Chapter 4 Planar Linkage Mechanisms
(平面连杆机构)
§ 4.1 Characteristics(特性) of Planar
Linkage Mechanisms
Linkage mechanisms are lower-pair mechanisms.
The main practical advantage of lower pairs over
higher pairs is:
(1)The contact pressure(压强) is lower.
(2) Better ability to trap(围圈) lubricant(润滑剂)
between enveloping(包容) surfaces.
(3) The lower pair elements are easy to manufacture.
As a result, the linkage is preferred(首选的) for
low wear and heavy load situations.
A planar four-bar(杆) mechanism is the
simplest planar linkage mechanism with one
degree of freedom. Four-bar mechanisms are
extremely(极端地) versatile(万能的) and
useful devices(设备). For the sake(缘故) of
simplicity(简单), designers should always first
try to solve their problem with this device.
BB
11
AA
22
CC
BB
33
11
22
CC
44
DD
AA
44
33
EE
§ 4.2 The Types of Four-bar Mechanisms
（四杆机构的类型）
1.Revolute four-bar mechanism(铰链四杆机构)
CC
BB
11
AA
22
BB
33
11
22
CC
44
DD AA
44
EE
33
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(铰链四杆机构), which
is the basic form of the four-bar mechanism.
B
Coupler
2
Side 1
link
A
Frame 4
C
3 Side
link
D
In a revolute four-bar mechanism, the links connected
to the frame are called side links(连架杆). Usually,
one of the side links is an input link, and the other side
link is an output link. The floating(漂浮的) link
couples(连接) the input to the output. The floating link
is therefore called the coupler(连杆).
If two links connected by a revolute can rotate 360o
relative to each other, the revolute is called a fully
rotating revolute(整周转动副); otherwise, a partially
rotating revolute(摆动副). The revolutes A and B are
fully rotating revolutes, while the revolutes C and D
are partially(部分地) rotating revolutes.
B
1
A
B
C
2
3
1
2
4
4
D
A
C 3
E
If a side link can rotate continuously(连续地) through
360o relative to the frame, it is called a crank(曲柄);
otherwise, a rocker(摇杆).
According to the types
of the two side links, the types of the revolute four-bar
mechanisms can be divided into
B
Coupler
2
Crank 1
A
C
3
Rocker
Frame 4
D
(a) Crank-rocker mechanism(曲柄摇杆机构):
one side link AB can rotate continuously through
360o relative to the frame while the other side link
DC just rocks(摇摆). Therefore, AB is a crank
while DC is a rocker. This mechanism is called a
crank-rocker mechanism.
B
Coupler
2
Crank 1
A
C
3
Rocker
Frame 4
D
Applications of crank-rocker mechanism
Applications of crank-rocker mechanism
The input link may be the crank or the rocker.
In the footoperated
sewing(缝纫)
machine, the
oscillation(摆动)
of the driving
rocker is
transformed into
the continuous
rotation of the
driven crank.
(b) Double-crank mechanism (双曲柄机构) : 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
C
If one crank rotates
at a constant speed,
2
the other crank will
B
3
rotate in the same
direction at a
1
varying(变化的)
4
speed.
A
D
Applications of double-crank mechanism
(c) Double-rocker mechanism(双摇杆机构)
Both the side links DA and CB can only rock(摇摆)
through a limited(有限的) angle relative to the frame.
Therefore, both DA and CB are rockers. This
mechanism is called a double-rocker mechanism.
C
B
1
A
2
3
4
D
C'
C
E
B
B'
E'
D
Q
Q
A
The crane(鹤式起重机) is a famous(著名的) 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(描绘)
a horizontal(水平) line.
如果两摇杆长度相等，则称为等腰梯形机构。
汽车前轮转向机构中的四杆机构ABCD即为
等腰梯形机构。
A
D
4
1
B
3
2
C
汽车转弯时，两前轮轴线的交点应始终落在
后轴线上，即：两前轮的转角是不等的
A
D
4
1
B
2
3
C
The three kinds of mechanism can transform
each other by following way:For the same
kinematic chain, different kinds of linkage
mechanisms will be generated(产生) by
holding different links fixed as the frame. Such
kinds of variations(变异) are called
inversions(倒置).
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.
D
B
1
A
1
A
A
BB
33
11
44
Crank-rocker
Crank-rocker
BB
1
CC
22
22
D
D
Crank-rocker
Double-crank
B
1
44
DD
33
44
A
A
Double-crank
Double-crank
CC
33
CC
22
A
DD
C
2
3
4
Double-rocker
D
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(曲柄
滑块机构).
Applications of slider-crank mechanism:
Internal combustion engine
Applications of slider-crank mechanism:
punch machine
If the extended(延伸) path of the centre of revolute C
goes through the centre A of the crankshaft, the
