Self-equilibrium and super-stability of truncated regular ...rspa.royalsocietypublishing.org/content/royprsa/468/2147/3323.full.pdf · Self-equilibrium and super-stability of truncated

Proc. R. Soc. A (2012) 468, 3323–3347doi:10.1098/rspa.2012.0260

Published online 25 July 2012

Self-equilibrium and super-stability of truncatedregular polyhedral tensegrity structures:

a unified analytical solutionBY LI-YUAN ZHANG1, YUE LI2, YAN-PING CAO1, XI-QIAO FENG1,*

AND HUAJIAN GAO3

1CNMM and AML, Department of Engineering Mechanics, and 2Institute ofNuclear and New Energy Technology, Tsinghua University,

Beijing 100084, People’s Republic of China3School of Engineering, Brown University, Providence, RI 02912, USA

In spite of their great importance and numerous applications in many civil, aerospace andbiological systems, our understanding of tensegrity structures is still quite preliminary,fragmented and incomplete. Here we establish a unified closed-form analytical solutionfor the necessary and sufficient condition that ensures the existence of self-equilibratedand super-stable states for truncated regular polyhedral tensegrity structures, includingtruncated tetrahedral, cubic, octahedral, dodecahedral and icosahedral tensegrities.

Keywords: tensegrity; truncated regular polyhedron; self-equilibrium; super-stability;analytical method

1. Introduction

Tensegrities refer to a class of light-weight and reticulated structures made of adiscontinuous set of axial compressive elements (bars) and a continuous set ofaxial tensile elements (strings) (Juan & Tur 2008; Feng et al. 2010). Owing totheir unique features and properties (Skelton et al. 2001), tensegrity structureshold promises for a wide variety of technologically important applications,ranging from architectural designs (Motro 2003; Rhode-Barbarigos et al. 2010),deployable aerospace structures (Tibert & Pellegrino 2002; Sultan 2009), smartactuators and sensors (Ali & Smith 2010; Moored et al. 2011), advanced materialsengineering (Luo & Bewley 2005; Fraternali et al. 2012), to molecular andcellular biomechanics (Ingber 1993; Luo et al. 2008; Pirentis & Lazopoulos2010). In particular, truncated polyhedral tensegrities, a subclass of tensegritystructures with topology based on polyhedrons with truncated vertices (Pugh1976; Murakami & Nishimura 2001; Li et al. 2010b), have been adopted in man-made geodesic domes (Fu 2005; Yuan et al. 2007) and models of cytoskeleton andother biological structures (Stamenovic & Ingber 2009; Ingber 2010). A key stepin the design of a tensegrity structure is a self-equilibrium analysis to determine,once the overall topology has been specified, conditions under which the structure*Author for correspondence ([email protected]).

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3324 L.-Y. Zhang et al.

not self-equilibrated states

self-equilibratedbut unstable states

self-equilibratedand stable states

self-equilibratedand super-stable

states

Figure 1. Relationships of different states of tensegrity structures. (Online version in colour.)

will be self-equilibrated in the absence of external loads. Various methods havebeen proposed for the self-equilibrium analysis based on the concept of forcedensity, defined as the ratio of internal force to the current length of an element(Tibert & Pellegrino 2003). While analytical approaches based on the nodal forceequilibrium have been developed for tensegrities with a relatively small number ofelements or high symmetry (Connelly & Terrell 1995; Sultan et al. 2001; Zhanget al. 2010), numerical methods are often needed for larger or more complexstructures (Estrada et al. 2006; Zhang & Ohsaki 2006; Tran & Lee 2010).

Following the self-equilibrium analysis, a stability analysis can be performed todetermine the conditions under which a self-equilibrated tensegrity structure isstable. A stable structure always tends to return to its equilibrated configurationwhen subject to an infinitesimal and conservative disturbance. In the presentpaper, we consider only the static stability of tensegrities, excluding the instabilityproblems under non-conservative follower forces (Langthjem & Sugiyama 2000).The stability of a tensegrity can be guaranteed by the positive definiteness of itstangent stiffness matrix, which is defined as the derivative of the external forcevector with respect to the nodal displacement vector (Schenk et al. 2007; Zhang &Ohsaki 2007). Furthermore, the structure is said to be super-stable if it is stablefor any level of force densities satisfying the self-equilibrium conditions withoutmaterial failure (Connelly & Back 1998; Juan & Tur 2008). The relationshipsof different states of tensegrities are shown in figure 1. For many importantapplications, tensegrities are required to be not only self-equilibrated and stablebut also super-stable.

