study on graphs of kinematic chain
DESCRIPTION
Theory of MachineTRANSCRIPT
STUDY ON GRAPHS OF KINEMATIC CHAIN
MD AAMIR SALAM 09mes 28
MD JAUHAR ALI 10mes 29
MD NASIM SEHAR 10mes 30
SHAMSHUN NAJAF 10mes 57
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Under the guidance of
Dr. ALI HASAN
INTRODUCTION
• Topological structures of kinematic chains can be represented by graphs.
• Graphs of kinematic chains can be enumerated systematically by using graph theory.
• There are a number of graphs but all of them are not suitable for construction of kinematic chains. Only graphs satisfying the structural characteristics are said to be feasible solutions.
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BASICS
• Link: Every part of the machine or mechanism which is having some relative motion with respect to some other part is called a link.
•Kinematic pair: The connection between the two link is always a joint or pair. A pair is called a kinematic pair if relative motion between the links is constrained motion.
•Kinematic Chain: If all the links are connected in such a way, such that first link connected to the last link and the relative motion between them is constrained then such chain is known as kinematic chain.
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DEFINITIONS:•GRAPH :A graph consists of a set of vertices (points) together with a set of edges or lines.The set of vertices is connected by the set of edges.
•EDGE: Edge of a graph connects two vertices called the end points.
• An edge is said to be incident with a vertex, if the vertex is an end point of that edge. The two end points of an edge are said to be adjacent.
• Two edges are adjacent if they are incident to a common vertex. For the (11, 10) graph shown in Figure below , e23 is incident at vertices 2 and 3. Edges e12, e23, and e25 are adjacent.
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•DEGREE OF VERTEX: The degree of a vertex is defined as the number of edges incident with that vertex. A vertex of zero degree is called an isolated vertex. We call a vertex of degree two a binary vertex, a vertex of degree three a ternary vertex, and so on.
•WALK : A sequence of alternating vertices and edges, beginning and ending with a vertex, is call a walk. A walk is called a trail if all the edges are distinct and a path if all the vertices.
•CIRCUIT: If each vertex appears once, except that the beginning and ending vertices are the same, the path forms a circuit or cycle. For the graph shown the sequence (2, e23, 3, e34, 4, e45, 5) is a path, whereas the sequence (2, e23, 3, e34, 4, e45, 5, e52, 2) is a circuit.
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CONNECTED GRAPH: Two vertices are said to be connected, if there exists a path from one vertex to the other. A graph G is said to be connected if every vertex in G is connected to every other vertex by at least one path.
•SUBGRAPH: A subgraph of G is a graph having all the vertices and edges contained in G. In other words, a subgraph of G is a graph obtained by removing a number of edges and/or vertices from G.
•COMPONENT: A graph G may contain several pieces, called components, each being a connected subgraph of G.
•PARALLEL EDGE: Two edges are said to be parallel, if the end points of the two edges are identical.
•MULTIGRAPH: . A graph is called a multigraph if it contains parallel edges.
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•SLING: A sling or self-loop is an edge that connects a vertex to itself.
•ISOMOPHISMS: Two graphs which contain the same number of graph vertices connected in the same way are said to be isomorphic. It follows that two isomorphic graphs must have the same number of edges, and the degrees of the corresponding vertices must be equal to one another.
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•PLANAR GRAPH: A graph is said to be embedded in a plane when it is drawn on a plane surface such that all edges are drawn as straight lines and no two edges intersect each other. A graph is planar if it can be embedded in a plane.
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MATRIX REPRESENTATION OF GRAPH: The topological structure of a graph can be conveniently represented in matrix form. The matrix representation makes analytical manipulation of graphs on a digital computer feasible.
•ADJACENCY MATRIX: A vertex-to-vertex adjacency matrix, A, is defined as follows:
ai,j =( 1, if vertex i is adjacent to vertex j,
0 ,otherwise (including i = j) ,
0 1 0 1 1
1 0 1 0 1
A= 0 1 0 1 1
1 0 1 0 0
1 1 1 0 0
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INCIDENCY MATRIX: An incidence matrix, B , is defined as a v × e matrix in which each row corresponds to a vertex and each column corresponds to an edge.
bi,j =( 1 if vertex i is an end vertex of edge j,
0 otherwise )
1 0 0 0 1 0 1
0 1 0 0 1 1 0
B= 0 0 1 1 0 1 0
0 0 0 1 0 0 1
1 1 1 0 0 0 0
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CIRCUIT MATRIX: A circuit matrix C, is defined as an l× e matrix in which each row corresponds to a circuit and each column denotes an edge.
1 1 0 0 1 0 0
C= 0 1 1 0 0 1 0
1 0 1 1 0 0 1
0 0 0 1 1 1 1
ci,j =( 1 if circuit i contains edge j 0 otherwise , )
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PATH MATRIX: A path matrix, T , is defined for storing the information about all paths that emanate from the root and terminate at the remaining vertices of a rooted tree.
T i,j =
( 1 if edge i lies on the path emanating from the root and terminating at vertex j+1
0 otherwise )
1 1 1 1T= 1 0 0 0 0 1 1 0 0 0 1 0
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STRUCTURAL REPRESENTATIONS OF MECHANISMS•The kinematic structure of a mechanism contains the essential information about which link is connected to which other link by what type of joint.
•The following assumptions are made for all methods of representation.
1. For simplicity, all parallel redundant paths in a mechanism be illustrated by a single path.
2. All joints are assumed to be binary. A multiple joint will be substituted by a set of equivalent binary joints.
3. Two mechanical components rigidly connected for the ease of manufacturing or assembling will be considered and
shown as one link.
