Mechanism Topology Analysis And Multi-configuration Motion Transformation Characteristics Of Rubik’s Cube Mechanism

The new mechanism principle is the basis of mechanical innovation.The Rubik’s Cube mechanism with variable topology and strong coupling lacks corresponding theoretical guidance,which seriously restricts the innovation and application of such mechanisms.This thesis aims to explore the mechanism and machine theory that have not been investigated in the Rubik’s Cube.The classical 3×3×3 Rubik’s Cube is taken as the object of study,and its topological structures analysis,degree of freedom analysis,mechanism attribute and property judgment,motion space analysis and description,forward transformation and inverse transformation analysis,as well as the study of construction principle and application assumption are carried out.These will establish the systematic mechanism theory of Rubik’s Cube mechanism and lay the foundation for their development and application.The main research contents of this paper are as follows:(1)The variable topology and multi-configuration characteristics of Rubik’s Cube mechanism are investigated.Specifically,the structural composition,the mapping relationship between topological structure and characteristic size,and the law of configurations change are analyzed.The different topological structures of Rubik’s Cube mechanism are represented by graph theory,and a mathematical description method of the adjacency matrix to describe the Cube topology is established,which will be helpful to guide the topological structure analysis and design of multi-configuration mechanisms containing scale information.(2)The kinematic diagrams of Rubik’s Cube element mechanism are drawn,and a hierarchical directed graph is proposed to represent the loop and motion flow direction of the mechanism.Based on the loop sequential closure method and the generalized kinematic pair equivalent method,a new method for over-constraint analysis of Rubik’s Cube mechanism is proposed.The motion and constraint relations in each loop are established by using screw theory.Then,the degrees of freedom of component in different configurations are analyzed by applying the modified G-K formula,and the correctness of the theoretical analysis is verified.(3)The attributes of Rubik’s cube mechanism are judged,and the properties of metamorphic and discontinuously movable,multi-loop coupling and module reconfigurable of Rubik’s Cube mechanism are obtained.Then,the basic concepts are rationally expanded from the perspective of component and kinematic pair.The four motion characteristics of the Rubik’s Cube mechanism are summarized.The feature position and orientation spaces of different sub-pieces are analyzed from the perspective of multiple sets,and the mathematical models of Rubik’s Cube feature geometry and motion are established by several kinds of modern mathematics tools.(4)The forward and inverse problems of Rubik’s Cube states transformation are investigated,and the mathematical model that can describe the transformation law between different states is established.On the basis of adjacency matrix representation of Rubik’s Cube topology,the transformation method and solution method of the forward representation problem of Rubik’s Cube transformation are constructed based on matrix theory,and the existence and solvability of the solution of Rubik’s Cube transformation motion plan are determined and proved based on group theory and a solution is given.(5)The mechanism construction theory based on shape body segmentation method and the structure construction theory based on tenon and mortise connection method are established from the two essential aspects of Rubik’s Cube movement and retention,which will guide the innovation of the new Rubik’s Cube.The relationship between theoretical research and application is established.Based on the function of Rubik’s Cube changing structure,the application of Rubik’s Cube mechanism in mechanical industry and other fields is popularized,and different application design cases are given.The above research results of Rubik’s Cube mechanism will enrich the existing mechanism theory,An in-depth understanding of the mechanical structure and performance characteristics of the Rubik’s Cube itself will promote the invention of the new products and the application of the mechanism in mechanical engineering.