During the past decade, much effort has been directed towards the self-equilibrium analysis, some also including structural stability, of truncatedregular polyhedral tensegrities. Based on the nodal force equilibrium conditionsand symmetric congruent operations, Murakami & Nishimura (2001) analysedthe self-equilibrated states of truncated regular dodecahedral and icosahedraltensegrities. Tibert & Pellegrino (2003) obtained an analytical self-equilibriumsolution for truncated tetrahedral tensegrities expressed in terms of forcedensities. Pandia Raj & Guest (2006) solved the self-equilibrated states oftruncated tetrahedral tensegrities based on a group representation theory, anddiscussed the super-stability of such structures by means of a symmetry-adapted force density matrix. Estrada et al. (2006) developed a multi-parameternumerical procedure to determine the self-equilibrated states and configurations

Proc. R. Soc. A (2012)



Truncated regular polyhedral tensegrity 3325

of truncated regular tetrahedral and icosahedral tensegrities by iterativelycalculating both the equilibrium geometry and the force densities. Recently, Liet al. (2010a) proposed a Monte Carlo form-finding method for truncated regulartetrahedral, dodecahedral and icosahedral tensegrities. This method can be usedto determine both the self-equilibrated configurations and the associated forcedensities in these structures.

In spite of the above-mentioned progress on both analytical and numericalmethods for specific truncated regular polyhedral tensegrities, the existinganalytical solutions are fragmented and lack a unified treatment. In addition,the stability of these structures remains a major concern that needs furtherinvestigations. The present study aimed to develop a unified solution for theself-equilibrated and super-stable states of all truncated regular polyhedraltensegrity structures.

The paper is organized as follows. Section 2 reviews some basic conceptsincluding regular polyhedra, truncated regular polyhedra and truncated regularpolyhedral tensegrity structures. Section 3 presents the theoretical frameworkfor analysing self-equilibrium and super-stability. Section 4 derives the self-equilibrium solutions of truncated regular tetrahedral, cubic, octahedral,dodecahedral and icosahedral tensegrities. Section 5 provides a unified solutionfor the self-equilibrated states of all truncated regular polyhedral tensegrities,and §6 considers the necessary and sufficient condition for the super-stability ofthe solution. Section 7 summarizes the main contributions of the present study.

2. Truncated regular polyhedral tensegrity structures

Truncated regular polyhedral tensegrity structures can be constructed fromtruncated regular polyhedra (Murakami & Nishimura 2001; Li et al. 2010b)following a few basic steps discussed below.

(a) Regular polyhedra

Regular polyhedra have faces that are congruent regular polygons assembledin the same way around each vertex. In this paper, our attention will be focusedon convex regular polyhedra to be used in constructing tensegrity structures viaa polyhedral truncation scheme. As shown in figure 2a, there are five types ofconvex regular polyhedra, sometimes referred to as Platonic solids (Cromwell1997). To facilitate subsequent discussions, we adopt the following definitions ofSchläfli symbol and duality for polyhedra (Coxeter 1973):

Definition 2.1. A regular polyhedron can be uniquely identified by the Schläflisymbol {n, m}, where n is the number of edges of each face and m is the numberof faces around each vertex.

Definition 2.2. Polyhedron {n, m} is said to be dual to polyhedron {m, n}.Based on definitions (2.1) and (2.2), the five regular polyhedra in figure 2a

can be denoted as {3, 3} for tetrahedron, {4, 3} for cube, {3, 4} for octahedron,{5, 3} for dodecahedron and {3, 5} for icosahedron. One can see that a cube andan octahedron, and a dodecahedron and an icosahedron, are dual-pairs, while atetrahedron is self-dual.





type tetrahedral octahedral dodecahedral icosahedralcubic

(a)polyhedron

(b)truncated

polyhedron

(c)critical truncated

polyhedron

(d)hyper-truncated

polyhedron

Figure 2. Regular and truncated platonic solids: (a) regular polyhedra, (b) truncated regularpolyhedra, (c) critical truncated polyhedra and (d) hyper-truncated polyhedra. Blue, remainingfaces; green, truncating faces.

