maximal minimal resolutions that is exact everywhere except at f0 where the cokernel is m such a resolution always exists if r is a noetherian n graded ring with r0 a field and m is a finitely generated z graded r module then a resolution may be constructed in a minimal way in which each fi is graded and each differential is homogeneous of degree 0 the rank of the ith free module in a minimal free resolution of m is an invariant of m called the ith betti number of m and denoted i m likewise r ij m the number of elements of degree j in a minimal set of homogeneous generators of fi is also an invariant of m called the i j th graded betti number of m in this paper we study modules with maximal graded betti numbers more precisely consider a set consisting of pairs r m where r is a noetherian n graded ring with r0 a field and m is a finitely generated graded r module is there some r m such that ij m ij m for every i and j and every r m there are trivial examples in which the answer is yes e g has only one element or no e g consists of all such pairs theorems 1 and 2 below give affirmative answers for certain sets defined by conditions on the hilbert series and the depths of r and m theorem 3 allows one to compute the maximal graded betti numbers for the pairs in the sets considered in theorems 1 and 2 throughout this paper k denotes a field and q the polynomial ring k x1 xn with the usual n grading given by degxi 1 the hilbert series of a finitely generated graded q module m is hm s dimk mis we define d lexicographic ideals and submodules in section 2 the poincare series of an r module m is p m s t ij m s t we write 4 for coefficient wise inequality of laurent series with coefficients in z

why pre tensioning stiffens cable systems it is proved that the potential energy of any pin bar assembly with totally tensioned members possesses strict minimum independently of the assembly topology geometry and magnitudes of member forces

identification of a crack in a beam based on the finite element method of a b spline wavelet on the interval the model based forward and inverse problems in the diagnosis of structural crack location and size by using the finite element method of a b spline wavelet on the interval fem bswi were studied first the crack and uncracked elements of bswi were built to solve the forward problem the first three frequencies influencing functions of normalized crack location and size are approximated by means of surface fitting techniques then the first three measured natural frequencies are employed as inputs of the functions the intersection of the three frequencies contour lines predicted the normalized crack location and size both the numerical and experimental studies verified the validity of the bswi elements in solving crack singular problems with high performance

cohomological degrees and hilbert functions of graded modules making use of the recent construction of cohomological degrees functions we give several estimates on the relationship between number of generators and degrees of ideals and modules with applications to hilbert functions they extend results heretofore known from generalized cohen macaulay local rings to nearly arbitrary local rings the rules of computation these functions satisfy enables comparison with castelnuovo mumford s regularity in the graded case as application we derive sharp improvements on predicting the outcome of effecting noether normalizations in tangent cones

articulated mechanism design with a degree of freedom constraint this paper deals with design of articulated mechanism using a truss based ground structure representation the proposed method can accommodate extremely large displacement by considering geometric non linearity in addition it can also control the mechanical degrees of freedom dof of the resultant mechanism by using a dof equation based on maxwell s rule the optimization is based on a relaxed formulation of an original integer problem and also involves developments directed at handling the redundancy inherent in the ground structure representation one planar test example is selected as the basis for the developments so as to compare the proposed method with other alternative approaches including a graph theoretical enumeration approach which guarantees the identification of the globally optimal solution also an inverter problem is treated where a continuation method is required in order to direct the optimization algorithm towards an integer solution copyright 2004 john wiley sons ltd

a monte carlo form finding method for large scale regular and irregular tensegrity structures we propose a monte carlo form finding method that employs a stochastic procedure to determine equilibrium configurations of a tensegrity structure this method does not involve complicated matrix operations or symmetry analysis works for arbitrary initial configurations and can handle large scale regular or irregular tensegrity structures with or without material geometrical constraints

initial shape finding and modal analyses of cyclic frustum tensegrity modules initial equilibrium and modal analyses of kenneth snelson s cyclic frustum tensegrity modules with an arbitrary number of stages are presented there are m 3 bars at each stage the maxwell number of the modules is 6 2m and is independent of the number of stages in the axial direction calladine s relations reveals that there are 2 5m infinitesimal mechanism modes for multi stage modules the necessary conditions for axial assembly of one stage modules with the same internal element forces are investigated one stage modules with geometrically similar frustum modules satisfy the necessary conditions for pre stressed configurations modal analyses were conducted to investigate the mode shapes of infinitesimal mechanism modes

tensegrity deployment using infinitesimal mechanisms abstract 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

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

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

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

path generation of articulated mechanisms by shape and topology variations in non linear truss representation this paper presents studies on an optimization based method for path generation of articulated mechanisms an extended truss ground structure approach is taken in which both the shape and topology of the truss are designed using cross sectional areas and nodal positions as design variables this leads to a technique for simultaneous type and dimensional synthesis of articulated mechanisms for the analysis part it is essential to control the mechanism configuration so that the mechanism remains within a given configuration space thus stabilizing the optimization process and resulting in realistic solutions this can be achieved by using the levenberg marquardt method the design method is illustrated by a number of design cases for both closed and open input and output paths

