Advanced automatic grouping for form-finding of tensegrity structures

An advanced automatic grouping method for form-finding of tensegrity structures is presented. In the proposed method, properties of self-equilibrium and stability in tensegrity structures can be obtained by using the force density method combined with a genetic algorithm. A constrained minimization problem is formulated using the standard deviation of the force density in the cables. As a result, the minimum number of member groups for tensegrity structures with automatic grouping can be obtained. This elicited regular tensegrity structures with uniform force density values. Moreover, the geometrical and mechanical parameters of tensegrity structures with multiple states of self-stress can be easily obtained by using the proposed method.     

Synthesis of tensegrity structures of desired shape using constrained minimization

There are two types of problems in tensegrity design: (i) form-finding when the tensegrity shape is not specified and (ii) synthesis when the tensegrity shape is specified. We address synthesis problems in this paper. We first formulated and solved an optimization problem to synthesize tensegrity structures of specified shape when the connectivity of the elements (bars and cables) is known a priori. We minimize the error in force-balance at the vertices in the desired equilibrium configuration by using force densities as the design variables. This constrained minimization problem enabled us to synthesize a known asymmetric tensegrity arch and a hitherto unknown tensegrity of biconcave shape similar to that of a healthy human red blood cell. We also extend the above method to a reduced order optimization problem for synthesizing complex symmetric tensegrity structures. Using this approach, we synthesized a truncated dodecahedron inside another truncated dodecahedron to emulate a nucleus inside a cell. We use a restricted global structure on an already available two-step mixed integer linear programming (MILP) topology optimization formulation to synthesize a non-convex tensegrity structure when only the coordinates are provided. We further improve this two-step MILP to a single-step MILP. We also present static analysis of a tensegrity structure by minimizing the potential energy with unilateral constraints on the lengths of the cables that cannot take compressive loads. Furthermore, we use this method to synthesize a tensegrity table of desired height and area under a predefined load. The prototypes of three synthesized tensegrities were made and validated.     

Analysis of target configurations under dead loads for cable-supported bridges

This paper presents a rigorous approach for analyzing the target configurations of cable-supported structures under dead loads by the Newton–Raphson method. A linearized equilibrium equation of a cable element, which includes the nodal coordinates and the unstrained element length as unknowns, is formulated using the analytical solution of an elastic catenary cable. An incremental equilibrium equation for a single cable is formed with the proposed equilibrium matrices of cable elements. The geometry of the target configuration of a cable-supported structure under dead loads is utilized to solve the incremental equilibrium equation. Detailed procedures to analyze the target configurations of suspension bridges and cable-stayed bridges are presented. The efficiency and the accuracy of the proposed method are demonstrated through numerical examples.     

The Galerkin element method applied to the vibration of rectangular damped sandwich plates

The Galerkin element method (GEM), which combines Galerkin orthogonal functions with the traditional finite element formulation, has previously been applied successfully to the vibration analysis of damped sandwich beams, and an improved iteration method was developed for its eigen solution. In the current paper, this promising method is extended to the vibration of damped sandwich plates. A quite different model is formulated which has both nodal coordinates and edge coordinates, while in the case of beams, there are only nodal coordinates. Displacement compatibility over the interfaces between the damping layer and the elastic layers is taken account of in order to ensure a conforming element and thereby guarantee good accuracy. The seed matrix method is proposed for simplifying the building of the element mass, stiffness and damping matrices. Numerical examples show that the application of the GEM to sandwich plate structures is computationally very efficient, while providing accurate estimates of natural frequencies and modal damping over a wide frequency range.     

