Transcript F=3n－(2P L +P h )

Chapter 2
Structural Analysis of Mechanisms
§2－1
Content and Object
§2－2
Composition of Mechanisms
§2－3 The Kinematic Diagram of a Mechanism
§2－4
Conditions for a Mechanism to Have a
Determined Motion
§2－5 Degree of Freedom of a Planar Mechanism
§2－6
Points for Attention during the
Calculation of DOF
§2－7 The Composition Principle and
Structural Analysis
§2－1 Content and Objective
1. Study of the composition of mechanism and method
of drawing kinematical diagram of the mechanisms.
2. Understanding of rules concerning the motion
certainty of mechanisms.
3. Study of Composition and Classification of the
Mechanisms.
§2－2 Composition of Mechanisms
一、Component or link
Component——Every unit that has independent
motion in mechanism is called a component.
Part——Every unit that has independent manufacture
in machine is called a part.
二、Kinematical Pair or joint
Kinematical Pair——A joint is a connection between
two or more links, which allows some motion between
the connected links.
1. Classification of kinematical pairs
1）By the number of constraints allowed at the joint
Constraint——independent motion will be constrained。
The number of constraints provides＝The number of DOF removes
Grade I pair、 Grade II pair 、 Grade III pair 、 Grade IV pair 、 Grade V pair .
2）By the type of contact between the elements,line,point,or surface
higher
pair
lower
pair
Higher pair——point or line contact
Lower pair——surface contact
3)By the relative motion between two components
revolute
pair
sliding
pair
planar pair
screw
pair
spherical
pair
spatial pair
4) By the relative motion between two components on
plane or space
2. Representation(Symbol) of a Kinematical Pair(GB 4460－84)
2
1
1
2
2
1
1
revolute
pair
2
1
2
1
2
2
1
1
1
2
2
1
2
1
2
sliding
pair
higher
pair
spherical
pair
2
1
screw
pair
三、 Kinematical Chain
Kinematical Chain——When a number of links are connected by means of
kinematical pairs, the resulting mobile system is a kinematical chain.
2
3
1
4
Planar chain
closed
chain
四、 Mechanism
open chain
Spatial
chain
Mechanism——A kinematic chain in which at least one link has been “grounded”,
or attached, to the frame of reference(which itself may be in motion).
2
C
2
B
3
3
1
1
A
D
4
4
§2－ 3 The Kinematic Diagram of a Mechanism
The Kinematic Diagram of a Mechanism—— In order to analyze an existing mechanism
or design a new mechanism, it is helpful to draw a simple diagram to indicate the
kinematic relationship between links. Since this diagram is used only to express the
relationship between the motions of links, it should be simple but provide all necessary
(not redundant) information determining the relative motion of all links. Such a diagram
is called the kinematic diagram of the mechanism.
Procedures：
1.
Run the mechanism slowly, study carefully the structure of the mechanism.
Determine the types of all kinematic pairs. Analyze the transmission route from the
driving link to the output link.
2. For a planar mechanism, all moving links move in parallel planes. Thus, a plane
parallel to these planes is chosen as a drawing plane.
3. Select a suitable scale μl , the factor of which is μl =actual length (m) /
length in diagram (mm)
4. Draw the diagram of the mechanism according to symbols and scale.
Example : Draw the kinematic diagram of the crusher
6
1
O
F
2
5
4
C
A
3
B
D
E
§2－4 Conditions for a Mechanism to Have a Determined Motion
four-bar
mechanism
F＝1
If one independent kinematic parameter is
given, the locations of all moving links will
be determined. If two independent
kinematic parameters are given, the
mechanism is broken.
five-bar
mechanism
Conditions for a Mechanism
to Have a Determined Motion:
the DOF of the mechanism
F＝1
F＝2
＝ the number of the driving
links
§2－5 Degree of Freedom of a Planar Mechanism
Link 2：3 degree of freedom（x，y, ）
y
2
If two links are connected by means of a revolute pair,
the number of DOF is１.
F ＝ unconstrained link DOF
1
－ removed DOF＝３－２＝１
number of moving links：n
planar higher pairs：Ph
DOF： 3n
x
planar lower pairs：Pl
number of constraints：2 PL＋Ｐh
the DOF of a planar mechanism：
F=3n－(2PL +Ph )
Example 2-1 Determine the DOF of four-bar mechanism
Solution：n = 3 Pl= 4 Ph= 0
F=3n - (2Pl+Ph) = 3×4 - 2×4 = 1
B
C
2
3
1
A
4
D
Example 2-2 Calculate the DOF of the cam mechanism
shown in Fig.
