Reshaping of tensegrities using a geometrical variation approach

Accepted Manuscript Reshaping of tensegrities using a geometrical variation approach K. Koohestani PII: DOI: Reference:

S0020-7683(15)00289-9 http://dx.doi.org/10.1016/j.ijsolstr.2015.06.025 SAS 8828

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International Journal of Solids and Structures

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Please cite this article as: Koohestani, K., Reshaping of tensegrities using a geometrical variation approach, International Journal of Solids and Structures (2015), doi: http://dx.doi.org/10.1016/j.ijsolstr.2015.06.025

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1

Reshaping of tensegrities using a geometrical variation

2

approach

3

K. Koohestani *

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Department of Structural Engineering, Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran

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Abstract

6

We propose a novel approach for the reshaping of tensegrities and pre-stressed pin-

7

jointed frameworks. Both free standing and restrained frameworks are studied. First, a

8

geometrical variation is introduced and considered as an initial geometrical change to a

9

self-equilibrated tensegrity. A new nonlinear set of equilibrium equations is then

10

established based on this variation and subsequently linearized using an expansion. The

11

linearization provides us with an underdetermined system of linear equations with a

12

rectangular Jacobian matrix which is derived in a very simple and compact matrix form.

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The system is finally solved iteratively using a Newton-Raphson method employing a

14

Moore-Penrose pseudo inverse operation to establish a new equilibrium state. The

15

method promises to generate very accurate results with a remarkable convergence rate.

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The outstanding features and performance of the proposed method are demonstrated

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and studied comprehensively through three examples.

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Keywords: Reshaping; Geometrical variation; linearization; Newton-Raphson method

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* Corresponding author. Tel.: +98 411 33392391; fax: +98 411 33356024.

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Email address: [email protected]; [email protected]

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1

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1. Introduction

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A tensegrity is a lightweight, pin-jointed framework with a rigidity which results from a

24

self-stressed equilibrium between cables and struts. In recent years, a wide range of

25

innovative applications have been developed for tensegrities due to their unique

26

characteristics. We briefly refer to applications including aerospace structures (Sultan,

27

2009; Tibert and Pellegrino, 2002), architectural and structural design (Motro, 2003;

28

Rhode-Barbarigos et al. 2010), molecules and cells (Stamenovic and Ingber, 2009),

29

biomechanics (Luo et al. 2008), smart systems (Ali and Smith, 2010; Moored et al. 2011)

30

and advanced engineering materials (Fraternali et al. 2012).

31

The analysis and design of tensegrity structures normally depend on two main variables

32

(considering that the connectivity is known) including geometry and the pre-stress level

33

of its elements. In the most general case, both the geometry and pre-stress level of the

34

elements are unknown. The solution process which simultaneously determines the self-

35

stressed equilibrium and the corresponding geometry is called form-finding. Form-

36

finding of tensegrities is usually performed through analytical and numerical methods.

37

The numerical methods are more practical for large and asymmetrical tensegrities

38

whilst the analytical methods are more appropriate for tensegrities with a high order of

39

symmetry. Form-finding of tensegrities has been widely studied using different

40

numerical methods. Motro (1984) employed the dynamic relaxation method and

41

Pellegrino (1986) suggested a nonlinear programming approach. Sultan et al. (1999)

42

proposed a reduced coordinate method and Masic et al. (2005) developed an algebraic

43

method based on invariant tensegrity transformations. Form-finding of tensegrity

44

structures using concepts of finite elements has been presented by Pagitz and Mirats Tur

45

(2009). Estrada et al. (2006), Tran and Lee (2010, 2013) and Zhang and Ohsaki (2006)

2

46

proposed numerical methods that employ either iterative eigenvalues or singular value

47

decompositions of the force density and equilibrium matrices. Koohestani and Guest

48

(2013) introduced a combined form of the equilibrium and geometrical compatibility

49

equations for analytical and numerical form-finding of tensegrity structures. Recently,

50

Zhang et al. (2014) developed an efficient stiffness matrix-based form-finding method

51

for tensegrities and form-finding of irregular tensegrities has been studied by Li et al.

52

(2010) using the Monte Carlo method. Rieffel et al. (2009) proposed an evolutionary

53

form-finding method, Koohestani (2012), Paul et al. (2005) and Xu and Luo (2010) used

54

genetic algorithms and Chen et al. (2012a, b) used colony systems. Also, Koohestani

55

(2013) suggested an unconstrained minimization approach to the simultaneous form-

56

finding and design of tensegrities with geometrical and force constraints.

57

Kanno (2013a) proposed a mixed integer linear programming approach to the

58

formation of tensegrities with kinematical indeterminacy (see also Kanno, 2012, 2013b

59

for kinematically determinate tensegrities). The method needs no information about the

60

topology of a tensegrity but requires a ground structure with fixed nodes. As a result,

61

new and irregular tensegrities (from the topological point of view) can be formed.

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Formation of irregular tensegrities has been studied by Zhang et al. (2006) using

63

dynamic relaxation method with kinetic damping. The method needs an initial geometry

64

but the configuration may be extended. Furthermore, during the form-finding process,

65

either the force or length of some elements can be fixed by an appropriate choice of

66

related stiffness.

67

A marching procedure for the form-finding of tensegrity structures has been presented

68

by Micheletti and Williams (2007a). They have also developed a method for the shape-

69

changing of tensegrities (Micheletti and Williams, 2007b). The method starts from an

70

initial known configuration and is based on the characterization of the rank-deficiency 3

71

manifold in the space of nodal coordinates. The change in shape is obtained using

72

element lengths as control parameters in compatibility equations. The compatibility

73

equations and orthogonality conditions form together a system of ordinary differential

74

equation and is solved efficiently with the Runge-Kutta integration method.

75

Symmetrical reconfiguration of tensegrity structures has been presented by Sultan et al.

76

(2002) employing nonlinear dynamics of tensegrities and symmetrical motions (see also

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Sultan, 2014 for tensegrity deployment by imposing motions along infinitesimal

78

mechanisms). There are a considerable number of studies related to the control and

79

deployment of tensegrity structures. In the most of these methods either cables or struts

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are served as actuators with adjustable pre-stress level and length i.e., shape changing is

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preformed through changing the length of elements employing efficient control

82

algorithms. However note that these interesting research topics may have some

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common points with those of our paper, but there are major differences in assumptions,

84

theory and objectives. We do not review these methods for the sake of brevity.

