A novel method for determining the feasible integral self-stress states for tensegrity structures

The form-finding analysis is a crucial step for determining the stable self-equilibrated states for tensegrity structures, in the absence of external loads. This form-finding problem leads to the evaluation of both the self-stress in the elements and the shape of the tensegrity structure. This paper presents a novel method for determining feasible integral self-stress states for tensegrity structures, that is self-equilibrated states consistent with the unilateral behaviour of the elements, struts in compression and cables in tension, and with the symmetry properties of the structure. In particular, once defined the connectivity between the elements and the nodal coordinates, the feasible self-stress states are determined by suitably investigating the Distributed Static Indeterminacy (DSI). The proposed method allows for obtaining feasible integral self-stress solutions by a unique Singular Value Decomposition (SVD) of the equilibrium matrix, whereas other approaches in the literature require two SVD. Moreover, the proposed approach allows for effectively determining the Force Denstiy matrix, whose properties are strictly related to the super-stability of the tensegrity structures. Three tensegrity structures were studied in order to assess and discuss the efficiency and accuracy of the proposed innovative method.

Analytical Form-Finding for Highly Symmetric and Super-Stable Configurations of Rhombic Truncated Regular Polyhedral Tensegrities

Tensegrities have exhibited great importance and numerous applications in many mechanical, aerospace, and biological systems, for which symmetric configurations are preferred as the tensegrity prototypes. Besides the well-known prismatic tensegrities, another ingenious group of tensegrities with high symmetry is the truncated regular polyhedral (TRP) tensegrities, including Z-based and rhombic types. Although Z-based TRP tensegrities have been widely studied in the form-finding and application issues, rhombic TRP tensegrities have been much less reported due to the lack of explicit solutions that can produce their symmetric configurations. Our former work presented a unified solution for the rhombic TRP tensegrities by involving the force-density method which yet cannot control structural geometric sizes and may produce irregular shapes. Here, using the structural equilibrium matrix-based form-finding method, we establish some analytical equations, in terms of structural geometric parameters and force-densities in elements, to directly construct the self-equilibrated, symmetric configurations of rhombic TRP tensegrities, i.e., tetrahedral, cubic/octahedral, and dodecahedral/icosahedral configurations. Moreover, it is proved, both theoretically and numerically, that all of our obtained rhombic TRP tensegrities are super-stable and thus can be stable for any level of the force-densities without causing element material failure, which is beneficial to their actual construction. This study helps to readily design rhombic tensegrities with high symmetry and develop novel biomechanical models, mechanical metamaterials, and advanced mechanical devices.

Of the matrics of the equilibrium of the rod formation system based on the principle the duality of the problems of structures mechanics

The formation of the equilibrium matrix of the core system in the matrix form is based on the use of the mechanical model of the system obtained by its discretization. The topological structure of the model is set using the graph and the accompanying incidence matrix. The matrix transformation of the vector of nodal displacements in combination with the extended incidence matrix allows determining the absolute elongations and distortions of each finite element. The composition of only two matrices (matrix of incidence and lengths of elements) and the skew vector leads to a geometric matrix characterizing the dependence of concentrated bending deformations in the calculated cross sections of the core system from the nodal displacements for a given load. Based on the duality principle, by transposing the geometric matrix, the equilibrium equation of the core system is derived in matrix form.

Shape Design of Cable-net for Parabolic Cylindrical Deployable Antenna

A method of cable-net shape design based on the equilibrium matrix method is proposed for a new parabolic cylindrical deployable antenna structure with fewer modules. And the inverse iteration method is adopted to find the shape of cable-truss structure with considering the truss deformation induced by cable tension. Firstly, the ideal geometrical configuration of the locally symmetric support cable is designed for the given truss. Then, the pretension distribution of the cable is solved by the equilibrium matrix method under the circumstance of the unchanged topology of cable-net structure, position of nodes and boundary condition. In addition, the inverse iteration method is adopted to find the shape of cable-truss structure. Finally, the validity of the method is verified by simulation analysis.

Thermodynamic Analysis of Ti3O5Nanoparticles Formed in Melt and Their Effects on Ferritic Steel Microstructure

Based on the Wagner’s formalism combined with mass conservation, a thermodynamic analysis method has been developed previously. This method enables the calculation of the equilibrium matrix composition, precipitate composition and precipitate total molar fraction for TixOy(s) in molten metal, which can be determined at any appropriate temperature. In this present study, the Ti3O5 phase precipitation and the quantitative relationship between the addition of Ti, O and Ti3O5 in the molten steel were studied using the thermodynamic model. Using the combined multipoint dispersion supply method, electromagnetic stirring and well-dispersed 5-nm Ti3O5 nanoparticles were fabricated in the ferrite matrix of the as-cast high-strength steel with 0.05 wt % Ti—0.002 wt % O. The as-cast microstructure was improved by the homogeneously dispersed Ti3O5 nanoparticles through heterogeneous nucleation and grain refinement.

Form-finding analysis for a new type of cable–strut tensile structures generated by semi-regular tensegrity

A tensegrity structure is a type of self-balancing tensile structure, which consists of tension cables surrounding compression struts. Based on the geometry and topology of the classic half-octahedron tensegrity, this article presents a form-finding analysis of semi-regular tensegrity units using singular value decomposition of the equilibrium matrix. We propose the design formulas for the unit geometric transformation, obtain its internal self-stress modes and inextensional mechanism modes, and verify its geometric stability. Then, we devise a design method and compute the overall feasible self-stress of a tensegrity torus. A novel cable–strut tensile structural system is generated through combining a tensegrity torus and a Levy-type cable dome. Finally, a physical model is constructed to verify the feasibility of this structural system. This work enriches existing forms of tensegrity structures and contributes to further practical applications of tensegrity systems.