mechanism is then called an in-line(对心) slidercrank mechanism , otherwise, an eccentric (or
offset)(偏置) slider-crank mechanism . The distance
from the crankshaft A to the path of the centre of the
revolute C is called the offset (偏置), denoted(标为)
as e
B
1
B
2
1
A
4
e
C
E
3
2
A
C
B B
B B
1 1 2 2
A A 4
4
1 1
C C
E Ee
e
2 2
A A
4 4
C 3C 3
3 3
B B
1 1 2 2
A A
4 4
B B
C C
3 3
1 1
e eA
2 2
A
4 4
3 3
C C
Both mechanisms in left are in-line slider-crank
mechanism , while the other two in right are eccentric
(or offset) slider-crank mechanism .
B
B
2
1
C
E
A
4
3
Rotating
(b)guide-bar
A
C
4
A
4
C
E
3
mechanism
2
1
2
(d)
Translating sliding-rod
mechanism
B
1
3
E
Crank and oscillating
block mechanism(曲柄
摇块机构).
B
B
2
1
A
1
A
C
E
4
B
1
3
2
C
4
(c)
3
A
2
4
B (b)
3
1
2
E A
4
(d)
C
E
C
E
3
B
2
1
A
4
3
C
E
If the crank BC is longer than the frame
BA, the guide-bar AE can rotate
continuously. It is a rotating guide-bar
mechanism
B
1
2
4
A
If the crank BC is shorter than the frame BA, the
guide-bar AE can only oscillate(摆动). The linkage
mechanism is called an oscillating guide-bar
mechanism(摆动导杆机构).
C
The quick-return(急回) mechanism in a shaper(牛头
刨床) is one of the applications of the oscillating
guide-bar mechanism
E
F
G
5
D
6
4
¦ Ø1 4
3
A
B
2
C
The hydraulic(液压) cylinder(油缸) is one of
applications of the crank and oscillating block
mechanism. The hydraulic cylinder is used widely in
practice. The self-tipping(自卸) vehicle(车辆) is an
example
C
3
C
3
1
A
4
2
4
2
4
A
B
1
3
1
B
A
C
2 B
Hand-operated well(井) pump(泵) mechanism is one
of applications of the translating sliding-rod
mechanism.
3. Replacing 2 revolute pairs with 2 sliding pair
there are two sliding pairs.
The output displacement
X of the translating(平动
的) guide-bar 3 is the
sine(正弦) function(函数)
ofthe input angle  of the
crank AB, i.e.
X=R*sin(). Thus, this
crank and translating
guide-bar mechanism is
often used as a sinusoid
generator(正弦发生器).
X
φ
B
2
1 R
ω
A
4
C
3
Crank and
translating guidebar mechanism
Applications of crank and translating guide-bar
mechanism
X
φ
2
B
2
1
1 R
ω
A
4
B
C
A
3
4
3
Crank and
translating
guide-bar
mechanism
If the link 3 is fixed as the
2 B
frame, then we get a double
sliding block mechanism(双
1
滑块机构）.
A
4
3
The right mechanism is called an elliptic trammel(椭
圆仪). This name comes from the fact that any point
on link AB traces(描绘) out an ellipse(椭圆
2
B
1
A
3
4
2
B
2
2
B
1
1
A
4
B
3
A
3
4
2
B
If the link 1 is fixed as the frame, one obtains a
mechanism known as the double rotating
1 block
mechanism (双转块机构) or Oldham
A
3
coupling(联轴器).
4
Double rotating block mechanism(中文称“十字滑块
1
联轴节”) or Oldham coupling is used to connect two
rotating shafts(轴) with parallel(平行) butAnon3
collinear(共线) axes.
4
2
B
1
A
4
3
If the link 4 is fixed as the frame, one obtains a
sinusoid generator(正弦发生器).
2
B
1
A
3
output displacement Y of the slider 3 is the tangent(正
切) function of the input angle  of the oscillating(摆
动) guide-bar AC, i.e. Y=L*tan(). Thus, this
oscillating guide-bar mechanism can be used as a
tangent generator
C
3
ω
1
A
φ
L
2
B
y
4
4. Enlarging(扩大) a revolute pair
B
1
2
1
A
4
C
3
A
E
B
4
2
3
C
The length LAB of the crank AB is determined
according to the kinematic requirements, while the
radii(半径) of the revolutes are determined by the
transmitted power(动力). Note: Enlarging a
revolute pair in no way changes the motion
relationship between any links
5. Interchanging guide-bar and sliding block
B
A
1
B
2
A
1
A 1
B
B
A 1
2
2
A 1
B
2
3
2
3
3
3
D
D
4
4
D
4
D
3
4
D
4
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(特性) Analysis of Fourbar Linkages
1 Condition for having a crank(Grashof Criterion准则)
In a revolute four-bar
C
mechanism, the input motion
2
is usually obtained through a
B
3
side link driven by an
1
electric motor directly or
4
indirectlythrough belt
A
mechanism or gears.
D
Therefore a designer must
ensure(确保) 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(从动件) rocker.
If the RRR Assur
C
group can be
b
assembled(装配) onto
the basic mechanism B
c
by the two outer
f
revolutes B and D, the
A
D
lengths of the three
sides in BCD must
obey (服从) the
triangle inequality(不
等式）
b  c  f

