Analysis Of Mobile Network Of Spherical 4R Linkages And Kinematics Of Rigid Origami

In this thesis,we explore the possibilities of constructing mobile networks of spherical 4R linkages,present the analysis of rigid origami patterns,and propose a kinematic synthesis for rigid origami of thick panels.This thesis is to analyse the kinematic properties of spherical 4R linkage firstly.According to the symmetrical characters,we build three types of mobile assemblies of four identical spherical 4R linkages,i.e.,the rotational symmetric type,the plane symmetric type and the two-fold symmetric type.Combined with the kinematic of spherical 4R linkage,the compatible conditions of these mobile assemblies are proposed.As the geometric parameters of the linkage are changed,the input-output relationships between the kinematic variables changes accordingly,we choose sixteen special alternative relationships to modify the compatible conditions of the assemblies.According to the new compatible conditions,the mobile assemblies of four different spherical 4R linkages are derived while the kinematic compatibility is always kept.Furthermore,rigid origami is a subset of origami and there is no exception for rigid origami where the sheet can neither be bent nor stretched except rotation about creases.With the paper treated as links and the creases as joints,thus the rigid origami pattern is a kind of network of spherical linkages.In order to get new rigid origami patterns by referring to mobile assemblies of spherical linkages,the further geometric condition should be added to make sure that the paper facets are flat.The tessellation of these assemblies gives larger scale origami patterns.This thesis not only provides the solutions for the mobile assemblies of spherical 4R linkages,but also shows the feasibility to design rigid origami patterns by studying the kinematic compatibility condition of spherical 4R linkage assemblies.When projecting the mobile assembly of four spherical 4R linkages on the flat plane to get rigid origami patterns,there are two possibilities for the folding creases,the mountain fold and the valley fold.Besides geometric design parameters,the mountainvalley fold assignments also affect the rigidity of flat foldable origami patterns.This thesis proposes a kinematic method to analyze rigidity and explores different rigid tessellations of the double corrugated patterns.By stacking a number of those tessellation patterns layer by layer,as a result,some types of 3D metamaterial are generated.When the single unit in the metamaterial folds and extends following the rigid motion,there will be a large deformation on the metamaterial.And due to the kinematic property of the single unit,the whole metamaterial exhibits negative Poisson’s ratios in two directions.And the kinematics of the pattern’s folding dominates the metamaterial’s structural mechanics.Metamaterials with negative Poisson’s ratios are invented whose deformation during the folding can be greatly changed by different mountain-valley assignments.The square-twist pattern and its metamaterials are also discussed to show the generalization of this method.Origami patterns are commonly created for a zero-thickness sheet.To apply them for real engineering applications where thickness cannot be disregarded,various methods were suggested,almost all of which involve tampering with idealised fold lines and their surrounds whereas the fundamental kinematic model where folding is treated as spherical linkages remains unchanged.This thesis establishes a novel and comprehensive kinematic synthesis for rigid origami of thick panels that is capable of reproducing motions kinematically equivalent to that of zero-thickness origami.The approach,proven to be effective for single and multiple vertex origami consisting of four,five and six creases,can be readily applied to engineering practices involving folding of thick panels.