Design and control of tensegrity robots for locomotion

The static properties of tensegrity structures have been widely appreciated in civil engineering as the basis of extremely lightweight yet strong mechanical structures. However, the dynamic properties and their potential utility in the design of robots have been relatively unexplored. This paper introduces robots based on tensegrity structures, which demonstrate that the dynamics of such structures can be utilized for locomotion. Two tensegrity robots are presented: TR3, based on a triangular tensegrity prism with three struts, and TR4, based on a quadrilateral tensegrity prism with four struts. For each of these robots, simulation models are designed, and automatic design of controllers for forward locomotion are performed in simulation using evolutionary algorithms. The evolved controllers are shown to be able to produce static and dynamic gaits in both robots. A real-world tensegrity robot is then developed based on one of the simulation models as a proof of concept. The results demonstrate that tensegrity structures can provide the basis for lightweight, strong, and fault-tolerant robots with a potential for a variety of locomotor gaits.     

Exploring new tensegrity structures via mixed integer programming

A tensegrity structure is a prestressed pin-jointed structure consisting of continuously connected tensile members (cables) and disjoint compressive members (struts). Many classical tensegrity structures are prestress stable, i.e., they are kinematically indeterminate but stabilized by introducing prestresses. This paper presents a procedure for generating various prestress stable tensegrity structures. This method is based on truss topology optimization and does not require connectivity relation of cables and struts of a tensegrity structure to be known in advance. Unlike the conventional form-finding methods, the locations of nodes are fixed throughout optimization. The optimization problem with the constraints expressing the definition of tensegrity structure, kinematical indeterminacy, and symmetry of configurations is formulated as a mixed integer linear programming (MILP) problem. Numerical experiments demonstrate that various tensegrity structures can be generated from one given initial structure by solving the presented MILP problems by using a few control parameters.     

Accuracy test of sensitivity analysis in the semi-analytic method with respect to configuration variables

Configuration optimization is a structural optimization method where the geometrical shape of the structures can be changed during the optimization process. Sensitivity informations are required in the general optimization and quite costly. Especially, they are extemely expensive in the structural optimization where the finite element analysis is utilized. Since the nodal coordinates are regarded as design variables in the configuration optimization, the sensitivities according to the nodal coordinates must be calculated. The characteristics of the configuration optimization is that the transformation matrix in the finite element analysis is a function of design variables. Thus the sensitivity calculation in the configuration optimization is even more complicated. For the efficient sensitivity calculations, various methods have been proposed. They are the analytic method (AM), overall finite difference method (OFD), and semi-analytic method (SM). The semi-analytic method consists of the forward and central difference approximation. This study has been conducted to choose an appropriate method by comparison based on the mathematical and numerical aspects. Some standard structural problems are selected for the evaluations.     

Ordering for bandwidth reduction

The bandwidth reduction of the stiffness and flexibility matrices of structures is achieved by ordering the elements of cocycle (nodal) and generalized cycle bases of the corresponding graph models, respectively. For nodal numbering, a two-step approach is used. In the first step the node-set of the graph model is decomposed into ordered subsets, and in the second step ordering within each ordered subset is performed. Six different algorithms are developed and implemented for rinding a suitable starting node for this ordering. The application of the proposed method for nodal ordering is extended to ordering the elements of generalized cycle bases for reducing the bandwidth of the corresponding flexibility matrices, and nodal numbering of finite element models for reducing the bandwidth of their stiffness matrices.     

A co-rotational formulation for nonlinear dynamic analysis of curved euler beam

A co-rotational finite element formulation for the dynamic analysis of a planar curved Euler beam is presented. The Euler-Bernoulli hypothesis and the initial curvature are properly considered for the kinematics of a curved beam. Both the deformational nodal forces and the inertial nodal forces of the beam element are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory in element coordinates which are constructed at the current configuration of the corresponding beam element. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the effectiveness of the proposed element and to investigate the effect of the initial curvature on the dynamic response of the curved beam structures.     