(b) Truncated regular polyhedra

A truncated regular polyhedron can be obtained from a regular polyhedron bytruncating each vertex with a plane perpendicular to the radius that connectsthe polyhedral centre to the vertex. There are five types of truncated regularpolyhedra, as shown in figure 2b. The truncated original faces and edges arereferred to as remaining faces and remaining edges, and the newly producedfaces and edges are called truncating faces and truncating edges, respectively.Each remaining edge is shared by two remaining faces, and each truncating edgeby one remaining face and one truncating face. Two special states of truncatedregular polyhedra are defined as follows:

Definition 2.3. A critical truncated polyhedron is the truncated polyhedronwhose remaining edges are of zero length.

Definition 2.4. A hyper-truncated polyhedron is the truncated polyhedronwhose remaining faces are of zero area.

For the truncated regular polyhedra in figure 2b, the corresponding criticaltruncated polyhedra are shown in figure 2c, where each vertex is located at themidpoint of an edge of the corresponding regular polyhedra in figure 2a. It isseen that a regular polyhedron and its dual counterpart have the same criticaltruncated polyhedron. Figure 2d illustrates the five hyper-truncated polyhedra,each of which has the same shape as the dual of the original regular polyhedronin figure 2a.





edge

vertex

bar

remaining-string1

2

3

4

truncating-string

node

(b)(a)

Figure 3. Truncated regular tetrahedral tensegrity: (a) the edges and vertices of a truncated regulartetrahedron, and (b) the strings, bars and nodes of the corresponding truncated regular tetrahedraltensegrity. In (b), four nodes in a Z -shaped elementary cell are numbered from 1 to 4, in which thecell consists of one bar (connecting nodes 1 and 4), one remaining-string (connecting nodes 2 and 3)and two truncating-strings (connecting nodes 1 and 2, and nodes 3 and 4, respectively). (Onlineversion in colour.)

(c) Truncated regular polyhedral tensegrities

A truncated regular polyhedral tensegrity structure can be constructed asfollows. Let the edges and vertices of a truncated regular polyhedron correspondto strings and nodes. The strings along the remaining edges are denoted asremaining-strings, and those along the truncating edges referred to as truncating-strings. A polygon formed by truncating-strings is a truncating polygon. In such astructure, the nodes of each truncating v-polygon (v is the number of edges) areconnected to the nodes of v adjacent truncating polygons by v bars that are addedinto the structure following the rules of Z -shaped elementary cells, referred to as aZ -based tensegrity (Pugh 1976; Li et al. 2010b). For example, figure 3a shows theedges and vertices of a truncated tetrahedron, and figure 3b illustrates the bars,strings and nodes of the corresponding truncated tetrahedral tensegrity, where aZ -shaped cell consisting of one bar (connecting nodes 1 and 4), one remaining-string (connecting nodes 2 and 3) and two truncating-strings (connecting nodes 1and 2 and nodes 3 and 4) is highlighted. In the following, we will investigatethe self-equilibrium and super-stability of all five types of truncated regularpolyhedral tensegrities.

3. Self-equilibrium and super-stability of tensegrity structures

The self-equilibrium and super-stability of a given tensegrity structure are usuallyinvestigated via the eigenvalues of its force density matrix, which is discussed fortruncated regular polyhedral tensegrities in this section.

(a) Force density matrix

The force density of element e connecting nodes i and j is defined as

qe(ij) = te(ij)

le(ij), (3.1)





which is sometimes also referred to as the stress or tension coefficient (Vassart &Motro 1999; Guest 2006; Schenk et al. 2007).

