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


Chapter 4 Planar Linkage Mechanisms ( ) 4.1 Characteristics( ) of Planar Linkage Mechanisms Linkage mechanisms are lower-pair mechanisms. – PowerPoint PPT presentation

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

Chapter 4 Planar Linkage Mechanisms (??????)
  • 4.1 Characteristics(??) of Planar Linkage
  • 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
  • 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

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
  • 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
(No Transcript)
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

  • 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.
(No Transcript)
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

  • 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(??????).
(No Transcript)
  • 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
  • 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

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

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(??)

  • 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
  • 1 Condition for having a crank(Grashof

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.
(No Transcript)
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

  • 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
  • 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.

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

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.

where tf and tS are the time durations for
the faster stroke and the slower stroke,
From the above, we can see that k is also the
time ratio of the slower stroke to the faster
  • 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

  • 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

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

  • 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-

  • ? 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

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
  • 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
  • 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
  • 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.
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.

  1. 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

(No Transcript)
(No Transcript)
  • 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
  • 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

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,?????????????????
???5?????xA?yA?a?k??? ??5??????????
N lt5 ?,???????
2 Function generation
  • Synthesis problem that involves(??)
    coordinating(??) the rotational and/or
    translational orientations(??) of the input and
    output is called function generation(????).

Supposea/a1, b/am, c/an, d/al?
????5?????,????????? ????????????N5?,???????? ??
N gt5 ?,??????,??????? ??N lt5 ?,
(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-?) ,

  • 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-1)/(k1)180o

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

  • 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

  • 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

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

  • 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(?

  • 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

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

  • 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

  • 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