A computational framework for the form-finding and design of tensegrity structures

A new computational framework is proposed for the form-finding and design of tensegrity structures with or without super-stability. The form-finding of tensegrities is formulated as two unconstrained minimisation problems where their objective functions are defined based on eigenvalues of a modified force density matrix. The Nelder–Mead simplex method is then used to solve the minimisation problems. Furthermore, another efficient method is suggested for the interactive form-finding and design of tensegrities with geometrical and force constraints. Examples of the form-finding of tensegrities are presented and the results obtained are compared and contrasted with those analytical results documented in the literature, to verify the accuracy and efficiency of the developed methods.     

Tensegrity applied to modelling the motion of viruses

A considerable number of viruses’structures have been discovered and more are expected to be identified.Different viruses’symmetries can be observed at the nanoscale level.The mechanical models of some viruses realised by scientists are described in this paper,none of which has taken into consideration the internal deformation of subsystems. The authors’models for some viruses’elements are introduced,with rigid and flexible links,which reproduce the movements of viruses including internal deformations of the subunits.     

Wave analysis and control of double cascade-connected damped mass-spring systems

This paper presents wave analysis and control for double cascade-connected damped mass-spring systems, whose mass is connected beyond the adjacent masses. The system is motivated by a cantilevered tensegrity beam supporting tensile and compressive forces. The wave solution is derived from a recurrent formula, and the properties of the propagation constants are precisely investigated. Elimination of reflected waves provides the impedance matching controller. We show that the impedance matching controller can be constructed from a similarity transformation of the characteristic impedance matrix by a matrix composed of the propagation constants. A numerical example of vibration control of a tensegrity beam is shown.     

On Characterizations of Rigid Graphs in the Plane Using Spanning Trees

We study characterizations of generic rigid graphs and generic circuits in the plane using only few decompositions into spanning trees. Generic rigid graphs in the plane can be characterized by spanning tree decompositions [5,6]. A graph G with n vertices and 2n − 3 edges is generic rigid in the plane if and only if doubling any edge results in a graph which is the union of two spanning trees. This requires 2n − 3 decompositions into spanning trees. We show that n − 2 decompositions suffice: only edges of G − T can be doubled where T is a spanning tree of G. A recent result on tensegrity frameworks by Recski [7] implies a characterization of generic circuits in the plane. A graph G with n vertices and 2n − 2 edges is a generic circuit in the plane if and only if replacing any edge of G by any (possibly new) edge results in a graph which is the union of two spanning trees. This requires decompositions into spanning trees. We show that 2n − 2 decompositions suffice. Let be any circular order of edges of G (i.e. ). The graph G is a generic circuit in the plane if and only if is the union of two spanning trees for any . Furthermore, we show that only n decompositions into spanning trees suffice.     

A numerical algorithm for tensegrity dynamics with non-minimal coordinates

This paper provides a numerical correction algorithm for implementation of the dynamics of tensegrity systems described by non-minimal coordinates. This correction algorithm corrects any numerical error that would violate the fixed-length bar constraints. A recursive form of the correction algorithm is proposed, and simulation results support the validity of the proposed scheme.     

Programmable 3D Hexagonal Geometry of DNA Tensegrity Triangles

Non-canonical interactions in DNA remain under-explored in DNA nanotechnology. Recently, many structures with non-canonical motifs have been discovered, notably a hexagonal arrangement of typically rhombohedral DNA tensegrity triangles that forms through non-canonical sticky end interactions. Here, we find a series of mechanisms to program a hexagonal arrangement using: the sticky end sequence; triangle edge torsional stress; and crystallization condition. We showcase cross-talking between Watson–Crick and non-canonical sticky ends in which the ratio between the two dictates segregation by crystal forms or combination into composite crystals. Finally, we develop a method for reconfiguring the long-range geometry of formed crystals from rhombohedral to hexagonal and vice versa. These data demonstrate fine control over non-canonical motifs and their topological self-assembly. This will vastly increase the programmability, functionality, and versatility of rationally designed DNA constructs.     

Geometric and material nonlinear analysis of tensegrity structures

A numerical method is presented for the large deflection in elastic analysis of tensegrity structures including both geometric and material nonlinearities.The geometric nonlinearity is considered based on both total Lagrangian and updated Lagrangian formulations,while the material nonlinearity is treated through elastoplastic stress-strain relationship.The nonlinear equilibrium equations are solved using an incremental-iterative scheme in conjunction with the modified Newton-Raphson method.A computer program is developed to predict the mechanical responses of tensegrity systems under tensile,compressive and flexural loadings.Numerical results obtained are compared with those reported in the literature to demonstrate the accuracy and efficiency of the proposed program.The flexural behavior of the double layer quadruplex tensegrity grid is sufficiently good for lightweight large-span structural applications.On the other hand,its bending strength capacity is not sensitive to the self-stress level.     

Tensegrity deployment using infinitesimal mechanisms

Kinematic properties of tensegrity structures reveal that an ideal way of motion is by using their infinitesimal mechanisms. For example in motions along infinitesimal mechanisms there is no energy loss due to linearly kinetic tendon damping. Consequently, a deployment strategy which exploits these mechanisms and uses the structure’s nonlinear equations of motion is developed. Desired paths that are tangent to the directions determined by infinitesimal mechanisms are constructed and robust nonlinear feedback control is used for accurate tracking of these paths. Examples demonstrate the feasibility of this approach and further analysis reveals connections between the power and energy dissipated via damping, infinitesimal mechanisms, speed of the motion, and deployment time.     

A novel method of determining the sole configuration of tensegrity structures

A novel analysis method is presented for form-finding of tensegrity structures. The spectral decomposition of the force density matrix and the singular value decomposition of the equilibrium matrix are performed iteratively to find the feasible sets of nodal coordinates and force densities. An algorithm of determining the sole configuration of free-form tensegrities is provided by specifying an independent set of nodal coordinates, which indicates the geometrical and mechanical properties of the structures can be at least partly controlled by the proposed method. Several illustrative examples are presented to demonstrate the efficiency and robustness in finding self-equilibrium configurations of tensegrity structures.     

An operative algebraic formulation for the unilaterally-constrained mechanical problem of smart tensegrities

Kinematics and statics of tensegrities are addressed by means of a novel algebraic formulation. The inequality constraints, associated to cable-type unilateral structural members, are explicitly enforced in the equilibrium and compatibility problems. Fundamental tensegrity properties (rigidity, pre-stressability, and stability) are focused by a novel structural perspective and algebraic criteria for their assessment are established. Some classical results are generalized to the case of tensegrity models involving both deformable and non-deformable structural members. An operative algorithm for the analysis of the large-displacement elastic tensegrity response is proposed, not limited by special requirements in terms of structural symmetries or member connectivity, and therefore resulting a general design tool. Exemplary applications highlight the effectiveness of the proposed approach for designing tensegrity structures endowed with smart global behavior related to the optimal tuning of structural stiffness.