c  f  b
b  f  c

The distance f is
a variable(可变的)
value during the
motion of the
mechanism.
b  c  f

---(4-1)
b  c  f
c  b  f

C
b
B
c
f
A
D
b  c  f max  a  d

 b  c  f min  d  a
c  b  f  d  a
min

b  c  f

b  c  f
c  b  f

C
b
B
c
f
A
D
b  c  f max  a  d

Suppose: d>a  b  c  f min  d  a
c  b  f  d  a
min

b  c  f

b  c  f
c  b  f

C
C2
b
C1
B
f
a
B2
A
B1
c
d
fmin =d-a
fmax =d+a
D
b  c  f max  a  d

 b  c  f min  d  a
c  b  f  d  a
min

a  d  b  c
a  c


a  b  c  d (4-2)  a  b (4-3)
a  c  b  d
a  d


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(剩
余) two links. This is called Grashof criterion(准则)
or the Condition for having a crank.
The Grashof criterion can be expressed as: LMAX
+LMIN < Lb +Lc. A linkage mechanism which satisfies
the Grashof criterion is sometimes called a Grashof
linkage mechanism.
C
C2
b
C1
B
f
a
B2
A
B1
c
d
fmin =d-a
fmax =d+a
D
If LMAX +LMIN >Lb +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(倒置) are double-rocker mechanisms
C
C"
B
B'
B"
A
C'
D
In a non-Grashof
linkage mechanism,
no link can rotate
through 360o relative
to any other link.
However, in a
Grashof doublerocker mechanism,
the coupler can
rotate 360o with
respect to other
links.
C
C"
B
B'
B"
C'
D
A
2
A
B
3
1
D
4
C
A well-known example
of the Grashof doubleA
2
rocker mechanism is the
5
swing(摇摆) mechanism ω
1
5
of a swing fan.
ω21
First, 21 can be
determined according
51. Then 1 can be
1
found in the single DOF
ω
1
mechanism 1-2-3-4
4
D
according to 21.
B
3
C
C'2
If LMAX +LMIN =Lb
B
C
+Lc, the centre lines
of the four links can
B1
D
become collinear(共 A
C1
线). At these
C
2
positions, the output
B2
behavior
may become indeterminate(不确定的). 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.
B'
C'
B
A
C
D
the change-points are handled by providing the
duplicate(复制的) linkage 90 out of phase(相位). As
a consequence(结果), each linkage carries the other
through its change-points so that the output
remains(保持) determinate(确定的) at all positions.
Crankrocker
Double-crank
Double-rocker
Table 4-1 Type criteria for the revolute fourbar mechanisms
Frame
Lmax+Lmin
<Lb+Lc
Grashof
Shortest link Double-crank
Opposite to Double-rocker
the shortest
link
Adjacent to Crank-rocker
the shortest
link
Lmax+Lmin
>Lb+Lc
NonGrashof
Doublerocker
Lmax+Lmin
=Lb+Lc
Changepoint
From the above, we know that the Grashof
criterion LMAX +LMIN < Lb +Lc is only a
necessary condition, not sufficient(充足的)
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.
B'
B
a
e
A
b
C
C'
2. Quick Return Characteristics（急回特性）
C2
C1
¦ Ø1
A
¦È
a
b
B2
d
¦ ·max
c
D
B1
C1DC2 is called the angular stroke(行程) of the
rocker, denoted as max. C1AC2 is called the crank
acute angle between the two limiting positions(极位夹
角), denoted as .
C2
C1
¦ Ø1
A
¦È
a
b
B2
d
¦ ·max
c
D
B1
If the crank rotates counter-clockwise(逆时针) at constant
speed, it will take a longer time for the rocker in its counterclockwise stroke than its clockwise stroke. The ratio(比值) of
the faster average(平均的) angular velocity f to the slower
one S is called the coefficient of travel speed variation(行程
速度变化系数), denoted as k.
 max t
ts
f
f
k=
= 
=
max
s
tf
ts
o
180  
= 180o