As discussed in §2c, the elements in a truncated regular polyhedral tensegritycan be classified into three kinds: truncating-strings, remaining-strings and bars.For each kind of elements, their force densities are assigned a specified value. Inaddition, the force density of all truncating-strings is set as unity because theself-equilibrium and super-stability conditions of a tensegrity structure dependonly on the normalized force densities. Let qs and qb denote, respectively, thedimensionless force densities of the remaining-strings and bars normalized by theforce density of the truncating-strings.

The self-equilibrium conditions of a tensegrity structure based on the forcedensity matrix can be expressed as (Zhang & Ohsaki 2006)

D · px = D · py = D · pz = 0, (3.2)

where px , py and pz are the nodal coordinate vectors in the x , y and z directions,respectively; D is the force density matrix of dimension nn × nn, nn being thetotal number of nodes in the structure.

The force density matrix D can be calculated according to the following scheme(Schenk et al. 2007):

Dij =⎧⎨⎩

−qe(ij) if i �= j and i is connected with j by element e,0 if i �= j and i does not connect with j ,− ∑

k �=j Dik if i = j ,(3.3)

where for the truncated regular polyhedral tensegrities qe(ij) should be taken as 1,qs, qb for truncating-strings, remaining-strings, bars, respectively. Alternatively,D can also be obtained from the following multiplication of matrices (Tibert &Pellegrino 2003):

D = CT · Q · C, (3.4)

where Q = diag(· · · , qe(ij), · · · ) is the diagonal matrix consisting of the forcedensities of all elements in the structure and C is the connectivity matrixdefined as

Cei =⎧⎨⎩

1 if i is the first node of element e,−1 if i is the second node of element e,0 otherwise.

(3.5)

The nodal sequences in each element can be arbitrarily chosen, but once this isdone, they should remain fixed throughout the analysis.

(b) Self-equilibrium conditions

If a tensegrity structure is to have a self-equilibrated state, the rank of its forcedensity matrix should satisfy the following inequality (Tibert & Pellegrino 2003;Schenk et al. 2007):

rank(D) ≤ nn − d − 1, (3.6)

or equivalently,null(D) ≥ d + 1, (3.7)





where rank(·) and null(·) denote the rank and the nullity of a matrix, respectively;d is the dimension of the structure: d = 3 for the three-dimensional truncatedregular polyhedral tensegrities under investigation.

The self-equilibrium analysis aims to determine the conditions that allowequation (3.7) to be satisfied. The eigenvalues of the force density matrix D canbe determined from its characteristic polynomial:

det(lI − D) = lnn + Pnn−1lnn−1 + · · · + P3l3 + P2l2 + P1l + P0 = 0, (3.8)

where I is the unit matrix, l is an eigenvalue of D and Pa = Pa(qs, qb) is apolynomial function of the force densities. If the inequality (3.7) with d = 3 holds,equation (3.8) can be rewritten as

det(lI − D) = l4(lnn−4 + Pnn−1lnn−5 + · · · + P4)

= lnn + Pnn−1lnn−1 + · · · + P4l4 = 0. (3.9)

Comparing equations (3.8) and (3.9), we obtain the following equivalent form ofequation (3.7) with d = 3:

P3 = P2 = P1 = P0 = 0. (3.10)

Note that there are no restrictions on the coefficients Pa for a ≥ 4. If null(D) = 4,we will have P4 �= 0, and if null(D) = nD > 4, more coefficients would vanish, i.e.PnD − 1 = · · · = P4 = 0.

For a self-equilibrated tensegrity structure without any nodes fixed, the sum ofall components in each row or column of the force density matrix should vanish,as can be seen from equation (3.3). Therefore, the force density matrix is alwayssingular, indicating that P0 = 0 is always true. Therefore, we only need to ensure

P3 = P2 = P1 = 0 (3.11)

for the self-equilibrium of a three-dimensional tensegrity, where P1, P2 andP3 are polynomials of the force densities in the structural elements. In whatfollows, these conditions will be investigated for the truncated regular polyhedraltensegrities. Note that equation (3.11) can only guarantee the existence of self-equilibrated states; the conditions for super-stability of these structure will befurther investigated.

