Transcript The Theory of Machines and Mechanisms Sun Zhihong

The Theory of Machines and
Mechanisms
Chapter 3. Planar Linkage and Its Design
Sun Zhihong
[email protected]
67792564-108
东华大学专用
作者： 孙志宏
Today’s topics
Basic knowledge of four-bar mechanism
1. The condition for having a crank（曲柄存在
的条件）
2. Pressure angle and transmission angle（压
力角和传动角）
3. Toggle positions and Dead-points（极位与
死点）
4. Quick return characteristics (急回运动特性)
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作者： 孙志宏
1. Condition for having a crank（曲柄存在的条件）
If there is a revolute pair  there could be a crank
Suppose a<d, if side link AB could revolute，it
must be collinear with the frame twice
From △B’C’D ：
a+d ≤ b + c
From △B”C”D：
b≤(d – a)+ c  a+b ≤ c + d
c≤(d –a)+ b
 a+ c ≤ b + d
C’
b
a≤ b, a≤c,
a≤d
AB is the shortest link
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B’
Aa
a
B”
C”
bc
d d- a
c
D
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1. Condition for having a crank（曲柄存在的条件）
 The sum of the shortest and the longest links
must be less than the sum of the remaining two
links (Grashof condition),
 The shortest link is the side link or the frame.
C
b
B
A
c
a
D
d
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– If the four-bar mechanism fits the Grashof
condition，there could be a crank.
– If assign different link as the frame, different
mechanisms could be got：
Crank-rocker1 Crank-rocker2 Double crank
曲柄摇杆1
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曲柄摇杆2
双曲柄
Double rocker
双摇杆
作者： 孙志宏
2. Pressure angle and transmission angle
Pressure angle α (压力角)：The acute angle between the
directions of the force F and the velocity of the point
receiving the force on the follower
Transmission angle γ(传动角) :The complement of the
pressure angle α is called transmission angle γ
Tangential force: F’= Fcosα
=Fsinγ
F”
F
C γα
Normal force: F”= Fcosγ
F’
B
A
D
γ↑, it is good for the
mechanism
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作者： 孙志宏
2. Pressure angle and transmission angle
The position of γmin：
When ∠BCD≤90°，
C
γ＝∠BCD
B
B
When ∠BCD>90°，
A A
F”
γ
F’
F”
F
γ
C Fα
F’
DD
γ＝180°- ∠BCD
When the driving crank and the frame link are
collinear, ∠BCD reaches its extreme，γmin will
occur.
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According the law of cosines：
∠B1C1D＝arccos[b2+c2-(d-a)2]/2bc
γ1＝∠B1C1D
If ∠B1C1D≤90°,
∠B2C2D＝arccos[b2+c2-(d+a)2]/2bc
γ2＝180°-∠B2C2D
If ∠B2C2D>90°,
γmin＝[∠B1C1D, 180°-∠B2C2D]min
The α and γ must be
drawn on the driven link
车门
F γ
γ2
C2
B2
A
a
B1
d
b
γ1 c
C1
D
α
v
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作者： 孙志宏
3. The toggle positions and the Dead points
In a crank-rocker mechanism，when the crank AB
and the coupler BC overlapping collinear or
extended collinear, the rocker CD reaches its two
limiting positions DC1 and DC2, they are called
toggle positions(极位)。
The angle between
DC1 and DC2 at the
two limiting positions
is denoted as θ(极位夹
角) 。
B
ω θ
A
B1
C
C2
C1
DD
B2
Crank-rocker mechanism
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作者： 孙志宏
3. The toggle positions and the Dead points
F
If the rocker is a
driver，then at its
limiting positions,
γ＝0
F
γ＝0
Transmission angle of the mechanism isγ＝0
These positions are called Dead-points (死点)
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3. The toggle positions and the Dead points
Methods to overcome dead-points ：
Stagger two sets of the same mechanism:
wheel driving mechanism of trains;
The inertial force of flywheel: Sewing machine
F’
B’
A’
E’
A
B
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C’
D’
G’
E
G
D
F
C
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Applications of Dead-points：
Landing mechanism in an airplane,
Clamping device on machine tools。
P
C
D
D
A
γ=0
B
B
C
B
工件
Landing
mechanism
F
A
B 2
2 C
11
A
P
C
33
D
D
4
T
Clamping
device
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4.Quick return characteristics(急回运动特性)
Crank rocker
mechanism
B
180°＋θ ω θ
A
B1
CC
C2
1
DD
B2
If the crank AB rotates 180°+θ counterclockwise at a constant angular speed of ω，
the rocker DC swing from C 1 D to C 2 D。
The time t1 it needs is
t1  (180   ) / 
The average speed V1 are：
V1  C1C2 t1  C1C2 /(180    )
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When the crank goes on rotating the rest 180°-θ，the rocker
will swing from DC2 back to DC1， then t2 and v2 are:
t2  (180    ) / 
C1
C2
V2  C1C2 t2
 C1C2 /(180    )
Obviously, t1 >t2 v2 > v1
B2
A
B1
D
180°-θ
Quick return property
C1C2 t2
t1
V2
180   


K

C1C2 t1
t2
V1
180   
When design a new mechanism，
usually time ratio K is assigned
first，then :
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K is called time
ratio(行程速比
系数)。
K 1
  180 
K 1
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Quick return property of slider-crank mechanism
180°＋θ
180°＋θ
θ
θ
180°-θ
180°-θ
Questions：
If the following mechanisms have quick return
property?
in-line crank mechanism(对心曲柄滑块机构)、
oscillating guide-bar mechanism (摆动导杆机构)
Application:：save the time of return stroke.
Example, shaper (牛头刨)。
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Page 238
8-6
Page 239
8-8, 8-9
东华大学专用
作者： 孙志宏