where tf and tS are the time durations for the
faster stroke and the slower stroke, respectively.
From the
C2
above, we can
C1
b
see that k is
c
max
¦
·
also the time
¦È
B2
a
ratio of the
A
D
d
¦ Ø1
slower stroke
to the faster
B1
stroke.
the counter-clockwise stroke of the follower rocker
should be the working stroke(工作行程), and the
clockwise stroke should be the return stroke(回程). If
the clockwise stroke is needed to be a working stroke,
then the rotation direction of the crank should be
reversed(倒转).
C2
C1
b
¦ Ø1
B1
A
¦È
a
B2
d
¦ ·max
D
c
C2
C1
B2
A
¦ ·max
¦ È=0¡ ã
D
B1
A crank-rocker mechanism with special dimensions
may not have quick return characteristics.if
a2+d2=b2+c2, then =0 and k=1. This crank-rocker
mechanism has no quick-return characteristics
B1
¦Ø
a
e
A
¦ ÈB2
C1
b
H
C2
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(向右) stroke than in its
leftward(向左) stroke. The coefficient k of the travel
speed variation, or the time ratio, is (180o+)/(180o-).
2
1
A
4
B1
C1
A
B2
C
3
E
3
C
Since, an in-line(对心) 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
D2
guide-bar CD occur when
D1
the driving crank AB is
A
perpendicular(垂直于) to
the oscillating guide-bar
B1
B2
CD.
Note: The limiting
max
positions of the follower
guide-bar CD do not occur
when the driving crank AB
C
is horizontal(水平的).
D1CD2 is the angular
D1
stroke max of the
follower. The acute angle
between AB1 and AB2 is
B1
. For this linkage
mechanism, 
happens(碰巧) to be
equal to max.
The
coefficient of travel
speed variation K, or the
time ratio, is
K=(180o+)/(180o-).
D2
A
B2
max
C
3 . Pressure Angle(压力角)  and Transmission
Angle(传动角) 
Fr
F
¦ æ Á
VC
F  F  cos
C
t
Ft
¦Ã
B
F  F  sin a
r
A
D
The acute angle(锐角) between the directions of the
force F and the velocity of the point receiving(接受)
the force on the follower is defined as the pressure
angle(压力角)  of the mechanism at that position.
Fr
F
¦ æ Á
C
F  F  cos
t
Ft
VC
¦Ã
B
F  F  sin a
r
A
D
Only the tangential(切向的) component(分量) F t can
create the output torque(转矩) on the driven link DC.
The radial(径向的) component F r only increases
pivot(枢轴) friction(摩擦) and does not contribute to
the output torque. For this reason, it is desirable that 
is not too great or  is not too small.
Fr
F
¦ æ Á
C
Ft
VC
VC
C
¦Ã
B
¦ÁF
¦Ã
B
A
D
A
The complement(余角) of the pressure angle  is
called transmission angle(传动角) . The
transmission angle  is also the acute angle(锐角)
between the coupler(连杆) and the follower. If
BCD<90o, then =BCD. If BCD >90o, then
=180o- BCD.
D
Fr
F
¦ æ Á
C
Ft
VC
¦Ã
B
A
D
 and  change during motion. The maximum
value of  should be less than the allowable(允许的)
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(极值) 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(挑选) the worst
case, i.e.,
min =min{min, min}.
C
C
γmin
'
A
B
D
B
A
γmin
''
D
Suppose: lAB=a, lBC=b, lCD=c and lAD=d
C
C
γmin
'
A
B
D
B
γmin
''
A
 '  B' C ' D
 "  B"C" D
b 2  c 2  (d  a)2
 arccos
2bc
b 2  c 2  (d  a) 2
 arccos
2bc
or
2
2
2
b

