Computers and Structures

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1 Computers and Structures () Contents lists available at ScienceDirect Computers and Structures journal homepage: Advanced form-finding of tensegrity structures Hoang Chi Tran, Jaehong Lee * Department of Architectural Engineering, Sejong University, 9 Kunja Dong, Kwangjin Ku, Seoul -, Republic of Korea article info abstract Article history: Received May 9 Accepted October 9 Available online November 9 Keywords: Tensegrity structure Single value decomposition Form-finding Eigenvalue problem A numerical method is presented for form-finding of tensegrity structures. The topology and the types of members are the only information that requires in this form-finding process. The eigenvalue decomposition of the force density matrix and the single 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 and robust in searching self-equilibrium configurations of tensegrity structures. Ó 9 Elsevier Ltd. All rights reserved.. Introduction The tensegrity structures first proposed by Fuller [] have been developed in recent years due to their innovative forms, lightweight and deployability. They belong to a class of free-standing pre-stressed pin-jointed cable strut system where contacts are allowed among the struts []. Their classification is presented as Class (where bars do not touch) and Class (where bars do connect to each other at a pivot) []. As a pioneering work of formfinding, so-called force density method was proposed by Schek [] for form-finding of tensile structures. Motro et al. [] presented the dynamic relaxation which has been reliably applied to tensile structures [] and many other non-linear problems. Vassart and Motro [] employed the force density method in symbolic form for searching new configurations. Recently, Masic et al. [], Zhang and Ohsaki [9] and Estrada et al. [] developed new numerical methods using a force density formulation. Micheletti et al. [] used a marching procedure for finding stable placements of a given tensegrity, and Zhang et al. [] employed a refined dynamic relaxation procedure for form-finding of nonregular tensegrity systems. Most recently, Rieffel et al. [] introduced an evolutionary algorithm for producing large irregular tensegrity structures. Tibert and Pellegrino [] presented a review paper for the existing methods for form-finding problem of tensegrity structures. The most recent review for this problem can be found in Juan and Tur []. In most available form-finding methods, the symmetry of structure and/or some of member lengths must be assumed known in advance. Moreover, element force density coefficients are solved in symbolic form. For example, (i) symmetric properties are * Corresponding author. Tel.: + ; fax: +. address: jhlee@sejong.ac.kr (J. Lee). employed in a group theory so as to simplify form-finding problem as presented in []; (ii) a number of member lengths are specified at the start in a dynamic relaxation procedure and non-linear programming, as presented in [,,]; however, these information may not always be available or easy to define at the beginning; and (iii) force density coefficients are considered as symbolic variables which cannot be applied for structures with large number of members []. In this paper, a numerical method is presented for form-finding of tensegrity structures. Both Classes and of tensegrity structures are considered. Tensegrities satisfying either stability (i.e., the tangent stiffness matrix is positive definite) or super stability (i.e., the geometrical stiffness matrix is positive definite) can be obtained by present form-finding procedure in a few of remarkable iterations, which is more efficient and versatile than other available methods only dealing with super stable tensigrities (e.g. [9,]) that are more restrictive in the real mechanical structures. The topology and the types of members, i.e. either compression or tension, are the only information that requires in this form-finding process. In other words, any initial nodal coordinate, symmetric properties, and member lengths are not necessary for the present form-finding. The force density matrix is derived from an incidence matrix and an initial set of force densities assigned from prototypes, while the equilibrium matrix is defined by the incidence matrix and nodal coordinates. The eigenvalue decomposition of the force density matrix and the single 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. The evaluation of the eigenvalues of tangent stiffness is also included for checking the stability of the tensegrity structures. -99/$ - see front matter Ó 9 Elsevier Ltd. All rights reserved. doi:./j.compstruc.9..