85 86

In this paper, we study the sensitivity of the equilibrium equations of a self-stressed

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tensegrity (free standing or restrained) with respect to a geometrical variation. The

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main aim of this paper is to establish a computational framework which enables us to

89

reshape a tensegrity based on an initial geometrical variation, while providing a new set

90

of force densities, nodal coordinates, and most importantly retains the self-stressed

91

state of the tensegrity. The main challenge is that we cannot necessarily form a self-

92

equilibrated tensegrity based on an arbitrarily selected geometrical variation. However,

93

it is possible to find an adequately close variation to the initially selected geometry

94

which corresponds to a self-stressed state. We address this challenge by proposing an

95

iterative approach which is based on a linearized form of the underdetermined 4

96

nonlinear system of equilibrium equations. A highly accurate solution with a very good

97

convergence rate is then obtained by using a Newton-Raphson method employing a

98

Moore-Penrose pseudo inverse operation. The method enables us to form a wide variety

99

of irregular tensegrities from those of regular or existing ones. Note that our method

100

only takes into consideration the change in the geometry of a tensegrity, therefore the

101

irregularity we pointed out is merely related to the geometry and subsequently in the

102

force density of elements but not topology of a tensegrity. Furthermore, this platform

103

may be viable for the design of tensegrities with special geometries or force

104

distributions. The performance of the method is comprehensively studied and tested

105

through three examples.

106

107

2. Self-equilibrium equations

108

This section describes two counterpart formulations of self-equilibrium equations for a

109

general three-dimensional tensegrity structure with n nodes and m elements. Consider

110

a typical node i which is adjacent (connected by an element) to all nodes belong to the

111

set Ω . The equilibrium of internal and external forces at node i in the x-, y- and z-

112

directions can be written as:

113

∑ f ik ( x i − x k ) / Lik = pix

(1)

k∈Ω

114

∑ f ik ( y i − y k ) / Lik = piy

(2)

k∈Ω

115

∑ f ik (z i − z k ) / Lik = piz

(3)

k∈Ω

5

116

where fik and Lik are the internal force and length of element ( i , k ) respectively and

117

pix , piy and piz are the external forces at node i in the x-, y- and z-directions. These

118

external forces are generally set to zero since tensegrities are self-equilibrated

119

structures. To form the equilibrium equations for a three-dimensional tensegrity, the

120

above equations must be defined and integrated for all nodes. This gives the force

121

density formulation of the equilibrium equations (Schek, 1974) as:

122

G[x , y , z] = [0, 0, 0] , G = BQBt

123

Here, G ∈R n×n is the force density matrix and Q = diag(q) , q = [q1 , q2 , K , qm ] t is the

124

diagonal matrix of force densities. Note that force density of an element, labeled by a

125

single number j , is defined as f j / L j . In Eq. (4), x = [x 1 , x 2 ,K , x n ] t , y = [ y1 , y2 ,K , yn ] t and

126

z = [z1 , z 2 ,K , z n ] t are vectors of the Cartesian coordinates of nodes. B = [bij ]n×m , known as

127

node-member incidence matrix, is defined as:

128

− 1 if i is the start node of element j  bij =  1 if i is the end node of element j 0 otherwise 

129

Equation (4) plays a crucial role in the form-finding of tensegrity structures. In general,

130

for a d-dimensional tensegrity in a state of self-stress, Eq. (4) must have at least d

131

independent solutions. However, due to the intrinsic rank deficiency of G , there is

132

always a trivial solution which is a vector of ones. Thus the minimum rank deficiency of

133

G for a d-dimensional tensegrity is d + 1 . This also means that G must have at least d + 1

134

zero eigenvalues. The stability of tensegrities may also be studied using the spectral

135

characteristics of the force density matrix, i.e. G (Zhang and Ohsaki, 2007). However,

136

this form of the equilibrium equations is appropriate only when finding a set of nodal

(4)

(5)

6

137

coordinates corresponding to a set of known force densities of the elements. For cases in

138

which the geometry of a tensegrity is known, whilst the vector of force densities is

139

unknown, another form of the equilibrium equations is used. Equation (6) gives the

140

relationships between the elements projected lengths and nodal coordinates as:

141

d x = B t x , d y = B t y , d z = Bt z

142

where d x , d y and d z are the vectors of the elements projected lengths in the x-, y- and z-

143

directions respectively. Substitution of the relationships given in Eq. (6) into Eq. (4)

144

yields three sets of equilibrium equations:

145

BQd x = B diag(d x )q = 0

(7-a)

146

BQd y = B diag(d y )q = 0

(7-b)

147

BQd z = B diag (d z )q = 0

(7-c)

148

By combining Eqs. (7a-c), an alternative set of equilibrium equations is expressed as:

149

B diag(d x ) Aq = 0 , A = B diag(d y )  B diag(d z )

150

where A ∈R dn×m (d is 2 or 3 for two or three-dimensional tensegrities) is the rectangular

151

equilibrium matrix. For cases where the geometry of a tensegrity is known, the

152

coefficient matrix in Eq. (8), A is known and a single state or multiple states of self-

153

stressed equilibrium can be identified according to the dimension of its null space. Note

154

that different bases can be calculated for the null space of a matrix (Stewart, 1973)

155

including triangular bases (normally obtained by Gauss elimination or LU factorization

156

methods) and orthogonal bases (normally obtained by singular value decomposition,

157

SVD).

(6)

(8)

7

158

159

3. Stability

160

In general, statically and kinematically indeterminate pin-jointed structures exhibit

161

different kinds of stability. However, in this paper the stability of a tensegrity is checked

162

after reshaping on the base of the pre-stress stability and super-stability. In general, a d-

163

dimensional tensegrity structure needs to satisfy the following three conditions to be

164

the super-stable (Zhang and Ohsaki, 2007).

165

(a) The force density matrix G has the minimum rank deficiency d + 1.

166

(b) G is positive semi-definite.

167

(c) The rank of the geometry matrix is d (d + 1)/2 or, equivalently, the member

168

directions do not lie on the same conic at infinity (Connelly, 1982).

169 170

It is useful to note that the first two conditions above play a crucial role in the study of

171

the stability of tensegrity structures since the third condition is usually satisfied (Zhang

172

and Ohsaki, 2012).