c

(
d

a
)
 "  1800  arccos
2bc
D
Fr
F
C
Ft
VC
C
VB
B
B
F
A
A
D
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!!
D
B
a
e
A
b
α
γ
C
VC
F
If the crank is an input link and the slider is an output,
then the acute angle(锐角) between the coupler BC
and the slider path is  at that position.  =90o- .
The extreme(极端的) values of  and , max and min ,
occur when the crank AB is perpendicular(垂直于) to
the slider path, i.e., max=90o- min= sin-1[(a+e)/b]
B'
b
a
e
A
αmax
γmin
C'
4 Toggle(肘节) 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(重
叠) collinear (AB1C1D) and extended collinear
C2
(AB2C2D).
C1
b
c
¦ Ø1
B1
A
a B2
d
D
C2
C1
b
c
A
a
B2
d
D
B1
if the rocker DC is a driver, then at its limiting
positions, the force applied to the follower AB passes
through the fixed pivot(枢轴) A of the follower.
Therefore, the output torque is zero regardless(不管)
of the amount of the input torque applied. In this sense,
the limiting positions are called dead points(死点).
However, if the link AB is a driver, then near the
limiting positions of the rocker DC, a small torque(转
矩) applied to the link AB can generate a huge(巨大的)
torque on the follower rocker DC. In this sense(意义),
the limiting positions are called toggle positions(肘节
位置).
C2
C1
¦ Ø1
A
B1
b
c
a
B2
d
D
C1
¦ Ø1
A
C2
C1
b
C2
b
c
a B2
d
c
A
D
B1
a B2
d
D
B1
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
ω
B1
A
e
θB2
C1
H
C2
toggle positions
B1
e
A
B2
C1
dead points
C2
Note: the limiting position of the rocker DC is
different from that where min may occurs
C
C
γmin
'
A
B
B
D
A
γmin
D
Obviously, the limiting positions of the slider are
different from that where min occurs.
B1
a
A
e
B2
b
C1
H
B'
a
e
A
b
min
max
C'
C2
克服死点的措施
• 利用惯性力
If a rocker or a slider is the driver, a flywheel(飞
轮) 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.
• 相同机构错位排列
the dead points are overcome by providing the
duplicate(复制的) linkage 90 out of phase(相位).
F’
G’
E’
G
E
F
P
C
B
D
A
In some circumstances(情况), the dead point is very
useful. An example of the application of a dead point
is the clamping device(夹具) on machine tools(机床).
The mechanism is at the dead point under the force
from the clamped work piece(工件).
=00
折叠家具机构
Shown is a
landing(着陆)
mechanism in
airplane. When the
wheel is at its
lowest position,
links BC and CD
are collinear.
A
C
D
B
Therefore the mechanism is at a dead point. A small
torque(转矩) on the link CD is enough to prevent(防
止) the link DC from rotating.
小结
要求：
1. 运动特性
 曲柄存在条件
 杆长条件
 最短杆条件
 急回特性
 极位夹角
 行程速比系数
2. 传力特性
 压力角和传动角
 死点
 正确理解和掌握平面机构
工作特性的有关概念；
 用有关工作特性检验机构
的运动和传力性能；
 运用有关概念设计性能优
良的机构。
§3-4 Dimensional Synthesis (综合)
Planar Linkage Mechanisms
• Synthesis: Qualitative synthesis and Quantitative
synthesis (定性综合和定量综合).
• Type Synthesis is a form of qualitative. It refers to
the definition of the proper type of mechanism
best suited to the problem(型综合).
• Dimensional Synthesis of a linkage is the
determination of the proportions(lengths) of the
links necessary to accomplish the desired motions
(尺寸综合).
Dimensions affecting(影响) the motion of the
mechanism are called kinematics dimensions(运动