2 H.C. Tran, J. Lee / Computers and Structures (). Formulation of self-equilibrium equations.. Fundamental assumptions In this study, the following assumptions are made in tensegrity structures: The topology of the structure in terms of nodal connectivity is known. Members are connected by pin joints. No external load is applied and the self-weight of the structure is neglected during the form-finding procedure. There are no dissipative forces acting on the system... Force density method for the self-equilibrium and free-standing tensegrity structures For a d-dimensional tensegrity structure with b members, n free nodes and n f fixed nodes (supports), its topology can be expressed by a connectivity matrix C s ð R bðnþn f Þ Þ as discussed in [,,9]. Suppose member k connects nodes i and jði < j), then the ith and jth elements of the kth row of C s are set to and, respectively, as follows: >< for p ¼ i; C sðk;pþ ¼ for p ¼ j; ðþ >: otherwise: If the free nodes are numbered first, then to the fixed nodes, C s can be divided into two parts as C s ¼½CC f Š; where Cð R bn Þ and C f ð R bn f Þ describe the connectivities of the members to the free and fixed nodes, respectively. Let x; y; zð R n Þ and x f ; y f ; z f ð R n f Þ denote the nodal coordinate vectors of the free and fixed nodes, respectively, in x, y and z directions. For a simple two-dimensional two-strut tensegrity structure as shown in Fig., which consists of six members (b ¼, four cables and two struts) and four nodes, the connectivity matrix Cð R Þ is given in Table. The equilibrium equations in each direction of a general pinjointed structure given by [] can be stated as C T QCx þ C T QC f x f ¼ p x ; C T QCy þ C T QC f y f ¼ p y ; C T QCz þ C T QC f z f ¼ p z ; ðþ ðaþ ðbþ ðcþ where p x ; p y and p z ð R n Þ are the vectors of external loads applied at the free nodes in x; y and z directions, respectively. The symbol, ðþ T, denotes the transpose of a matrix or vector. And Qð R bb Þ is diagonal square matrix, calculated by () () () () () Fig.. A two-dimensional two-strut tensegrity structure. The thick and thin lines represent the struts and cables, respectively. () Table The incidence matrix of the -D two-strut tensegrity structure. Q ¼ diagðqþ; where qð R b Þ suggested in [] is the force density vector, defined by q ¼fq ; q ;...; q b g T in which each component of this vector is the force f i to length l i ratio q i ¼ f i =l i known as force density or self-stressed coefficient in []. Without external loading, Eq. () can be rewritten neglecting the self-weight of the structure as Dx ¼ D f x f ; Dy ¼ D f y f ; Dz ¼ D f z f ; ðþ ðþ ðaþ ðbþ ðcþ where matrices Dð R nn Þ and D f ð R nn f Þ are, respectively, given by D ¼ C T QC; D f ¼ C T QC f or by D ¼ C T diagðqþc; D f ¼ C T diagðqþc f : ðaþ ðbþ ðaþ ðbþ It is noted that when external load and self-weight are ignored, a tensegrity system does not require any fixed node, and the tensegrity geometry is defined by the relative position of the nodes, and the system can be considered as free-standing, forming a rigid-body free in space [,]. In this context, Eq. (b) and (b) vanish, and Eq. () becomes: Dx ¼ ; Dy ¼ ; Dz ¼ ; C Member/node () () () () () () ð9aþ ð9bþ ð9cþ where D known as force density matrix [,] or stress matrix [ 9] can be written directly by [,,9] without using Eq. () or () as follows: q k if nodes i and j are connected by member k; >< P D ði;jþ ¼ q k for i ¼ j; kx >: otherwise ðþ in which X denotes the set of members connected to node i. For example, for the two-dimensional two-strut tensegrity structure in Fig., D can be written explicitly from Eq. () as q þ q þ q q q q q q þ q þ q q q D ¼ q q q þ q þ q q : q q q q þ q þ q ðþ From Eq. (), it is obvious that D is always square, symmetric and singular with a nullity of at least since the sum of all components