A unified formulation of small-strain corotational finite elements: I. Theory

This paper presents a unified theoretical framework for the corotational (CR) formulation of finite elements in geometrically nonlinear structural analysis. The key assumptions behind CR are: (i) strains from a corotated configuration are small while (ii) the magnitude of rotations from a base configuration is not restricted. Following a historical outline the basic steps of the element independent CR formulation are presented. The element internal force and consistent tangent stiffness matrix are derived by taking variations of the internal energy with respect to nodal freedoms. It is shown that this framework permits the derivation of a set of CR variants through selective simplifications. This set includes some previously used by other investigators. The different variants are compared with respect to a set of desirable qualities, including self-equilibrium in the deformed configuration, tangent stiffness consistency, invariance, symmetrizability, and element independence. We discuss the main benefits of the CR formulation as well as its modeling limitations.     

Control‐oriented modeling and deployment of tensegrity–membrane systems

This study addresses control‐oriented modeling and control design of tensegrity–membrane systems. Lagrange's method is used to develop a control‐oriented model for a generic system. The equations of motion are expressed as a set of differential‐algebraic equations (DAEs). For control design, the DAEs are converted into second‐order ordinary differential equations (ODEs) based on coordinate partitioning and coordinate mapping. Because the number of inputs is less than the number of state variables, the system belongs to the class of underactuated nonlinear systems. A nonlinear adaptive controller based on the collocated partial feedback linearization (PFL) technique is designed for system deployment. The stability of the closed‐loop system for the actuated coordinates is studied using the Lyapunov stability theory. Because of system complexity, numerical tests are used to conduct stability analysis for the dynamics of the underactuated coordinates, which represents the system's zero dynamics. For the tensegrity–membrane systems studied in this work, analytical proof of zero dynamics stability remains an open theoretical problem. An H_∞ controller is implemented for rapid stabilization of the system at the final deployed configuration. Simulations are conducted to test the performance of the two controllers. The simulation results are presented and discussed in detail. Copyright © 2016 John Wiley & Sons, Ltd.     

A practical approach for nonlinear analysis of tensegrity systems

Tensegrity systems are lightweight structures composed of cables and struts. The nonlinear behavior of tensegrity systems is critical; therefore, the design of these types of structures is relatively complex. In the present study, a practical and efficient approach for geometrical nonlinear analysis of tensegrity systems is proposed. The approach is based on the point iterative method. Static equilibrium equations are given in nodes for subsystems, thus the maximum unknown displacement number in each step is three. Pre-stress forces in the system are taken into account in a tangent stiffness matrix, while similar calculations are carried out for each node in the system which has a minimum of one degree of freedom. In each iteration step, the values found in previous steps are used. When it reaches permissible tolerance of calculation, final displacements and internal forces are obtained. The structural behavior of the tensegrity systems were evaluated by the proposed method. The results show that the method can be used effectively for tensegrity systems.     

Absolute nodal coordinate plane beam formulation for multibody systems dynamics

A new plane beam dynamic formulation for constrained multibody system dynamics is developed. Flexible multibody system dynamics includes rigid body dynamics and superimposed vibratory motions. The complexity of mechanical system dynamics originates from rotational kinematics, but the natural coordinate formulation does not use rotational coordinates, so that simple dynamic formulation is possible. These methods use only translational coordinates and simple algebraic constraints. A new formulation for plane flexible multibody systems are developed utilizing the curvature of a beam and point masses. Using absolute nodal coordinates, a constant mass matrix is obtained and the elastic force becomes a nonlinear function of the nodal coordinates. In this formulation, no infinitesimal or finite rotation assumptions are used and no assumption on the magnitude of the element rotations is made. The distributed body mass and applied forces are lumped to the point masses. Closed loop mechanical systems consisting of elastic beams can be modeled without constraints since the loop closure constraints can be substituted as beam longitudinal elasticity. A curved beam is modeled automatically. Several numerical examples are presented to show the effectiveness of this method.     

