Topological Synthesis and Analysis of Geared Linkages Based on Matrix Powers

1989 ◽  
Author(s):  
D. G. Olson ◽  
A. G. Erdman ◽  
D. R. Riley
Keyword(s):  
Adjacency Matrix ◽  
Digital Computer ◽  
Visual Inspection ◽  
Kinematic Chain ◽  
New Method ◽  
Graph Representation ◽  
Kinematic Chains ◽  
Degree Matrix ◽  
Topological Synthesis

Abstract A new method for transforming pin-jointed kinematic chains into geared linkages is introduced. The method utilizes the graph representation in the form of the adjacency matrix and the “degree matrix” [20], and the powers of these matrices. The method involves first determining the feasible locations for assigning gear pairs in a kinematic chain, followed by determining which of the choices are distinct, and finally, determining the distinct possible ways of assigning the ground link for each distinct “geared kinematic chain” so formed. Because the method is based on matrix manipulations and does not rely on visual inspection, it is easily implemented on a digital computer. The method is applied to an example class of geared mechanism, the single-dof geared seven-bar linkages.

Combined Graph Layout Algorithms for Automated Sketching of Kinematic Chains

2012 ◽  
Author(s):  
Martín A. Pucheta ◽  
Nicolás E. Ulrich ◽  
Alberto Cardona
Keyword(s):  
Computational Cost ◽  
Kinematic Chain ◽  
Initial Position ◽  
Graph Representation ◽  
Combinatorial Algorithms ◽  
Graph Layout ◽  
Kinematic Chains ◽  
Aesthetic Characteristics ◽  
Edge Crossings ◽  
Independent Loops

The graph layout problem arises frequently in the conceptual stage of mechanism design, specially in the enumeration process where a large number of topological solutions must be analyzed. Two main objectives of graph layout are the avoidance or minimization of edge crossings and the aesthetics. Edge crossings cannot be always avoided by force-directed algorithms since they reach a minimum of the energy in dependence with the initial position of the vertices, often randomly generated. Combinatorial algorithms based on the properties of the graph representation of the kinematic chain can be used to find an adequate initial position of the vertices with minimal edge crossings. To select an initial layout, the minimal independent loops of the graph can be drawn as circles followed by arcs, in all forms. The computational cost of this algorithm grows as factorial with the number of independent loops. This paper presents a combination of two algorithms: a combinatorial algorithm followed by a force-directed algorithm based on spring repulsion and electrical attraction, including a new concept of vertex-to-edge repulsion to improve aesthetics and minimize crossings. Atlases of graphs of complex kinematic chains are used to validate the results. The layouts obtained have good quality in terms of minimization of edge crossings and maximization of aesthetic characteristics.

scholarly journals Structural synthesis of plane kinematic chain inversions without detecting isomorphism

2021 ◽  
Vol 12 (2) ◽  
pp. 1061-1071
Author(s):  
Jinxi Chen ◽  
Jiejin Ding ◽  
Weiwei Hong ◽  
Rongjiang Cui
Keyword(s):  
Mechanism Design ◽  
Adjacency Matrix ◽  
Synthesis Method ◽  
Kinematic Chain ◽  
Degree Of Freedom ◽  
Synthesis Methods ◽  
Creative Design ◽  
Structural Synthesis ◽  
Kinematic Chains

Abstract. A plane kinematic chain inversion refers to a plane kinematic chain with one link fixed (assigned as the ground link). In the creative design of mechanisms, it is important to select proper ground links. The structural synthesis of plane kinematic chain inversions is helpful for improving the efficiency of mechanism design. However, the existing structural synthesis methods involve isomorphism detection, which is cumbersome. This paper proposes a simple and efficient structural synthesis method for plane kinematic chain inversions without detecting isomorphism. The fifth power of the adjacency matrix is applied to recognize similar vertices, and non-isomorphic kinematic chain inversions are directly derived according to non-similar vertices. This method is used to automatically synthesize 6-link 1-degree-of-freedom (DOF), 8-link 1-DOF, 8-link 3-DOF, 9-link 2-DOF, 9-link 4-DOF, 10-link 1-DOF, 10-link 3-DOF and 10-link 5-DOF plane kinematic chain inversions. All the synthesis results are consistent with those reported in literature. Our method is also suitable for other kinds of kinematic chains.