3 H.C. Tran, J. Lee / Computers and Structures () 9 in any row or column is zero for any tensegrity structure. Different from the matrix D of the cable net, which is always positive definite [], D of the tensegrity structure is semi-definite due to the existence of struts as compression members, with q k <. Consequently, it cannot be invertible. For simplicity, Eq. (9) can be reorganized as D½xyzŠ¼½Š: ðþ However, by substituting Eq. () into Eq. (9) the equilibrium equations of the tensegrity structure can be expressed as C T diagðqþcx ¼ ; ðaþ C T diagðqþcy ¼ ; ðbþ C T diagðqþcz ¼ : ðcþ Eq. () can be reorganized as Aq ¼ ; ðþ where Að R dnb Þ is known as the equilibrium matrix in [], defined by C T diagðcxþ B A ¼ C T diagðcyþ A: ðþ C T diagðczþ Eq. () presents the relation between force densities and nodal coordinates, while Eq. () shows the relation between projected lengths in x; y and z directions, respectively and force densities. Both Eqs. () and () are linear homogeneous systems of self-equilibrium equations with respect to nodal coordinates and force densities, respectively.. Requirement on rank deficiency conditions of force density and equilibrium matrices Let q be the vector of force density and C be the incidence matrix of a d-dimensional tensegrity structure in self-equilibrium. It is well known that the set of all solutions to the homogeneous system of Eq. () is the null space of D. The dimension of this null space or rank deficiency of D is defined as n D ¼ n r D ; ðþ where r D ¼ rankðdþ. It is obvious that vector I ¼f; ;...; g T ð R n Þ, is a solution of Eq. () since the sum of the elements of a row or a column of D is always equal to zero. The most important rank deficiency condition related to semi-definite matrix D of Eq. () is defined by n D P d þ : ðþ This condition forces Eq. () to yield at least d useful particular solutions [] which exclude the above vector I due to degenerating geometry of tensegrity structure [,9]. These d particular solutions form a vector space basis for generating a d-dimensional tensegrity structure. Therefore, the minimum rank deficiency or nullity of D must be ðd þ Þ for configuration of any tensegrity structure embedding into R d, which is equivalent to the maximum rank condition of D proposed by [ 9,] as follows: maxðr D Þ¼n ðd þ Þ: ðþ Similarly, the set of all solutions to the homogeneous system of Eq. () lies in the null space of A. Let n A denote dimension of null space of the equilibrium matrix A which is computed by n A ¼ b r A ; ð9þ where r A ¼ rankðaþ. The second rank deficiency condition which ensures the existence of at least one state of self-stress can be stated as s ¼ n A P ; ðþ where s is known as the number of independent states of self-stress, while the number of infinitesimal mechanisms is computed by m ¼ dn r A, as presented in [,]. It is clear that Eq. () allows Eq. () to create at least one useful particular solution []. In short, based on these two rank deficiency conditions, Eqs. () and (), the proposed form-finding procedure searches for self-equilibrium configurations that permit the existence of at least one state of self-stress in the structure. It should be noted that these two are necessary but not sufficient conditions which have to be satisfied for any d-dimensional tensegrity structure to be in a self-equilibrium state [,,]. The sufficient conditions for tensegrity or self-stressed pin-jointed structures can be found in [,].. Form-finding process The proposed form-finding procedure only needs to know the topology of structure in terms of the incidence matrix C, and type of each member, i.e. either cable or strut which is under tension or compression, respectively. Based on element type, the initial force density coefficients of cables (tension) are automatically assigned as + while those of struts (compression) as, respectively, as follows: 9T < = q ¼ þ fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} þ þ fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} : ðþ : ; cables struts Subsequently, the force density matrix D is calculated from q by Eq. (). After that, the nodal coordinates are selected from the eigenvalue decomposition of the matrix D which is discussed in the next section. These nodal coordinates are substituted into Eq. () to define force density vector q by the single value decomposition of the equilibrium matrix A which is also presented in the next section. The force density