symmetric prismatic tensegrity structures part ii symmetry adapted formulations this paper presents analytical formulations for the symmetry adapted equilibrium force density and geometrical stiffness matrices for prismatic tensegrity structures with dihedral symmetries an analytical expression for the infinitesimal mechanisms is also derived

impulsive control of sticking motion in van der pol one sided constraint system in this paper the impulsive control method is used to dominate sticking motion in van der pol one sided constraint system we find that the method can restrain sticking motion effectively and the method can also induce sticking motion for special parameters in addition numerical results show that the method is stable even for high levels of multiplicative noise or additive noise

symmetry adapted equilibrium matrices a number of authors have previously used group representation theory to block diagonalise the stiffness matrix of a symmetric structure this paper describes how similar techniques can be used to block diagonalise the equilibrium matrix of a symmetric structure this is shown to provide useful insight into the static and kinematic response of such systems in particular it simplifies finding and classifying states of self stress and mechanisms as well as reducing the computational effort required for a force method analysis

matrix analysis of statically and kinematically indeterminate frameworks abstract the paper is concerned with the structural mechanics of assemblies of bars and pinjoints particularly where they are simultaneously statically and kinematically indeterminate the physical significance of the four linear algebraic vector subspaces of the equilibrium matrix is examined and an algorithm is set up which determines the rank of the matrix and the bases for the four subspaces in particular this algorithm gives full details of any states of self stress and modes of inextensional deformation which an assembly may possess a scheme is devised for the segregation of inextensional modes into rigid body modes up to six of these may be allowed by the foundation constraints and internal mechanisms in some circumstances a state of self stress may impart first order stiffness to an inextensional mode a matrix method for detecting this effect is devised and it is shown that if there is no state of self stress which imparts first order stiffness to a given mode then that mode can undergo rather large distortion which involves either zero change in length of the bars or possibly changes in length of third or higher order in the displacements the significance of negative stiffness as indicated by the matrix method is discussed the paper contains simple examples which illustrate all of the main points of the work

initial prestress design and optimization of tensegrity systems based on symmetry and stiffness abstract this study mainly focuses on initial prestress design of tensegrity structures with given shapes for symmetric structures with multiple self stress modes two efficient methods respectively applying singular value decomposition on the constrained equilibrium matrix and performing eigenvalue analysis without manual classifications of member types are proposed to compute the independent states of self stress retaining full symmetry for structures with multiple symmetric states of self stress a novel optimization model for the purpose of maximizing the global rigidity of the structure is presented to determine the combination coefficients of the independent symmetric self stress states a penalty function is employed to consider both hard and soft constraints including stability conditions unilateral property of members and prestress distribution evenness the fitness ranking based genetic algorithms with and without incorporating isolation niche technique are developed to solve such optimization problem several numerical examples are tested to demonstrate the efficiency of the proposed methods

group theoretic exploitations of symmetry in computational solid and structural mechanics the use of group theory in simplifying the study of problems involving symmetry is a well established approach in various branches of physics and chemistry and major applications in these areas date back more than 70 years within the engineering disciplines the search for more systematic and more efficient strategies for exploiting symmetry in the computational problems of solid and structural mechanics has led to the development of group theoretic methods over the past 40 years this paper reviews the advances made in the application of group theory in areas such as bifurcation analysis vibration analysis and finite element analysis and summarizes the various implementation procedures currently available illustrative examples of typical solution procedures are drawn from recent work of the author it is shown how the group theoretic approach through the characteristic vector space decomposition enables considerable simplifications and reductions in computational effort to be achieved in many cases group theoretic considerations also allow valuable insights on the behaviour or properties of a system to be gained before any actual calculations are carried out copyright 2009 john wiley sons ltd