173

A tensegrity that does not meet above conditions is not super-stable but may still be

174

stable. However, for these cases, the stability can be investigated based on a pre-

175

stress/stiffness ratio of elements and spectral characteristics of the tangent stiffness

176

matrix (sum of the linear elastic stiffness and geometrical stiffness matrices). The reader

177

may refer to Ohsaki and Zhang (2006) for the necessary and sufficient conditions for the

178

stability of pin-jointed structures, including tensegrities.

179

A kinematically indeterminate tensegrity is pre-stress stable if the state of self-stress

180

stiffens all the infinitesimal mechanisms (excluding rigid body motions). The left null

181

space of the equilibrium matrix A forms a basis for the infinitesimal mechanisms of a

8

182

tensegrity. This basis can effectively be calculated employing the singular value

183

decomposition (SVD) as follows.

184

[U, V , W ] = SVD ( A )

185

where the vectors corresponding to infinitesimal mechanics (displacement modes that

186

do not elongate elements) are those vectors in U , entirely denoted by matrix U m , that

187

satisfies A t U m = 0 .

188

The stability of a tensegrity can be assessed by checking the positive definiteness of its

189

tangent stiffness matrix. However if an infinitesimal mechanism is applied to this matrix,

190

the terms involving equilibrium matrix are vanished (see Estrada et al. 2007, and notes

191

about second order rigidity in Connelly and Whiteley, 1996) therefore the only term

192

which may impart stiffness is a section of geometric stiffness, i.e. I d ⊗ G . The pre-stress

193

stability of a tensegrity is then checked by evaluating eigenvalues of the following

194

quadratic form.

195

Λ = U tm (I d ⊗ G)U tm

196

Note that this is the case for a tensegrity with a single state of self-stress. For multiple

197

states of self-stress the process is more complicated (Calladine and Pellegrino, 1991;

198

Pellegrino and Calladine, 1986).

199

In order to ensure that self-stress state stiffens infinitesimal mechanisms and exerts

200

positive work, all eigenvalues of the above-mentioned quadratic form must be positive

201

excluding 3 or 6 zero eigenvalues corresponding to rigid body motions in 2 or 3-

202

dimensional spaces.

203

The method proposed in this paper does not check the stability of a tensegrity during

204

the reshaping process, however, since we normally work based on small geometric

205

variations, the method normally preserves the stability. In section 7 we demonstrate this

(9)

(10)

9

206

feature through numerical examples but formation of an unstable tensegrity after

207

reshaping implies that the geometrical variation considered is not appropriate and must

208

be modified.

209

210

4. Definition of a geometrical variation

211

This section describes geometrical variations and how these variations affect the

212

equilibrium equations. Consider we have a tensegrity in a state of self-stress where its

213

nodal coordinates and force density of elements are denoted by vectors x , y , z and q

214

respectively. The equilibrium equations for this tensegrity can be written based on Eq.

215

(4) as follows:

216

(B diag(q)B t )[x , y , z] = [0, 0, 0]

217

Now let a variation in geometry be imposed on the tensegrity by applying changes

218

(denoted by ∆x , ∆y , ∆z ) to the nodal coordinates. The new set of equilibrium equations

219

can be written as:

220

(B diag(q)Bt )[x + ∆x , y + ∆y , z + ∆z] ≠ [0, 0, 0]

221

Clearly, we generally lose the equilibrium at all nodes, i.e. there will be out-of-balance

222

forces at all nodes. It is not possible to find a new set of force densities ( q + ∆q ) as given

223

in Eq. (13) since there is no one-to-one correspondence between an arbitrarily selected

224

geometry and a set of self-equilibrated force densities in tensegrities.

225

(B diag(q + ∆q)Bt )[x + ∆x , y + ∆y , z + ∆z] ≠ [0, 0, 0]

(11)

(12)

10

(13)

226

Only special geometries with their corresponding force densities provide a state of self-

227

stressed equilibrium. In this paper we aim to find a close variation to the predefined

228

geometry in such way that we could find a corresponding set of force densities ( q' ) and

229

retain the self-stressed equilibrium. This process is mathematically represented as

230

follows:

231

(B diag(q' )B t )[x + ∆x + ∆x' , y + ∆y + ∆y' , z + ∆z + ∆z' ] = [0, 0, 0]

232

In Eq. (14) [x + ∆x + ∆x' , y + ∆y + ∆y' , z + ∆z + ∆z' ] represents a new geometry close to that

233

generated by the initial variation in the nodal coordinates.

(14)

234

235

5. Proposed computational framework (free standing tensegrities)

236

The equilibrium equations given in Eq. (11) can be expanded by considering that the

237

three vectors of the nodal coordinates are joined together to make a single vector of

238

3n × 1 as follows:

239

B diag(q )B t  f (q , x , y , z ) =   

B diag(q )B

t

  x  0      y  = 0 B diag(q )B t   z  0

(15)

240

In this new form of equilibrium equations f (q , x , y , z ) is a vector-valued function

241

consisting of equilibrium equations of all nodes in the x-, y- and z- directions as given in

242

Eq. (16).

11

243

 f 1 x  0      M  M   f nx  0      f 1 y  0 f (q , x , y , z ) =  M  =  M       f ny  0  f  0  1z     M  M       f nz  0

244

For example, f 1 x is the equilibrium equation in node 1 and in the x- direction. This vector

245

is clearly a function of the force density of all members and nodal coordinates of the

246

framework. Since we are investigating the case of a self-stressed equilibrium, the right

247

hand vector must be a zero vector.

248

We partition the node-member incidence matrix ( B ) into n rows as follows:

249

b1    b B =  2  M    b n 

250

where, b i is the ith row of B . According to the above definitions, the equilibrium

251

equation of node i in x-, y-, and z- directions can be expressed as:

(16)

(17)

252

f ix = b i diag(q )B t x

(18-a)

253

f iy = b i diag(q)B t y

(18-b)

254

f iz = bi diag(q )Bt z

(18-c)

255

The equilibrium equations defined in Eq. (15) are an underdetermined nonlinear system

256

of equations. The system is underdetermined since there are 3n equations with 3n + m

12

257

variables. We linearize this system of equations using the expansion process given in Eq.

258

(19).

259

f (q + ∆q , x + ∆x , y + ∆y , z + ∆z) ≈ f (q , x , y , z ) + J (q , x , y , z ) Ψ

260

In Eq. (19) J is the matrix of all first-order partial derivatives of the vector valued

261

function f (q , x , y , z ) , known as the Jacobian matrix, and Ψ is a vector of small variations

262

in the variables.

