The Duality between Planar Kinematics and Statics

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• The Duality between Planar Kinematics and Statics
• Dr. Shai, Tel-Aviv, Israel
• Prof. Pennock, Purdue University

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(No Transcript)
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• The Outline for the tutorial
• A Study of the Duality between Kinematics and
  Statics
• Two new concepts for statics that are derived
  from kinematics
• Equimomental line and face force.
• Transforming theorems and rules between
  kinematics and statics.
• Characterizing and finding dead center positions
  of mechanisms and the stability of determinate
  trusses.
• Correlation between Instant Centers and
  Equimomental Lines.
• Graph theory duality principal and the dual of
  linkages trusses.
• Detailed example of the face force and the
  procedure for deriving the face force.
• Transforming Stewart platforms into serial robots
  and vice versa.
• Checking singularity through the duality
  transformation.
• Applying the duality transformation for
  systematic conceptual design.
• Discussion and suggestion for future research in
  this area

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Kinematics Statics


The absolute equimomental line is the line where
the moment produced by a force is zero.
The absolute instant center is the point in a
link where the linear velocity is zero.
The linear velocities of points located at a
distance r from the absolute instant center are
the same.
The moments produced by a force located a
distance r from the absolute equimomental line
are the same.
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The Linear Velocities Field
The Moment Field
The field of the linear velocities produced by
the angular velocity. The linear velocity is zero
at the absolute instant center.
The field of the moments produced by the force.
The moment is zero along the absolute
equimomental line.
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Relative instant center.
Relative equimomental line.
I12
r1
r2
The relative equimomental line of two forces is
the line where the forces produce the same moment.
The relative instant center of two links is the
point in both links where the angular velocities
produce the same linear velocity.
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Arnold-Kennedy Theorem
Dual Kennedy Theorem
I23
I20
I30
m13
m30
m10
m12
m23
I12
m20
I10
I13
The Dual Kennedy Theorem. For any three forces,
their three relative equimomental lines must
intersect at the same point.
The Arnold Kennedy Theorem. For any three
links, their three relative instant centers must
lie on the same line.
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Correlation between Instant Centers and
Equimomental Lines.
I
II
III
I10
I20
I12
I12
I12
I12
I12
I12
II
mAB
mA0
mAB
mB0
I
mAB
III
mAB
mAB
mAB
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The idea behind the transformation of Kinematic
systems (Linkages) into Static systems
(determinate trusses)

• Each engineering system can be represented into
  mathematical model based on graph theory.
• There are mathematical relations between the
  graph representations such as the graph theory
  duality.
• For example, the representations of linkages and
  trusses were found to be dual. Thus, linkages and
  determinate trusses are dual systems.
• The following slides will show the process of
  constructing dual engineering systems on the
  basis of the graph theory duality principle.