when is a symmetric pin jointed framework isostatic maxwell s rule from 1864 gives a necessary condition for a framework to be isostatic in 2d or in 3d given a framework with point group symmetry group representation theory is exploited to provide further necessary conditions this paper shows how for an isostatic framework these conditions imply very simply stated restrictions on the numbers of those structural components that are unshifted by the symmetry operations of the framework in particular it turns out that an isostatic framework in 2d can belong to one of only six point groups some conjectures and initial results are presented that would give sufficient conditions in both 2d and 3d for a framework that is realized generically for a given symmetry group to be an isostatic framework

shape analysis via oriented distance functions the object of this paper is twofold we first present constructions which induce topologies on subsets of a fixed domain or hold all d in rn by using set parameterized functions in an appropriate function space second we study the role of the family of oriented distance functions also known as algebraic or signed distance functions in the analysis of shape optimization problems they play an important role in the introduction of topologies which retain the classical geometric properties associated with sets convexity exterior normals mean curvature ck boundaries etc

symmetry and states of self stress in triangulated toroidal frames abstract a symmetry extension of maxwell s rule for rigidity of frames shows that every fully triangulated torus has at least six states of self stress with a well defined set of symmetries related to rotational and translational representations in the point group in contrast in trivalent polyhedral toroidal frames i e the duals of the deltahedra the rigid body motions plus the mechanisms are at least equal in number to the bars and their symmetry representation contains that of a set of vector slides along the bars

double link expandohedra a mechanical model for expansion of a virus double link expandohedra are introduced each is constructed from a parent polyhedron by replacing all faces with rigid plates adjacent plates being connected by a pair of spherically jointed bars powerful symmetry techniques are developed for mobility analysis of general double link expandohedra and when combined with numerical calculation and physical model building they demonstrate the existence of generic finite breathing expansion motions in many cases for icosahedrally symmetric trivalent parents with pentagonal and hexagonal faces only fullerene polyhedra the derived expandohedra provide a mechanical model for the experimentally observed swelling of viruses such as cowpea chlorotic mottle virus ccmv a fully symmetric swelling motion a finite mechanism is found for systems based on icosahedral fullerene polyhedra with odd triangulation number t 31 and is conjectured to exist for all odd triangulation numbers

a symmetry extension of maxwell s rule for rigidity of frames a symmetry extension of maxwell s rule for rigidity of frames is found this rule subsumes and strengthens maxwell s 1864 rule by requiring that the internal and external degrees of freedom of a pin jointed structure are not only numerically equal but are also equisymmetric a number of special cases of maxwell s original rule are studied to show the improved insight that the symmetry adapted version can give

symmetric prismatic tensegrity structures part i configuration and stability this paper presents a simple and efficient method to determine the self equilibrated configurations of prismatic tensegrity structures nodes and members of which have dihedral symmetry it is demonstrated that stability of this class of structures is not only directly related to the connectivity of members but is also sensitive to their geometry height radius ratio and is also dependent on the level of self stress and stiffness of members a catalogue of the structures with relatively small number of members is presented based on the stability investigations

detection of finite mechanisms in symmetric structures abstract using group representation theory it is possible to block diagonalise the equilibrium matrix of a symmetric structure this analysis can identify the symmetry properties of any states of self stress or mechanisms present in the structure this paper will show that in some cases this linear analysis combined with symmetry arguments can show that particular mechanisms of a symmetric structure must be finite

finite motions from periodic frameworks with added symmetry recent work from authors across disciplines has made substantial contributions to counting rules maxwell type theorems which predict when an infinite periodic structure would be rigid or flexible while preserving the periodic pattern as an engineering type framework or equivalently as an idealized molecular framework other work has shown that for finite frameworks introducing symmetry modifies the previous general counts and under some circumstances this symmetrized maxwell type count can predict added finite flexibility in the structure in this paper we combine these approaches to present new maxwell type counts for the columns and rows of a modified orbit matrix for structures that have both a periodic structure and additional symmetry within the periodic cells in a number of cases this count for the combined group of symmetry operations demonstrates there is added finite flexibility in what would have been rigid when realized without the symmetry given that many crystal structures have these added symmetries and that their flexibility may be key to their physical and chemical properties we present a summary of the results as a way to generate further developments of both a practical and theoretic interest

