Essam Lauibi Esmail

Planetary gear trains (PGTs) with one or more degrees of freedom (DOFs) have numerous uses in PGT-based mechanisms. The majority of the currently available synthesis methods have focused on 1-DOF PGTs, with only a few investigations on multi-DOF PGT synthesis. The method for synthesizing 7-link 3-DOF PGMs is outlined. All possible link assortments are produced, labeled spanning trees are generated, and potential geared graphs are constructed. The guidelines for including geared edges and how to synthesize geared graphs are outlined. Vertex-degree arrays are generated to validate the geared graphs. Isomorphic geared graphs are identified by comparing the isomorphic identification numbers of geared graphs with the same spanning tree. Fractionated geared graphs are identified using the reachability matrix method. The new method has a straightforward algorithm. In contrast to what is reported in the literature, the results of the synthesis of 7-link 3-DOF PGMs show that there are seven non-fractionated mechanisms. MATLAB programs are used to acquire the vertex-degree arrays.

Graphs are an effective tool for planetary gear trains (PGTs) synthesis and for the enumeration of all possible PGTs for transmission systems. In the past fifty years, considerable effort has been devoted to the synthesis of PGTs. To date, however, synthesis results are inconsistent, and accurate synthesis results are difficult to achieve. This paper proposes a systematic approach for synthesizing PGTs depending on spanning trees and parent graphs. Trees suitable for constructing rooted graphs are first identified. The parent graphs are then listed. Finally, geared graphs are discovered by inspecting their parent graphs and spanning trees. To precisely detect spanning trees, a novel method based on two link assortment equations is presented. Transfer vertices and edge levels are detected without the use of any computations. This work develops the vertex matrix of the rooted graph, and its distinctive equation is used to arrange the vertex degree arrays according to the vertex levels and eliminate the arrays that violate the distinctive equations. The precise results of the 5-link geared graphs are confirmed to be 24. The disparity between the recent and previous synthesis results can be attributed to the fact that the findings of the current method, which employs rooted graphs, are more comprehensive than those obtained with graphs lacking multiple joints. A novel algorithm for detecting structural isomorphism is proposed. By comparing the vertex degree listings and gear strings, non-isomorphic geared graphs are obtained. The algorithm is simple and computationally efficient. The graph representation is one-to-one with the vertex degree listing and gear string representation. This allows for the storage of a large number of graphs on a computer for later use

This study aims to develop new structures for planetary gear-cam mechanisms (PGCMs) using a newly modified creative design methodology. To begin, the design requirements and constraints are outlined based on the kinematics of PGCMs already in use. The structural synthesis theory is then used to make 110 PGCMs by adding different joints to spanning trees to create viable specialized chains. Each viable chain is then turned into its matching PGCM. Furthermore, two illustrated examples are provided, and the design's feasibility is validated through computer simulation using LINKAGE software. The new solutions can provide a wider variety of non-uniform motion, which is necessary for driving variable speed input mechanisms. In addition, the results validated the methodology as an effective tool for developing new variations of PGCMs in addition to improving the existing ones.

A locked chain is an immovable sub-chain within a kinematic chain. The detection of immovable sub-chains is a critical step in the structural synthesis of planetary gear trains (PGTs). This paper presents a simplified method for the identification of locked chains in PGTs based on the vertex-edge mobility criterion. The vertex-edge mobility criterion is formulated, and the detection method is proposed. A connectivity matrix is developed for the analytical determination of locked chains in multi-planet gear trains. The algorithm is suitable for planetary gear trains with any degree of freedom (DOF). For the automatic identification of locked chains, a MATLAB program is developed. The only input to the program is the adjacency matrix. The result of this work is beneficial for the automation of the structural synthesis of planetary gear trains.

For a given number of links and degrees of freedom, several methods were proposed to enumerate the entire set of planetary gear trains. Acceptable solutions necessitate the detection of embedded locked sub-chains during the enumeration process. Previously published methods detect degenerate structures via combinational analysis or manual decomposition, with the process becoming more involved as the number of links grows. A matrix-based method for inspecting the presence of locked sub-chains is proposed in this work. The new method is based on the sequential removal of binary strings from rooted-parent graphs. A sub-graph is said to be locked if it becomes a structure with zero degrees of freedom under single or repeated application of binary strings deletion. The new degeneracy detection method is very efficient, comprehensive, and systematic for the automatic detection of locked sub-chains during the structural synthesis of PGTs. The locked sub-chain is uniquely determined regardless of the complexity of the system or its graph representation. Several examples are presented including some counterexamples and unexplored cases. A MATLAB program with the adjacency matrix as the only input is developed for the automatic detection of locked sub-chains.

This paper introduces some basic concepts of graph theory needed to elucidate the structural topology of planetary gear trains. This paper proposes a novel method for the structural decomposition of PGTs with the aim of dividing the structure into several independent substructures with single DOF. To make the analytical manipulation of graphs on a computer possible, various matrix representations are employed. This makes the detection of fundamental geared entities of PGTs by an algebraic method easier. Based on the graph-matrix approach, the definition of FGE is explicitly expanded to include triangular loops and thus expanding the method to include the synthesis of more complex multi-planet PGTs.

In this paper, an algebraic method is introduced to discretize any planetary gear train (PGT) into fundamental circuits, and fundamental geared entities (FGEs). This is accomplished by using the concepts of graph theory and a vertex-to-edge incidence matrix. The proposed algorithm relies on relationships among the circuit matrix elements so as to determine the FGEs. Type-8001 PGT is used to demonstrate the procedure. The approach is also applicable for multi-planet PGTs. The other advantage of this method is to analyze the structure of PGTs in a fully automated way.

In this paper, a matrix notation is presented to decompose the structure of any planetary gear train (PGT) into its constituted fundamental geared entities (FGEs). The representation of the PGTs by matrices motivates the need for a method for discretizing PGTs into FGEs. This paper proposes a novel method for the structural decomposition of PGTs with the aim of dividing the structure into several independent substructures with single DOF. This is accomplished by determining the transfer vertices and second level vertices in matrix form. Although there are a large number of existing analysis methods, there is no method yet that entirely performs the kinematic analysis of multi-planet gear trains automatically in the computer in a simple way. Understanding the topological properties of existing PGTs is very useful for benchmark synthesis to explore all possible designs that are based on existing ones. Also, any type of FGEs with any number of links can be investigated without knowing the exact size of gears.

The mechanical efficiency is a computed value for comparing the performance of the multi degrees-of-freedom geared transmissions of hybrid vehicles. Most of the current methods for estimating gear trains mechanical efficiency require the decomposition of gear transmissions in basic structural elements or planetary gear units (PGU). These are two degrees-of-freedom components whose mechanical efficiency has a deep influence on the overall device. The authors (E.L.E., E.P.) already evidenced that, under certain kinematic conditions, the classic Radzimovsky’s formulas, widely accepted for computing the mechanical efficiency of PGUs, are not adequate. In this paper, more general and reliable formulas for computing the mechanical efficiency are deduced. The proposed formulas herein, exploiting the concept of potential or virtual power, evidence the dependency between kinematics and efficiency. A numerical example compares our results with previous work on the subject.