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

on the collapse of locally isostatic networks we examine the flexibility of periodic planar networks built from rigid corner connected equilateral triangles such systems are locally isostatic since for each triangle the total number of degre

extrapolation locking and its sanitization in koiter s asymptotic analysis abstract this paper shows that the fem implementation of koiter s asymptotic method w t koiter on the stability of elastic equilibrium 1970 ph d thesis delft 1945 english transl nasa tt f10 883 1967 affdl tr70 25 outlined by casciaro et al finite element asymptotic analysis of slender elastic structures a simple approach int j num meth eng 35 1992 1397 1426 provides accurate and reliable results in the critical and post critical analysis of non linear elastic structures care however does have to be taken in implementing apparently minor details to avoid locking effects which adversely affect accuracy and which can destroy the method reliability as the effects related to the finite element interpolation have been discussed before this paper focuses on the non linear locking due to the use implicit in the method of finite distance extrapolations within this scope it is shown that perturbation algorithms based on compatible formulations can imply a strong critical and post critical locking when analysing structures characterized by high stiffness ratios in the presence of moderate pre critical rotations on the contrary perturbation algorithms based on independent extrapolations of displacements and stresses furnish reliable results in excellent agreement with those provided by step by step analysis at a small fraction of its computational cost

mixed formulation and locking in path following nonlinear analysis abstract the arc length riks strategy has rapidly become a standard tool for path following analysis of nonlinear structures due to its theoretical ability to surpass limit points the aim of this paper is to show that the failures in convergence that are occasionally experienced are not related to proper defects of the algorithm but come from a subtle locking effect intrinsic to the nonlinear nature of the problem as a consequence its sanitization has to be pursued within a reformulation of the structural model the use of a mixed stress displacement variant of the algorithm in particular appears very promising in this respect the topic is discussed with reference to the analysis of nonlinear frames using a mixed version of the nonlinear beam model discussed in 39 it is shown that with no extra computational cost and only a minor modification in coding with respect to a purely compatible formulation it is possible to achieve a noticeable improvement in convergence and a real gain in both computational time and overall robustness of the algorithm

computation of kinematic paths and bifurcation points this article deals with the kinematic simulation of movable structures that go through special configurations of kinematic bifurcation as they move a series of algorithms are developed for structures that can be modelled using pin jointed bars and that admit a single parameter motion these algorithms are able to detect and locate any bifurcation points that exist along the path of the structure and at each bifurcation point can determine all possible motions of the structure the theory behind the algorithms is explained and the analysis of a simple example is discussed in detail then a simplified version of the particular problem that had motivated this work the simulation of the folding and deployment of a thin membrane structure forming a solar sail is analysed for the particular cases that are considered it is found that the entire process is inextensional but a detailed study of the simulation results shows that in more general cases it is likely that stretching or wrinkling will occur

static and dynamic analyses of tensegrity structures part 1 nonlinear equations of motion abstract in order to present basic equations for static and dynamic analyses of a class of truss structures called tensegrity structures large deformation kinematics and kinetics were presented in both eulerian and lagrangian formulations the two sets of equations of motion yield the same values even if different stress and strain measures were employed for their computation the eulerian formulation was implemented in an updated lagrangian finite element code using newton s method with consistently linearized equations of motion by utilizing the linearized lagrangian equations of motion at pre stressed initial configurations harmonic modal analyses of a three bar tensegrity module and a six stage tensegrity beam were conducted in the second part of the paper linearized equations were utilized to investigate the equilibrium configurations of basic tensegrity modules and the stiffness of pre stressed tensegirty structures

determination of mechanism s order for kinematically and statically indetermined systems abstract this paper deals with the mechanisms in kinematically and statically indeterminate reticulated systems knowledge of length variation amplitude for members in association with an assigned mechanism allows determination of mechanism order this is a fundamental characteristic of these systems mainly for stability considerations we submit on one part simple tests allowing distinction between order one mechanisms and mechanisms of higher order and on the other part an algorithm giving access to order exact value for all mechanisms associated with a given reticulated system with this algorithm order one mechanisms and higher order mechanisms are identified simple examples are given in the text and illustrate these aspects in conclusion we submit a stop criterion for the algorithm which gives access to the finite mechanisms for most of constructive reticulated systems

reshaping of tensegrities using a geometrical variation approach abstract we propose a novel approach for the reshaping of tensegrities and pre stressed pin jointed frameworks both free standing and restrained frameworks are studied first a geometrical variation is introduced and considered as an initial geometrical change to a self equilibrated tensegrity a new nonlinear set of equilibrium equations is then established based on this variation and subsequently linearized using an expansion the linearization provides us with an underdetermined system of linear equations with a rectangular jacobian matrix which is derived in a very simple and compact matrix form the system is finally solved iteratively using a newton raphson method employing a moore penrose pseudo inverse operation to establish a new equilibrium state the method promises to generate very accurate results with a remarkable convergence rate the outstanding features and performance of the proposed method are demonstrated and studied comprehensively through three examples

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

reduction numbers of ideals abstract we give general bounds for the reduction numbers of ideals in arbitrary noetherian rings and multiplicity dependent bounds for m primary ideals in a noetherian local ring r m in the case of polynomial rings over fields the bound is a non elementary function with four levels of exponentiation for primary ideals the bound is linear in the samuel multiplicity of the ideal finally we extend these techniques to generic complete intersections of dimension one