finite symmetric systems and their analysis abstract symmetry of linear mechanical systems permits one to substantially reduce the effort in their analysis classification and structural analysis of the finite linear symmetric systems models are studied in this paper there are two variants of the symmetry approach mechanical and algebraic in accordance with the former a symmetric system is replaced by one or several small nonsymmetric subsystems which are subjected to special loads obtained from the initial set the total response of the original symmetric system is found by special superposition of partial responses of these subsystems the algebraic approach is based on the explicit block diagonal decomposition of the matrix equation corresponding to a symmetric system while both approaches have the same efficiency the latter is easier to implement and describe it is presented here the condtions under which the symmetry technique may be utilized do not include symmetry of the applied loads specifically symmetry of those loads which form the right side of the associated equations nevertheless if the loads are symmetric the efficiency of the symmetry approach substantially increases group theory which is widely used in this paper is the mathematical tool for the study of symmetry and all necessary notions are introduced

lower order symmetric mechanism modes and bifurcation behavior of deployable bar structures with cyclic symmetry abstract kinematic bifurcation is generally unavoidable for deployable structures stating at one critical point along the compatibility path a deployable structure could exhibit unexpected motions it can either keep transforming along the primary path or get into one of the bifurcation paths in this study a group theoretic approach which is applicable for bifurcation analyses on symmetric deployable bar structures is developed with the aid of symmetry groups and their irreducible representations the key issue is to decompose the compatibility matrix of a deployable structure into block matrices in the symmetry adapted coordinate system and then extract new mechanism modes with lower order symmetries these symmetric mechanisms are associated with independent bifurcation paths and can identify the feasibility of the paths further using the prediction correction algorithm the lower order symmetric mechanisms are utilized to lead the structure smoothly transform into expected bifurcation paths to verify the effectiveness of the proposed approach several 2d linkages and 3d deployable structures are analyzed which possess inherent cyclic symmetries numerical results indicate that the proposed approach can effectively trace the complex and independent bifurcation paths

an orthogonal self stress matrix for efficient analysis of cyclically symmetric space truss structures via force method abstract the analysis of structures is normally carried out through displacement method while the force method is considered as an alternative approach for this purpose and used on occasion the generation of compatibility conditions the transpose of self stress matrix is one of the major and complicated parts of any structural analysis using force method in this paper an efficient method is proposed for producing orthogonal self stress matrix related to space truss structures with cyclic symmetry this is actually performed by eigen decomposition of a special matrix having the same null basis as in equilibrium matrix then the advantages of the obtained compatibility conditions are demonstrated with respect to different formulations such as standard force method eigen force method and integrated force method finally the efficiency of the presented method is comprehensively compared with three well known numerical methods and tested on a set of practical examples the results indicate clearly the significant superiority of the proposed approach in terms of both computational time and the accuracy of the results

a class of expandable polyhedral structures a class of expandable polyhedral structures the expandohedra consisting of prismatic faces linked along edges by hinged plates provides a model for the swelling of viruses the finite breathing mode of the expandohedra apparent in the model and from geometric arguments is not however detected by elementary counting techniques symmetry extended mobility criteria show that the breathing motion is part of a set of face mechanisms for many expandohedra numerical analysis using the singular value decomposition of the compatibility matrix confirms the completeness of the deductions from general symmetry theorems derived for expandohedra

the discrete degree of metric spaces and the lipschitz continuity of distance transforms abstract a quantitative characteristic named the discrete degree of a metric space is introduced in terms of such a characteristic relations between the lipschitz continuity of distance transforms and the properties of the metrics on spaces are discussed