263

(19)

 ∆q    ∆x Ψ=   ∆y     ∆z 

(20)

264

The Jacobian matrix can be calculated using:

265

 ∂f J=  ∂q

266

where its expanded form is as follows:

267

 ∂f 1 x  ∂q  1  M  ∂f nx  ∂q  1  ∂f 1 y  ∂q  1 J= M  ∂f ny  ∂q  1  ∂f 1z  ∂q  1  M  ∂f nz  ∂q  1

∂f ∂x

∂f ∂y

∂f 1 x ∂qm M ∂f nx L ∂qm ∂f 1 y L ∂qm M ∂f ny L ∂qm ∂f 1z L ∂qm M ∂f nz L ∂qm L

∂f  ∂z 

∂f 1 x ∂x 1 M ∂f nx ∂x 1 ∂f 1 y ∂x 1 M ∂f ny ∂x 1 ∂f 1 z ∂x 1 M ∂f nz ∂x 1

(21)

∂f 1 x ∂x n M ∂f nx L ∂x n ∂f 1 y L ∂x n M ∂f ny L ∂x n ∂f 1z L ∂x n M ∂f nz L ∂x n L

∂f 1 x ∂y 1 M ∂f nx ∂y 1 ∂f 1 y ∂y 1 M ∂f ny ∂y 1 ∂f 1z ∂y 1 M ∂f nz ∂y 1

∂f 1 x ∂y n M ∂f nx L ∂y n ∂f 1 y L ∂y n M ∂f ny L ∂y n ∂f 1z L ∂y n M ∂f nz L ∂y n L

13

∂f 1 x ∂z 1 M ∂f nx ∂z 1 ∂f 1 y ∂z 1 M ∂f ny ∂z 1 ∂f 1z ∂z 1 M ∂f nz ∂z 1

L

L L

L L

L

∂f 1 x  ∂z n  M  ∂f nx  ∂z n  ∂f 1 y  ∂z n   M  ∂f ny  ∂z n  ∂f 1 z  ∂z n   M  ∂f nz  ∂z n 

(22)

268

Based on the symbolic definition of the equilibrium equation of a typical node i, we

269

calculate all the derivatives, simplify the results, and integrate them simultaneously. We

270

may then express the Jacobian of f (q , x , y , z) in a very simple form according to the

271

known matrices defined earlier. This simple and compact form is given in Eq. (23).

272

J = [ A I 3 ⊗ (B diag(q )B t ) ]

273

We next adopt a Newton-Raphson scheme in order to iteratively solve the nonlinear

274

system of equations given in Eq. (15) as follows.

275

f ( q j + 1 , x j + 1 , y j +1 , z j +1 ) = f ( q j , x j , y j , z j ) + J ( q j , x j , y j , z j ) Ψ j

276

where,

(23)

(24)

277

∆q j   j ∆x Ψj = j  ∆y   j  ∆z 

278

q j +1 = q j + ∆q j

(26-a)

279

x j +1 = x j + ∆x j

(26-b)

280

y j +1 = y j + ∆y j

(26-c)

281

z j +1 = z j + ∆z j

(26-d)

282

Our aim is to find a new state of self-stressed equilibrium by imposing a moderately

283

small variation on the geometry of the tensegrity. We start the iterative process given in

284

Eq. (24) from an equilibrium state so that f (q 0 , x 0 , y 0 , z 0 ) = 0 . Clearly by imposing a slight

285

variation on the geometry (applying Ψ 0 ) we lose the equilibrium at all nodes. This can

(25)

14

286

be mathematically represented through Eq. (27) where p j is the out-of-balance force

287

vector in the jth iteration.

288

f (q j , x j , y j , z j ) = p j

289

Our problem is to find a geometry close to the predefined one in such a way that it

290

enables us to eliminate all out-of-balance forces i.e. p j = 0 . Hence we first calculate an

291

approximate variation in the force density of elements ( ∆q 0 ) using the following

292

equation. Note that at this stage the geometry of the tensegrity is known.

293

∆q 0 = ( A 0 )+ ( −I 3 ⊗ (B diag(q 0 )B t Ψ 0 )

294

In Eq. (28) ( A 0 )+ is the Moore-Penrose pseudo-inverse (Golub and Van Loan, 1996) of

295

the coefficient matrix of the equilibrium equations ( A ) that was calculated for the initial

296

geometry. It should be noted that we have an underdetermined system of equations

297

therefore a pseudo-inverse need to be used to calculate ∆q 0 . Due to the linearization and

298

since the solution is not unique the equilibrium will not be established at this step,

299

therefore:

300

f (q j +1 , x j +1 , y j +1 , z j +1 ) ≠ 0

301

We now have access to a set of variations of the force densities as well as the nodal

302

coordinates. The iterative process commences at this point with the aim of reducing the

303

out-of-balance force vector as far as is required. We first calculate the rectangular

304

Jacobian matrix and the out-of-balance force vector using the current information of

305

force densities and nodal coordinates, after which better approximations to the

306

geometrical and force variables are calculated using the following equation:

(27)

j = 1, K , k

(28)

(29)

15

307

Ψ j = ( J (q j , x j , y j , z j ))+ (−f (q j , x j , y j , z j ))

308

where the plus sign on the Jacobian matrix again refers to a pseudo-inverse operator.

309

The infinity norm of the new out-of-balance force vector is then calculated and

310

compared with a predefined very small value ε as represented in Eq. (31). The above

311

process is iterated until convergence is achieved.

312

f (q j +1 , x j +1 , y j +1 , z j +1 )

≤ε inf

(30)

ε = 1 × 10 −12

(31)

313

After convergence, we obtain a new set of force densities and nodal coordinates which

314

provide us with a new configuration in a state of self-stressed equilibrium. Note that if

315

the initial variation in geometry is considered moderately small the method normally

316

converges very fast and the final geometry is close to the initial one. However, if a very

317

large variation in geometry is considered the method may converge to a degenerated

318

geometry. As a result, large geometrical variations must be imposed through iterative

319

imposing of small variations and by using this approach, it is also possible to find self-

320

equilibrated configurations even for large geometrical variations.