Identification of kinematic chains and distinct mechanisms using extended adjacency matrix

2007 ◽  
Vol 221 (1) ◽  
pp. 81-88 ◽  
Author(s):  
A Mohammad ◽  
R A Khan ◽  
V P Agrawal
Keyword(s):  
Adjacency Matrix ◽  
Kinematic Chain ◽  
Theoretic Approach ◽  
Kinematic Chains ◽  
Graph Theoretic ◽  
The Past ◽  
Graph Theoretic Approach ◽  
The Family ◽  
Eigen Value ◽  
Eigen Values

Development of the methods for generating distinct mechanisms derived from a given family of kinematic chains has been persued by a number of researchers in the past, as the distinct kinematic structures provide distinct performance characteristics. A new method is proposed to identify the distinct mechanisms derived from a given kinematic chain in this paper. Kinematic chains and their derived mechanisms are represented in the form of an extended adjacency matrix [EA] using the graph theoretic approach. Two structural invariants derived from the eigen spectrum of the [EA] matrix are the sum of absolute eigen values EA∑ and maximum absolute eigen value EAmax. These invariants are used as the composite identification number of a kinematic chain and mechanism and are tested to identify the all-distinct mechanisms derived from the family of 1-F kinematic chains up to 10 links. The identification of distinct kinematic chains and their mechanisms is necessary to select the best possible mechanism for the specified task at the conceptual stage of design.

The Novel Characteristic Representations of Kinematic Chains and Their Applications

2005 ◽  
Author(s):  
Z. Huang ◽  
H. F. Ding ◽  
Y. Cao
Keyword(s):  
Adjacency Matrix ◽  
Degree Sequence ◽  
Object Oriented ◽  
Kinematic Chain ◽  
Object Oriented Programming ◽  
The Novel ◽  
Topological Graphs ◽  
Kinematic Chains ◽  
Set Up ◽  
Oriented Programming Language

In this paper, based on perimeter topological graphs of kinematic chains, many novel topological concepts including the synthetic degree-sequence, the characteristic adjacency matrix and the characteristic representation code of kinematic chain are proposed. Both the characteristic adjacency matrix and the characteristic representation code are unique for any kinematic chain and easy to be set up. Therefore a quite effective isomorphism identification method is presented depending on the characteristic adjacency matrix. It high effectiveness is proved by many examples. With object-oriented programming language, a program which can sketch topological graphs of kinematic chains has been developed based on the characteristic representation code. Finally, an application software system establishing the atlas database of topological graphs is introduced. And some functions about the atlas database are also presented in this paper.

Generation and Sketching of Generalized Kinematic Chains

2008 ◽  
Author(s):  
Chiu-Fan Hsieh ◽  
Yii-Wen Hwang ◽  
Hong-Sen Yan
Keyword(s):  
Computer Program ◽  
Adjacency Matrix ◽  
Kinematic Chain ◽  
Kinematic Chains ◽  
A Chain

An algorithm of generalized kinematic chains and its computer program are developed in this paper. By this program, users can give the number of links and joints and then the link assortments and contracted link assortments can be calculated. The synthesis of multiple link adjacency matrix (MLAM) and the cut-link diagnosis are proposed to produce effectively the generalized kinematic chains. The algorithm can automatically determine the feature of a chain, which is connected, closed, non-isomorphism, without any cut-link (or cut-joint), and with simple joint only. Then, it can be called a generalized kinematic chain. Finally, various given number of links and joints, the nice looking atlas of generalized kinematic chains can also be generated. The developed computer program could help designers to be able to study and compare different devices in a very basic way.

An Algorithm for Automatic Sketching of Planar Kinematic Chains

1985 ◽  
Vol 107 (1) ◽  
pp. 106-111 ◽  
Author(s):  
D. G. Olson ◽  
T. R. Thompson ◽  
D. R. Riley ◽  
A. G. Erdman
Keyword(s):  
Graph Theory ◽  
Line Graph ◽  
Kinematic Chain ◽  
Graph Representation ◽  
Synthesis Process ◽  
Line Graphs ◽  
Type Synthesis ◽  
Kinematic Chains ◽  
Synthesis Of Mechanisms ◽  
Computer Aided

One of the problems encountered in attempting to computerize type synthesis of mechanisms is that of automatically generating a computer graphics display of candidate kinematic chains or mechanisms. This paper presents the development of a computer algorithm for automatic sketching of kinematic chains as part of the computer-aided type synthesis process. Utilizing concepts from graph theory, it can be shown that a sketch of a kinematic chain can be obtained from its graph representation by simply transforming the graph into its line graph, and then sketching the line graph. The fundamentals of graph theory as they relate to the study of mechanisms are reviewed. Some new observations are made relating to graphs and their corresponding line graphs, and a novel procedure for transforming the graph into its line graph is presented. This is the basis of a sketching algorithm which is illustrated by computer-generated examples.

