Author's Accepted Manuscript

Form-finding of tensegrity structures witharbitrary strut and cable members

Seunghye Lee, Jaehong Lee

PII: S0020-7403(14)00170-2DOI: http://dx.doi.org/10.1016/j.ijmecsci.2014.04.027Reference: MS2721

To appear in: International Journal of Mechanical Sciences

Cite this article as: Seunghye Lee, Jaehong Lee, Form-finding of tensegritystructures with arbitrary strut and cable members, International Journal ofMechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2014.04.027

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Form-finding of tensegrity structures with arbitrary

strut and cable members

Seunghye Lee1, Jaehong Lee∗

Department of Architectural Engineering, Sejong University98 Kunja Dong, Kwangjin Ku, Seoul 143-747, Korea

Abstract

An advanced form-finding method for tensegrity structures is presented. Inthe present study, determining an appropriate candidate for strut membersusing the discontinuity condition of struts is performed. Additionally, theself-equilibrium and stability properties of tensegrity structures are obtainedby using the force density method combined with a genetic algorithm. Theproposed algorithm is effectively capable of searching for appropriate strutcandidates with limited information in nodal connectivity. Appropriate mem-ber types and a feasible set of the nodal coordinates and the force densitiescan be found efficiently. Based on numerical examples, it is found that theproposed method is very efficient and robust in searching the self-equilibriumconfiguration of tensegrity structures.

Keywords: Tensegrity structures, Form-finding, Force density method,Genetic algorithm

1. Introduction

A tensegrity is a structure that consists of a set of discontinuous com-pressive components inside a set of continuous tensile components to definea stable volume in space (Pugh, 1976). A key step of the tensegrity struc-ture design is the determination of their equilibrium configuration, knownas form-finding (Tibert et al., 2003). There have been many studies on the

∗Professor, corresponding author. Tel.:+82-2-3408-3287; fax:+82-2-3408-3331Email address: jhlee@sejoing.ac.kr (Jaehong Lee)

1PhD student

Preprint submitted to International Journal of Mechanical Sciences May 9, 2014

form-finding of tensegrity structures, such as Masic et al. (2005), Estradaet al. (2006) and Li et al. (2010a). However, as initial input data, mostform-finding methods are required to specify the nodal connectivity, the ini-tial force densities and the types of members (i.e., either strut or cable).These initial values tend to dominate the final stability properties of tenseg-rity structures. Finally, it still remains as a challenging problem to find alabelling of tensegrity members (Ehara et al., 2010).

In this regard, the present study aims at determining an appropriate can-didate for strut members using the discontinuity condition of struts. Addi-tionally, the proposed method is performed to determine the self-equilibriumand stability properties of the tensegrity structures using the force densitymethod combined with a genetic algorithm. It is noteworthy that the pro-posed method requires only nodal connectivity as initial input data. Theforce density method is widely used in the form-finding of tensegrity litera-ture (Schek, 1974; Pandia et al., 2006; Zhang et al., 2006; Tran et al., 2010).Most form-finding methods using the force density method require the sym-metry properties that can be used to systematically reduce the number offorce density variables, equilibrium equations and geometrical variables (Juanet al., 2008). A different approach is to use a genetic algorithm. Several stud-ies have researched the form-finding methods of tensegrity structures usingthe genetic algorithms for searching self-equilibrium state (Paul et al., 2005;Xu et al., 2010; Yamamoto et al., 2011; Koohestani, 2012). Most recently, afinite element method (Klinka et al., 2012) and a structural stiffness matrix(Zhang et al., 2014) have been introduced for the form-finding of irregular orlarge-scale tensegrities.

In this paper, a force density method combined with a genetic algorithmis applied to the process of obtaining initial force densities and a single in-tegral feasible set of force densities. First, strut candidate groups that meetthe discontinuity condition of struts are selected from all the elements. Sec-ond, based on the candidates of strut members, upper and lower bounds offorce density for strut and cable members are set limits [−1, 0] and [0, 1], re-spectively. Finally, the form-finding process is performed using a constrainedminimisation problem which is based on the smallest eigenvalues of the forcedensity matrix and a standard deviation of the force density in the cablesuntil a set of force densities satisfies the required rank deficiencies of the forcedensity and equilibrium matrices.