exact block diagonalization of large eigenvalue problems for structures with symmetry we consider large eigenvalue problems for skeletal structures with symmetry we present an algorithm based upon a novel combination of group theoretic ideas and substructuring techniques that block diagonalizes such systems exactly and efficiently the procedure requires only the structural matrices of a repeating substructure together with the symmetry modes which are obtained from symmetry considerations alone we first present a simple paradigmatic example and then follow with several non trivial applications involving large lattice structures

prismatic tensegrity structures with additional cables integral symmetric states of self stress and cable controlled reconfiguration procedure abstract a simple energy method is put forward to determine the prestress distribution for symmetric tensegrity structures with multiple states of self stress and a class of prismatic tensegrities with additional cables are introduced to show the accuracy of presented method for the purpose of modifying structural shape as well as mechanical properties a cable controlled reconfiguration procedure is subsequently proposed for these structures by defining the length adjustments as the control parameters the reconfiguration procedure is regarded as a quasi static process consisting of a sequence of equilibrium configurations with varying control parameters then the nonlinear iterative algorithm based on the tangent stiffness of the structure is presented to simulate and follow this reconfiguration process by way of example a particular class of reconfigurations coined symmetrical reconfigurations are investigated carefully and the key features as well as the potential applications are given

analysis of symmetric or partially symmetric structures this paper combines group theoretic methods with substructuring techniques to treat the symmetric or partially symmetric structures subjected to arbitrary loads first for a given substructure module whose symmetry group is g it gives a method to find the generalized displacements associated with irreducible representations of g based on some concepts presented in this paper and then points out that for these displacements the analysis problem for a substructure module can be divided into a series of uncoupled subproblems for each subproblem the triangularization as well as condensation can be carried out individually finally a simplified calculation formula of condensed stiffness coefficients for the original displacements is established in which only the coefficients relating to the displacements in the basic region are necessary to be calculated a series of computer programs has been developed by the methods presented in this paper 1 4 these programs enable us to reduce the cpu time and save the memory noticeably

distance transforms for real valued functions a set in a metric space gives rise to its distance function that associates with every point its distance to the nearest point in the set this function is called the distance transform of the original set in the same vein given a real valued function f we consider the expected distances from any point to a level set of f taken at a random height this produces another function called a distance transform of f such transforms are called grey scale distance transforms to signpost their differences from the binary case when sets or their indicators give rise to conventional distance functions basic properties of the introduced grey scale distance transform are discussed the most important issue is the uniqueness problem whether two different functions may share the same distance transform we answer this problem in a generality completely sufficient for all practical applications in imaging sciences the full scale problem remains open

structural computations with the singular value decomposition of the equilibrium matrix abstract the singular value decomposition of the equilibrium matrix makes it possible to answer any question of a static kinematic or static kinematic nature for any structural assembly within a unified computational framework the paper introduces this decomposition relates it to the physical properties of a given assembly and derives formulae for stresses displacements etc a simple example illustrates the calculations

when is a symmetric body bar structure isostatic body bar frameworks provide a special class of frameworks which are well understood generically with a full combinatorial theory for rigidity given a symmetric body bar framework this paper exploits group representation theory to provide necessary conditions for rigidity in the form of very simply stated restrictions on the numbers of those structural components that are unshifted by the symmetry operations of the framework we give some initial results and conjectures for when these conditions are also sufficient for rigidity

maxwell laman counts for bar joint frameworks in normed spaces abstract the rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of euclidean bar joint frameworks in this paper we generalise this tool and introduce a rigidity matrix for bar joint frameworks in arbitrary finite dimensional real normed vector spaces using this new matrix we derive necessary maxwell laman type counting conditions for a well positioned bar joint framework in a real normed vector space to be infinitesimally rigid moreover we derive symmetry extended counting conditions for a bar joint framework with a non trivial symmetry group to be isostatic i e minimally infinitesimally rigid these conditions imply very simply stated restrictions on the number of those structural components that are fixed by the various symmetry operations of the framework finally we offer some observations and conjectures regarding combinatorial characterisations of symmetric isostatic bar joint frameworks in r 2 p where the unit ball p is a quadrilateral