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

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 numerical analysis of infinitesimal mechanisms the paper presents a numerical algorithm based on koiter s theory of the elastic stability for detecting the order of infinitesimal mechanisms i e kinematically indeterminate systems of pin jointed bars in cases of one degree of indeterminacy the algorithm improves in terms of computational simplicity and efficiency an analogous algorithm proposed by salerno in 1992 this is shown to be due to the vanishing of the terms higher than the third order of the asymptotic expressions of the energy owing to the use of the green strain measure and a mixed displacement and stress formulation of the problem moreover the proposed algorithm is able to provide a correct definition of mechanism in cases of several degrees of indeterminacy mainly for structures like those first studied by connelly and servatius in 1994 which the paper will treat in depth

on computational procedures for the force method it is known that the matrix force method has certain advantages over the displacement method for a class of structural problems it is also known that the force method when carried out by the conventional gauss jordan procedure tends to fill in the problem data making the method unattractive for large size sparse problems this poor fill in property however is not necessarily inherent to the method and the sparsity may be maintained if one uses what we call the turn back lu procedure the purpose of this paper is two fold first it is shown that there exist some close relationships between the force method and the least squares problem and that many existing algebraic procedures to perform the force method can be regarded as applications extensions of certain well known matrix factorization schemes for the least squares problem secondly it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint included in this unification is the turn back lu procedure which was originally proposed by topcu in his thesis 8 it is explained why this procedure tends to produce sparse and banded self stress and flexibility matrices with small band width some computational results are presented to demonstrate the superiority of the turn back lu procedure over the other schemes considered in this paper

a new approach to the analytical and numerical form finding of tensegrity structures we develop a new formulation for the form finding of tensegrity structures in which the primary variables are the cartesian components of element lengths both an analytical and a numerical implementation of the formulation are described each require a description of the connectivity of the tensegrity with the iterative numerical method also requiring a random starting vector of member force densities the analytical and numerical form finding of tensegrity structures is demonstrated through six examples and the results obtained are compared and contrasted with those available in the literature to verify the accuracy and viability of the suggested methods

advanced form finding for cable strut structures a numerical method is presented for form finding of cable strut structures the topology and the types of members are the only information that is required in this form finding process dummy members are used to transform the cable strut structure with supports into self stressed system without supports the requirement on rank deficiencies of the force density and equilibrium matrices for the purpose of obtaining a non degenerate d dimensional self stressed structure has been explicitly discussed 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 which satisfy the minimum required rank deficiencies of the force density and equilibrium matrices respectively based on numerical examples it is found that the proposed method is very efficient robust and versatile in searching self equilibrium configurations of cable strut structures

further remarks on first order infinitesimal mechanisms abstract this paper analyses a particular structural assembly that has two independent inextensional mechanisms and two independent states of self stress it is shown that for certain values of a variable bar length this assembly is a first order infinitesimal mechanism and yet it cannot be stiffened by the imposition of a state of prestress

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

a direct approach to design of geometry and forces of tensegrity systems in the process of designing a tensegrity system some constraints are usually introduced for geometry and or forces to ensure uniqueness of the solution because the tensegrity systems are underdetermined in most cases in this paper a new approach is presented to enable designers to specify independent sets of axial forces and nodal coordinates consecutively under the equilibrium conditions and the given constraints to satisfy the distinctly different requirements of architects and structural engineers the proposed method can be used very efficiently for practical applications because only linear algebraic equations are to be solved and no equation of kinematics or material property is needed some numerical examples are given to show not only efficiency of the proposed method but also its ability of searching new configurations

on solving first kind integral equations using wavelets on a bounded interval the conventional method of moments mom when applied directly to integral equations leads to a dense matrix which often becomes computationally intractable to overcome the difficulties wavelet bases have been used previously which lead to a sparse matrix the authors refer to mom with wavelet bases as wavelet mom there have been three different ways of applying the wavelet techniques to boundary integral equations 1 wavelets on the entire real line which requires the boundary conditions to be enforced explicitly 2 wavelet bases for the bounded interval obtained by periodizing the wavelets on the real line and 3 wavelet like basis functions furthermore only orthonormal on bases have been considered the present authors propose the use of compactly supported semi orthogonal so spline wavelets specially constructed for the bounded interval in solving first kind integral equations they apply this technique to analyze a problem involving 2d em scattering from metallic cylinders it is shown that the number of unknowns in the case of wavelet mom increases by m 1 as compared to conventional mom where m is the order of the spline function results for linear m 2 and cubic m 4 splines are presented along with their comparisons to conventional mom results it is observed that the use of cubic spline wavelets almost diagonalizes the matrix while maintaining less than 1 5 of relative normed error the authors also present the explicit closed form polynomial representation of the scaling functions and wavelets

statical and dynamical response of cable nets abstract the statical and the dynamical behaviour of cable nets is studied fully constrained and underconstrained cable nets are considered the conditions for initial stability and the prestressing of the nets are formulated a method for the analysis of the nodal displacements and the change in the internal forces in the cables due to static loads is proposed the cases where linear algebra can be used are discussed the method is extended to consider the dynamical behaviour of the net its free dynamical response and forced vibration are investigated

