Research On The Prestress-Stability Of Linkages And Analysis And Design Of Tensegrity Structures

Mechanisms with only infinitesimal mobility are called infinitesimal mechanisms or shaky structures.Due to lack of essential mobility or effective stiffness,these mechanisms lie in the fringe between structure engineering and mechanism theory.According to mechanism theory,these mechanisms possess prestress or preload without applying external load for the existence of redundant constraints.Currently,mechanism theory mainly focus on the kinematics of infinitesimal mechanisms such as mobility,shakiness and singularity etc.and the research on prestress of linkages is rather limited.Structural mechanics shows that some infinitesimal mechanisms can be stiffened by the prestress,which is also called prestress-stability.A matrix method to determine prestress-stability of pin-jointed structures was presented by the scholars from both structural engineering and mathematics.The discovery of prestress-stability of pin-jointed structures laid the foundation of the design and analysis of prestressed mechanisms such as tensegrity structures.Tensegrity structures,which are also called tensegrity mechanisms in certain literatures,compose of cables in tensile and struts in compression.Tensegrity structures can be prestrained and form an integrity though possessing a large number of mobility.In order to reveal the prestress effect of linkages and relationship of analysis and design between linkages and tensegrity,in this thesis,based on the fusion of mechanism theory and structural mechanics,the prestress-stability of pin-jointed structures is generalized to general linkages.The method of prestress-stability analysis of general multi-loop linkages is established with screws and the concept of prestressed linkages is presented for the first time,which is further verified by a prestress experiment of spherical linkages.And planar beam theory is used to design the pure-torque prestress of spherical linkages.With the foregoing research and the dual transformation between pin-jointed structure and linkages,this thesis delves deeper into the research of tensegrity structures.The method of prestress-stability analysis in screws and the designing method based on linkage transformation of triangulated tensegrity is presented.With this,the dual transformation of prestressed spherical linkage and the Grünbaum polygonal tensegrity is revealed and a new tensegrity with multiple prestress varieties is also presented.The content of this thesis is summarized as following.Firstly,the matrix method of prestress-stability of general multi-loop linkages is established.The definition of the order of infinitesimal linkages is given based on the theory of kinematic tangent cone.A quadratic form of potential energy function is constructed to determine the prestress-stability of general multi-loop linkages based on second-order kinematics in screws and matrix representations of topological graph.The definitiveness of the quadratic form indicates prestress can stiffen the mechanism modes,hence prestress-stable,and the corresponding linkages are called prestressed linkages.This matrix condition is also the sufficient condition to determine mechanisms with only first-order mobility.The prestressed linkages are further grouped as prestressed linkages of pure bending,pure twisting,pure longitudinal force and general wrench.Several infinitesimal linkages and prestressed linkages are analyzed,and a 3-UU linkage which is a first-order infinitesimal mechanism but is not prestress-stable is presented.Secondly,in order to further reveal the physics of the given quadratic form,the prestress realization of the pure-torques of spherical-linkages is investigated.The planar-beam theory is then used to design the pre-bending deformation of prestressed spherical linkages.The concepts of curvature patterns and curvature-strain patterns are used to characterize the self-equilibrium states,which is further verified by an experiment.Thirdly,based on truss-linkage transformation and the foregoing matrix method,this thesis presents the method of prestress-stability analysis in screws and the designing method based on linkage-transformation of triangulated tensegrity structures.This method show that triangulated tensegrity structures formed by a set of triangles or tetrahedrons can be transformed into closed-loop linkages,thus,mechanism method can be used to analyze and design triangulated tensegrity structures.The linkage transformation method is exemplified by the prestress-stability analysis of a set of tensegrity prisms and is also compared with the method of structural mechanics.The dual transformation between the spherical prestressed linkage and the classical Grünbaum polygon tensegrity structures is discovered.Fourthly,a novel tensegrity with multiple prestress-varieties is presented through the linkage transformation of tensegrity triplex.Because the axes of the dual linkage of the new tensegrity are perpendicular to that of triplex,this new tensegrity structure is coined as the ortho-triplex.Self-stress analysis shows that there are eight kinds of selfstress distributions for this kind of tensegrity.The phenomenon of existing of different prestress-distributions in the same topology of tensegrity with single self-stress state is called prestress-variety.Finally,a scaled energy function is formulated to distinguish the prestress-stability of different prestress-varieties.The aim of this thesis is to extend the scope of prestressed structures and to provide new prestressed structures and designing methods from the point of linkages in mechanism theory,thus fusing the methods from both mechanism theory and structural mechanics.