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 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.

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.

The effect of planetary gear/star gear on the transmission efficiency of closed differential double helical gear train

2020 ◽  
Vol 234 (21) ◽  
pp. 4215-4223
Author(s):  
Cheng Wang
Keyword(s):  
Power Flow ◽  
Power Transmission ◽  
Flow Direction ◽  
Load Capacity ◽  
Research Work ◽  
Planetary Gear ◽  
Transmission Efficiency ◽  
Helical Gear ◽  
Gear Train ◽  
Graph Representation

The research of transmission efficiency is of great significance for reducing energy consumption and improving the performance of the device. Researchers have done a lot of work on the calculation of transmission efficiency. However, in the present research work, the quantity of planetary gear/star gear is usually not considered and only a planetary gear/star gear is adopted in the gear transmission efficiency. In practice, in order to increase the stiffness and load capacity of gear train, a plurality of planetary gear/star gears is adopted. The closed differential double helical gear train has been widely used in many fields, such as the main reducer of aircraft engine, lifting mechanism, and the power transmission system of marine ships. Therefore, in this paper, the closed differential double helical gear train is taken as the research object and the effect of planetary gear/star gear on the transmission efficiency is analyzed. Firstly, according to the structure of closed differential double helical gear train, related kinematic analysis is given. Secondly, a graph representation is used to characterize the closed differential double helical gear train. According to the theory of virtual power, the power flow direction of closed differential double helical gear train is determined and the value of split power is obtained. According to the input and output values described in graph representation of closed differential double helical gear train, the formula of transmission efficiency is derived and the effects of planetary gear/star gear on transmission efficiency are analyzed. Finally, an illustrative example shows that compared with the theoretical value, the difference considering the effect of planetary gear/star gear on the transmission efficiency of closed differential double helical gear train is two percentage points.

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.

scholarly journals A calculation method of transmission efficiency for RV reducer

2021 ◽  
Vol 9 (4B) ◽  
Author(s):  
Cheng Wang ◽  
Keyword(s):  
Calculation Method ◽  
Industrial Robot ◽  
Planetary Gear ◽  
Transmission Efficiency ◽  
Gear Train ◽  
Graph Representation ◽  
Design Parameters ◽  
Virtual Power ◽  
Power Theory ◽  
Rv Reducer

RV reducer is the core component of industrial robot. It is of great significance to raise the transmission efficiency of RV reducer for improving the transmission performance of industrial robot. RV reducer belongs to the 2K-V type planetary gear train and consists of an equal speed ratio mechanism. Therefore, according to the structural characteristics of RV reducer, the virtual power theory and split power theory are adopted, and a calculation method of transmission efficiency for RV reducer is proposed. Firstly, the structure of a common RV reducer is introduced. Related structural analysis, kinematic analysis, and loaded analysis are given. Secondly, RV reducer is represented by the method of graph theory. By the virtual power theory and split power theory, the power flow direction is determined, and the values of split powers are calculated. Finally, according to the graph representation for RV reducer and the calculation principle of meshing power loss of planetary gear, the power losses of meshing gear pairs are calculated, respectively. According to the input and output values described in graph representation of RV reducer, the formula of transmission efficiency for RV reducer is derived. The calculation of transmission efficiency for RV-40E reducer is considered as an example. The result is compared with previous work on the subject, and the relations between design parameters and transmission efficiency of RV reducer are discussed.

Dynamic Characteristics of Double Helical Planetary Gear Train With Tooth Friction

2015 ◽  
Author(s):  
Fengxia Lu ◽  
Rupeng Zhu ◽  
Haofei Wang ◽  
Heyun Bao ◽  
Miaomiao Li
Keyword(s):  
Degrees Of Freedom ◽  
Sliding Friction ◽  
Planetary Gear ◽  
Gear Train ◽  
Variable Step Size ◽  
Step Size ◽  
Planetary Gear Train ◽  
Gear Mesh ◽  
Meshing Stiffness ◽  
Nonlinear Dynamics Model

A new nonlinear dynamics model of the double helical planetary gear train with 44 degrees of freedom is developed, and the coupling effects of the sliding friction, time-varying meshing stiffness, gear backlashes, axial stagger as well as gear mesh errors, are taken into consideration. The solution of the differential governing equation of motion is solved by variable step-size Runge-Kutta numerical integration method. The influence of tooth friction on the periodic vibration and nonlinear vibration are investigated. The results show that tooth friction makes the system motion become stable by the effects of the periodic attractor under the specific meshing frequency and leads to the frequency delay for the bifurcation behavior and jump phenomenon in the system.

Dynamics of the Gear Train Set in Wind Turbine

2011 ◽  
Vol 697-698 ◽  
pp. 701-705
Author(s):  
D.D. Ji ◽  
Y.M. Song ◽  
J. Zhang
Keyword(s):  
Wind Turbine ◽  
Dynamic Model ◽  
Harmonic Balance Method ◽  
Planetary Gear ◽  
Gear Train ◽  
Dynamic Responses ◽  
Multi Stage ◽  
Lumped Parameter ◽  
Cartesian Coordinate ◽  
The Impact

A lumped-parameter dynamic model for gear train set in wind turbine is proposed to investigate the dynamics of the speed-increasing gear box. The proposed model is developed in a universal Cartesian coordinate, which includes transversal and torsional deflections of each component, time-varying mesh stiffness, gear profile errors and external excitations. By solving the dynamic model, a modal analysis is performed. The results indicate that the modal properties of the multi-stage gear train in wind turbine are similar to those of a single-stage planetary gear set. A harmonic balance method (HBM) is used to obtain the dynamic responses of the gearing system. The responses give insight into the impact of excitations on the vibrations.

Analysis of Models for Torsional Vibration of Shaft System With Planetary Gearing

1997 ◽  
Author(s):  
Jinghui Sun ◽  
Lee Liu ◽  
William N. Patten
Keyword(s):  
Torsional Vibration ◽  
Planetary Gear ◽  
Gear Train ◽  
Mathematical Treatment ◽  
Dynamic Parameters ◽  
Mass Model ◽  
Planar Model ◽  
Shaft System ◽  
Effective Dynamic ◽  
Single Mass

Abstract The kinematics of planetary gearing are complex; thus, making it difficult to build an effective dynamic model. In this paper, a single-mass model of a planetary gear and shaft system is developed to study the torsional vibration of the mechanism. Two new models of the system are proposed: (a) a fictitious co-planar model and (b) an equivalent shaft model. The results from the calculations and analyses using these models indicate that: 1) the single-mass model and the general rotary model are both limited, either mathematically or geometrically; 2) the fictitious co-planar model includes all of the geometric and dynamic parameters of the general rotary model, and it can be connected with the shaft system easily; and 3) using a mathematical treatment, the equivalent shaft model is demonstrated to be the most useful and most effective model for the calculation of torsional vibration of a shaft and planetary gear train.

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