(c) Super-stability conditions

A tensegrity structure is said to be super-stable if it is stable for any levelof force densities satisfying self-equilibrium without inducing material failure(Connelly & Back 1998; Juan & Tur 2008). For a super-stable tensegrity,increasing the force densities tends to stiffen and stabilize the structure. Here,we consider the super-stability of truncated regular polyhedral tensegrities.According to Connelly (1999) and Zhang & Ohsaki (2007), the conditions ofsuper-stability for a truncated regular polyhedral tensegrity structure are:

— (i) the strings have positive force densities, and the bars have negative forcedensities;

— (ii) the force density matrix is positive semi-definite;— (iii) the nullity of the force density matrix is exactly 4; and





— (iv) there are no affine (infinitesimal) flexes of the structure, or equivalently,the rank of the structural geometry matrix is 6.

The above conditions of super-stability can be understood as follows. Thestability of a tensegrity structure requires that its tangent stiffness matrix Kbe positive definite (Schenk et al. 2007). According to Zhang & Ohsaki (2007),K can be decomposed into a sum of the linear stiffness matrix KM and thegeometrical stiffness matrix KG = D ⊗ I3, i.e. K = KM + KG, where ⊗ is theKronecker product symbol (Williamson et al. 2003) and I3 is the third-rankidentity matrix. The linear stiffness matrix KM is positive semi-definite for alltensegrities consisting of conventional elements with positive axial stiffness, or inother words, dT · KM · d ≥ 0 for any nodal displacement vector d.

While condition (i) is just the definition of tensegrity, condition (ii) isnecessary for super-stability of the structure. If the force density matrix Dis not positive semi-definite, the geometrical stiffness matrix KG = D ⊗ I3 willhave negative eigenvalues. In this case, the geometrical stiffness matrix KG willbecome dominant over the linear stiffness matrix KM at sufficiently large forcedensities in the elements, in which case the tangent stiffness matrix K ceases tobe positive definite and the structure loses its stability.

Condition (iii) guarantees that the solution satisfies the self-equilibriumconditions and is also stable. If the nullity of the force density matrix is largerthan 4, there will be multiple equilibrated configurations that render the structureunstable (Schenk et al. 2007). Conditions (iii) and (iv) together ensure thatthere exists no nodal displacement vector d under which dT · KM · d = 0 anddT · KG · d = 0 are simultaneously satisfied. Thus, the positive definiteness of thetangent stiffness matrix, K = KM + KG, can be guaranteed even if both KM andKG are positive semi-definite.

Among conditions (i)–(iv), (i) can be satisfied by setting qs > 0 and qb < 0.Conditions (ii) and (iii) can be checked by the sign of the minimum eigenvalueand the total number of zero-eigenvalues of the force density matrix, respectively.Condition (iv) will be examined by the rank of the structural geometry matrixdefined by Zhang & Ohsaki (2007). As we will further show in §6, for the truncatedregular polyhedral tensegrities under study, condition (iv) is automaticallysatisfied once conditions (i)–(iii) are met.

4. Self-equilibrium analysis

In this section, we analyse the self-equilibrium of truncated regulartetrahedral, cubic, octahedral, dodecahedral and icosahedral tensegrities based onequation (3.11). The self-equilibrium conditions will be expressed as polynomialsof qs and qb, and those satisfying the super-stability conditions will be identifiedand used as a basis to construct a unified solution for the self-equilibrated andsuper-stable truncated regular polyhedral tensegrities.

(a) Truncated regular tetrahedral tensegrity

A truncated regular tetrahedral tensegrity structure has 12 truncating-strings,six remaining-strings, six bars and 12 nodes, accompanied by a 12 × 12 force





density matrix whose components can be obtained from equation (3.3) or (3.4).Here and in the sequel, the detailed expressions for the components of the forcedensity matrix are omitted for the sake of simplicity. Substituting the obtainedforce density matrix into equation (3.8), we find the following expressions of P1,P2 and P3 for the truncated tetrahedral tensegrities:

P1 = −576[2(q2

s qb + qsq2b) + 2(q2

s + q2b) + 6qsqb + 3(qs + qb)

]3, (4.1)

P2 = 48P2,4[2(q2


s + q2b) + 6qsqb + 3(qs + qb)

]2(4.2)

and P3 = −4P3,4[2(q2


s + q2b) + 6qsqb + 3(qs + qb)

], (4.3)

where P2,4 and P3,4 are lengthy polynomials that do not affect self-equilibriumand are omitted in the paper.

