第二类抗干涉齿轮集机构复合齿轮的最小层数

Minimum composite gear levels of typeⅡ Counter-Meshing Gears mechanism

抗干涉齿轮集(CMG)机构是一种精巧的密码鉴别机构,可用于确保要害系统保证性的机械组合锁中。CMG机构的密码鉴别功能取决于两个配对的、多层密码齿轮固定堆叠而成的编码复合齿轮。复合齿轮层数的最小化有利于工程应用优化;对复合齿轮最小层数问题的认识,也是基于CMG机构的机械组合锁的关键科学问题之一。此前已证明第一类CMG机构复合齿轮的最小层数为3;对于有一个独特的指纹特征的第二类CMG机构,其最小层数的问题更为复杂。由于任意一个"NA+NB"的解锁符号序列都可以简并为"MAB"(M≤N)的形式且保持编码二维迷宫映射图中关键陷阱格点(CTG)的色数不变,且CTG互斥约束关系随着"MAB"持续增/减任意数量的"3AB"呈现出周期性,因此首先应用二维迷宫映射变换和CTG互斥的"十字叉"判据,将第二类CMG机构"3AB"(或"6AB"),"4AB"和"5AB"三种基本模式中复合齿轮最小层数的证明过程转换为求解无向图G(V,E)顶点着色的色数,得到其色数分别为3,5和6。进一步得到最终结论,第二类CMG机构必须细分为与简并解锁符号序列对应的三种模式"(3n)AB","(3n+1)AB"和"(3n+2)AB"(n是自然数1,2,…),最小齿轮层数分别为3,5和6。这意味着第二类CMG机构复合齿轮的最小层数受控于解锁符号序列结构的周期性。最后还给出了最小层数第二类CMG机构快捷编码的"三模板着色方法"。

Counter-Meshing Gears （CMG） mechanism is an elaborate discrimination mechanism which can be used in me- chanical combination locks for high-consequence system surety applications. Discrimination function of the CMG mechanism is de- fined by its two composite gears with several fixed stacked coded gear levels, which are coupled at each gear level. Level minimiza- tion 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 type I CMG mechanism is 3 has been reported previously. But for the type 1] CMG 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 ＂M AB＂（M≤N） 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 ＂M AB＂ continually gets lengthened or shortened with any amount ＂3AB＂, the 2-D maze mapping transform and the Cross Criterion which defines CTGs pairwise het- ero-level constraints are used firstly to convert the minimum gear levels, proving process into getting the vertex coloring chromat- ic 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 ＂SAB＂, 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）DAB＂, ＂（3n＋1）AB＂ and ＂（3nl＋2）AB＂（rl is the natural number 1, 2, ...）, with minimum gear level of 3, 5 and 6 respec- tively. 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.