matrix D is then updated by Eq. (). The process is iteratively calculated for searching a set of nodal coordinates ½x yzš and force density vector q until the rank deficiencies of Eqs. () and () are satisfied, which forces Eqs. () and () become true. In this context, at least one state of self-stress can be created, s P. In this study, based on required rank deficiencies from Eqs. () and () the form-finding process is stopped as n D ¼ d þ ; n A ¼ ; ðaþ ðbþ where n D and n A are minimum required rank deficiencies of the force density and equilibrium matrices, respectively... Eigenvalue decomposition of force density matrix The square symmetric force density matrix D can be factorized as follows by using the eigenvalue decomposition []: D ¼ UKU T ; ðþ where Uð R nn Þ is the orthogonal matrix (UU T ¼ I n, in which I n R nn is the unit matrix) whose ith column is the eigenvector basis / i ð R n Þ of D. Kð R nn Þ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e., K ii ¼ k i. The eigenvector / i of U corresponds to eigenvalue k i of K. The eigenvalues are in increasing order as k k k n : ðþ It is clear that the number of zero eigenvalues of D is equal to the dimension of its null space. Let p be the number of zero and

4 H.C. Tran, J. Lee / Computers and Structures () negative eigenvalues of D. There are two cases need to be considered. The first one is p n D, and the other is p > n D. Case. The first n D orthonormal eigenvectors of U are directly taken as potential nodal coordinates ½x y zšu ¼½/ / / n Š: ðþ D The algorithm then iteratively modifies the force density vector q as small as possible to make the first n D eigenvalues of D become null as k i ¼ ði ¼ ; ;...; n DÞ: ðþ All the projected lengths Lð R bn D Þ of U along n D directions for b members are computed by L ¼ CU ¼½ðC/ Þ ðc/ Þ ðc/ n ÞŠ D to remove one vector / i among n D eigenvector bases of U if ðþ C/ i ¼ ðþ or which / i causes detjl d ðl d Þ T j¼; ð9þ where L d ð R b Þ which indicates the vector of lengths of b members from any combination of d eigenvectors among n D above eigenvector bases in d-dimensional space (assuming d ¼ ) is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L d ¼ ðc/ i Þ þðc/ j Þ þðc/ k Þ : ðþ Eq. () shows / i is linearly dependent with the vector I while Eq. (9) proves at least one of b members in d-dimensional structure has a zero length. If there is no / i which satisfies Eqs. () and (9), the first three eigenvectors of U are chosen as nodal coordinates ½x yzš for -dimensional tensegrity structure. Accordingly, D will finally have the required rank deficiency n D without any negative eigenvalue. It implies D is positive semidefinite, and any tensegrity structure falling into this case is super stable regardless of material properties and level of self-stress coefficients [ 9,]. Case. Where p > n D, the rank deficiency may be forced to be larger than requirement or enough but D may not be positive semi-definite during iteration. Additionally, the proposed form-finding procedure will evaluate the tangent stiffness matrix of the tensegrity structure which is given in [,,,9] as follows: K T ¼ K E þ K G ; where K E ¼ Adiag ea l K G ¼ I D A T ; ðþ ðaþ ðbþ in which K E is the linear stiffness matrix, K G is the geometrical stiffness matrix induced by pre-stressed or self-stressed state; e; a and l R b denote the vectors of Young s moduli, cross-sectional areas and pre-stressed lengths of b members of the tensegrity structure, respectively; I ð R Þ and are the unit matrix and tensor product, respectively. If the tangent stiffness matrix is positive definite, then the structure is stable when its rigid-body motions are constrained; i.e., the quadratic form of K T is positive with respect to any non-trivial motion d as d T ðk T Þd > ðþ or eigðk T Þ¼½k P k P P k dn r > fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} positive stiffness k ¼ k ¼¼k r ¼ Š; fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} rigid-body motions ðþ where r ¼ dðd þ Þ= is the number of independent rigid-body motions. Using this criterion, stability of any pre-stressed or tensegrity structure can be controlled