Equations (4.1)–(4.3) suggest that the self-equilibrium conditions inequation (3.11) are reduced to just one condition

(q2s qb + qsq2

b) + (q2s + q2

b) + 3qsqb + 32(qs + qb) = 0, (4.4)

for truncated tetrahedral tensegrities. This relation has been derived by Tibert &Pellegrino (2003) and Pandia Raj & Guest (2006) using different methods.

(b) Truncated regular cubic tensegrity

A truncated regular cubic tensegrity structure has 24 truncating-strings, 12remaining-strings, 12 bars and 24 nodes, with a 24 × 24 force density matrixgiven by equation (3.3) or (3.4). In this case, the expressions of P1, P2 and P3 inequation (3.8) are found to be

P1 = −9216(qs + qb) [qsqb + 2(qs + qb) + 3]2 [2qsqb + 3(qs + qb)]3

× [3(q2


s + q2b) + 8qsqb + 3(qs + qb)

]3, (4.5)

P2 = 768P2,6 [qsqb + 2(qs + qb) + 3] [2qsqb + 3(qs + qb)]2

× [3(q2


s + q2b) + 8qsqb + 3(qs + qb)

]2(4.6)

and P3 = −64P3,6 [2qsqb + 3(qs + qb)]

× [3(q2


s + q2b) + 8qsqb + 3(qs + qb)

], (4.7)

where P2,6 and P3,6 are lengthy polynomials that do not affect self-equilibrium.Combining equations (4.5)–(4.7) with (3.11) results in the following two

equations,

qsqb + 32(qs + qb) = 0 (4.8)

and

(q2s qb + qsq2

b) + 23(q2

s + q2b) + 8

3qsqb + (qs + qb) = 0, (4.9)

for the self-equilibrated states of a truncated regular cubic tensegrity. As we willfurther show in §6, a certain range of force densities satisfying equation (4.9)can ensure both self-equilibrium and super-stability. In contrast, it can be shown





0 50 100 150

0 0(a) (b)

(c) (d)0 50 100 150

10

–9

–6

–3

0

–20

–15

–10

–5

0

–9

–6

–3

0

–5

0

–0.5

–1.0

–1.5

forc

e de

nsity

of

bars

, qb

forc

e de

nsity

of

bars

, qb

force density of remaining-strings, qsforce density of remaining-strings, qs

min

imum

eig

enva

lue

of D

min

imum

eig

enva

lue

of D

–0.5

–1.0

–1.5

force density of bars qbminimum eigenvalue of D

force density of bars qb

minimum eigenvalue of D

force density of bars qbforce density of bars qb

minimum eigenvalue of Dminimum eigenvalue of D

50 1050

–1.32

–1.38

–1.44

–1.50

–1.2

–1.6

–2.0

–2.4

–10

–15

–20

3(qs + qb) = 0

2qsqb + 3

(qs + qb) + 2 = 02

qsqb +

(qsqb + qsqb) +2 (qs + qb)2 22 5 +5

5

3(5 + 5)15 +5 10

5 qsqb + (qs + qb) = 0

(qsqb + qsqb) +2 (qs + qb)2 22 5 +5

5

3(5 + 5) 3 +25

5qsqb + (qs + qb) = 0++

Figure 4. Self-equilibrium solutions with the minimum eigenvalue of force density matrix beingnegative, hence violating the positive semi-definite condition of super-stability for truncatedregular (a) cubic, (b) octahedral, (c) dodecahedral and (d) icosahedral tensegrities. (Online versionin colour.)

that no solution of equation (4.8) can be super-stable. To see this, let us rewriteequation (4.8) as

qb = − 3qs

2qs + 3. (4.10)

Super-stability requires that the strings have positive force densities, and the barshave negative force densities, i.e. qs > 0 and qb < 0. Equation (4.10) would demandqb to decrease from 0 to −3/2 as qs increases from 0 to infinity. In this case, onecan numerically determine the minimum eigenvalue of the force density matrix asalways negative, as shown in figure 4a. Therefore, the solution from equation (4.8)cannot satisfy the positive semi-definite condition of super-stability.