On static analysis of tensile structures with sliding cables: the frictional sliding case

In some structural systems, such as cable structures, membranes and tensegrity structures, the use of sliding cables allows to reduce the number of elements required to be controlled during tensioning or activation. However, using sliding cables modifies the structural behavior of tensile structures since it alters the distribution of axial forces in structural members. This has been experienced in structures with continuous cables under the assumption of frictionless sliding. However, sliding-induced friction can further alter the behavior of the system. An enhancement of the static analysis of tensile structures with sliding-induced friction is investigated in this paper. In the proposed formulations, the finite-element analysis method and the dynamic relaxation method are combined with a linear complementary approach. Sliding-induced friction is integrated in the formulations through the consideration of the Euler–Eytelwein equation. The importance of considering sliding-induced friction in the static analysis of tensile structures is demonstrated through a series of examples, where it is shown that friction significantly affects the mechanical behavior of the structures. The examples also reveal that the proposed formulations do not affect the computational time of the static analyses.

     

Probabilistic dynamic analysis with ADINA

In structural analysis the extreme values of the structural response quantities are of primary interest. In the case of linear structures under stationary Gaussian random exicitations a probabilistic analysis provides the expected extreme values as well as confidence intervals in a mathematically rigorous way. The analysis becomes particularly simple in modal coordinates, if the damping matrix decouples. All quantities required for the probabilistic analysis are then readily obtained from the modal covariance matrices. The evaluation of these matrices by analytical integration is discussed. This method is computationally very effective and maintains full accuracy in the dynamic properties of the structural model. The implementation of the probabilistic analysis in ADINA is outlined. As an illustration, the seismic analysis of a structure under multiple support random excitation is presented.     

Force density method for simultaneous optimization of geometry and topology of trusses

A new method of simultaneous optimization of geometry and topology is presented for plane and spatial trusses. Compliance under single loading condition is minimized for specified structural volume. The difficulties due to existence of melting nodes are successfully avoided by considering force density, which is the ratio of axial force to the member length, as design variable. By using the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members, the compliance and structural volume are expressed as explicit functions of force density only. After obtaining optimal cross-sectional area, nodal locations, and topology, the cross-sectional areas and nodal coordinates are further optimized using a conventional method of nonlinear programming. Accuracy of the optimal solution is verified through examples of plane trusses and a spatial truss. It is shown that various nearly optimal solutions can be found using the proposed method.     

A 3D Finite Element Method for Flexible Multibody Systems

An efficient finite element (FE) formulation for the simulation of multibody systems is derived from Hamilton's principle. According to the classical assumptions of multibody systems, a large rotation formulation has been chosen, where large rotations and large displacements, but only small deformations of the single bodies are taken into account. The strain tensor is linearized with respect to a co-rotated frame. The present approach uses absolute coordinates for the degrees of freedom and forms an alternative to the floating frame of reference formulation that is based on relative coordinates and describes deformation with respect to a co-rotated frame. Due to the modified strain tensor, the present formulation distinguishes significantly from standard nodal based nonlinear FE methods. Constraints are defined in integral form for every pair of surfaces of two bodies. This leads to a small number of constraint equations and avoids artificial stress singularities. The resulting mass and stiffness matrices are constant apart from a transformation based on a single rotation matrix for each body. The particular structure of this transformation allows to prevent from the usually expensive factorization of the system Jacobian within implicit time--integration methods. The present method has been implemented and tested with the FE-package NGSolve and specific 3D examples are verified with a standard beam formulation.     

Finite elements for post-buckling analysis. I—The W-formulation

This paper presents a convenient formulation for the stability analysis of structures using the finite element method. The main assumptions are linear elasticity, a linear fundamental path, and the existence of distinct critical loads (i.e. no coupling between buckling modes occurs). The formulation developed is known as W-formulation, in which the energy is written in terms of a sliding set of incremental coordinates measured with respect to the fundamental path. In the presentation developed here, the only ingredients required to carry out the analysis are the strain-displacement and the constitutive matrices at the element level. The present formulation is compared with the so called V-formulation, in which the displacements refer to the unloaded state. It is shown that under the present assumptions of linear fundamental path, the advantages of the V-formulation are lost and both approaches are similar. An example of a circular plate under in-plane loading illustrates the procedures. Part II of this paper deals with the application to the post buckling analysis of plate assemblies made of composite materials.