The proposed method can find the appropriate member type and feasi-ble set of the nodal coordinates and the force densities efficiently. Also, the

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present method does not require any symmetry condition of tensegrity struc-tures. The fitness function using the standard deviation can be significant increating uniform force density values. Several numerical examples of tenseg-rity structures are presented to demonstrate the efficiency and accuracy ofthe proposed method. The paper is structured as follows. Section 2 will de-scribe the selection process of member type using the discontinuity conditionof strut members. In Sections 3 and 4, the formulation of self-equilibriumequations and the form-finding process will be presented, respectively. Thenumerical results and discussions will be analysed in Section 5 and Section 6will conclude this paper.

2. Selection of member types

2.1. Separation between struts

According to Pugh’s definition (Pugh, 1976) of tensegrity, no connectionis allowed between two compression members. In this study, the tensegritystructures in which each node has only one strut and a few cables are consid-ered. In such a design approach, a d-dimensional tensegrity structure withbs compression members and n free nodes has the following equation (Li etal., 2010b).

n = 2bs (1)

For a tensegrity structure with b total members can be expressed by aconnectivity matrix C (∈ R

b×n) as discussed in (Schek, 1974). If member kconnects nodes i and j (i < j), then the ith and jth elements of the kth rowof a connectivity matrix C are set to 1 and -1, respectively, as follows:

C(k,p) =

⎧⎨⎩

1 for p = i−1 for p = j0 otherwise

(2)

A simple two-dimensional two-strut tensegrity structure shown in Fig.1will be used as an example. The structure consists of six members (b = 6)and four nodes (n = 4). The connectivity matrix C (∈ R

6×4) is given inTable 1. As the discontinuity condition of struts, if element 1 is a strutmember, nodes 1 and 3 which are connected by element 1 should not beshared with any other strut. In other words, the elements connected to any

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node with nonzero value in the first row of the connectivity matrix C arenot strut members. Finally, only element 4 can be a strut member. Threekinds of strut candidates which satisfy the discontinuity condition of strutsare obtained as shown in Fig.2. If member k connects nodes i and j (i < j),then the discontinuity condition of struts is formulated as

bk : The member k is a strut. (k = 1, 2, · · · , b) (3)

bl :

{if C(l,i) = 0 and C(l,j) = 0 ⇒ The member l is a strut.

if C(l,i) �= 0 or C(l,j) �= 0 ⇒ The member l is a cable.(4)

(l = 1, 2, · · · , b and l �= k)

The tensegrity structure with selected strut members requires a verifica-tion process through equilibrium equations and the form-finding method.

2.2. Assignment of members

As the second stage of the tensegrity design, member types should bespecified in order to perform the form-finding process. If the set of strutcandidates is obtained from Eq.(3) and Eq.(4), each member is assigned byusing upper and lower bounds of a force density for the algorithm. The forcedensity method uses a linear equation in the nodal coordinates; this equationcan be linearised using Eq.(5), known as force density (Tibert et al., 2003).

qk =fklk

(5)

where any member k has a member force fk and a length of element lk (k =1, 2, · · · , b).

For a member k of the tensegrity structure, upper and lower bounds ofqk are defined as follows.

{if member k is a strut : − 1 ≤ qk < 0

if member k is a cable : 0 < qk ≤ 1(6)

In Eq.(6), the negative(positive) number part of a force density denotesthat member is a strut(cable). The upper limit and lower limit in Eq.(6) willbe used for the genetic algorithm which is presented in the section 4.

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3. Equilibrium equations

Let x,y, z (∈ Rn) denote the nodal coordinate vectors of the free node,

in x, y and z directions. When the external load and self-weight are ignored,the equilibrium equations can be written as follows:

D[x y z] = CTdiag(q)C[x y z] = [0 0 0] (7)

where D (∈ Rn×n) is the force density matrix (Tibert et al., 2003; Estrada

et al., 2006), or stress matrix (Connelly, 1982).Instead of using the connectivity matrix C and force density vector q,

the force density matrix D can be written directly by Vassart et al. (1999);Connelly et al. (1995) as

D(i,j) =

⎧⎪⎪⎨⎪⎪⎩−qk if nodes i and j are connected by member k∑k∈Ω

qk for i = j

0 otherwise

(8)

in which Ω denotes the set of members connected to node i. Eq.(8) indicatesthat the force density matrix D is always square and symmetric. Using asecond term of Eq.(7), the equilibrium equation can be expressed as