by checking eigenvalues of tangent stiffness matrix of the structure [,]. In short, the best scenario of configuration in -dimensional space is formed by three best candidate eigenvectors selected from the first fourth eigenvector bases which corresponding to the first fourth smallest eigenvalues, respectively. These eigenvalues will be gradually modified to be zero by the proposed iterative form-finding algorithm. In other words, the proposed formfinding procedure has repeatedly approximated equilibrium configuration such that D½x y zš½ Š: ðþ.. Single value decomposition of the equilibrium matrix The equilibrium matrix A is computed by substituting the set of approximated nodal coordinates [x y z] from Eq. () into Eq. (). In order to solve linear homogeneous system (Eq. ()) the single value decomposition [] is carried out on the equilibrium matrix A: A ¼ UVW T ; ðþ where Uð R dndn Þ¼½u u u dn Š and Wð R bb Þ¼½w w w b Š are the orthogonal matrices. Vð R dnb Þ is a diagonal matrix with non-negative single values of A in decreasing order as r P r P P r b P : ðþ As indicated in Eq. (b), the iterative form-finding algorithm is successful in case of n A ¼. Accordingly, there are also two cases for s during the iterative form-finding procedure: Case. s ¼, there exists no null space of A. In other words, the right single value ðr b Þ of A in V is not equal to zero. It denotes that Eq. () has no nonzero force density vector q as a solution. In this case, if the right single vector basis ðw b Þ in W corresponding to smallest singular value ðr b Þ in V is used as the approximated q, the sign of q may not match with that of q. Thus, all columns of W employed to compute a vector q that best matches q are scanned by form-finding procedure. The procedure stops sign-finding until the sign of all components of w j ðj ¼ b; b ;...; Þ is identical to that of q, i.e. signðw j Þsignðq Þ. That vector w j is directly taken as the approximated q. In doing so, the form-finding procedure defines the approximated q that matches in signs with q, such that Aq : ðþ Case. s ¼, it is known [] that the bases of vector spaces of force densities and mechanisms of any tensegrity structure are calculated from the null spaces of the equilibrium matrix. In this case, the matrices U and W from Eq. () can be expressed, respectively, as U ¼½u u u ra jm m dn ra Š ð9aþ W ¼½w w w b jq Š ð9bþ where the vectors mð R dn Þ denote the mð¼ dn r A Þ infinitesimal mechanisms; and the vector q ð R b Þ matching in signs

5 H.C. Tran, J. Lee / Computers and Structures () with q is indeed the single state of self-stress which satisfies the homogeneous Eq. (). In summary, the eigenvalue decomposition of force density matrix D and the single value decomposition of the equilibrium matrix A are performed iteratively to find the feasible sets of nodal coordinates [x y z] and force density vector q which satisfy the minimum required rank deficiencies of the force density and equilibrium matrices as presented in Eq. (), respectively. Let the coordinates of node i be denoted as p i ¼½x i y i z i ŠR. It should be noted that since the tensegrity structure without fixed nodes is a free body in the space, the force density vector q does not change under affine transformation [,9,] which transforms p i to p i as follows: p i ¼ p i T þ t; ðþ where Tð R Þ and t ð R Þ represent affine transformation which is a composition of rotations, translations, dilations, and shears. It preserves collinearity and ratios of distances; i.e., all points lying on a line are transformed to points on a line, and ratios of the distances between any pairs of the points on the line are preserved []. Hence, numerous geometries in self-equilibrium are derived by Eq. () with the same force density vector q. For simplicity, in this paper, it is assumed that t = and T is considered as B C T A: ðþ Since the tensegrity structure should satisfy the self-equilibrium conditions, the vector of unbalanced forces e f ð R dn Þ defined as follows can be used for evaluating the accuracy of the results: e f ¼ Aq: ðþ The Euclidean norm of e f is used to define the design error