Experimental and numerical study on the self-stress design of tensegrity systems

Tensegrity systems as kinematically and statically indeterminate pin-jointed systems are characterized by mechanisms and self-stress states. Unlike the other reticulated systems, in tensegrity systems, unilateral behavior of cables causes some problems in determining the basis of compatible self-stress states. At the present study, self-stress design of tensegrity systems is presented. Experimental study on two 3×3×0.7 m tensegrity grids was conducted to verify the accuracy and validity of the numerical method. Using supporting constraints, an effective method for the implementation of self-stress states in a much reduced number of stages is proposed and calibrated. Considering the results of the present study, the self-stress design of these systems can be improved to obtain specific desired behavior.     

Structural morphology of tensegrity systems

The coupling between form and forces, their structural morphology, is a key point for tensegrity systems. In the first part of this paper we describe the design process of the simplest tensegrity system which was achieved by Kenneth Snelson. Some other simple cells are presented and tensypolyhedra are defined as tensegrity systems which meet polyhedra geometry in a stable equilibrium state. A numerical model giving access to more complex systems, in terms of number of components and geometrical properties, is then evoked. The third part is devoted to linear assemblies of annular cells which can be folded. Some experimental models of the tensegrity ring which is the basic component of this “hollow rope” have been realized and are examined.     

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.     

Self-stress design of tensegrity grid structures with exostresses

A numerical method is presented for initial self-stress design of tensegrity grid structures with exostresses, which is defined as a linear combination of the coefficients of independent self-stress modes. A discussion on proper division of the number of member groups for the purpose of existence of a single integral feasible self-stress mode has been explicitly given. Dummy elements to transform the tensegrity grid structure with statically indeterminate supports into self-stressed pin-jointed system without supports are employed. The unilateral properties of the stresses in cables and struts are taken into account. Evaluation of the stability for the structure is also considered. Several numerical examples are presented to demonstrate the efficiency and robustness in searching initial single integral feasible self-stress mode for tensegrity grid structures.     

A direct probabilistic approach to solve state equations for nonlinear systems under random excitation

In this paper, a direct probabilistic approach (DPA) is presented to formulate and solve moment equations for nonlinear systems excited by environmental loads that can be either a stationary or nonstationary random process. The proposed method has the advantage of obtaining the response’s moments directly from the initial conditions and statistical characteristics of the corresponding external exci-tations. First, the response’s moment equations are directly derived based on a DPA, which is completely independent of the It?/filtering approach since no specific assumptions regarding the correlation structure of excitation are made. By solving them under Gaussian closure, the response’s moments can be obtained. Subsequently, a multiscale algo-rithm for the numerical solution of moment equations is exploited to improve computational efficiency and avoid much wall-clock time. Finally, a comparison of the results with Monte Carlo (MC) simulation gives good agreement. Furthermore, the advantage of the multiscale algorithm in terms of efficiency is also demonstrated by an engineering example.     

A linear approach to generalized minimum variance controller design for MIMO nonlinear systems

Designing minimum variance controllers (MVC) for nonlinear systems is confronted with many difficulties. The methods which are able to identify MIMO nonlinear systems are scarce, and linear models are not accurate in modeling nonlinear systems. In this paper, Vector ARX (VARX) models are proposed for designing MVC and generalized minimum variance controller (GMVC) for linear and nonlinear systems, and the accuracy of these models in approximating the nonlinear MIMO system is studied. However, the VARX is a linear model. It is shown that this model can identify some kinds of nonlinear systems with any desired accuracy. Therefore, the controller designed by the VARX is accurate, even for these nonlinear systems. The proposed controller is tested on a both linear system and a nonlinear four-tank benchmark process. In spite of the simplicity of designing GMVCs for the VARX models, the results show that the proposed method is accurate and implementable.     

Topology design of tensegrity structures via mixed integer programming

This paper presents a numerical method for finding a tensegrity structure based on the ground structure method. We first solve a mixed integer programming (MIP) problem which maximizes the number of struts over the self-equilibrium condition of axial forces and the discontinuity condition of struts. Subsequently we solve the minimization problem of the number of cables in order to remove redundant self-equilibrium modes, which is also formulated as an MIP. It is regarded to be advantageous that our method does not require any connectivity information of cables and struts to be known in advance, while the obtained tensegrity structure is guaranteed to satisfy the discontinuity condition of struts rigorously.     

A comprehensive dynamic model for class-1 tensegrity systems based on quaternions

In this paper we propose a new dynamic model, based on quaternions, for tensegrity systems of class-1. Quaternions are used to represent orientations of a rigid body in the 3-dimensional space eliminating the problem of singularities. Moreover, the equations based on quaternions allow to perform more precise calculations and simulations because they do not use trigonometric functions for the representation of angles. We present a thorough introduction of tensegrities and the current state of research. We also introduce the quaternions and provide in the appendix some important details and useful properties. Applying the Euler–Lagrange approach we derive a comprehensive dynamic model, first for a simple rigid bar in the space and, at last, for a class-1 tensegrity system. We present two model forms: a matrix and a vectorial form. The first more compact and easier to write, the latter more suitable to apply the tools and the theory based on vector fields.     

A unified approach to direct and inverse boundary layer solutions

A unified approach is presented for solving the two-dimensional incompressible boundary layer equations. Solutions are obtained for direct and inverse options using the same equation formulation by a simple interchange of boundary conditions. A modified form of the mechul function scheme obtains inverse solutions with specification of transformed wall shear, skin friction coefficient or displacement thickness distributions. Direct solutions may be obtained without altering the block tridiagonal structure of the system by simply requiring no corrections on the streamwise pressure gradient parameter. Fourth-order spline discretization approximates normal derivatives with two- and three-point backward differences approximating streamwise derivatives, yielding a fully implicit solution method. The resulting spline/finite difference equations are solved by Newton-Raphson iteration together with partial pivoting. The results of the study demonstrate the importance of proper linearization of all equations. The successful use of spline discretization is also tied to the use of strong two-point boundary conditions at the wall for cases involving reversed flow. Numerical solutions are presented for several non-similar flows and compared with published results.     

