Applying Rigidity Theory Methods for Topological Decomposition and Synthesis of Gear Train Systems

2012 ◽  
Author(s):  
Maria Terushkin ◽  
Offer Shai
Keyword(s):  
Synthesis Method ◽  
Building Blocks ◽  
The Body ◽  
Computational Method ◽  
Gear Train ◽  
Graph Representation ◽  
Rigidity Theory ◽  
Graph Representations ◽  
Machine Theory ◽  
Gear Trains

This paper introduces a novel way to augment the knowledge and methods of rigidity theory to the topological decomposition and synthesis of gear train systems. A graph of gear trains, widely reported in the literature of machine theory, is treated as a graph representation from rigidity theory—the Body-Bar graph. Once we have this Body-Bar graph, methods and theorems from rigidity theory can be employed for analysis and synthesis. In this paper we employ the pebble-game algorithm, a computational method which allows determination of the topological mobility of mechanisms and the decomposition of gear trains into basic building blocks—Body-Bar Assur Graphs. Once we gain the ability to decompose any gear train into standalone components (Body-Bar Assur Graphs), this paper suggests inverting the process and applying the same method for synthesis. Relying on rigidity theory operations (Body-Bar extension, in this case), it is possible to construct all of the Body-Bar Assur Graphs, meaning the building blocks of gear trains. Once we have these building blocks at hand, it is possible to recombine them in various ways, providing us with a topological synthesis method for constructing gear trains. This paper also introduces a transformation between the Body-Bar graph and other graph representations used in mechanisms, thus leaving room for the application of the proposed synthesis and decomposition method directly to known graph representations already used in machine theory.

A New Technique Based on Loops to Investigate Displacement Isomorphism in Planetary Gear Trains

2002 ◽  
Vol 124 (4) ◽  
pp. 662-675 ◽  
Author(s):  
V. V. N. R. Prasad Raju Pathapati ◽  
A. C. Rao
Keyword(s):  
Graph Isomorphism ◽  
Planetary Gear ◽  
Degree Of Freedom ◽  
Gear Train ◽  
Graph Representation ◽  
Kinematic Structure ◽  
Graph Representations ◽  
Planetary Gear Trains ◽  
Gear Trains ◽  
A New Technique

The most important step in the structural synthesis of planetary gear trains (PGTs) requires the identification of isomorphism (rotational as well as displacement) between the graphs which represent the kinematic structure of planetary gear train. Previously used methods for identifying graph isomorphism yielded incorrect results. Literature review in this area shows there is inconsistency in results from six link, one degree-of-freedom onwards. The purpose of this paper is to present an efficient methodology through the use of Loop concept and Hamming number concept to detect displacement and rotational isomorphism in PGTs in an unambiguous way. New invariants for rotational graphs and displacement graphs called geared chain hamming strings and geared chain loop hamming strings are developed respectively to identify rotational and displacement isomorphism. This paper also presents a procedure to redraw conventional graph representation that not only clarifies the kinematic structure of a PGT but also averts the problem of pseudo isomorphism. Finally a thorough analysis of existing methods is carried out using the proposed technique and the results in the category of six links one degree-of-freedom are established and an Atlas comprises of graph representations in conventional form as well as in new form is presented.

A Single Universal Construction Rule for the Structural Synthesis of Mechanisms

2016 ◽  
Author(s):  
Elad Hahn ◽  
Offer Shai
Keyword(s):  
Synthesis Method ◽  
Building Blocks ◽  
Construction Rule ◽  
Synthesis Methods ◽  
Rigidity Theory ◽  
Structural Synthesis ◽  
Graph Representations ◽  
Synthesis Of Mechanisms ◽  
Different Types ◽  
Universal Construction

In the field of structural synthesis of mechanisms several synthesis methods have been developed using different approaches. One of the more interesting approaches was that of bottom-up construction via the combination of modular structural groups, known as Assur groups. This approach is combined with new graph representations of mechanisms taken from rigidity theory, capable of representing all the different types of planar and spatial mechanisms. With the strong mathematical base of rigidity theory, a new synthesis method is proposed based on Assur groups, which are reformulated in terms of graph theory and renamed Assur Graphs. Using a single universal construction rule, Assur Graphs of different types and of any number of links are constructed, creating a complete set of building blocks for the synthesis of feasible mechanisms. As its name implies, the single universal construction is applicable for mechanisms of all types of joints and links, for planar or spatial motion.