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FUNCTIONAL SCHEMATIC REPRESENTATION•Functional schematic representation refers to the most familiar cross-sectional drawing of a mechanism.
•Shafts, gears, and other mechanical elements are drawn as such.
•For clarity and simplicity, only those functional elements that are essential to the structural topology of a mechanism are shown.
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STRUCTURAL REPRESENTATION•In a structural representation, each link of a mechanism is denoted by a polygon whose vertices represent the kinematic pairs.
•Specifically, a binary link is represented by a line with two end vertices.
• A ternary link is represented by a cross-hatched triangle with three vertices.
• A quaternary link is represented by a cross-hatched quadrilateral with four vertices,
•Plain vertices shown in denote revolute joints, whereas solid vertices denote gear pairs.
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GRAPH REPRESENTATION•In this, the vertices denote links and edges denote joints of a mechanism.
•To distinguish the differences between various pair connections, the edges can be labelled or coloured.
•The gear pairs in a gear train can be represented by thick edges and the turning pairs by thin edges.
•The graph of a mechanism is defined similarly with only one addition; the vertex denoting the fixed link is labelled accordingly, usually with two small concentric circles.
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B =
Graph representation and incidence matrix of the planetary gear set
PLANETARY GEAR SET AND ITS KINEMATIC REPRESENTATIONS
1 0 0 1 00 0 1 1 10 1 1 0 01 1 0 0 1
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WATT MECHANISM AND ITS KINEMATIC REPRESENTATIONS
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STUDY ON SIX BAR AND EIGHT BAR KINEMATIC CHAIN FOR DEGREE OF FREEDOM, f =1
SIX BAR KINEMATIC CHAIN HAVING FOUR BINARY LINKS AND TWO TERNARY LINKS.
FIGURE (1) FIGURE (2)
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STRUCTURAL INVARIANTS [SACP] AND [MACP] FOR FIG-1
0 1 0 1 1 01 0 1 0 0 1
Aij = 0 1 0 1 0 01 0 1 0 0 01 0 0 0 0 10 1 0 0 0 0 FIGURE 1
GRAPH
ADJACENCY MATRIX OF THE GRAPH
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STRUCTURAL INVARIANTS [SACP] AND [MACP] FOR FIG-2
0 0 0 1 1 1 0 0 1 0 1 1
Aij = 0 1 0 1 0 01 0 1 0 0 01 1 0 0 0 01 1 0 0 0 0
FIGURE 2
GRAPH
ADJACENCY MATRIX OF THE GRAPH
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•The set of structural invariant for kinematic chain Shown in Fig.1, derived from matrix using software MAT LAB are:
[SACP] = 12, [MACP] = 6 and for Fig.2 are
[SACP] = 21, [MACP] = 9
•Our method reports that kinematic chain shown in Fig.1 and Fig.2 are non-isomorphic as the values of structural invariants [SACP] and [MACP] are different.
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EIGHT BAR KINEMATIC CHAIN HAVING FOUR BINARY AND FOUR TERNARY LINKS.
FIGURE 3
FIGURE 4
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STRUCTURAL INVARIANTS [SACP] AND [MACP] FOR FIG-3
0 1 0 1 0 0 0 0
1 0 1 0 1 0 0 0
0 1 0 1 0 0 1 0
Aij = 1 0 1 0 0 0 0 1
0 1 0 0 0 1 0 0
1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1
0 0 0 10 0 1 0 FIGURE 3 GRAPH
ADJACENCY MATRIX OF GRAPH
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STRUCTURAL INVARIANTS [SACP] AND [MACP] FOR FIG-4
0 1 0 0 0 1 1 0
1 0 1 0 0 0 0 1
0 1 0 1 0 0 0 0
Aij= 0 0 1 0 1 0 01
0 0 0 1 0 1 1 0
1 0 0 0 1 0 0 0
1 0 0 0 1 0 0 0
0 1 0 1 0 0 0 0
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•The set of structural invariant for kinematic chain Shown in Fig.3, derived from matrix using software MAT LAB are:
[SACP] = 36, [MACP] = 18 and for Fig.4 are
[SACP] = 52, [MACP] = 25
•Our method reports that kinematic chain shown in Fig.3 and Fig.4 are non-isomorphic as the values of structural invariants [SACP] and [MACP] are different.
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CONCLUSION:
•To help the designer at the conceptual stage of design, method of representation of structures in the form of matrices are presented. This representation minimises the possibility of duplicate characterization for structurally different mechanisms. Furthermore, for convenience of computer programming, the kinematic structure of a kinematic chain is represented by a kinematic graph and the graph is represented in matrix form. The matrix representation is useful for identification and classification of mechanisms.
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REFERENCES• Aas Mohammad, V.P. Agrawal., 1999. Identification and Isomorphism of Kinematic Chains and Mechanisms. 11th ISME Conference 197-202.
• Agrawal V.P., Yadav J.N., Pratap C.R., 1996. Mechanism of Kinematic Chain and the Degree of Structural Similarity based on the concept of Link – path Code. Mech. and Mach. Theory 31(7), 865 – 871.
•Agrawal V.P., Yadav J.N., Pratap C.R., 1996. Mechanism of Kinematic Chain and the Degree of Structural Similarity based on the concept of Link – path Code. Mech. and Mach. Theory 31(7), 865 – 871.
•Yan,H.S., 1998, Creative Design of Mechanical Devices, Springer-Verlag, Singapore.
• .Kong F.G.,Q.li,and W.J,”AN artificial neutral network approach to mechanism kinematic chain is0morphism identification,”Mechanism and Machine Theory,34(2),PP271-283,1999.
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