学尺寸). LAB between the centres of the two
holes in the coupler is the only kinematics
dimension.
BB
A
A
(a)
(a)
A
A
(b)
(b)
BB
Dimensional synthesis(综合) of a mechanism
is the determination of the kinematic
dimensions necessary to achieve(获得) 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(刚体导引)
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 铸造车间的翻箱机构
Example 2 热处理炉门机构
Three types of synthesis tasks(2)
2.Path Generation (轨迹生成器)
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.
Blender
Crane
Three types of synthesis tasks(3)
3. Function Generation(函数发生器).
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(刚体导引)
F1 A revolute fourF2
bar mechanism
ABCD is to be
F3
designed to guide
E2
a line segment(段)
AB on the coupler
E1
passing through
three specified
positions E1F1,
E3
E2F2 , …and EiFi.
(1)The fixed pivots(铰链) 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
C1
1) 刚体作连杆，选定
其上二 活动铰链，
即定连杆长 lBC，定
比例 尺l作图；
C2
B1
B2
3
l
m
 l  BC (
)
BC mm
2) 活动铰链相对于
固定铰链的运动轨
迹为圆；
3) 用三点定心法确
定二固定铰链D,C。
C3
B
D
A
4) 计算待求杆长
lAB=AB· l m;
lCD=CD· l m;
lAD=AD· l m;
(2) The fixed pivots have not been determined
Points B1 and
C1 can be
chosen
arbitrarily(任
意地) on the
first position
of the coupler.
F2
F3
C1
E2
E1
B1
E3
F1
The shape of the quadrilateral(四边形) 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.
F1
F2
F3
C1
E2
C2
C3
B2
E1
B1
E3
B3
Since the locus(轨迹) 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.
F1
F2
F3
C1
E2
C2
C3
B2
E1
A
B1
E3
B3
Similarly, bisect(平分) C1C2 and C2C3. The
intersection(交点) of the two bisectors(平分线) is
F1
F2
the fixed pivot D.
F3
C1
E2
C2
C3
B2
E1
A
B1
E3
B3
D
The accuracy(精度) of the graphical methods
by hands is insufficient(不够的). However, the
accuracy of the graphical methods is good
enough if AutoCAD is used.
C1(F1)
A
B2( E2)
D
B1(E1)
C2( F2 )
C3( F3)
B3(E3)
It can be seen that the synthesized mechanism cannot
move the coupler through all three specified positions
in a continuous(连续的) motion cycle
C1(F1)
B2( E2)
B'3
A
D
B1(E1)
C2( F2 )
C3( F3)
B3(E3)
C1(F1)
A
B1(E1)
B2( E2)
D
C'3
C2( F2 )
C3( F3)
B3(E3)
For this reason, the mechanism must be checked
after synthesis to see whether the assembly
mode(装配模式) of the Assur group remains the
same in a continuous motion cycle. This check
is called consistency(一致性) 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.
B
1
e
2
A
C
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
N1
M2
N2
M1
Mi
Ni
左侧杆组
右侧杆组
设计要求：要求连杆上某点M能占据一系列的预
定位置Mi(xMi, yMi) 且连杆具有相应的转角θ2i 。
设计思路： 建立坐标系Oxy，将四杆机构分为左
侧双杆组和右侧双杆组分别讨论。
左侧双杆组分析：
由矢量封闭图得
OA  ABi  Bi M i  OM i  0
写成分量形式为
xA  a cos1i  k cos(   2i )  xMi  0

y A  a sin 1i  k sin(    2i )  yMi  0 
消去θ1i整理得
( x  y  x  y  k  a ) / 2  x A xMi  y A yMi
2
Mi
2
Mi
2
A
2
A
2
2
 k ( x A  xMi ) cos(   2i )  k ( y A  yMi ) sin(    2i )  0
式中有5个待定参数：xA、yA、a、k、γ。
可按5个预定位置精确求解。
N <5 时，可预选参数数目
当预定连杆位置数N=3：可预选参数xA、yA
( x  y  x  y  k  a ) / 2  x A xMi  y A yMi
2
Mi
2
Mi
2
A
2
A
2
2
 k ( x A  xMi ) cos(   2i )  k ( y A  yMi ) sin(    2i )  0
X 0  A1i X 1  A2i X 2  A3i  0
代入连杆
三组位置
参数
X0、X1、X2
xBi  xMi  k cos(   2i )

yBi  yMi  k sin(    2i ) 
右侧杆组分析：同上
右侧杆组
b  ( xBi  xci )  ( yBi  yci ) 

