Hind A. Nafeh - Department of Mechanical Engineering, University of Al-Qadisiyah, Diwaniyah, ۵۸۰۰۱, Iraq‎
Essam L. Esmail - Department of Mechanical Engineering, University of Al-Qadisiyah, Diwaniyah, ۵۸۰۰۱, Iraq‎
Sajad H. Abdali - Department of Mechanical Engineering, University of Al-Qadisiyah, Diwaniyah, ۵۸۰۰۱, Iraq‎

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 ۵-link geared graphs are confirmed to be ۲۴. 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.

displacement graph, link assortment array, rooted parent graph, pseudo-isomorphism graph, structure synthesis, spanning ‎tree‎