minimized by choosing

the location of the fixed pivot A on the arc C1P

suitably.

- Obviously, ?AC2Dgt ?min . Therefore, the position

of the fixed pivot A can not be too low

- If k, ?max and two of the dimensions a, b and c

are known, the mechanism can be designed

analytically

Analytical Synthesis

Suppose b?AC2C1, if qy/2,then

d1 ifqlty/2,then d-1?

- (2) Offset slider-crank mechanism Suppose

that stroke H, time ratio k, and offset e are

known, design it graphically.

- This mechanism, however, can be easily designed

analytically with some equations derived as

follows.

- In ?C1AC2 , according to the cosine rule,

H2(b-a)2(ba)2-2(b-a)(ba)cos? 2 b2

(1- cos?)2 a2 (1 cos?) - According to the sine rule (b-a)/sin(?

AC2C1)H/sin?

- sin(? AC2C1) (b-a) sin ? /H In right triangle

AMC2, sin(?AC2C1)e/(ba) Therefore, e(ba)

sin(? AC2C1) (b2-a2)sin ?/H

- If k (from which ? can be determined) and any two

of the four parameters (H, e, b and a) are known,

then the other two unknowns can be calculated by

solving former Eqs. simultaneously(???). - H22 b2 (1- cos?)2 a2 (1 cos?)
- e (b2-a2)sin ?/H

(3) Oscillating guide-bar mechanism The angle ?

happens to be equal to the angular stroke ?max

of the guide-bar CD.

- Suppose that LAC and k are known.
- ?max ? (k-1)/(k1) 180o.
- In right triangle ABC, LAB LACsin(?max

/2).

3 Path Generation

- It is often desired to synthesize a linkage

mechanism so that a point on the coupler will

move along a specified path. This synthesis

problem is called path generation.

Example1 ?????

Example2 ?????

Analytical Synthesis

???????????????????????M,???????????????????? ???

????????????????????????????

????

????

????

????9?xA?yA? xD?yD ? a?c?e?f?g? ???????9?????????

?

The path generated by the point on the coupler is

called a coupler curve and the generating point

is called the coupler point.

- The atlas(??) of four-bar coupler curves consists

of a set of charts(??) containing

approximately(??) 7300 coupler curves of

crank-rocker mechanisms

- The small circle on coupler curve shows relative

position of coupler point on coupler. Each dash

on coupler curves represents 5o of input crank

rotation

- Arc E1EE2 of the coupler curve approximates a

circular arc. - A connecting link EF with a length equal to the

radius of this arc is added.

The output link GF will dwell, while coupler

point E moves through points E1, E, and E3.

5 Limitations(??) of Linkage Mechanisms

- Suppose that the output link DC is required to

rotate through an angle ? oi from its initial

position DC0 when the input link AB rotates ? oi

from its initial position AB0. In other words,

the revolute four-bar linkage is to be

synthesized to generate a given function ?oi f

(?oi).

- Suppose that the linkage is used to

coordinate(??) the rotational angle of the input

and output for five positions, i.e. - ?oi f (?oi), (i1, 2, 3, 4, 5). Putting these

five specified relationships between ?oi and ?oi

into ?oi f (a, b, c, ?o, ?o ,?oi) , - one obtains five equations as follows.

- Since there are only up to five independent

design variables in this synthesis problem, at

most five equations can be solved

simultaneously(?????). Therefore this linkage can

coordinate exactly only up to five relationships

between the input angle and the output angle. At

other positions, there will be some error (called

structural error????) between the actual function

and the required function

- Suppose that a revolute four-bar linkage ABCD is

to be designed so that a coupler point E will

pass through an ideal curve (dashed curve). It

can be shown that the actual coupler curve (solid

curve) can pass exactly through up to nine points

on the ideal curve.

- From the last examples, we can see that a linkage

mechanism can match the function exactly at only

a limited number of positions. At other

positions, there will be structural errors

- If the number of the required positions is larger

than 3, the algebraic(??) synthesis method often

leads to a set of non-linear equations containing

transcendental(???) functions of the unknown

angles. Also, the method cannot really control

?max, ?min, Grashof?s criterion, and the

structural error between the two precision

points. Optimization(??) methods

are now widely used in the synthesis of linkages