CTdiag(q)Cx = 0 (9a)

CTdiag(q)Cy = 0 (9b)

CTdiag(q)Cz = 0 (9c)

Eq.(9) can be reorganized as

Aq = 0 (10)

where A (∈ Rdn×b) is known as the equilibrium matrix, defined by

A =

⎛⎝ CTdiag(Cx)

CTdiag(Cy)CTdiag(Cz)

⎞⎠ (11)

Eq.(7) shows the relationship between force density matrix D and nodalcoordinates, and Eq.(10) illustrates the relationship between the equilibriummatrix A and force densities. Additionally, the vector of unbalanced forces

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εf (∈ Rdn) defined as follows can be used to evaluate the accuracy of the

results:

εf = Aq (12)

The Euclidean norm of εf is used to define the design error ε as

ε =√(εf )Tεf (13)

4. Form-finding process

4.1. Formulation

The proposed form-finding procedure only needs to know the nodal con-nectivity of structure for a form-finding procedure. Based on obtained ele-ment type which is mentioned earlier, the initial force densities are automat-ically assigned using random seeds of the genetic algorithm which is satisfiedwith the upper and lower limit. Subsequently, the force density matrix Dis caculated from q0 by Eq.(7). After that, the nodal coordinates are se-lected from the eigenvalue decomposition of the matrix D. The square forcedensity matrix D can be factorized as follows by using the eigenvalue decom-position(Meyer, 2000).

D = ΦΛΦT (14)

where Φ (∈ Rn×n) is the orthogonal matrix (ΦΦT = In, in which In ∈

Rn×n is the unit matrix) whose ith column is the eigenvector basis φi (∈ R

n)of D. Λ (∈ R

n×n) is the diagonal matrix whose diagonal elements are thecorresponding eigenvalues, i.e., Λii = λi. The eigenvector φi of Φ correspondsto eigenvalue λi of Λ. The eigenvalues are in increasing order as

λ1 ≤ λ2 ≤ · · · ≤ λn (15)

It is clear that the number of zero eigenvalues of D is equal to the dimen-sion of its null space. The form-finding procedure for the tensegrity structuresrequires the rank deficiency conditions of force density and equilibrium ma-trices. From Eq.(8), it is obvious that D is always square, symmetric andsingular with a nullity of at least one since the sum of the elements of the row

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or column of the force density matrix (D) always equals zero for any tenseg-rity structure (Tran et al., 2010). In a d-dimensional tensegrity structure,the rank deficiency of D has at least d useful particular solutions. Therefore,the rank deficiency condition is defined as

nD ≥ d+ 1 (16)

where nD is the dimension of null space or rank deficiency of D.The second rank deficiency condition is related to the dimension of null

space of the equilibrium matrix A. The dimension of null space of the equi-librium matrix A is identical to ”s”, known as the number of independentstates of self-stress. A tensegrity structure ensures the existence of at leastone state of self-stress and can be stated as

s = nA ≥ 1 (17)

Where the number of zero and negative eigenvalues of D are larger thanminimum required rank deficiencies of the force density matrices (Eq.(16)),the proposed form-finding procedure will evaluate the tangent stiffness ma-trix of the tensegrity structure which is given in (Ohsaki et al., 2006) forevaluation of stable or super-stable state as follows:

KT = KE +KG (18)

in whichKE is the linear stiffness matrix andKG is the geometric stiffnessmatrix induced by pre-stressed or self-stressed state (Guest, 2006). If thetangent stiffness matrix is positive definite, then the structure is stable. Alsoif the force density matrix D has the minimum rank deficiency, D is positivesemi-definite and the number of independent rigid-body motions is d(d+1)/2,then these conditions lead structure to super stable regardless of materialsand prestress levels (Zhang et al., 2007).

4.2. Genetic algorithm

In this work, a genetic algorithm was used to evolve the range of feasi-ble sets of the nodal coordinates and the force densities. Firstly, the forcedensity matrix is calculated from the initial force density vector. A genetic

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algorithm is then used to obtain the initial force density values that lead tothe force density matrix to satisfy Eq.(7). The fitness function consists ofthree variables, α, β and γ.