321

322

6. Proposed computational frame work (restrained tensegrities)

323

The method proposed to this point has been formulated for free standing tensegrities;

324

however it can be generalized in order to reform general pre-stressed pin-jointed

325

frameworks with some restrained nodes. Here, we provide another version of the

326

proposed method for this type of framework. First we consider that all nodes of a pin-

327

jointed pre-stressed framework are divided into two sets, including free and restrained

328

nodes. All variables related to the free and restrained nodes are identified by adding the

16

329

letters f and r as subscripts. Accordingly, we partition the node-member incidence

330

matrix into two parts: the first part, denoted by B f , corresponds to the free nodes; and

331

the second part, denoted by B r , corresponds to the restrained nodes. These are

332

expressed as follows:

333

B  B= f  Br 

334

According to this partitioning, the equilibrium equations given in Eq. (15) need to be

335

modified in order to establish the self-stressed equilibrium at all free nodes. Eq. (33)

336

gives the new set of equilibrium equations as:

337

B f diag(q )B t  f f (q , x , y , z ) =   

338

The linearized form of the equilibrium equations for free nodes is also as follows:

339

f f (q j +1 , x j +1 , y j +1 , z j +1 ) = f f (q j , x j , y j , z j ) + J f (q j , x j , y j , z j ) Ψ jf

340

For this set of equations we again calculate the rectangular Jacobian matrix where its

341

simplified representation is given in Eq. (35):

342

J f = [ A f I3 ⊗ (B f diag(q)B tf ) ]

343

Note that in Eq. (35) A f is the coefficient matrix of the equilibrium equations related to

344

the free nodes which is mathematically represented as:

345

B f diag(d x )   A f = B f diag(d y ) , d x = B t x , d y = B t y , d z = B t z B f diag(d z )  

(32)

B f diag(q )B

t

  x  0      y  = 0 t B f diag(q )B   z  0

(33)

(34)

(35)

17

(36)

346

The solution process for these kinds of frameworks is almost identical to that presented

347

earlier with, however, some slight changes in the variables. We start from a self-stressed

348

state (only related to free nodes) i.e. f f (q 0 , x 0 , y 0 , z 0 ) = 0 . Then the first variation in the

349

force densities is calculated as follows:

350

∆q 0 = ( A 0f ) + ( −I3 ⊗ (B f diag(q 0 )B tf Ψ 0f )

351

in which

(37)

352

 ∆q j   j ∆x j Ψ f =  fj  ∆y   jf   ∆z f 

353

x jf +1 = x jf + ∆x jf ,

∆x rj = 0 ,

x rj +1 = x 0r

(39-a)

354

y jf +1 = y jf + ∆y jf ,

∆y rj = 0 , y rj +1 = y 0r

(39-b)

355

z jf +1 = z jf + ∆z jf ,

∆z rj = 0 , z rj +1 = z 0r

(39-c)

356

Note that in the above relationships x 0r , y 0r and z 0r are the nodal coordinates of the

357

restrained nodes that are fixed during the iterations (i.e. the geometrical variations

358

related to these nodes are all zero).

359

The new vector of iterated variations (including variations in the force densities and

360

geometrical variations) is then calculated through Eq. (40):

361

Ψ jf = ( J f (q j , x j , y j , z j ))+ ( −f f (q j , x j , y j , z j ))

362

The condition for convergence is also defined as:

(38)

(40)

18

363

f f (q j +1 , x j +1 , y j +1 , z j +1 )

≤ε inf

ε = 1 × 10 −12

(41)

364

365

7. Examples

366

7.1. Example 1

367

In Fig. 1(a) a two-dimensional tensegrity is shown. We study this tensegrity with respect

368

to two cases. In the first case there are no restrained nodes i.e. the tensegrity is free

369

standing, while in the second case the horizontal strut is removed and its two end nodes

370

are restrained (Fig. 1(c)).

371

First case: In this case the tensegrity has 6 nodes and 9 elements and is free standing. An

372

analytical solution for a self-stressed state of this tensegrity is available in the literature

373

(Koohestani and Guest, 2013). In Table 1 the corresponding force density of elements

374

are provided. Clearly, the initial configuration is regular with two axis of symmetry

375

whilst the force densities of the elements are packed into three groups. Now we assume

376

a random set of nodal geometrical variations as given in Table 2 and seek to find a new

377

self-stressed state through the proposed method. The method converges in only 5

378

iterations at which point a new self-stressed state is reached. The new set of force

379

densities (all elements have a different force density value) and nodal coordinates are

380

given in Tables 1-2. The corresponding configuration is depicted in Fig. 1(b) where the

381

new tensegrity is completely irregular. Note that even though the initial nodal variations

382

are assumed to be very large (in comparison with the initial nodal coordinates) the

383

method efficiently converged after only 5 iterations and generated a feasible solution.

384

The convergence history (infinity norm of the out-of-balance nodal forces vs. iteration

385

number) of the numerical procedure is shown in Table 3. In this case, we have 19

386

investigated both pre-stress stability and super-stability of the tensegrity. The results

387

provided in Tables 4-5 (based on the context given in Section 3) verify that the stability

388

is preserved during reshaping process.

389

390

Table 1

391

Set of force densities before and after imposing nodal geometrical variations (example 1,

392

case 1) Element Number 1 2 3 4 5 6 7 8 9

Initial force density ( q 0 ) 2.0000 1.0000 2.0000 2.0000 1.0000 2.0000 -1.0000 -1.0000 -1.0000

Final force density ( q 5 ) 1.5349 1.1844 2.0471 2.4802 1.1893 1.7575 -0.9973 -0.8673 -1.0543

393

394

Table 2

395

Set of nodal coordinates before and after imposing nodal geometrical variations

396

(example 1, case 1) Initial nodal coordinates (self-stressed)

Nodal variations

Final nodal coordinates (self-stressed)

x0

y0

∆x 0

∆y 0

x5

y5

-1.0000 -1.0000 -2.0000 2.0000 1.0000 1.0000

1.5275 -1.5275 0.0000 0.0000 1.5275 -1.5275

0.9138 0.7067 0.5578 0.3134 0.1662 0.6225

0.9879 0.1704 0.2578 0.3968 0.0740 0.6841

-0.2208 -0.2055 -1.3824 2.2706 1.2975 1.5209

2.3323 -1.3794 0.3598 0.3323 1.7633 -0.8372

397

20

398 399

Fig. 1 a) initial self-stressed configuration (free standing); b) configuration after