(c) Truncated regular octahedral tensegrity

A truncated regular octahedral tensegrity structure has 24 truncating-strings,12 remaining-strings, 12 bars and 24 nodes, with a 24 × 24 force density matrix.In this case, the coefficients P1, P2 and P3 in equation (3.8) are

P1 = −9216(qs + qb + 2) [qsqb + 2(qs + qb)]2 [2qsqb + 3(qs + qb) + 4]3

× [3(q2


s + q2b) + 6qsqb + 2(qs + qb)

]3, (4.11)

P2 = 768P2,8 [qsqb + 2(qs + qb)] [2qsqb + 3(qs + qb) + 4]2

× [3(q2


s + q2b) + 6qsqb + 2(qs + qb)

]2(4.12)





and P3 = −64P3,8(qs + qb + 2) [2qsqb + 3(qs + qb) + 4]

× [3(q2


s + q2b) + 6qsqb + 2(qs + qb)

], (4.13)

where P2,8 and P3,8 are lengthy polynomials that are irrelevant for the presentanalysis on self-equilibrium.

Combining equations (4.11)–(4.13) with (3.11) leads to two equations:

qsqb + 32(qs + qb) + 2 = 0 (4.14)

and

(q2s qb + qsq2

b) + 23(q2

s + q2b) + 2qsqb + 2

3(qs + qb) = 0, (4.15)

which ensure the self-equilibrium of truncated regular octahedral tensegrities.While a certain range of force densities from equation (4.15) can satisfy bothself-equilibrium and super-stability, no solution of equation (4.14) can meet thepositive semi-definite condition of super-stability, as shown in figure 4b.

(d) Truncated regular dodecahedral tensegrity

A truncated regular dodecahedral tensegrity structure has 60 truncating-strings, 30 remaining-strings, 30 bars and 60 nodes, with a 60 × 60 force densitymatrix. The corresponding coefficients P1, P2 and P3 in equation (3.8) are

P1 = −38880 [qsqb + (qs + qb)]4 [qsqb + 3(qs + qb)]

4

× [2q2

s q2b + 10(q2


s + q2b) + 24qsqb + 9(qs + qb)

]5

×[10(q2

s qb + qsq2b) + 2(5 + √

5)(q2s + q2

b) + 2(15 + √5)qsqb

+ 3(5 + √5)(qs + qb)

]3 ×[10(q2

s qb + qsq2b)

+ 2(5 − √5)(q2

s + q2b) + 2(15 − √

5)qsqb

+ 3(5 − √5)(qs + qb)

]3, (4.16)

P2 = 25920P2,12 [qsqb + (qs + qb)]3 [qsqb + 3(qs + qb)]

3

× [2q2

s q2b + 10(q2


s + q2b) + 24qsqb + 9(qs + qb)

]4

×[10(q2

s qb + qsq2b) + 2(5 + √

5)(q2s + q2

b) + 2(15 + √5)qsqb

+ 3(5 + √5)(qs + qb)

]2

×[10(q2

s qb + qsq2b) + 2(5 − √

5)(q2s + q2

b) + 2(15 − √5)qsqb

+ 3(5 − √5)(qs + qb)

]2(4.17)





and P3 = −17280P3,12 [qsqb + (qs + qb)]2 [qsqb + 3(qs + qb)]

2

× [2q2

s q2b + 10(q2


s + q2b) + 24qsqb + 9(qs + qb)

]3

×[10(q2

s qb + qsq2b) + 2(5 + √

5)(q2s + q2

b) + 2(15 + √5)qsqb

+ 3(5 + √5)(qs + qb)

]

×[10(q2

s qb + qsq2b) + 2(5 − √

5)(q2s + q2

b) + 2(15 − √5)qsqb

+ 3(5 − √5)(qs + qb)

], (4.18)

where P2,12 and P3,12 are lengthy polynomials that are irrelevant for the presentanalysis.