2
2
d  ( x A  xD )  ( y A  yD ) 
2
根据左右杆
组各参数有：
2
2 Function generation
Synthesis problem that involves(涉及)
coordinating(协调) the rotational and/or
translational orientations(方位) of the input
and output is called function generation(函数
发生).
(1)按预定的两连架杆对应位置设计
AB + BC = AD + DC
Suppose:a/a=1,
b/a=m,
c/a=n, d/a=l。
m cos 2i  l  n cos( 3i   0 )  cos(1i   0 ) 消去θ2i

m sin  2i  n sin(  3i   0 )  sin( 1i   0 )

P
0
cos(1i   0 )  n cos( 3i   0 )  (n / l ) cos( 3i   0  1i   0 )
 (l 2  n 2  1  m2 ) /( 2l )
P2
P1
cos(1i   0 )  P0 cos( 3i   0 )  P1 cos( 3i   0  1i   0 )  P2
cos(1i   0 )  P0 cos( 3i   0 )  P1 cos( 3i   0  1i   0 )  P2
将两连架杆的已知对应角代入上式，列方程组求解
方程共有5个待定参数，根据解析式可解条件：
★当两连架杆的对应位置数N=5时，可以实现精确解。
★当N >5 时，不能精确求解，只能近似设计。
★当N <5 时， 可预选尺度参数数目N0=5-N，故有无穷多解。
(2 )Design of Quick Return Mechanisms
Crank-rocker mechanism C1
C2
Suppose that the length c
of the follower rocker,
¦ ·max
the angular stroke max,
and the coefficient(系数)
c
k of the travel speed
variation have been
D
specified. A crank-rocker
mechanism is to be
designed
When the driving crank AB runs at a constant speed,
the coefficient k= (180o+) / (180o-) ,
k 1