400

imposing nodal geometrical variations (free standing); c) initial self-stressed

401

configuration (restrained); and d) configurations after imposing nodal geometrical

402

variations (restrained)

403

404

Table 3

405

Infinity norm of the out-of-balance force vector at different iterations (example 1)

f f

inf

inf

(case 1)

1 0.6955

2 0.0836

(case 2)

0.9999

0.0240

Iteration number 3 4 3.6025e-4 3.4055e-9 3.9703e-5

406 21

4.8408e-11

5 7.1054e-15 1.7764e-15

407

408

Table 4

409

Infinitesimal mechanisms and the corresponding work done by the state of self-stress

410

before and after reshaping (example 1, case 1) Before reshaping Node

Free standing (1st infinitesimal mode)

Free standing (2nd infinitesimal mode)

Free standing (3rd infinitesimal mode)

Free standing (4th infinitesimal mode)

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

eig (U tm (I3

⊗ G)U m ) =

[0.0,0.0,0.0,1.3333] ux uy

-0.0222 0.1331 0.0555 0.0555 -0.0222 0.1331 -0.6126 0.1413 -0.2357 -0.2357 -0.6126 0.1413 0.0935 0.5003 0.2969 0.2969 0.0935 0.5003 0.0299 -0.3121 -0.1411 -0.1411 0.0299 -0.3121

-0.0356 -0.0356 -0.0865 -0.5275 -0.5784 -0.5784 0.0173 0.0173 -0.2295 0.2035 -0.0432 -0.0432 -0.2343 -0.2343 -0.3675 0.2164 0.0832 0.0832 -0.5360 -0.5360 -0.4240 -0.0919 0.0201 0.0201

411

412

413

414 22

After reshaping eig (Utm (I3 ⊗ G)U m ) = [0.0,0.0,0.0,1.8061] ux uy

-0.0991 0.3004 0.1132 0.1132 -0.2346 0.3974 -0.3871 0.3329 -0.0045 -0.0016 -0.4413 0.3577 -0.4615 -0.1946 -0.3197 -0.3163 -0.3280 -0.3067 0.0819 0.3579 0.2286 0.2336 0.3198 0.1631

-0.1487 -0.1470 -0.2737 -0.2736 -0.5101 -0.4558 0.2823 0.2853 0.0570 0.4367 0.1377 0.2064 -0.3049 -0.3038 -0.3885 0.0593 0.0514 0.0532 -0.3315 -0.3304 -0.4179 0.2446 0.3033 0.2898

415

Table 5

416

Eigenvalues of the force density matrix before and after reshaping (example 1, case 1)

λ1 λ2 λ3 λ4 λ5 λ6

Eigenvalues before reshaping 0.0000 0.0000 0.0000 2.0000 6.0000 6.0000

Eigenvalues after reshaping 0.0000 0.0000 0.0000 2.5232 5.1391 6.8867

417

418

Second case: In this case the tensegrity has 6 nodes and 8 elements and is not free

419

standing. The initial geometry is the same as first case but we have removed the

420

horizontal strut and restrained both ends. The initial self-stressed state remains

421

constant during this process. Now we apply a new set of nodal geometrical variations

422

and try to find a corresponding self-stressed state using our method. The method again

423

successfully determines a set of force densities corresponding to the initial variations in

424

5 iterations. The results are summarized in Tables 6-7. The final shape and history of the

425

convergence are depicted in Fig. 1(d) and Table 3, respectively. In this case, again the

426

stability is preserved during reshaping; see Table 8 for the results in more detail.

427

428

429

430

431

23

432

Table 6

433

Set of force densities before and after imposing nodal geometrical variations (example 1,

434

case 2) Element Number 1 2 3 4 5 6 7 8

Initial force density ( q 0 ) 2.0000 1.0000 2.0000 2.0000 1.0000 2.0000 -1.0000 -1.0000

Final force density ( q 5 ) 0.7939 0.8369 2.2859 3.6742 1.5160 1.2771 -0.4931 -1.4056

435

436

437

Table 7

438

Set of nodal coordinates before and after imposing nodal geometrical variations

439

(example 1, case 2) Initial nodal coordinates (self-stressed)

Nodal variations

Final nodal coordinates (self-stressed)

x0

y0

∆x 0

∆y 0

x5

y5

-1.0000 -1.0000 -2.0000 2.0000 1.0000 1.0000

1.5275 -1.5275 0.0000 0.0000 1.5275 -1.5275

0.5306 0.8324 0.0000 0.0000 0.2992 0.4526

0.4226 0.3596 0.0000 0.0000 0.4243 0.4294

-0.4558 -0.0520 -2.0000 2.0000 1.2471 1.4577

1.9224 -1.1951 0.0000 0.0000 1.9091 -1.1877

440

441

442

443

24

444

Table 8

445

Infinitesimal mechanisms and the corresponding work done by the state of self-stress

446

before and after reshaping (example 1, case 2) Before reshaping Node

Second case (restrained)

1 2 5 6

eig

(U tm (I 3

⊗ G)U m ) = 1.2000

After reshaping eig (U tm (I 3 ⊗ G)U m ) = 1.3197

ux

uy

ux

uy

0.4183 -0.4183 0.4183 -0.4183

-0.2739 -0.2739 0.2739 0.2739

-0.4607 0.2864 -0.4650 0.2893

0.3700 0.4668 -0.1834 -0.1321

447

448

7.2. Example 2

449

In this example reshaping of a truncated tetrahedron tensegrity is investigated. This

450

tensegrity is one of the most well-known and has been widely studied. The tensegrity

451

has 12 nodes and 24 elements (including 18 cables and 6 struts) and is free standing. A

452

set of force densities corresponding to a self-stressed state is extracted from an

453

analytical solution available in Tibert and Pellegrino (2002) and summarized in Table 9.

454

We also calculate the nodal coordinates of this tensegrity using the approach introduced

455

in Koohestani and Guest (2013). The configuration obtained and the nodal coordinates

456

(multiplied by 10) are depicted in Fig. 2(a) and provided in Table 10. The reshaping of

457

this tensegrity is carried out based on the random set of nodal geometrical variations

458

given in Table 10. The proposed method converges after 9 iterations (the history is

459

summarized in Table 11) and forms a new irregular truncated tetrahedron with a

460

feasible set of force densities, as shown in Fig. 2(b). Note that all elements now have

461

different force densities. The corresponding nodal coordinates are provided in Table 10.