analysis of prestressed mechanisms abstract a new theory is presented for the matrix analysis of prestressed structural mechanisms made from pin jointed bars the response of a prestressed mechanism to any external action is decomposed into two almost separate parts which correspond to extensional and inextensional modes a matrix algorithm which treats these two modes separately is developed and tested it is shown that the equilibrium requirements for the assembly in its initial configuration as well as in deformed configurations which are obtained through infinitesimal inextensional displacements can be fully described by a square equilibrium matrix it is also shown that any set of extensional nodal displacements has to satisfy some equilibrium conditions as well as standard compatibility equations and that the resulting system of linear equations defines a square kinematic matrix theoretical as well as experimental evidence supporting this approach is given in the paper two simple experiments which were of crucial importance in arriving at the equilibrium conditions on the extensional displacements are described the interaction between the two modes of action of a prestressed mechanism is discussed together with a rapidly converging iterative procedure to handle it a study of the non linear effect by which the self stress level in a statically indeterminate assembly rapidly increases if an inextensional mode is excited supported by further experimental results concludes the paper this work is relevant to the analysis of most cable systems pneumatic domes fabric roofs and tensegrity frameworks

koiter s analysis of thin walled structures by a finite element approach summary this paper refers to the analysis of the postbuckling behaviour of thin walled structures by means of an asymptotic approach based on a finite element implementation of koiter s non linear theory of instability the analysis has been accomplished by using the following assumptions i the structure is described as an assemblage of flat slender rectangular panels ii a non linear kirchhoff type plate theory is used to model each panel iii hc finite elements discretization is used iv linear and quadratic extrapolations are assumed for the fundamental and the postbuckling paths respectively v multimodal buckling is considered and vi imperfection sensitivity analysis is performed in both multimodal and monomodal form based on the steepest descent path criterion several numerical results are presented and discussed comparisons with numerical solution obtained by standard incremental codes are given which show the accuracy and reliability of the proposed approach

adaptive force density method for form finding problem of tensegrity structures a numerical method is presented for form finding of tensegrity structures eigenvalue analysis and spectral decomposition are carried out iteratively to find the feasible set of force densities that satisfies the requirement on rank deficiency of the equilibrium matrix with respect to the nodal coordinates the equilibrium matrix is shown to correspond to the geometrical stiffness matrix in the conventional finite element formulation a unique and non degenerate configuration of the structure can then be obtained by specifying an independent set of nodal coordinates a simple explanation is given for the required rank deficiency of the equilibrium matrix that leads to a non degenerate structure several numerical examples are presented to illustrate the robustness as well as the strong ability of searching new configurations of the proposed method

zero stiffness tensegrity structures tension members with a zero rest length allow the construction of tensegrity structures that are in equilibrium along a continuous path of configurations and thus exhibit mechanism like properties equivalently they have zero stiffness the zero stiffness modes are not internal mechanisms as they involve first order changes in member length but are a direct result of the use of the special tension members these modes correspond to an infinitesimal affine transformation of the structure that preserves the length of conventional members they hold over finite displacements and are present if and only if the directional vectors of those members lie on a projective conic this geometric interpretation provides several interesting observations regarding zero stiffness tensegrity structures

static and dynamic analyses of tensegrity structures part ii quasi static analysis linearized lagrangian equations developed in the first part of the paper were employed for static analyses of cyclic cylindrical tensegrity modules linearized equilibrium equations at natural configurations were used to investigate initial shape static and kinematic indeterminancy pre stress and infinitesimal mechanism modes and the sensitivity analysis of initial geometry linearized equilibrium equations at pre stressed initial configurations were utilized to investigate pre stress stiffening and to distinguish first order mechanisms from higher order mechanisms to estimate critical loads for bar buckling and cable slacking nonlinear equilibrium equations were employed to compute element forces further the equivalence between the twist angle theorem obtained from a geometrical consideration and the equilibrium analysis was established for cyclic cylindrical tensegrity modules it is concluded that infinitesimal mechanism modes and pre stresses characterize the static and dynamic response of tensegrity structures

singularity and kinematic bifurcation analysis of pin bar mechanisms using analogous stiffness method abstract based on the analogy between bifurcation of equilibrium paths in structures and kinematic bifurcation of mechanisms this paper proposes an analogous stiffness method to detect the singularity and kinematic bifurcation of mechanisms the analogous stiffness in mechanisms is first defined as the derivative of the state variable with respect to the controlling variable by investigating the value of analogous stiffness the singularity can be classified into output singularity input singularity and architectural singularity and the kinematic characteristics of free joints at corresponding singularity configurations are expounded the singular and kinematic bifurcation points of mechanisms can then be determined by solving analogous stiffness equations and compatibility equations simultaneously following that the analytical criterion for finite motion of corresponding free joints at singularity configurations is derived from the second order analysis of compatibility equations the efficiency of the proposed method is finally illustrated by three typical examples

a symmetry treatment of danzerian rigidity for circle packing a symmetry extended mobility criterion that incorporates danzer s concept of bar and joint assemblies is derived and applied to the spherical circle packing problem the known scalar counting rule

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

around the eakin sathaye theorem abstract we present a generalisation of the eakin sathaye theorem on reductions to the case of complete and so of joint reductions in the sense of rees applications are given generalisations are also indicated for work of lipman eakin sathaye and sally vasconcelos that fed into or resulted from the original version of the eakin sathaye theorem finally the result is reinterpreted as a multiplicative normalisation theorem