Minimize : αβγ (19)

Subject to :

i)

∣∣∣∣qsi − qsjqsi

∣∣∣∣ ≤ ε0 (qsi , qsj ∈ Γs) (20a)

ii)

∣∣∣∣qci − qcjqci

∣∣∣∣ ≤ ε0 (qci , qcj ∈ Γc) (20b)

in which

α =s∑

i=1

|λi| (21)

β =b∑

j=1

1

|qj| (22)

γ =

√√√√√√√m∑k=1

(qk)2

m−

⎛⎜⎜⎝

m∑k=1

qk

m

⎞⎟⎟⎠

2

(23)

In Eq.(19), α is the sum of the first d + 1 smallest eigenvalues of theforce density matrix as suggested by Koohestani (2012). The term β is avariable as suggested by Koohestani (2012) that significantly increases thevalue of the fitness function related to the force densities with zero or near-zero values. This feature ensures that the result value converges to the forcedensities near the upper and lower bounds in the minimisation problem with

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a genetic algorithm. In Eq.(19), γ is the standard deviation of the forcedensity in the cables and m denotes the number of cables. The role of γ isto make uniform the force density values of cables so that the value of thefitness function can be minimized. This logic controls any force densitiesthat may be abnormally increased. Finally, this effective variable is used notonly to significantly improve the behaviour of the fitness function, but alsoto provide uniformity to the force density values.

In Eq.(20), Γ denotes the total set of the force density, Γc is the set of theforce density for cable members, and Γs is the set of the force density for strutmembers. The term ε0 is used to define the tolerance. Eq.(20a) and Eq.(20b)are optional constraints for achieving the regular shapes of the tensegrities.In other words, the force density value of arbitrary members can be drawnevenly using these equations if the members are assigned to the same class.However, unlike other methods using grouping of elements that reduce thetotal number of variables, this method does not. Also the proposed methoddoes not use additional condition (such as a symmetry condition). Sinceindividual force density results of each elements can be obtained by adjustingthe tolerance value which induces any grouping automatically, this methodis flexible and versatile than prior existing methods. Tensegrity structurescan be highly efficient in terms of regular form. The best situation is one inwhich the tensegrity structure is divided into two groups (cables and struts).Eventually, these constraints affect the convergence speed improvement ofthe algorithm.

For a simple two-dimensional two-strut tensegrity structure in Fig.1, theforce density vector which is obtained by using the proposed form-findingmethod is given in Table 2. Firstly, three strut candidates can be obtainedas shown in Fig.2 and each member is given upper and lower bound. Next,a total of three results for three cases can be obtained. All results are nor-malized with respect to the force density coefficient of the cable. If an in-appropriate strut candidate which does not satisfy the equilibrium equation(Eq.(7)) is chosen, the genetic algorithm process stops and moves on to thenext candidate. In this process, all of the cases are verified using the roughdesign error in the five iteration.

As shown in Table 2, all three results of the proposed method are in per-fect agreement with the corresponding results of the previous study (Ohsakiet al., 2006). It is noteworthy that the proposed method does not use anysymmetry and grouping conditions. In all three cases, the final geometrieshave the same arrangement with Fig.3 and show a square shape. In the case

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of Fig.2(a) and Fig.2(b), the nodal coordinates change the position to satisfythe self-equilibrium condition. In other words, the selected strut membersare rearranged to make the stable configuration of the structure. In this ex-ample, since all three results are same, all results can be selected as the finalsolution.

5. Numerical examples

5.1. 2D Hexagonal tensegrity

An initial nodal connectivity of the two-dimensional hexagonal tensegritystructure which was studied by Tibert et al. (2003) and Estrada et al. (2006)is shown in Fig.4. This structure, which is composed six nodes and nine el-ements, is herein used for verification purpose. The only initial informationis the nodal connectivity which is employed to select appropriate candidatesfor strut members using the discontinuity condition of struts. The structureobtained has only one self-stress state (s = 1) and one infinitesimal mech-anism (m = 1) when its rigid-body motions are constrained indicating it isstatically and kinematically indeterminate (Tibert et al., 2003).