462

We also check the viability of our results by calculating the first four eigenvalues of the 25

463

force density matrix all of which are of the order 1e-15. This also verifies the super-

464

stability of the tensegrity after reshaping (see Table 12).

465

466

Table 9

467

Set of force densities before and after imposing nodal geometrical variations (example

468

2) Element Number 1 2 3 4 5 6 7 8 9 10 11 12

Initial force density ( q 0 ) 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Final force density ( q 9 ) 0.5307 1.4747 1.0284 0.6277 1.7149 0.6160 0.8636 1.1055 0.6438 1.1197 0.6853 1.7006

Element Number 13 14 15 16 17 18 19 20 21 22 23 24

469

470

471

472

473

474

475

476

26

Initial force density ( q 0 ) 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -0.5746 -0.5746 -0.5746 -0.5746 -0.5746 -0.5746

Final force density ( q 9 ) 1.0084 0.8532 0.7765 0.7037 1.5288 0.8610 -0.4720 -0.4614 -0.4604 -0.7455 -0.4642 -0.6265

477

Table 10

478

Set of nodal coordinates before and after imposing nodal geometrical variations

479

(example 2) Initial nodal coordinates (self-stressed) y0 1.0386 -1.0281 -0.2698 -1.5127 -2.6805 -2.4113 -1.0585 1.0128 0.2283 2.6958 1.5542 2.4313

x0 2.2633 2.4396 1.0871 0.5608 0.9890 -0.9941 -2.6605 -2.7455 -1.9136 -0.6831 0.2657 1.3914

z0 1.6190 1.3466 2.7509 -2.4939 -0.8118 -1.4210 0.7895 0.5083 2.2601 -1.0431 -2.5171 -0.9874

Nodal variations ∆x 0 0.1920 0.1389 0.6963 0.0938 0.5254 0.5303 0.8611 0.4849 0.3935 0.6714 0.7413 0.5201

∆y 0 0.3477 0.1500 0.5861 0.2621 0.0445 0.7549 0.2428 0.4424 0.6878 0.3592 0.7363 0.3947

∆z 0 0.6834 0.7040 0.4423 0.0196 0.3309 0.4243 0.2703 0.1971 0.8217 0.4299 0.8878 0.3912

Final nodal coordinates (self-stressed) x9 2.4249 2.7984 1.7289 0.7617 1.4453 -0.5446 -1.8948 -2.0784 -1.4931 -0.0684 0.9128 1.8562

y9 1.4323 -0.8165 0.3056 -1.3020 -2.6153 -1.6724 -0.7495 1.5274 0.7654 3.1395 2.1836 2.8104

z9 2.2074 2.0798 3.0658 -2.4350 -0.3283 -1.1597 0.9753 0.7517 3.2343 -0.6298 -1.6022 -0.5569

480

481

482

Table 11

483

Infinity norm of the out-of-balance force vector in different iterations (example 2)

f

inf

1 0.7111

2 0.0440

Iteration number 3 4 5 6.2670e-5 1.400e-10 1.1990e-10

484

27

… …

9 2.41e-14

485 486

Fig. 2 a) initial truncated tetrahedron tensegrity; and b) new form after imposing nodal

487

geometrical variations

488

489

Table 12

490

Eigenvalues of the force density matrix for the truncated tetrahedron tensegrity before

491

and after reshaping

λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 λ12

Eigenvalues before reshaping 0.0000 0.0000 0.0000 0.0000 3.0000 3.0000 3.0782 3.0782 3.0782 4.6234 4.6234 4.6234

492

493

28

Eigenvalues after reshaping 0.0000 0.0000 0.0000 0.0000 2.0685 2.3620 2.4700 3.4987 3.6714 4.5320 4.7996 5.8231

494

7.3. Example 3

495

In Fig. 3(a) a 12-plex cylindrical tensegrity is shown. This tensegrity has 24 nodes and

496

36 elements including 24 cables and 12 struts. All 12 nodes that are located in x-y plane

497

(z=0) are considered to be restrained. For this type of tensegrity (in general for an n-

498

plex) an analytical solution corresponding to a state of self-stressed equilibrium is

499

available (Tibert and Pellegrino, 2002). The self-stressed equilibrium is reached when

500

all nodes at the top of the tensegrity are rotated about the z-axis by an angle

501

θ = (π / 2) − (π / n) = 5π / 12 . The force density of cables and struts are accordingly

502

obtained as 1, 2 sin(π / 12) and − 2 sin(π / 12) as given in Table 13. We impose a set of

503

nodal geometrical variations to the free nodes (nodes at z=2) as defined in Table 14. The

504

proposed method successfully forms an irregular tensegrity as shown in Fig. 3(b) in only

505

5 iterations.

506 507 508 509

Fig. 3 a) an initial restrained 12-plex cylindrical tensegrity; and b) new form after imposing nodal geometrical variations to top free nodes

29

510

511

Note that this outstanding performance is achieved even though the assumed variations

512

are moderately large. The final nodal coordinates and infinity norm of the out-balance

513

force vector are provided in Tables 14-15. This tensegrity has one infinitesimal

514

mechanism which has been stiffened by the single state of self-stress (see the results

515

given in Table 16 before and after reshaping).

516 517 518

Table 13

519

Set of force densities before and after imposing nodal geometrical variations (example

520

3) Element Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Initial force density ( q 0 ) 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.5176 0.5176 0.5176 0.5176 0.5176 0.5176

Final force density ( q 5 ) 1.0128 0.9922 0.9714 0.9291 0.8672 0.5920 0.6583 0.5887 0.5743 0.5774 0.5973 0.7432 1.4110 0.9939 0.9311 0.8907 0.9162 0.4359

Element Number 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

521

522

30

Initial force density ( q 0 ) 0.5176 0.5176 0.5176 0.5176 0.5176 0.5176 -0.5176 -0.5176 -0.5176 -0.5176 -0.5176 -0.5176 -0.5176 -0.5176 -0.5176 -0.5176 -0.5176 -0.5176

Final force density ( q 5 ) 0.3890 0.7026 0.6701 0.6660 0.6696 1.2051 -0.9788 -0.9235 -0.8859 -0.8792 -0.5599 -0.4712 -0.6780 -0.6662 -0.6636 -0.6732 -1.1046 -1.2430