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

detection of crack location and size in structures using wavelet finite element methods abstract a methodology to detect crack location and size is presented which takes advantage of wavelet finite element methods wfem in the modal analysis for singularity problems like a cracked beam first the beam is discretized into a set of wavelet finite elements and then the natural frequencies of the beam with various crack locations and sizes are accurately obtained the frequency response functions function of crack location and size are approximated by means of surface fitting techniques measured natural frequencies are used in a crack detection process and the crack location and size can be identified by finding the point of intersection of three frequency contour lines the experimental data from a free free beam studied by silva are employed to verify the accuracy of the method the present method is general and can be easily extended to complex structures with multiple cracks

design of planar articulated mechanisms using branch and bound this paper considers an optimization model and a solution method for the design of two dimensional mechanical mechanisms the mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously the underlying mechanical analysis model is based on a truss representation allowing for large displacements for mechanisms undergoing large displacements elastic stability is of major concern we derive conditions modeled by nonlinear matrix inequalities which guarantee that a stable equilibrium is found and that buckling is prevented the feasible set of the design problem is described by nonlinear differentiable and non differentiable constraints as well as nonlinear matrix inequalities to solve the mechanism design problem a branch and bound method based on convex relaxations is developed to guarantee convergence of the method two different types of convex relaxations are derived the relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub problems encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality

stability conditions for tensegrity structures stability conditions for tensegrity structures are derived based on positive definiteness of the tangent stiffness matrix which is the sum of the linear and geometrical stiffness matrices a necessary stability condition is presented by considering the affine motions that lie in the null space of the geometrical stiffness matrix the condition is demonstrated to be equivalent to that derived from the mathematical rigidity theory so as to resolve the discrepancy between the stability theories in the fields of engineering and mathematics furthermore it is shown that the structure is guaranteed to be stable if the structure satisfies the necessary stability condition and the geometrical stiffness matrix is positive semidefinite with the minimum rank deficiency for non degeneracy

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

first order infinitesimal mechanisms abstract this paper discusses the analytical conditions under which a pin jointed assembly which has s independent states of self stress and m independent mechanisms tightens up when its mechanisms are excited a matrix algorithm is set up to distinguish between first order infinitesimal mechanisms which are associated with second order changes of bar length and higher order infinitesimal or finite mechanisms it is shown that in general this analysis requires the computation of s quadratic forms in m variables which can be easily computed from the states of self stress and mechanisms of the assembly if any linear combination of these quadratic forms is sign definite then the mechanisms are first order infinitesimal an efficient and general algorithm to investigate these quadratic forms is given the calculations required are illustrated for some simple examples many assemblies of practical relevance admit a single state of self stress s 1 and in this case the algorithm proposed is straightforward to implement this work is relevant to the analysis and design of pre stressed mechanisms such as cable systems and tensegrity frameworks

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

on the evaluation of statical kinematic stiffness matrix for underconstrained structural systems abstract because of singularity of the elastic stiffness matrix analysis of underconstrained systems requires the comprehensive stiffness matrix comprising both kinds of first order structural stiffness elastic and statical kinematic the latter accounts for the role of the member forces induced in the system by the equilibrium part of the applied load as the source of the system resistance to the perturbation part of the same load evaluation of the sought member forces using orthogonal load resolution into equilibrium and perturbation components is awkward and computationally expensive a special modification of the singular elastic stiffness matrix simplifies the procedure and makes it amenable to the conventional tools of structural analysis the efficiency and accuracy of the procedure is illustrated by a detailed numerical example

the prestressability problem of tensegrity structures some analytical solutions in this paper we formulate the general prestressability conditions for tensegrity structures these conditions are expressed as a set of nonlinear equations and inequalities on the tendon tensions several examples of tensegrity structures for which the prestressability conditions can be analytically solved are then presented

geometry the leading parameter for the poisson s ratio of bending dominated cellular solids control over the deformation behaviour that a cellular structure shows in response to imposed external forces is a requirement for the effective design of mechanical metamaterials in particular those with negative poisson s ratio this article sheds light on the old question of the relationship between geometric microstructure and mechanical response by comparison of the deformation properties of bar and joint frameworks with those of their realisation as a cellular solid made from linear elastic material for ordered planar tessellation models we find a classification in terms of the number of degrees of freedom of the framework model first in cases where the geometry uniquely prescribes a single deformation mode of the framework model the mechanical deformation and poisson s ratio of the linearly elastic cellular solid closely follow those of the unique deformation mode the result is a bending dominated deformation with negligible dependence of the effective poisson s ratio on the underlying material s poisson s ratio and small values of the effective young s modulus second in the case of rigid structures or when geometric degeneracy prevents the bending dominated deformation mode the effective poisson s ratio is material dependent and the young s modulus view the mathml sourcee cs large all analysed structures of this type have positive values of the poisson s ratio and large values of view the mathml sourcee cs third in the case where the framework has multiple deformation modes geometry alone does not suffice to determine the mechanical deformation these results clarify the relationship between mechanical properties of a linear elastic cellular solid and its corresponding bar and joint framework abstraction they also raise the question if in essence auxetic behaviour is restricted to the geometry guided class of bending dominated structures corresponding to unique mechanisms with inherently low values of the young s modulus