Enumeration of kinematic chains and mechanisms

2008 ◽  
Vol 223 (4) ◽  
pp. 1017-1024 ◽  
Author(s):  
R Simoni ◽  
A P Carboni ◽  
D Martins
Keyword(s):  
Group Theory ◽  
Kinematic Chain ◽  
Equivalence Classes ◽  
New Method ◽  
Complete List ◽  
Planar Mechanisms ◽  
Group Of Automorphisms ◽  
Kinematic Chains ◽  
New Concepts

A problem still unsolved in kinematics is the enumeration of a complete list of kinematic chains and mechanisms without isomorphisms and without degenerate chains that operate in any screw system. In this paper, a method for the enumeration of kinematic chains without isomorphisms and degenerate chains for all screw systems and a new method of enumeration of kinematic chain inversions (i.e. mechanisms) based on group theory techniques are presented. New concepts of the group theory are introduced and a new method of enumeration of inversions is presented. Kinematic inversions are related to the symmetries of the chain which can be identified analysing the corresponding graph. The symmetry of a graph can be identified for the group of automorphisms of the graph and its orbits provides sets of vertices (links) that are in the same equivalence classes, i.e. they have the same properties of symmetry. The main definitions of group theory and examples of application of the new method of enumeration of inversions are presented. New results are obtained and divided in two classes: original results in non-planar screw systems (λ≠3) and results in agreement with a previously published list for planar kinematic chains and planar mechanisms (inversions). Two tables (1 and 3) are provided, which are up-to-date lists of kinematic chains and mechanisms for several screw systems.

A New Isomorphism Index and its Applications to Kinematic Chains and Mechanisms

2000 ◽  
Author(s):  
K. Vijayananda ◽  
T. S. Mruthyunjaya
Keyword(s):  
Digital Computer ◽  
Kinematic Chain ◽  
Kinematic Chains ◽  
Driving Mechanisms ◽  
Visual Comparison ◽  
Novel Method

Abstract In this paper a novel method for the identification of the structure of a kinematic chain has been proposed. By expressing in discrete steps the process that usually goes with the visual comparison of two kinematic chains, a reliable and fast computational test for the detection of isomorphism is developed and successfully implemented on a digital computer. No counterexamples for the test have been encountered among kinematic chains and graphs. Also the new isomorphism index directly detects distinct mechanisms that can be derived from the chain. With slight modifications to the method multiple jointed kinematic chains and the driving mechanisms are also identified.

What Are the Structures of the Octet Rule Obeying All-Carbon Species Cx (2 ≤ x ≤ 7 and Larger x)?

2017 ◽  
pp. 1-45
Author(s):  
Kori D. McDonald ◽  
Evelyn O. Ojo ◽  
Joel F. Liebman
Keyword(s):  
Adjacency Matrix ◽  
Visual Inspection ◽  
Structural Diversity ◽  
New Method ◽  
Eigenvalues And Eigenvectors ◽  
Carbon Species ◽  
Carbon Structures ◽  
Octet Rule

With most carbon structures still unknown and undiscovered, it becomes increasingly important to find a way to discover, characterize, and understand them. This paper discusses the possible structures for all-carbon species in which each carbon obeys the octet rule. The number and structural diversity of such compounds strongly increases with the number of carbons: C2, 1; C3, 1; C4, 3; C5, 6; C6, 15. Only some of the C7 species were drawn -- merely 23 isomers were given. To guarantee structural uniqueness, names and visual inspection appear to be insufficient. Instead, a new method, using the eigenvalues and eigenvectors of the structure's adjacency matrix and modified matrices, was introduced and then employed. With this we hoped to gain a better understanding of what was chemically reasonable and realizable for our produced structures.

Research on Kinematic Chain Isomorphism Identification Method Based on the Standardization Adjacent Matrix

2010 ◽  
Vol 43 ◽  
pp. 514-518 ◽  
Author(s):  
Mao Zhong Ge ◽  
Jian Yun Xiang ◽  
Yong Kang Zhang
Keyword(s):  
Graph Theory ◽  
Kinematic Chain ◽  
New Method ◽  
Kinematic Chains ◽  
Identification Method ◽  
Isomorphism Identification

In order to solve a baffling problem of kinematic chain isomorphism identification, proceeded from the isomorphism’s principles of graph theory, a new method for detecting isomorphism among planar kinematic chains using the standardization adjacent matrix is presented in this paper. The general course of adjacent matrix standardization processing and numbering principle of node are introduced, the implementation of this new method is illustrated with an example, it is showed that this new method can be accurately and effectively performed.

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