523

Table 14

524

Set of nodal coordinates before and after imposing nodal geometrical variations

525

(example 3) Initial nodal coordinates (self-stressed) y0 0.2588 -0.2588 -0.7071 -0.9659 -0.9659 -0.7071 -0.2588 0.2588 0.7071 0.9659 0.9659 0.7071 1.0000 0.8660 0.5000 0.0000 -0.5000 -0.8660 -1.0000 -0.8660 -0.5000 0.0000 0.5000 0.8660

x0 0.9659 0.9659 0.7071 0.2588 -0.2588 -0.7071 -0.9659 -0.9659 -0.7071 -0.2588 0.2588 0.7071 0.0000 0.5000 0.8660 1.0000 0.8660 0.5000 0.0000 -0.5000 -0.8660 -1.0000 -0.8660 -0.5000

z0 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Nodal variations ∆x 0 0.9659 0.9659 0.7071 0.2588 -0.2588 -0.7071 -0.9659 -0.9659 -0.7071 -0.2588 0.2588 0.7071 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

∆y 0 0.2588 -0.2588 -0.7071 -0.9659 -0.9659 -0.7071 -0.2588 0.2588 0.7071 0.9659 0.9659 0.7071 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

∆z 0 0.0000 0.4000 0.8000 1.2000 1.6000 2.0000 2.0000 1.6000 1.2000 0.8000 0.4000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Final nodal coordinates (self-stressed) x5 1.8470 1.8604 1.4015 0.5891 -0.3901 -1.3571 -2.0386 -2.0389 -1.5278 -0.5836 0.5111 1.4101 0.0000 0.5000 0.8660 1.0000 0.8660 0.5000 0.0000 -0.5000 -0.8660 -1.0000 -0.8660 -0.5000

y5 0.4999 -0.4283 -1.2634 -1.7823 -1.8571 -1.4884 -0.4717 0.4724 1.4298 1.9995 1.9999 1.4278 1.0000 0.8660 0.5000 0.0000 -0.5000 -0.8660 -1.0000 -0.8660 -0.5000 0.0000 0.5000 0.8660

z5 2.0152 2.3279 2.6827 3.0661 3.4826 4.0777 4.0954 3.6000 3.1962 2.8037 2.4252 2.0445 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

526

527

528

Table 15

529

Infinity norm of the out-of-balance force vector in different iterations (example 3)

f

inf

1 0.6941

Iteration number 2 3 4 0.0811 8.8969e-4 4.0463e-7

530

31

5 2.0372e-14

531

Table 16

532

Infinitesimal mechanisms and the corresponding work done by the state of self-stress

533

before and after reshaping Before reshaping eig (Utm (I 3

Node ux

1 2 3 4 5 6 7 8 9 10 11 12

0.0673 -0.0673 -0.1838 -0.2511 -0.2511 -0.1838 -0.0673 0.0673 0.1838 0.2511 0.2511 0.1838

After reshaping

⊗ G)U m ) = 0.2173 uz uy

-0.2511 -0.2511 -0.1838 -0.0673 0.0673 0.1838 0.2511 0.2511 0.1838 0.0673 -0.0673 -0.1838

-0.1255 -0.1255 -0.1255 -0.1255 -0.1255 -0.1255 -0.1255 -0.1255 -0.1255 -0.1255 -0.1255 -0.1255

eig (U tm (I 3 ⊗ G)U m ) = 0.1936 ux uz uy

-0.0676 0.0657 0.1769 0.2433 0.2507 0.1982 0.0752 -0.0754 -0.2071 -0.2811 -0.2758 -0.1931

0.2522 0.2452 0.1769 0.0652 -0.0672 -0.1982 -0.2808 -0.2815 -0.2071 -0.0753 0.0739 0.1931

0.1245 0.0979 0.0810 0.0705 0.0643 0.0600 0.0737 0.0724 0.0822 0.0955 0.1109 0.1273

534

535

536

8. Concluding remarks

537

The generation of irregular tensegrities through conventional form-finding methods is a

538

challenging research topic. We have proposed a novel numerical approach in order to

539

address this issue through reshaping of existing regular/irregular tensegrities. The

540

reshaping method uses an underdetermined linearized form of the nonlinear

541

equilibrium equations and a geometrical variation approach to form an irregular self-

542

equilibrated tensegrity. The performance of the proposed method has been studied

543

through the examples of free standing and restrained tensegrities. The method promises

544

to generate very accurate results within an iterative computational framework with a

545

remarkable efficiency and very good convergence rate. Since the geometrical variation

32

546

can be applied either in a random or completely guided manner, both random irregular

547

configurations and tensegrities with some specific geometrical properties can be

548

achieved, making the reshaping method very attractive for designing real world

549

tensegrities with special characteristics for particular applications. Our method does not

550

check the stability of a tensegrity during the reshaping process, however, since we work

551

based on small geometric variations, the stability of a tensegrity is normally preserved

552

during reshaping.

553

554

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Zhang, J.Y., Ohsaki, M., 2006. Adaptive force density method for form-finding problem of tensegrity structures. Int. J. Solids Struct. 43, 5658–5673. Zhang, J.Y., Ohsaki, M., 2007. Stability conditions for tensegrity structures. Int. J. Solids Struct. 44, 3875–3886. Zhang, J.Y., Ohsaki, M., 2012. Self-equilibrium and stability of regular truncated tetrahedral tensegrity structures. J. Mech. Phys. Solids. 60, 1757–1770. Zhang, L.Y., Li, Y., Cao, Y.P., Feng, X.Q., 2014. Stiffness matrix based form-finding method of tensegrity structures. Eng. Struct. 58, 36-48. 555

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Figures Captions

570

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Fig. 1 a) initial self-stressed configuration (free standing); b) configuration after

572

imposing nodal geometrical variations (free standing); c) initial self-stressed

573

configuration (restrained); and d) configurations after imposing nodal geometrical

574

variations (restrained)

575

576

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Fig. 2 a) initial truncated tetrahedron tensegrity; and b) new form after imposing nodal

578

geometrical variations

579

580

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Fig. 3 a) an initial restrained 12-plex cylindrical tensegrity; and b) new form after

582

imposing nodal geometrical variations to top free nodes

583

38

584

Highlights

585


We propose a novel iterative approach for the reshaping of tensegrities

586


Reshaping is performed using a geometrical variation approach

587


An underdetermined linearized form of nonlinear equilibrium equations is used

588


The method generates very accurate results with a remarkable convergence rate

589

39