static and dynamic characterization of regular truncated icosahedral and dodecahedral tensegrity modules static and dynamic properties of a pair of dual spherical tensegrity modules invented by buckminster fuller are investigated they are regular truncated icosahedral and dodecahedral tensegrity modules the computation of the maxwell number and the use of calladine s relation reveal that regular truncated icosahedral and dodecahedral tensegrity modules possess 55 infinitesimal mechanism modes a reduced equilibrium matrix is presented for the initial shape finding to economically impose the existence of a pre stress mode both the initial shape and the corresponding pre stress mode are analytically obtained by using graphs of the icosahedral group and the reduced equilibrium matrix for both icosahedral and dodecahedral modules the maximum values of the cable tension is always less than the absolute value of bar compression in order to classify a large number of infinitesimal mechanism modes modal analyses are conducted infinitesimal mechanism modes have the stiffness due to pre stress and are associated with lowest natural frequencies their natural frequencies increase proportionally to the square root of the amplitude of pre stress it is found that there are only 15 distinct natural frequencies associated with the infinitesimal mechanism modes

mixed formulation in koiter analysis of thin walled beams abstract the paper extends the asymptotic mixed formulation proposed in g garcea g salerno sanitizing of locking in koiter perturbation analysis through mixed formulation fourth world congress on computational mechanics buenos aires argentina 1988 g garcea g salerno r casciaro comput methods appl mech engrg 180 1 2 1999 137 167 to structures composed of an assemblage of flat slender elastic panels the same type of structure has already been investigated in a d lanzo g garcea r casciaro int j numer methods engrg 38 1995 2325 2345 a d lanzo g garcea int j numer methods engrg 39 1996 3007 3031 with a compatible formulation and using the usual hypothesis of negligible precritical rotations in order to avoid the extrapolation locking see also g garcea g a trunfio r casciaro comput methods appl mech engrg 165 1 4 1998 137 167 produced by the high axial to flexural stiffness ratio of the panels the use of a mixed formulation provides a consistent way of eliminating the extrapolation locking without introducing ad hoc approximations it is thus both reliable and accurate when combined with a quadratic strain displacement relationship as presented here it gives zero fourth order energy variations further computational simplifications and a natural linearization of the eigenvalue problem required by the buckling analysis an extensive numerical testing shows the effectiveness and robustness of the proposed approach

singular configurations of structural systems abstract a singular configuration of a structural system is characterized by rank deficiency of the equilibrium matrix and kinematic matrix the rank is lower than both the number of degrees of freedom and the number of constraints such configurations exist only in systems that are not geometrically invariant underconstrained structural systems most interesting among them are systems with infinitesimal mobility which attracted attention of many prominent researchers this paper puts the entire issue in a different perspective by addressing a critical yet so far unexplored aspect of singular configurations their realizability it turns out that the only generic physically realizable type of a singular configuration is a system with first order infinitesimal mobility and even this cannot be constructed without inducing prestress of finite magnitude all other singular configurations unprestressable first order mechanisms higher order mechanisms and singular configurations of finite mechanisms are unrealizable moreover short of exact or symbolic calculation they are also noncomputable and are just formal analytical constructs

computational analysis of multi stepped beams and beams with linearly varying heights implementing closed form finite element formulation for multi cracked beam elements abstract the model where the cracks are represented by means of internal hinges endowed with rotational springs has been shown to enable simple and effective representation of transversely cracked slender euler bernoulli beams subjected to small deflections it namely provides reliable results when compared to detailed 2d and 3d models even if the basic linear moment rotation constitutive law is adopted this paper extends the utilisation of this model as it presents the derivation of a closed form stiffness matrix and a load vector for slender multi stepped beams and beams with linearly varying heights the principle of virtual work allows for the simple inclusion of an arbitrary number of transverse cracks the derived at matrix and vector define an exact finite element for the utilised simplified computational model the presented element can be implemented for analysing multi cracked beams by using just one finite element per structural beam member the presented expressions for a stepped beam are not exclusively limited to this kind of height variation as by proper discretisation an arbitrary variation of a cross section s height can be adequately modelled the accurate displacement functions presented for both types of considered beams complete the derivations all the presented expressions can be easily utilised for achieving computationally efficient and truthful analyses

reduction of equilibrium compatibility and flexibility matrices in the force method this paper develops analytical expressions and numerical for reducing the structural matrices which are used in the force method of structural analysis two simple examples illustrate the calculations for pin jointed trusses two advanced applications a beam cross consisting of two coplanar beams which are linked by a shear connector and an equilibrium linear stress finite element are also presented

borel fixed ideals and reduction number abstract the aim of this paper is to study the relationship between the reduction number and borel fixed ideals in all characteristics especially it is shown that r r i r r i lex where i lex denotes the unique lex segment ideal whose hilbert function is equal to that of i this solves a recent question by conca

effective insights into the geometric stability of symmetric skeletal structures under symmetric variations abstract geometric stability is a necessary criterion to guarantee stable equilibrium in engineering structures however we generally encounter enormous calculations to examine the geometric stability when we make variations on the geometry or the connectivity of a given kinematically and statically indeterminate structure this study describes how symmetry is utilized to enhance the mobility and geometric stability analysis of symmetric skeletal structures symmetry extended mobility distinguishes representations of the internal mechanisms and self stress states from relative mobility based on their inherent symmetries using group theoretic method thus it acts as an efficient tool to evaluate the order of internal mechanisms that may be indistinguishable by traditional structural approaches further it is used to gain effective insights into the mobility and geometric stability of a symmetric skeletal structure with symmetrically perturbed connectivity or geometry the first order changes of symmetry extended mobility are deduced to describe the changes induced by the variations of nodal coordinates members and kinematic constraints respectively examples are given to verify the correctness and effectiveness of the proposed method we show that the geometry or connectivity of kinematically indeterminate symmetric skeletal structures can be altered while at the same time retaining geometric stability and some or all of the original symmetry the results have potential application in the design of novel deployable structures

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

extended truss theory with simplex constraints this paper traces a way of generalization of the classical truss theory in addition to the kinematic constraint expressing the distance between two nodes connected by a bar element other similar constraints involving three and four nodes are introduced derived from energy principles a general tangent stiffness formulation is given possible mechanical interpretations as well as problems of pre stressing are also discussed

on the physical realizability of singular structural systems abstract analytical models of structural systems allow for exceptional singular geometric configurations characterized by rank deficiency of the equilibrium and kinematic matrix the feasibility of physical and numerical realization of such configurations depends on the type of singularity generic vs nongeneric it turns out that some interesting theoretically predicted and thoroughly studied types of singular configurations systems with simultaneous statical and kinematic indeterminacy unprestressable first order mechanisms all higher order mechanisms singular configurations of finite mechanisms and kinematically mobile closed polyhedral surfaces are nongeneric hence physically unrealizable and noncomputable except for exact or symbolic calculation thus in spite of their sometimes remarkable theoretical features these systems and configurations are just purely formal constructs moreover their attempted implementation would produce a generic prototype with essentially different properties including structural response a few of the somewhat unexpected implications of this observation are discussed and a complete set of analytical criteria for the four statical kinematic types of realizable structural systems is presented

on the rigidity of triangulated hyperbolic paraboloids the paper investigates the rigidity of two classes of three dimensional pin jointed assemblies which describe triangulated surfaces of hyperbolic paraboloidal shape type 1 with straight boundaries and type 2 with parabolic boundaries the two arrangements have identical horizontal plans the investigation begins with a brief historic review of techniques for detecting lack of rigidity the behaviour of both type 1 and type 2 assemblies depends on the number n of bars on each edge of the structure and although both geometric arrangements satisfy maxwell rule 3 j b where j is the number of joints and b is the number of bars for any value of n all type 1 assemblies with n even and greater than 2 and type 2 assemblies with n 1 are in fact not rigid this result is first proved for some specific cases by the zero load test and then for arbitrary values of n the equilibrium equations of each joint are written down in terms of a stress function and are then assembled in an equilibrium matrix whose rank is found by general matrix manipulations the number of inextensional mechanisms and static redundancies is also obtained the rigidity of assemblies consisting of four interconnected type 1 hyperbolic paraboloidal sheets is also discussed and it is concluded that they are equally prone to the same kind of misbehaviour

orthogonal load resolution and statical kinematic stiffness matrix the structural behavior of underconstrained systems is explored with an emphasis on statically indeterminate ones the roles of the elastic stiffness and the statical kinematic stiffness induced by prestress or an equilibrium load are investigated and compared the linearized response of a prestressed underconstrained system is shown to go beyond the simple notion of the first order stiffness induced by prestress and independent of the member stiffnesses some interesting features of structural behavior under a general load and stability under an equilibrium load are revealed and illustrated with an example of a quasi variant pin bar system finally a comprehensive stiffness matrix with coefficients representing various sources of first order stiffness is introduced to replace when necessary the conventional elastic stiffness matrix the comprehensive matrix is valid for structural systems of any type and is always invertible

multi objective optimization for prestress design of cable strut structures abstract stiffness of a cable strut structure is dependent on the level and distribution of prestress although a higher prestress level enhances overall stiffness of the structure this condition may demand larger member sectional areas and material cost the effect of fabrication and installation error should also be incorporated in the prestress design this paper presents a new optimization method for prestress design of cable strut structures a multi objective optimization problem is formulated and solved to obtain preferred coefficient vectors of prestress modes with the following four objective functions 1 minimize average area of members 2 maximize minimum eigenvalue of stiffness matrix 3 minimize prestress variance of cables and 4 minimize maximum eigenvalue of error sensitivity matrix pareto optimal solutions are obtained by nsga ii among the pareto optimal solutions the most preferred solution is selected using promethee ii two examples are presented to illustrate the proposed process the results are compared with those by existing methods of optimization with a single objective function significance of each objective function is also evaluated by the number of remaining pareto optimal solutions after removal of the objective function

