An Application of Graph Theory for the Detection of Degenerate Structures in Planetary Gear Trains

Abstract

One of the most important steps in the structural synthesis of planetary gear trains is to eliminate degenerate structures. First, the graph theory is used to represent planetary gear trains (PGTs). Second, a procedure is developed to identify fundamental geared entities (FGEs). Further, the single-planet FGEs are shown to have one-DOF and, therefore, cannot constitute a degenerate structure. It is this that allows a significant reduction in the calculation in relation to other methods of diagnosing degenerate structures. Third, using the concepts of FGEs and the notation of the associated adjacency matrix, an algorithm is developed for the detection of degenerate structures in PGTs. The algorithm is based on the fact that any degenerate structure is a PGT formed by two fundamental geared entities with common edges and/or vertices equal to or more than 3. Forth, the concept of connectivity between single-planet FGEs is introduced and a simple, straightforward approach for deducting the connectivity matrix from the adjacency matrix is developed. The new vertex-edge mobility criterion does not require combinatorial analysis. Besides, the method is applicable to one and multiple degrees of freedom PGTs, it is also applicable to multi-planet PGTs and complex PGTs, including contrary examples found in the literature.

Genetically compatible graphs for planetary gear train synthesis

“…In third step, geared graphs are detected from parent graphs and spanning trees. The rooted graph of an N-link, F-DOF PGT comprises (𝑁 + 1) vertices and (2 𝑁 − 𝐹 − 1) edges; hence, the number of independent loops can be obtained from 𝐿 = 𝑒 − 𝑣 + 1 [25][26][27][28]. The link assortment arrays of a spanning tree and a parent graph are expressed as [𝑉 1 , 𝑉 2 , 𝑉 3 , … , 𝑉 𝑚 ] and [𝑉 2 , 𝑉 3 , … , 𝑉 𝑚 ] , respectively.…”

“…Shanmukhasundaram and colleagues [19] examined the recursive, non-recursive, and acyclic graph-based approaches in depth. Graph theory enabled the modeling of PGT kinematic structure as well as the advancement of structure synthesis methodologies [20][21][22][23][24][25][26][27][28]. In this paper, geared graphs for a given number of links and DOF are enumerated using a method that makes use of the correspondence between spanning-tree graphs and parent graphs.…”

Research on Kinematics and Efficiency Calculation of Binary Logic Planetary Gearbox Based on Graph Theory

“…H Xue et al [17] used graph theory to analyze the kinematics, static force and power flow of planetary gear systems. EL Esmail et al [18] used graph theory to represent planetary gear trains (PGT), developed a procedure to identify fundamental leverage entities (FGE) and developed an algorithm to detect the degenerate structure of PGTs using the concept of FGEs and the notation of related adjacency matrices. VR Shanmukhasundaram et al [19] proposed a graph theory-based method for detecting degenerate planetary gear train diagrams in enumeration sets.…”