Minimum composite gear levels of type Ⅱ Counter-Meshing Gears mechanism

Counter-Meshing Gears (CMG) mechanism is an elaborate discrimination mechanism which can be used in mechanical combination locks for high-consequence system surety applications. Discrimination function of the CMG mechanism is defined by its two composite gears with several fixed stacked coded gear levels, which are coupled at each gear level. Level minimization of the CMG composite gear assembly is desired for engineering optimization, and the knowledge of which is based on insight into the physics behind the CMG mechanism. A conclusion that the minimum composite gear level of the typeⅠ CMG mechanism is 3 has been reported previously. But for the typeⅡCMG mechanism with a unique finger print feature, its minimum gear level problem is far more complicated. As the arbitrary Unlocking Symbol Sequence in the NA+NB form can be degenerated to the MAB(MN) form whereas maintains the same chromatic number of the Critical Trap Grids(CTGs) in the Coding-oriented 2-D maze map, and those CTGs pairwise hetero-level constraints present periodicity when the MAB continually gets lengthened or shortened with any amount 3AB, the 2-D maze mapping transform and the Cross Criterion which defines CTGs pairwise hetero-level constraints are used firstly to convert the minimum gear levels, proving process into getting the vertex coloring chromatic number of the undirected graph G(V, E) corresponding to the three basic patterns of the type Ⅱ CMG mechanism, i.e. 3AB(or 6AB), 4AB and 5AB, and the resulted chromatic number is 3, 5 and 6 respectively. And we finally conclude that the type Ⅱ CMG mechanism must be subdivided into three patterns corresponding to the degenerated Unlocking Symbol Sequenceas (3n)AB, (3n+1)AB and (3n+2)AB(n is the natural number 1, 2, ), with minimum gear level of 3, 5 and 6 respectively. That also means, as to the type Ⅱ CMG mechanism, the minimum composite gear levels is periodically dominated by the Unlocking Symbol Sequence structure.And the 3-templates coloring method is presented for easy and rapid coding for the type Ⅱ CMG mechanism with the minimum composite gear levels.