Firstly, six kinds of strut candidates can be obtained as shown in Fig.5.All cases are satisfied with the discontinuity condition of struts. Each strutmember is set upper and lower bound to [-1,0]. As is the case with thetwo-dimensional two-strut tensegrity structure, the six candidates all showthe same results, with only a difference of convergence speed among them.After all, there is the choice problem among the results. In this example,the last candidate case (Fig.5(f)) is chosen due to being able to satisfy thestopping criteria ε = 0.001 of the fitness function at first in which the caseconverged after 98 generations. And the corresponding first five eigenvaluesare [1.60, 1.60, −5.20×10−3, −1.80×10−3, 8.00×10−17]. In this example,Eq.(20a) is used to reduce the number of iterations. In Table 3, a comparisonof the force density values of the previous study (Tran et al., 2010) and theforce density values of the proposed method with selected (7), (8) and (9)strut elements is presented. The force density results of the proposed methodare in good agreement with those of the previous study.

The associated stable configuration of the structure is plotted in Fig.6.The force density matricesD is positive semi-definite which leads structure tosuper stable (Tran et al., 2010; Koohestani , 2013). The nodal coordinateschange position in order to satisfy the self-equilibrium condition and theselected strut members are rearranged to make the stable configuration of

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the structure. The force density result values of other strut candidate cases inFig.5 also tend to reach a self-equilibrium state. All cases tend to rearrangeselected struts in order to satisfy the stable condition of the structure. Thisindicates that a suitable set of the stable properties is easily obtained byusing fewer numbers of iteration.

In order to show the effect of the new term, gamma in Eq.(23), we made acomparison between the two cases where gamma is used or not. Fig.7 showsthe comparison of the performance according to the effect of the standard de-viation term in fitness function for the two-dimensional hexagonal tensegritystructure. We can see that the minimum fitness function value rapidly tendsto zero with the increasing number of evaluation generations than those ofthe case without gamma.

5.2. 3D Truncated tetrahedral tensegrity

The initial topology of truncated tetrahedral tensegrity has 12 nodes and24 members as shown Fig.8. Truncated tetrahedral tensegrity was inventedby Fuller (1962), and this structure is mentioned in many studies on form-finding tensegrity structures (Tibert et al., 2003; Koohestani, 2012; Estradaet al., 2006). The obtained structure possesses one state of self-stress (s = 1)and 7 infinitesimal mechanisms (m = 7) (Tibert et al., 2003).

Nine strut candidate groups which satisfy the discontinuity condition ofstruts can be obtained as shown in Table 4 and each member is set upperand lower bound. The form-finding process was performed through severalchanges in the range of other cable and strut members. Firstly, 300 maxi-mum generations were conducted in order to observe a tendency. In this case,the obtained force density results tend to be grouped into three types. Asfor the case of the previous example, the nine cases all show the same resultthat arrives at a self-equilibrium state of tensegrity structures. However, thefitness value does not reach the design error (ε = 0.001) within 300 gener-ations even both Eq.(20a) and Eq.(20b) constraints are used. Nonetheless,the single integral feasible self-stress states are obtained. For comparison,as suggested by many researchers, the force densities in the truncating-edgecables (qt) are held as unity (Linkwitz et al., 1971). The obtained forcedensities in the vertical cables (qv) and the force densities in the struts (qb)are shown in Fig.9. We found 6 cables with qv = 0.47 and 6 struts withqb = −0.35. This result satisfies the super-stable condition and completelylies in the super-stable path of the analytical solution (Tibert et al., 2003)and other literature (Koohestani , 2013).

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To obtain more precise comparison, a fixed grouping condition which leadthree design variables should be used. In Table 5, a comparison of the fitnessvalue and the first five eigenvalues according to the number of variables forthe two-dimensional hexagonal tensegrity structure is presented. In the casewhere two constraints are used (12 variables), a fitness value of 5.03×10−1 wasobtained in the 300 generation and the corresponding first five eigenvaluesare [2.70, − 4.20 × 10−2, 1.10 × 10−2, 2.40 × 10−3, − 1.30 × 10−16]. Bycontrast, in the case where a grouping condition is used, a fitness valueof 1.80 × 10−3 was obtained in the 300 generation and the correspondingfirst five eigenvalues are [2.80, 3.80 × 10−4, 3.80 × 10−4, 3.80 × 10−4, −1.90 × 10−16]. This comparison shows that by increasing the total numberof variables, the number of iterations increases (Koohestani , 2013). Toincrease the convergence rate, thus grouping condition of elements shouldbe appropriately used. Therefore, in proposed method, minimum number ofmember groups are easily obtained automatically with the constraints. Thena grouping condition can be used to reduce the design variables and obtainthe fitness value which satisfy the design error.