A Unified Concept for the Graph Representation of Constraints in Mechanisms

2014 ◽  
Author(s):  
Andreas Müller ◽  
Offer Shai
Keyword(s):  
Rigid Bodies ◽  
The Body ◽  
Graph Representation ◽  
The Novel ◽  
Topological Graphs ◽  
Mixed Graph ◽  
Mixed Constraint ◽  
Constraint Graph ◽  
Special Cases ◽  
Machine Theory

There are two established approaches to represent constraints: the body-bar (BB) and the bar-joint (BJ) graph that can be used in machine theory. They are referred to as topological graphs as they describe the relation between members of a mechanism. It is known, however, that in many cases these graphs are not unique. Hence any method for kinematic analysis or mobility determination that is based on these topological graphs is prone to failures. In this paper a generalized and unified concept for the representation of constraints in mechanisms is introduced. It is first shown in which situations BB and BJ representations fail to correctly represent the mechanism. The novel constraint graph is then derived starting from the most general model of constrained rigid bodies. It is shown how BB and BJ graphs result as special cases. Therefore the new graph representation is called the ‘mixed graph’. It is further shown how this novel mixed constraint graph allows for computation of the correct generic (topological) mobility, and thus overcomes the problems of BB and BJ representations.

THE SYNTHESIS OF MECHANISM SYSTEMS USING A MECHANISM CONCEPT LIBRARY

2010 ◽  
Vol 34 (1) ◽  
pp. 151-163 ◽  
Author(s):  
Feng-Ming Ou ◽  
Hong-Sen Yan ◽  
Ming-Feng Tang
Keyword(s):  
Synthesis Method ◽  
Building Blocks ◽  
Graph Representation ◽  
Synthesis Approach ◽  
Synthesis Of Mechanism

This paper presents an approach for synthesizing all possible mechanism systems of kinematic building blocks in a mechanism concept library. The kinematic building blocks are defined as SISO primitive mechanisms, and their serial and/or parallel combinations are expressed as corresponding out-trees based on graph representation. By representing the constructive building blocks as labeled vertices and their possible combination relationships as directed edges, the synthesis approach is developed by adopting graph enumeration theorem. An illustrative example of four kinematic building blocks, including two crank-rocker linkages and two slider-crank mechanisms, is provided to validate the presented approach. The result shows that all feasible mechanism systems can be obtained effectively by following the synthesis method and which provides more alternatives in the library during design or re-design of mechanisms.

Extension of Graph Theory to the Duality Between Static Systems and Mechanisms

2005 ◽  
Vol 128 (1) ◽  
pp. 179-191 ◽  
Author(s):  
Offer Shai ◽  
Gordon R. Pennock
Keyword(s):  
Flow Line ◽  
Line Graph ◽  
Planetary Gear ◽  
Gear Train ◽  
Graph Representation ◽  
Static System ◽  
Graph Representations ◽  
Dual Transformation ◽  
Serial Robot ◽  
The Stability

This paper is a study of the duality between the statics of a variety of structures and the kinematics of mechanisms. To provide insight into this duality, two new graph representations are introduced; namely, the flow line graph representation and the potential line graph representation. The paper also discusses the duality that exists between these two representations. Then the duality between a static pillar system and a planar linkage is investigated by using the flow line graph representation for the pillar system and the potential line graph representation for the linkage. A compound planetary gear train is shown to be dual to the special case of a statically determinate beam and the duality between a serial robot and a platform-type robot, such as the Stewart platform, is explained. To show that the approach presented here can also be applied to more general robotic manipulators, the paper includes a two-platform robot and the dual spatial linkage. The dual transformation is then used to check the stability of a static system and the stationary, or locked, positions of a linkage. The paper shows that two novel platform systems, comprised of concentric spherical platforms inter-connected by rigid rods, are dual to a spherical six-bar linkage. The dual transformation, as presented in this paper, does not require the formulation and solution of the governing equations of the system under investigation. This is an original contribution to the literature and provides an alternative technique to the synthesis of structures and mechanisms. To simplify the design process, the synthesis problem can be transformed from the given system to the dual system in a straightforward manner.

A New Graph Representation for the Automatic Kinematic Analysis of Planetary Spur-Gear Trains

1992 ◽  
Vol 114 (1) ◽  
pp. 196-200 ◽  
Author(s):  
Cheng-Ho Hsu ◽  
Kin-Tak Lam
Keyword(s):  
Computer Program ◽  
Kinematic Analysis ◽  
Degrees Of Freedom ◽  
Spur Gear ◽  
Gear Train ◽  
Graph Representation ◽  
Kinematic Structure ◽  
Rotational Displacement ◽  
Interactive Computer Program ◽  
Gear Trains

The purpose of this paper is to propose a new graph representation to represent the kinematic structure of a planetary spur-gear train efficiently. Based on the graph representation, the kinematic analysis of planetary spur-gear trains is largely simplified. An interactive computer program is developed for the kinematic analysis of planetary spur-gear trains with any number of degrees of freedom. By only inputting the graph representation of a planetary spur-gear train and the data for the mating gear pairs, all possible fundamental circuits are determined and the rotational displacement equations are derived and solved automatically.