Optimal design of structures subjected to follower forces

Summary The optimal design of columns, consisting of segments with arbitrary thickness and length, subjected to follower forces is considered in this paper. The solution is presented for the case of cross modal interaction of a vibrating system. Since the resulting boundary value problem is nonselfadjoint only approximate solutions are generally possible. Herein, for the solution the method of the generalised functional and the transfer matrix technique have been used. As typical examples, the solutions of Beck's, Reut's, Leipholz's and Hanger's column are investigated.

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Übersicht Behandelt wird die optimale Auslegung von Stäben, die aus Abschnitten beliebiger Querschnittsfläche und Länge bestehen und durch Folgelasten beansprucht sind. Eine Lösung wird angegeben unter Berücksichtigung des Zusammenwirkens der Eigenformen des Systems. Da das zugehörige Randwertproblem nicht selbstadjungiert ist, sind im allgemeinen nur Näherungslösungen angebbar. Hier werden das Verfahren eines generalisierten Funktionais sowie das Übertragungsmatrizenverfahren verwendet. Als typisches Beispiel werden die Lösungen zu den Stäben von Beck, Reut, Leipholz und Hauger betrachtet.

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Fellow of the Alexander von Humboldt Foundation 1978/1979, visiting the University of Hannover     

A direct approach for simplifying nonlinear systems with external periodic excitation using normal forms

Nonlinear Dynamics - In this article, we present a straightforward methodology to obtain the normal forms of nonlinear systems subjected to external periodic excitation. Moreover, the nonlinear...     

Geometrical nonlinear elasto-plastic analysis of tensegrity systems via the co-rotational method

An efficient finite element formulation is presented for geometrical nonlinear elasto-plastic analyses of tensegrity systems based on the co-rotational method. Large displacement of a space rod element is decomposed into a rigid body motion in the global coordinate system and a pure small deformation in the local coordinate system. A new form of tangent stiffness matrix, including elastic and elasto-plastic stages is derived based on the proposed approach. An incremental-iterative solution strategy in conjunction with the Newton-Raphson method is employed to obtain the geometrical nonlinear elasto-plastic behavior of tensegrities. Several numerical examples are given to illustrate the validity and efficiency of the proposed algorithm for geometrical nonlinear elasto-plastic analyses of tensegrity structures.     

A direct approach for continuous topology optimization subject to admissible loading

In the present paper, a method is proposed for topology optimization of continuum structures subject to static and plastic admissibility conditions relative to a prescribed load. A key feature of the method is that, using a finite-element discretization, the form of the resulting topology optimization problem is similar to that of the direct static approach of the limit analysis problem. The proposed method is formulated in plane strain using Tresca materials and is illustrated on example problems taken from the literature.     

A dispersion force approach to modelling the effect of lift forces on fibre dispersion

Wood fibres suspended in air were studied whilst flowing through a throttle. Measurements of volume fraction and velocity were compared with results from a two-fluid model simulation. Results from earlier work have shown that the lift forces acting on the individual fibres are of importance for the dispersion of volume fraction. Since the orientation of the individual fibres cannot be determined, a dispersion force was utilised to model the dispersing effect of the lift forces. The addition of a lift dispersion model significantly improved the agreement between measurement and model data.     

A tentative approach to the direct simulation of drag reduction by polymers

An efficient technique for drag reduction uses dilute solutions of a few p.p.m. of polymers. A possible reduction in drag of up to 80% is achieved. Several experimental observations have been made which tend to indicate that the polymers modify the turbulence structures within the buffer layer. Flow visualisations have shown that the changes consist of a weakening of the strength of the streamwise vortices. Existing literature reveals no attempts of numerical simulation of this phenomenon. In this paper an approach is pursued by using a constitutive equation which relates the elongation viscosity to the local properties of the flow. According to this model this viscosity is large in zones where the amount of strain rate is greater than the amount of vorticity, and is zero when the vorticity exceeds the strain rate. Simulations have been performed in a “minimal channel” to give good resolution with a limited number of grid points. The accuracy of the method is tested by comparison with the results of other techniques. For simulations with polymers, quantitative comparisons cannot be made, but the results reproduce the qualitative outputs of the experiments. The mean streamwise velocity is modified in the buffer layer; the peak of the streamwise turbulent intensity, in wall units, increases and its maximum moves far from the wall; and the vertical turbulent intensity is largely reduced in the wall region. An interesting outcome from both the simulation and the experiments is that the strength of the longitudinal vortices is reduced when the polymers are present.     

A generalized approach to tracking the nominals of mechanical systems

Tracking inputs and trajectories of mechanical systems as control objects is analyzed with an emphasis on the objects having the number of external inputs different from the number of their degrees of freedom (DOFs). The problems of the synthesis of the realizable nominals and control laws were considered especially for the case where the number of DOFs of the control object is larger than the number of its external inputs. The difference between control laws for nominals tracking and regulation laws is demonstrated.     

A describing function approach to the design of robust limit-cycle controllers

The design of robust limit-cycle controllers is introduced for autonomous systems with separable SISO nonlinearities. The objective is to design a controller to secure specified robust oscillation amplitude and frequency. The method consists of quasi-linearization of the nonlinear element via a Describing Function (DF) approach and then shaping the loop to reach desired limit-cycle characteristics. As the DF method is used, loop shaping takes place in the Nyquist plot. An example is given to illustrate the robustness of the controlled system to uncertainties in the linear subsystem model.