The obtained super-stable configuration of the truncated tetrahedrontensegrity structure is plotted in Fig.10. The geometry follows Z-based pat-terns. In the Z-based form of a truncated tetrahedron, the cables lie alongthe edges of the structure and the struts are the diagonals connecting the ver-tices of the truncated tetrahedron (Zhang et al., 2012). Finally, this indicatesthat strut candidate groups and a suitable set of super-stable properties areeasily obtained using the proposed method with only the nodal connectivity.

5.3. 3D Truncated icosahedral tensegrity

The regular truncated icosahedral tensegrity has 60 nodes and 120 el-ements that consist of 30 identical compression members and 90 identicaltension cables. More detail on the connectivity of this tensegrity is reportedin Murakami et al. (2001). After the proposed form-finding procedure, allresults formed to have one self-stress state (s = 1) and 55 infinitesimal mech-anism modes, which is in agreement with the assumptions (Murakami et al.,2001).

Firstly, the combination of 120 elements taken 30 struts at a time withoutrepetition can be obtained as

120C30 =120!

30! 90!(24)

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In mathematics, a combination is a method of selecting several things out of alarger group. Only two valid strut candidates which satisfy the discontinuitycondition of struts can be obtained as shown in Table 6. A case of hugeelements has rather less strut candidates. Accordingly, the larger number ofelements of tensegrity structures, the more difficult it is to select appropriatecandidate which satisfy the discontinuity condition of struts. Both casesshow the same results of a self-equilibrium state of tensegrity structures.Setting 60 truncated edge cables (qt) as unity, the relation between qv andqb can be obtained. However, as stated previous example, it is clear that thenumber of iterations increases and the convergence rate reduces by increasingthe total number of variables (Koohestani , 2013). Consequently, if anyregular form of a tensegrity structure in necessary, the optional conditioncan be applied to the fitness functions. In other words, the specific solutionscan be derived in a versatile manner using the appropriate constraints or agrouping condition. In this example, the generalized multiple equilibriumpaths is obtained by using a grouping condition (three design variables).This result completely lies in the super-stable or non-super-stable path ofthe analytical solution (Tibert et al., 2003) and other literature (Koohestani, 2013) as shown in Fig.11. Also, the obtained super-stable configuration ofthe truncated icosahedral tensegrity structure is plotted in Fig.12 and thegeometry follows Z-based patterns.

6. Conclusion

In this study, an advanced form-finding method which can determine ap-propriate candidates for strut members, and the self-equilibrium and stabil-ity properties of tensegrity structures is presented. Firstly, candidate groupwhich satisfies the discontinuity condition of struts is selected from the topol-ogy of the tensegrity. Secondly, upper and lower bounds of force density forthe members are set. Finally, the form-finding process is performed by usingthe force density method combined with a genetic algorithm. The proposedmethod requires nodal connectivity only as initial input data. For all thenumerical examples, all strut candidates show the same results, with only adifference of convergence speed among them. Also, the selected strut mem-bers are rearranged to make a stable configuration of the structure. Theproposed algorithm is effectively capable of searching for appropriate strutcandidates with limited information in the nodal connectivity.

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Acknowledgement

This research was supported by National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology throughNRF2010-0019373 and 2012R1A2A1A01007405.

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Zhang, J.Y., Ohsaki, M., Adaptive force density method for form-findingproblem of tensegrity structures, International Journal of Solid and Struc-tures 43(2006) 5658-5673

Zhang, J.Y., Ohsaki, M., Stability conditions for tensegrity structures, Inter-national Journal of Solids and Structures 44(2007)3875-3886

Zhang, J.Y., Ohsaki, M., Self-equilibrium and stability of regular truncatedtetrahedral tensegrity structures, Journal of the Mechanics and Physics ofSolids 60 (2012) 1757-1770

Zhang L.Y., Li Y., Cao Y.P., Feng X.Q., Stiffness matrix based form-findingmethod of tensegrity structures, Engineering Structures 58 (2014) 36-48

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Figure 1: A initial nodal connectivity of the two-dimensional two-strut tensegrity structure

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(a) (b) (c)

Figure 2: ; The candidate groups of strut members for the two-dimensional two-struttensegrity structure (Thick lines indicate struts) (a) (1) and (4), (b) (2) and (3), (c) (5)and (6) elements