as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ e f ðe f Þ T : ðþ Two set of parameters which are nodal coordinates and force density vector can be simultaneously defined by proposed form-finding procedure through following algorithm. Algorithm Step : Define C by Eq. () for the given topology of tensegrity structure. Step : Specify the type of each member to generate initial force density vector q by Eq. (). Set i ¼. Step : Calculate D i using Eq. (). Step : Carry out Eq. () to define ½x y zš i through Eq. (). Step : Determine A i by Eq. (). Step : Perform Eq. () to define q iþ through Eq. (). Step : Define D iþ with q iþ by Eq. (). If Eq. () is satisfied, the solutions exist. Otherwise, set i ¼ i þ and return to Step. Step : If D is positive semi-definite, go to Step. Otherwise, specify material property, force density coefficient for each member based on force density vector found. Step 9: If Eq. () is fulfilled, go to Step. Otherwise, set i ¼ i þ and return to Step. Step : The process is terminated until Eq. () has been checked. The final coordinates and force density vector are the solutions. Otherwise, set i ¼ i þ and return to Step.. Numerical examples Numerical examples are presented for several tensegrity structures using Matlab Version. (Ra) []. Based on algorithm () () () () () (9) () Fig.. A two-dimensional hexagonal tensegrity structure. Table The force density coefficients of the -D hexagonal tensegrity structure. Force density coefficients Tibert and Pellegrino [] () () Estrada et al. [] Present q q... q q Fig.. The obtained geometry of the two-dimensional hexagonal tensegrity structure. () () () () () () Fig.. A two-dimensional four-strut tensegrity structure () () Fig.. The obtained geometry of the two-dimensional four-strut tensegrity structure.

6 H.C. Tran, J. Lee / Computers and Structures () Table Eigenvalues of the tangent stiffness matrix of the -D four-strut tensegrity structure. Axial stiffness k k k k k k k e i a i ¼ developed, both nodal coordinates and force density vector are simultaneously defined with limited information of nodal connectivity and the type of the each member. A hexagon, expandable octahedron and truncated icosahedron belong to Class tensegrity structure, while the remainings are of Class... Two-dimensional tensegrity structures... Hexagon The initial topology of the hexagonal tensegrity structure comprising three struts and six cables (Fig. ) which was studied by Tibert and Pellegrino [] and Estrada et al. [] is herein used for verification purpose. Initial nodal coordinates, member lengths and force density coefficients are unknown in advance. The only () () () 9 () (9) () () () () () () () () (9) () () () () () () () () () () () (9) () () () () Fig.. An expandable octahedron tensegrity structure. information is the incidence matrix C and the type of each member which is employed to automatically assign the initial force density vector by proposed form-finding procedure as q ¼fq q ¼ ; q q 9 ¼ g T : ðþ The obtained force density vector normalized with respect to the force density coefficient of the cable, as presented in Table, agrees well with those of Tibert and Pellegrino [] and Estrada et al. []. The associated stable configuration of the structure is plotted in Fig.. The form-finding procedure converges in only one iteration with the design error ¼ :. The structure obtained has only one self-stress state ðs ¼ Þ and one infinitesimal mechanism ðm ¼ Þ when its three rigid-body motions are constrained indicating it is statically and kinematically indeterminate []. The force density matrix D is positive semi-definite, and the structure is certainly super stable regardless of materials and Design error (ε) Table The force density coefficients of the expandable octahedron tensegrity structure. Force density coefficients Tibert and Pellegrino [] Estrada et al. [] Present q q... q q.... Number of iterations toward self equilibrium Fig.. The convergence of the proposed iterative algorithm for the expandable octahedron. a b Fig.. The obtained geometry of the expandable octahedron tensegrity structure. (a) Top view, (b) perspective view.