the set of forces that ideal trusses or wire webs under tension can support abstract the problem of determining those multiplets of forces or sets of force multiplets acting at a set of points such that there exists a truss structure or wire web that can support these force multiplets with all the elements of the truss or wire web being under tension is considered the two dimensional problem where the points are at the vertices of a convex polygon is essentially solved each multiplet of forces must be such that the net anticlockwise torque around any vertex of the forces summed over any number of consecutive points clockwise past the vertex must be non negative and one can find a truss structure that supports under tension and only supports those force multiplets in a convex polyhedron of force multiplets that is generated by a finite number of force multiplets each satisfying the torque condition progress is also made on the problem where only a subset of the points are at the vertices of a convex polygon and the other points are inside in particular in the case where only one point is inside an explicit procedure is described for constructing a suitable truss if one exists an alternative recipe to that provided by guevara vasquez et al 2011 based on earlier work of camar eddine and seppecher 2003 is given for constructing a truss structure with elements under either compression or tension that supports an arbitrary collection of balanced forces at the vertices of a convex polygon finally some constraints are given on the forces that a three dimension truss or wire web under tension must satisfy

how to recognize the order of infinitesimal mechanisms a numerical approach structures presenting kinematical inderterminacy are usually called mechanisms this paper is entirely concerned with assemblies which reveal themselves to be mechanisms at a null value of the load among them a first distinction is made between infinitesimal and finite ones the former being characterized by one or several directions of lower but not zero stiffness whereas the latter show at least one finite admissible displacement for which none of the bars undergoes any elongation moreover there exists the possibility to make a further distinction among the infinitesimal mechanisms according to which is the order of the stiffness along the direction considered above the way of evaluating this order is to perform a local analysis of the strain energy of the assembly once the displacement field is parametrized in terms of a suitable parameter by means of a finite element technique this analysis can be easily performed through the numerical approach presented in this report

large displacement analysis of elastic pyramidal trusses the paper examines the equilibrium stability problem for a simple class of elastic space trusses in the shape of a regular pyramid joints located at the vertices of the base polygon are fixed while the joint at the apex is subjected to a proportionally increasing load acting in either the vertical direction in the horizontal plane or along a generic oblique direction exact closed form solutions are derived for each load condition under the common hypotheses of linear material law small or moderate axial deformation in bars and large nodal displacements despite their seeming simplicity these mechanical systems exhibit a wide variety of post critical responses not exhausted by the classical snapping and bifurcation phenomena in addition to regular primary and secondary branches the equilibrium paths may include neutral branches namely branches entirely composed of bifurcation or limit points besides their immediate theoretical interest these branches are particularly difficult to handle by the standard numerical procedures of non linear analysis so the given solutions may represent severe benchmark tests

generic initial ideals of artinian ideals having lefschetz properties or the strong stanley property for a standard artinian k algebra a r i we give equivalent conditions for a to have the weak or strong lefschetz property or the strong stanley property in terms of the minimal system of generators of gin i using the equivalent condition for the weak lefschetz property we show that some graded betti numbers of gin i are determined just by the hilbert function of i if a has the weak lefschetz property furthermore for the case that a is a standard artinian k algebra of codimension 3 we show that every graded betti number of gin i is determined by the graded betti numbers of i if a has the weak lefschetz property and if a has the strong lefschetz respectively stanley property then we show that the minimal system of generators of gin i is determined by the graded betti numbers respectively by the hilbert function of i

generic combinatorial rigidity of periodic frameworks abstract we give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks the characterization is a true analogue of the maxwell laman theorem from rigidity theory it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms to prove our rigidity theorem we introduce and develop periodic direction networks and z 2 graded sparse colored graphs

algorithms for three dimensional rigidity analysis and a first order percolation transition a fast computer algorithm the pebble game has been used successfully to analyze the rigidity of two dimensional 2d elastic networks as well as of a special class of 3d networks the bond bending networks and enabled significant progress in studies of rigidity percolation on such networks application of the pebble game approach to general 3d networks has been hindered by the fact that the underlying mathematical theory is strictly speaking invalid in this case we construct an approximate pebble game algorithm for general 3d networks as well as a slower but exact algorithm the relaxation algorithm that we use for testing the new pebble game based on the results of these tests and additional considerations we argue that in the particular case of randomly diluted central force networks on bcc and fcc lattices the pebble game is essentially exact using the pebble game we observe an extremely sharp jump in the largest rigid cluster size in bond diluted central force networks in 3d with the percolating cluster appearing and taking up most of the network after a single bond addition this strongly suggests a first order rigidity percolation transition which is in contrast to the second order transitions found previously for the 2d central force and 3d bond bending networks while a first order rigidity transition has been observed previously for bethe lattices and networks with chemical order here it is in a regular randomly diluted network in the case of site dilution the transition is also first order for bcc lattices but results for fcc lattices suggest a second order transition even in bond diluted lattices while the transition appears massively first order in the order parameter the percolating cluster size it is continuous in the elastic moduli this and the apparent nonuniversality make this phase transition highly unusual

a movable pair of tetrahedra when we draw all the diagonals of all the faces of a cube we obtain the edges of two regular tetrahedra keeping one of these fixed we move the other under the condition that the originally intersecting edges of the two tetrahedra should still remain coplanar i e intersect are parallel or coincide we determine all such finite motions these will be seen to constitute one dimensional and two dimensional smooth manifolds we also deal with the infinitesimal degrees of freedom of the motions of our mechanism in several positions the number of infinitesimal degrees of freedom is not one or two but is three this is connected with the bifurcation phenomenon of the solutions