Solution： n = 2
Pl= 2
Ph=1
3
F = 3n - ( 2Pl + Ph )
= 3×2 - 2×2 - 1 = 1
2 ω
2
1
§2－6 Points for Attention during the Calculation of DOF
一、Calculate the number of kinematic pairs correctly
1. Compound Hinge——more than two components will be joined to each
other by revolute pairs at a single location.
components：
3
m
revolute pairs： 2
m-1
2
2
3
1
3
1
Example2-3 Calculate the DOF of the mechanism
shown in Fig.
D
Solution compound hinge：B、C、D、E
n=7
Pl= 10
4
3
Ph= 0
8
F = 3n - (2Pl+Ph)
= 3×7 - 2×10 = 1
6 E
F
C
B
2
7
A
1
5
2.
When two links are connected by more than one parallel sliding pair, only one
sliding pair can be counted during the calculation, others must not be counted.
B
B’
3. When two links are connected by more than one revolute pair whose axes
coincide, only one of the revolutes must be counted during the calculation.
4 . When two links are connected by more than one higher pair whose common
normals passing through the points of contact coincide, only one of the higher
pairs can be counted during the calculation.
n2 n
1
A A’
n1
n2
n1
n2
n1
n2
二、Passive DOF (local DOF)
F = 3×3 - 2×2 - 1 = 2
Passive DOF—— Such a DOF which does not change the
output motion of the mechanism is called a passive DOF.
 Solution 1： F = 3n -（ 2Pl + Ph ) - F’
= 3×3 - 2×3 - 1 -1 = 1
 Solution 2：delete the passive DOF by welding
the roller to the follower.
F=3×2 - 2×2 - 1= 1
三、Redundant Constraints(Void Constrain)
Redundant Constraints——In many cases, more than one constraint may have exactly the
same kinematic function. In these cases, only one of the constraints should be counted
during the calculation. Others are called redundant constraints, which must not be counted
during the calculation.
B
E
A
A
F
C
B
C
D
D
F = 3n -（ 2Pl – Ph )
= 3×4 - 2×6 = 0
 Solution 1：degree of void constrain
should be subtracted from degree of
constrain of the mechanism.
F = 3n -（ 2pl - ph - p’) – F’
= 3×4 – ( 2×6-1)-0=1
Solution 2 ：delete void constrain.
F = 3n -（ 2Pl – Ph )
= 3×3 - 2×4 = 1
Redundant constraints occur in many
situations as described below：
1.
When the locus of a point is a straight
line, adding one link with one fixed guide
way parallel to the straight line and one
revolute with its centre at that point will
create a redundant constraint.
B
C
E
2. When the distance between two points on
two links remains constant during the
motion of the mechanism, adding one link A
and two revolutes with their centers at these
two point will create a redundant constraint.
3. Symmetrical structure or duplicated
structure
F
D
Example2-4 Calculate the DOF of the mechanism shown
in Fig.
Compound
hinge
Solution:
n=8
Pl= 11
Ph= 1
F = 3n - (2Pl+Ph)
= 3×8 – (2×11 + 1 )
=1
Void
constrain
Passive
DOF
§2－7 The Composition Principle and Structural Analysis
一、Composition Principle of Mechanisms
We learned that in any mechanism which has a determined motion, the
number of drivers must be equal to the DOF of the mechanism.
6
F＝0。
6
1
1
2
5
4
2
5
3
3
4
Assur groups—— If the DOF of each group is zero and no group can be divided
further into two or more zero-DOF groups, then such groups are called Assur
groups.
Composition principle of mechanism—— we can see that any mechanism which
has a determined motion, can be assembled from a basic mechanism by
connecting Assur groups to the determined links using outer pairs, group by
6
1
group.
2
5
5
2
3
4
3
4
二、Classification of Planar Mechanism
Condition of Assur group:
F＝3n－2Pl－Ph＝0
In a lower-pair Assur group
F＝3n－2Pl＝0
3n ＝ 2Pl
n
Pl
2
3
4
6
6
9
Group type
gradeⅡ
grade Ⅲ
grade IV
Type of gradeⅡ
B
B
1
2
1
C
A
2
A
B
B
2
1
C
A
2
2
1
A
B
C
C
1
A
C
Type of grade Ⅲ
A
1
B
C
2
3
D
E
4
F
A
1
B
C
E
E
2
4
3
1
F
D
A
C 2
3
B
D
4
F
The grade of a mechanism is defined as the highest
grade of the Assur group in the mechanism. The basic
mechanism is sometimes called the grade I mechanism, e.
g., a ceiling fan (consisting of only a single rotating link)
is a grade I mechanism.
三、 Structural Analysis of the Planar Mechanism
Structural Analysis——The purpose of structural analysis is to
disconnect the Assur groups from the mechanism and to determine
their types and assembly order.
Example 2-5 Carry out the structural analysis for the mechanism in Fig.
1）Link 1 is driver.
6
1
2
5
3
4
2）Link 5 is driver.
GradeⅡ mechanism
6
1
2
5
3
4
Grade Ⅲ mechanism