7 H.C. Tran, J. Lee / Computers and Structures () prestress levels [ 9,]. Consequently, the proposed form-finding procedure with limited information about the incidence matrix and element prototype is indeed capable of finding a self-equilibrium stable tensegrity structure by imposing the two necessary rank deficiency conditions.... Four-strut tensegrity As a next example, consider a two-dimensional four-strut tensegrity structure comprising four struts and four cables (Fig. ). Similarly, the only information is the incidence matrix C and the type of each member which is employed to automatically assign the initial force density vector by proposed form-finding procedure as q ¼fq q ¼ ; q q ¼ g T : ðþ The obtained force density vector normalized with respect to the force density coefficient of the cable is as follows: q ¼fq q ¼ ; q q ¼ g T : ðþ The associated configuration of the structure whose nodal coordinates are obtained and presented in Fig.. The form-finding Table Main specification in form-finding procedure of the expandable octahedron tensegrity structure at every iteration. Iteration q q q q l s=l c procedure converges in only one iteration with the design error ¼ :. The structure achieved has only one self-stress state ðs ¼ Þ and no infinitesimal mechanism ðm ¼ Þ after constraining its three rigid-body motions indicating it is statically indeterminate and kinematically determinate []. Contrary to all the other approaches, the proposed form-finding does not require the force density matrix positive semi-definite. Particularly in this problem, the force density matrix D is negative semi-definite (possessing one negative eigenvalue) implying that the structure is not super stable. Accordingly, tangent stiffness of the structures has been investigated using Eq. (). For simplicity, all members are assumed to have the same axial stiffness e i a i ¼. The force density coefficients of the members are particularly set as Eq. (). The nonzero eigenvalues of K T after neglecting first three zero ones corresponding to three rigid-body motions are listed in Table which shows the smallest is.9; i.e., the structure is definitely stable... Three-dimensional tensegrity structures... Expandable octahedron The tensegrity structure consisting of struts and cables (Fig. ) was investigated first by Tibert and Pellegrino [] and then Design error (ε) Table The force density coefficients of the truncated icosahedron tensegrity structure. Force density coefficients Estrada et al. [] Present q q.. q q 9.. q 9 q... Number of iterations toward self equilibrium Fig.. The convergence of the proposed iterative algorithm for the truncated icosahedron. a b Fig. 9. The obtained geometry of the truncated icosahedron tensegrity structure. (a) Top view, (b) perspective view.

8 H.C. Tran, J. Lee / Computers and Structures () by Estrada et al. [] using symbolic analysis and numerical method, respectively. The calculated force density vector after normalizing with respect to the force density coefficient of the cable, as presented in Table, agrees well with those of Tibert and Pellegrino [] and Estrada et al. []. The associated stable configuration of the structure is presented in Fig.. All of struts and cables have the same length, respectively. The length ratio between struts and cables is l s =l c ¼ : coinciding with the one ðl s =l c ¼ :99Þ stated by Estrada et al. []. The form-finding procedure converges in iterations. The design error convergence is described in Fig. with ¼ : 9. In order to illustrate the proposed form-finding procedure, main specification at every step of iteration is presented in Table. As can be seen from this table, for the practical purpose it seems the proposed algorithm sufficiently converges at iteration, but for the demonstration of the efficiency all iterations are displayed. The structure obtained has only one self-stress state ðs ¼ Þ and one infinitesimal mechanism ðm ¼ Þ except for its six rigid-body motions. Accordingly, it belongs to statically and kinematically indeterminate structure []. The force density matrices D is positive semi-definite which leads structure to super stable regardless of materials and prestress levels [ 9,].... Truncated icosahedron More complicated example is the truncated icosahedron tensegrity structure with struts and 9 cables which was analyzed by Estrada et al. []. The calculated force density vector after normalizing with respect to the force density coefficient of the cable, as presented in Table, agrees well with that of Estrada et al. []. The associated stable configuration of the structure is plotted in Fig. 9. The design error ðþ shown in Fig. which exposes the convergence of the proposed form-finding in 9 iterations (a little bit more effective in convergence in comparison with iterations presented by []) is: 9. The obtained structure possesses one state of self-stress ðs ¼ Þ and infinitesimal mechanisms ðm ¼ Þ excluding its six rigid-body motions. The force density matrix D is positive semi-definite which leads structure to super stable regardless of materials and prestress levels [ 9,]. It is clear that the introduction of single prestress stiffens all infinitesimal mechanisms to make the structure stable.... Octahedral cell The system shown in Fig. has five struts and eight cables. Four of five struts form a quadrangular polygon. The calculated force density vector after normalizing with respect to the force density coefficient of the cable is: q ¼fq q ¼ ; q 9 q ¼ ; q ¼ g T : ðþ The associated configuration of the structure whose nodal coordinates are obtained and displayed in Fig.. The form-finding procedure converges in only one iteration with the design error ¼ :9. The structure obtained has only one state of self-stress ðs ¼ Þ and no infinitesimal mechanism ðm ¼ Þ after constraining its six rigid-body motions indicating it is statically indeterminate and kinematically determinate []. Similar to the force density matrix D of the four-strut tensegrity, that of the octahedral cell is also negative semi-definite (possessing one negative eigenvalue) implying that the structure is not super stable. Accordingly, tangent stiffness of the structures has been investigated using Eq. (). For simplicity, all members are assumed to have the same axial stiffness e i a i ¼. The force density coefficient of each member is directly set as Eq. (). The nonzero eigenvalues of K T after neglecting first six zero ones corresponding to six rigid-body motions are listed in Table which shows the smallest is.; i.e., the structure is mechanically stable. () () () () () () (9) () () () () () ()... Tensegrity dome A three-dimensional tensegrity dome displayed in Fig. has six struts and cables. The calculated force density vector after normalizing with respect to the force density coefficient of the cable is: q ¼fq q ¼ :; q q ¼ :; q q ¼ :; q q ¼ :g T : ðþ Fig.. An octahedral cell tensegrity structure. Fig. shows the obtained self-equilibrium configuration. The form-finding procedure converges in only one iteration with the design error ¼ :9. The structure obtained has only one a b Fig.. The obtained geometry of the octahedral cell tensegrity structure. (a) Top view, (b) perspective view.

9 H.C. Tran, J. Lee / Computers and Structures () Table Eigenvalues of the tangent stiffness matrix of the octahedral cell tensegrity structure. Axial stiffness k k k k k k k k k 9 k k k e i a i ¼ () () () () () () (9) () () () () 9 () () () () () () () Fig.. A three-dimensional six-strut tensegrity dome structure. a b Fig.. The obtained geometry of the three-dimensional six-strut tensegrity dome structure (a) Top view, (b) Perspective view. state of self-stress ðs ¼ Þ and four infinitesimal mechanisms ðm ¼ Þ after constraining its six rigid-body motions indicating it is statically and kinematically indeterminate []. The force density matrix D is positive semi-definite which leads structure to super stable regardless of materials and prestress levels [ 9,].. Concluding remarks The advanced form-finding procedure for both Classes and tensegrity structures has been proposed. The force density matrix is derived from the incidence matrix and initial set of force densities formed by the vector of type of member forces. The elements of this vector consist of unitary entries + and for members in tension and compression, respectively. The equilibrium matrix is defined by the incidence matrix and nodal coordinates. The eigenvalue decomposition of the force density matrix and the single value decomposition of the equilibrium matrix are performed iteratively to find the range of feasible sets of nodal coordinates and force densities which satisfy the required rank deficiencies of the force density and equilibrium matrices, respectively. Any assumption about initial nodal coordinates, symmetric properties, member lengths and the positive semi-definite condition of the force density matrix is not necessary in the proposed form-finding procedure. A rigorous definition is given for the required rank deficiencies of the force density and equilibrium matrices that lead to a stable non-degenerate d-dimensional tensegrity structure. The eigenvalues of the tangent stiffness are also evaluated for checking the stability of the tensegrity structures. In the numerical examples, a very good convergence of the proposed method has been shown for two-dimensional and three-dimensional tensegrity structures. The proposed algorithm is strongly capable of searching novel configurations with limited information of topology and the member s type. As a natural extension of this research, form-finding with more complicated constraints awaits further attention. Acknowledgments This research was supported by a grant (code#-r&d-b) from Cutting-edge Urban Development Program funded by the

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