Automatic Detection of Embedded Structure in Planetary Gear Trains

1997 ◽  
Vol 119 (2) ◽  
pp. 315-318 ◽  
Author(s):  
Cheng-Ho Hsu ◽  
Yi-Chang Wu
Keyword(s):  
Computer Program ◽  
Automatic Detection ◽  
Planetary Gear ◽  
Gear Train ◽  
Graph Representation ◽  
Structural Synthesis ◽  
Planetary Gear Train ◽  
Planetary Gear Trains ◽  
Gear Trains ◽  
Embedded Structure

The detection of embedded structure is one of important steps in the structural synthesis of planetary gear trains. The purpose of this paper is to develop a computer program for the automatic detection of embedded structure in planetary gear trains. First, the graph representation of a planetary gear train is used to clarify the kinematic structure. Next, the concept of fundamental circuit is applied to derive an algorithm for the detection of embedded structure in a planetary gear train. Using the notation of adjacency matrix, an interactive computer program has been developed such that embedded structure in a planetary gear train can be automatically analyzed by only entering the corresponding graph.

The Detection of Embedded Structure in Planetary Gear Trains

1996 ◽  
Author(s):  
Cheng-Ho Hsu ◽  
Jin-Juh Hsu ◽  
Yi-Chang Wu
Keyword(s):  
Computer Program ◽  
Planetary Gear ◽  
Gear Train ◽  
Graph Representation ◽  
Structural Synthesis ◽  
Planetary Gear Train ◽  
Interactive Computer Program ◽  
Planetary Gear Trains ◽  
Gear Trains ◽  
Embedded Structure

Abstract The detection of embedded structure is one of important steps in the structural synthesis of planetary gear trains. The purpose of this paper is to develop a computer program for the automatic detection of embedded structure in planetary gear trains. First, the graph representation of planetary gear trains are used to clarify the kinematic structure. Next, a method which is based on the concept of fundamental circuits for the detection of embedded structure in a planetary gear train. Using the notation of adjacency matrix, an interactive computer program has been developed such that embedded structure in a planetary gear train can be automatically analyzed by only entering the corresponding graph.

A New Graph Representation for the Automatic Kinematic Analysis of Planetary Spur-Gear Trains

1989 ◽  
Author(s):  
C.-H. Hsu ◽  
K. T. Lam
Keyword(s):  
Computer Program ◽  
Kinematic Analysis ◽  
Degrees Of Freedom ◽  
Spur Gear ◽  
Gear Train ◽  
Graph Representation ◽  
Kinematic Structure ◽  
Rotational Displacement ◽  
Interactive Computer Program ◽  
Gear Trains

Abstract The purpose of this paper is to propose a new graph representation to represent the kinematic structure of a planetary spur-gear train efficiently. Based on the graph representation, the kinematic analysis of planetary spur-gear trains is largely simplified. An interactive computer program is developed for the kinematic analysis of planetary spur-gear trains with any numbers of degrees of freedom. By only inputting the graph representation of a planetary spur-gear trains and the data of the mating gear pairs, all possible fundamental circuits are determined and the rotational displacement equations are derived and solved automatically.

Epicyclic gear trains for automotive automatic transmissions

2000 ◽  
Vol 214 (5) ◽  
pp. 523-532 ◽  
Author(s):  
C-H Hsu ◽  
J-J Hsu
Keyword(s):  
Adjacency Matrix ◽  
Gear Train ◽  
Graph Representation ◽  
Structural Synthesis ◽  
Automatic Transmissions ◽  
Epicyclic Gear ◽  
Systematic Methodology ◽  
Gear Trains ◽  
Configuration Synthesis ◽  
Admissible Graph

Epicyclic gear trains, which contain at least four coaxial links, are widely used in automatic transmissions. This paper presents a systematic methodology for the synthesis of epicyclic gear trains for automotive automatic transmissions. The graph representation of epicyclic gear trains is applied to develop an approach for the structural synthesis of the graphs of admissible epicyclic gear trains. Next, for each admissible graph the adjacency matrix of an epicyclic gear train is defined to develop an algorithm for the configuration synthesis of epicyclic gear trains. Using the proposed algorithm, all possible epicyclic gear trains for automotive automatic transmissions are systematically synthesized. In this paper, admissible epicyclic gear trains with up to ten links have been successfully synthesized. The results of work will lead to the creation of new multispeed automotive automatic transmissions.

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