A matrix-based method for detection of degenerate structures in planetary gear trains

Abstract

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.