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Constructing the dual graph from the original
graph
Original graph
1
3
3
A
C
1
II
5
2
4
II
O
Dual graph
2
4
I
III
I
5
III
O
B
D
O
6
7
7
6
IV
IV
8
8
Reference face.
Cutset is a set of edges so that if removed from
the graph, the graph becomes disconnected.
A circuit is a closed path.
Edges 3, 4 and 5 form a circle in dual graph.
Edges 1, 2 and 7 form a cutset in dual graph.
Each circuit in the original graph corresponds to
a cutset in the dual graph, and vice-versa. If
two faces are adjacent in the original graph then
their corresponding vertices are adjacent in the
dual graph.
A face in the graph is a circuit without inner
edges.
Face I corresponds to the vertex I.
Face II corresponds to the vertex II.
Face III corresponds to the vertex III.
Face IV corresponds to the vertex IV.
Each face in the original graph corresponds to a
vertex in the dual graph.
Reference face O corresponds to the reference
vertex O.
Two faces are adjacent if they have at least one
edge in common.
Each cutset in the original graph corresponds to
a circuit in the dual graph, and vice versa.
Edges 3, 4 and 5 constitute a cutset in the
original graph.
Edges 1, 2 and 7 constitute a circuit in the
original graph.
Every two adjacent faces correspond to two
adjacent vertices in the dual graph. The edge
common to these two faces corresponds to the edge
that connects the vertices in the dual graph.
Vertices II and O are adjacent.
Vertices O and III are adjacent.
Vertices I and IV are adjacent.
Faces O and I are adjacent.
Vertices O and I are adjacent.
Faces I and II are adjacent.
Vertices I and II are adjacent.
Faces II and O are adjacent.
Faces II and III are adjacent.
Vertices II and III are adjacent.
Faces O and III are adjacent.
Faces III and IV are adjacent.
Faces I and IV are adjacent.
Vertices III and IV are adjacent.
Faces O and IV are adjacent.
Vertices O and IV are adjacent.
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Constructing the Dual of a Linkage
Augmenting the geometry to the graph.
Constructing the dual graph.
The meaning of a directed edge in the dual
graph eltt,hgt is the flow (force) acting upon
the head vertex (joint) by the edge (rod). The
force in rod 4 acts upon the ground in this
orientation.
Constructing its topology.
Kinematic system.
O
O
Constructing the corresponding graph
B
3
3
2
4
A
B
(CCW)
The topology arrow and the force arrow are in the
same direction -gt compression.Inverse directions
- tension The type is compression.
A
(CCW)
3
(CW)
2
4
2
4
I
I
Two choices?
The direction of the force in rod 4.
The external force acts upon joint I
The force in rod 3 acts upon the ground in this
orientation.
4
(CW)
2
Vertex I corresponds to joint I.
The direction of the force in rod 3.
(CW)
(CW)
O2
O4
O
The type is tension.
O2
O4
O3
1
Adding the geometry.
1
1
Link 2 is the driving link.
Vertex O2 corresponds to joint O2.
Vertex A corresponds to joint A.
Edge 2 is the potential source that corresponds
to the driving link 2.
Edge 3 corresponds to link 3.
Faces I and O are adjacent.
Edge 4 is common to the two adjacent faces I and
O thus the dual edge 4 is between the two
adjacent vertices I and O.
The potential source, edge 2, is between the two
adjacent faces I and O in the original graph.
Therefore, in the dual graph it corresponds to
the flow source and it is between the two
adjacent vertices I and O.
The relative linear velocity corresponds to the
potential difference.
The relative velocity of link 2 corresponds to
the potential difference of edge 2.
The relative velocities of links 3 and 4
correspond to the potential differences of edges
3 and 4.
Edge 3 is common to the two adjacent faces I and
O thus the dual edge 3 is between the two
adjacent vertices I and O.
O4
For consistency, the direction of the edge in the
dual graph is defined by rotating the edge in the
original graph in CCW direction.
Vertices B and O4 correspond to joints B and O4,
respectively.
We can contract the edges with potential
difference equal to zero.
The kinematic analysis yields the magnitudes and
directions of the angular and relative linear
velocities.
Edges 4 and 1 correspond to link 4 and the fixed
link 1, respectively.
Reference face O corresponds to reference vertex
O.
3
4
The angular velocity in CW corresponds to
compressing force.
Potential differences in edges 3 and 4 correspond
to flows in edges 3 and 4.
Potential differences in the original graph
correspond to flows in the dual graph.
Potential differences of edge 2 corresponds to
the flow in edge 2.
The angular velocity is CCW which corresponds to
a tension.
Face I corresponds to vertex I.
Building the corresponding truss.
I
4
O4
(tension)
(compression)
3
(compression)
O3
The dual graph.
The corresponding truss.
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We obtain the dual systems.
Since linear velocity is associated with a joint
in the linkage, its dual variable is associated
with the face in the truss
We have systematically developed a new variable
in statics
Joint
Face

What is a counterpart to absolute linear velocity
of the joint
At first, what is this absolute velocity?

B
O3
3
The relative linear velocity of the input link
corresponds to the external force.
The relative linear velocity of the link 3
corresponds to the force in the rod 3.
The relative linear velocity of the link 4
corresponds to the force in the rod 4.
1.The absolute linear velocity has a property of
potential.
Velocity
Force
A
Why?
On the other hand we know that velocity
corresponds to force.
O4
4
Absolute linear velocity corresponds to face
force.
Velocity of a joint
Face Force
2
3
Because, We can give any absolute velocities to
the links, and they will satisfy the rule of
velocities (vectors KVL).
4
O2
O4
I
1
1
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Force in rod 1 is equal to the subtraction of
face forces FA by FO.
Force in rod 2 is equal to the subtraction of
face forces P by FA.
Force in rod 3 is equal to the difference of face
forces FA and FB.
Same direction corresponds to compression.
Opposite direction corresponds to tension.
In the same manner, locate the forces in the
other rods.
An arbitrary point on equimomental line mPA
An arbitrary point on the equimomental line mPB
Face force FP acts in the face P.
Face force FA acts in the face A.
Face force FB acts in face B.
The moment produced by the forces P and FA upon
the equimomental line mPA.
The moment produced by the forces P and FB on the
equimomental line mPB.
Each force in the rod is the difference of the
right and the left face forces (Right and left
defined according to the direction of the arrow
in the edge).
Absolute equimomental line mBO has to be
determined .
The equimomental lines that will locate mBO.
The circuit corresponds to the vertex.
The circuit corresponds to the vertex.
Set arbitrary directions of the edges.
In the same manner we can find the reaction R.
mPB
Thus we obtain the face force FB.
Thus we obtain the face force FA.
mPO
mBO
P
compression
O
2
mPA
mRO
mPO
A
P
R
mAO
4
tension
1
compression
compression
3
B
mPB
mBR
mPA
O
compression
mBR
5
A
mAB
B
R
mAB
mAO
mRO
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Finding and characterization of the dead center
positions of the mechanism.
O
2
7
1
4
4
5
3
O
2
3
5
7
O
A
C
1
6
6
2,3
5,7
A
C
6
In this case, the faces B and O (i.e., the
reference vertex) of the truss have the same face
force which indicates that
(ii) links 5 and 7 are collinear.
(i) links 2 and 3 are collinear, and
These two conditions ensure that the mechanism is
in a dead center position.
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Another examples to find dead positions of the
mechanism by Face Force.
Given mechanism topology
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Duality relation between stability and mobility
Due to links 1 and 9 being located on the same
line
By means of the duality transformation, checking
the stabiliy of trusses can be replaced by
checking the mobility of the dual linkage.