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.

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