 180 
k 1
C1
AC1  b  a
AC2  b  a
Therefore,
C2
¦ Ø1
AC2  AC1 B
1
a
2
AC2  AC1
b
2
A
¦ aÈ
b
B2
d
¦ · max c
D
By a well-known
geometrical
theorem(几何定
理), for any point
Ai
Ai on the arc C1P
of a circle,
C1AiC2 is
constant. If
PC1C2 =90o, then
PC2is the diameter
of the circle.
C1
C2
90¡-ã
O
P
• 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)/(k+1)*180o
C1
C2
¦ ·max
c
D
• Through C1
construct a line
perpendicular(垂直
于) 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.
C1
C2
¦ ·max
c
¦È
D
P
Choose a suitable
point on the arc
C1P as the fixed
pivot A. Measure
the distances
A
AC1and AC2. The a
actual lengths, a
B1
and b, of the crank
AB and the
coupler BC can be
calculated.
a= (AC2-AC1)/2
b=(AC2+AC1)/2
C1
C2
b
¦ È B ¦ ·max
2
¦È
D
P
c
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(无穷多) 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, AC2D> min . Therefore, the position of
the fixed pivot A can not be too low
C1
C2
b
¦ È B ¦ ·max
2
A
a
¦È
B1
D
P
c
If k, max and two of the dimensions a, b and c
are known, the mechanism can be designed
analytically
C1
C2
b
¦ È B ¦ ·max
2
A
a
¦È
B1
D
P
c
Analytical Synthesis
c sin( / 2)
r  loc1 
sin 
Suppose b=∠AC2C1 ，
if q≥y/2,then d=+1;
ifq＜y/2，then d=-1。
g  lOD  c sin  (  / 2)/ sin 
lC1C 2
2c sin  sin( / 2)
sin   2r sin  
sin 
sin 
l
2c sin(   ) sin( / 2)
l AC 2  b  a  C1C 2 sin(   )  2r sin(   ) 
sin 
sin 
a  c sin( / 2)sin(   )  sin  / sin 
b  c sin( / 2)sin(   )  sin  / sin 
l AC1  b  a 
d  r 2  g 2  2rg cos(2    )
(2) Offset slider-crank mechanism
Suppose that stroke H, time ratio k, and offset e are
known, design it graphically.
O
B1
A
e
M
a B2
¦È
b
90¡¦-ã È
C1
H
C2
This mechanism, however, can be easily
designed analytically with some equations
derived as follows.
O
B1
A
e
M
a B2
¦È
b
90¡¦-ã È
C1
H
C2
In C1AC2 , according to the cosine rule,
H2=(b-a)2+(b+a)2-2*(b-a)*(b+a)*cos
=2* b2 *(1- cos)+2* a2 *(1+ cos)
According to the sine rule (b-a)/sin( AC2C1)=H/sin
O
B1
A
e
M
a B2
¦È
b
90¡¦-ã È
C1
H
C2
sin( AC2C1)= (b-a) sin  /H
In right triangle AMC2, sin(AC2C1)=e/(b+a)
Therefore, e=(b+a) * sin( AC2C1) =(b2-a2)sin /H
O
B1
A
e
M
a B2
¦È
b
90¡¦-ã È
C1
H
C2
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(同时地).
H2=2* 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.
D2
D1
Suppose that LAC and k are
known.
A
max =  =(k-1)/(k+1)
*180o.
In right triangle ABC,
LAB= LAC*sin(max /2).
¦È
B1
B2
¦ ·max
C
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 鹤式起重机
Example2 搅拌器机构
Analytical Synthesis
左侧杆组
右侧杆组
设计要求：确定机构的各
尺度参数和连杆上的描点
位置M，使该点所描绘的
连杆曲线与预定的轨迹相
符。
设计思路：分别按左侧杆
组和右侧杆组的矢量封闭
图形写出方程解析式。
2
2
2
2
(
x

x
)

(
y

y
)

e

f
 2[e( x  x A )  f ( y  y A )] cos 2
A
A
联
立  2[ f ( x  x A )  e( y  y A )] sin  2  a 2
求 ( x  x )2  ( y  y )2  g 2  f 2  2[g( x  x )  f ( y  y )] cos 
D
D
D
D
2
解
 2[f ( x  x D )  g( y  yD )] sin 2  c2
( x  x )2  ( y  y )2  e2  f 2  2[e( x  x )  f ( y  y )]cos2 
2[ f ( x  x )  e( y  y )]sin2  a 2
( x  xD )2  ( y  yD )2  g 2  f 2  2[ f ( y  yD )  g ( x  xD )]cos2 
2[ f ( x  xD )  g ( y  y D )]sin  2  c
2
待定参数9个：xA、yA、 xD、yD 、 a、c、e、f、g。
故最多也只能按9个预定点进行精确设计
The path generated by the point on the coupler is
called a coupler curve and the generating point
is called the coupler point.
C
E
Y
B
A
O
X
D
b
a
c
d
b/a=3, c/a=3.5, d/a=2
The atlas(图谱) of four-bar coupler curves consists of
a set of charts(图表) containing approximately(大约)
7300 coupler curves of crank-rocker mechanisms
b
a
c
d
b/a=3, c/a=3.5, d/a=2
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.
E
B
E2
A
C
E1
G
F
D
5 Limitations(局限) of Linkage Mechanisms
Ci
C0
Bi
¦ È0i
A
B0
b
¦Õ
0i
c
¦ Õ0
a
¦ È0
D
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(协调) the rotational angle of the
input and output for five positions, i.e.
oi = f (oi), (i=1, 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.
 01  f ( a, b, c, 0 ,  0 , 01 )

 
  f ( a, b, c, ,  , )
0
0
05
 05
Since there are only up to five independent
design variables in this synthesis problem, at
most five equations can be solved
simultaneously（同时进行地）. 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结构误差)
between the actual function and the required
function
C
E
Y
B
A
O
D
X
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.
C
E
Y
B
A
O
D
X
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(代数) synthesis method
often leads to a set of non-linear equations
containing transcendental(超越的) 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(优化) methods are now
widely used in the synthesis of linkages