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Figure 3: The obtained geometry of the two-dimensional two-strut tensegrity structure

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Figure 4: A initial nodal connectivity of the two-dimensional hexagonal tensegrity struc-ture

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(a) (b) (c)

(d) (e) (f)

Figure 5: ; The candidate groups of strut members for the two-dimensional hexagonaltensegrity structure (Thick lines indicate struts) (a) (1), (3) and (5), (b) (1), (4) and (9),(c) (2), (4) and (6), (d) (2), (5) and (7), (e) (3), (6) and (8), (f) (7), (8) and (9) elements

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Figure 6: The obtained geometry of the two-dimensional hexagonal tensegrity structure(Note that (1), (3) and (5) elements are struts.)

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0.0

2.0

4.0

6.0

8.0

10.0

0 10 20 30 40 50 60 70

Fitness function :

Fitness function :

Figure 7: Comparison of the performance according to the effect of the standard deviationterm in fitness function for the two-dimensional hexagonal tensegrity structure

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Figure 8: A initial nodal connectivity of the three-dimensional truncated tetrahedrontensegrity structure

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-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.0 0.5 1.0 1.5 2.0

q b

qv

Analytical solution

Koohestani K.(2013)

Proposed method

Super stable line

Figure 9: Comparison between the results of qv versus qb for the three-dimensionaltruncated tetrahedral tensegrity structure

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Figure 10: The obtained geometry of the three-dimensional truncated tetrahedron tenseg-rity structure

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-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.0 0.5 1.0 1.5 2.0

q b

qv

Koohestani K.(2013)

Proposed method

Super stable line(Analytical solution)

Figure 11: Comparison between the results of qv versus qb for the three-dimensionaltruncated icosahedral tensegrity structure

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Figure 12: The obtained geometry of the three-dimensional truncated icosahedral tenseg-rity structure

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Table 1: The incidence matrix of the two-dimensional two-strut tensegrity structure

C

Member / node 1 2 3 4

1 1 0 -1 0

2 0 1 -1 0

3 1 0 0 -1

4 0 1 0 -1

5 1 -1 0 0

6 0 0 1 -1

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Table 2: The force density coefficients of the two-dimensional two-strut tensegrity struc-ture

Member Force density

Ohsaki et al. (2006) Strut candidate

(1) and (4) (2) and (3) (5) and (6)

1 1.0 -1.0 1.0 1.0

2 1.0 1.0 -1.0 1.0

3 1.0 1.0 -1.0 1.0

4 1.0 -1.0 1.0 1.0

5 -1.0 1.0 1.0 -1.0

6 -1.0 1.0 1.0 -1.0

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Table 3: The force density coefficients of the two-dimensional hexagonal tensegrity struc-ture

Member Force density

Tran et al. (2010) Present study

1 1.0 1.0

2 1.0 1.0

3 1.0 1.0

4 1.0 1.0

5 1.0 1.0

6 1.0 1.0

7 -0.5 -0.5

8 -0.5 -0.5

9 -0.5 -0.5

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Table 4: The candidate groups of strut members for the three-dimensional truncatedtetrahedron tensegrity structure

Case Strut members

(1) 1 9 10 21 22 24

(2) 1 11 14 15 17 22

(3) 3 5 7 10 14 16

(4) 6 8 10 13 15 16

(5) 6 12 13 15 18 23

(6) 7 10 13 14 16 24

(7) 13 14 15 16 17 18

(8) 13 18 19 21 23 24

(9) 19 20 21 22 23 24

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Table 5: Comparison of the fitness value and first five eigenvalues according to constraintsor the number of variables for the two-dimensional hexagonal tensegrity structure

Case Fitness value First five eigenvalues Force density

The case 5.03× 10−1 2.70 qt = 1.00

where two constraints are used −4.20× 10−2 qv = 0.47

(24 variables) 1.10× 10−2 qb = −0.352.40× 10−3

−1.30× 10−16

The case 1.80× 10−3 2.80 qt = 1.00

where a grouping condition is used 3.80× 10−4 qv = 0.41

(3 variables) 3.80× 10−4 qb = −0.323.80× 10−4

−1.90× 10−16

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Table 6: The candidate groups of strut members for the three-dimensional truncatedicosahedral tensegrity structure

Case Strut members

(1) 61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

(2) 91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

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