Substituting equations (4.16)–(4.18) into (3.11) leads to the followingfive equations:

qsqb + (qs + qb) = 0, (4.19)

qsqb + 3(qs + qb) = 0, (4.20)

q2s q

2b + 5(q2


s + q2b) + 12qsqb + 9

2(qs + qb) = 0, (4.21)

(q2s qb + qsq2

b) + 5 + √5

5(q2

s + q2b) + 15 + √

55

qsqb + 3(5 + √5)

10(qs + qb) = 0

(4.22)

and (q2s qb + qsq2

b) + 5 − √5

5(q2

s + q2b) + 15 − √

55

qsqb + 3(5 − √5)

10(qs + qb) = 0.

(4.23)

It can be shown that, while a certain range of force densities fromequation (4.23) can satisfy both self-equilibrium and super-stability, no solutionsfrom equations (4.19)–(4.22) can satisfy super-stability for the following reasons.Equations (4.19)–(4.21) lead to force density matrices with at least five zero-eigenvalues and thus do not meet the condition of nullity equal to 4, and theminimum eigenvalue of the force density matrix from equation (4.22) is alwaysnegative, as shown in figure 4c.

(e) Truncated regular icosahedral tensegrity

A truncated regular icosahedral tensegrity structure has 60 truncating-strings,30 remaining-strings, 30 bars and 60 nodes, with a 60 × 60 force density matrix.The corresponding coefficients P1, P2 and P3 are

P1 = −38880 [qsqb + (qs + qb) + 1]4 [qsqb + 3(qs + qb) + 5]4

× [2q2

s q2b + 10(q2


s + q2b) + 16qsqb + 5(qs + qb)

]5

×[10(q2

s qb + qsq2b) + 2(5 + √

5)(q2s + q2

b) + 6(5 + √5)qsqb





+ 5(3 + √5)(qs + qb)

]3

×[10(q2

s qb + qsq2b) + 2(5 − √

5)(q2s + q2

b) + 6(5 − √5)qsqb

+ 5(3 − √5)(qs + qb)

]3, (4.24)

P2 = 25920P2,20 [qsqb + (qs + qb) + 1]3 [qsqb + 3(qs + qb) + 5]3

× [2q2

s q2b + 10(q2


s + q2b) + 16qsqb + 5(qs + qb)

]4

×[10(q2

s qb + qsq2b) + 2(5 + √

5)(q2s + q2

b) + 6(5 + √5)qsqb

+ 5(3 + √5)(qs + qb)

]2

×[10(q2

s qb + qsq2b) + 2(5 − √

5)(q2s + q2

b) + 6(5 − √5)qsqb

+ 5(3 − √5)(qs + qb)

]2(4.25)

and P3 = −17280P3,20 [qsqb + (qs + qb) + 1]2 [qsqb + 3(qs + qb) + 5]2

× [2q2

s q2b + 10(q2


s + q2b) + 16qsqb + 5(qs + qb)

]3

×[10(q2

s qb + qsq2b) + 2(5 + √

5)(q2s + q2

b) + 6(5 + √5)qsqb

+ 5(3 + √5)(qs + qb)

]

×[10(q2

s qb + qsq2b) + 2(5 − √

5)(q2s + q2

b) + 6(5 − √5)qsqb

+ 5(3 − √5)(qs + qb)

]. (4.26)

The P2,20 and P3,20 are lengthy polynomials that are irrelevant for the presentanalysis.

Substituting equations (4.24)–(4.26) into (3.11) leads to the following fiveequations:

qsqb + (qs + qb) + 1 = 0, (4.27)

qsqb + 3(qs + qb) + 5 = 0, (4.28)

q2s q

2b + 5(q2


s + q2b) + 8qsqb + 5

2(qs + qb) = 0, (4.29)

(q2s qb + qsq2

b) + 5 + √5

5(q2

s + q2b) + 3(5 + √

5)5

qsqb + 3 + √5

2(qs + qb) = 0 (4.30)

and (q2s qb + qsq2

b) + 5 − √5

5(q2

s + q2b) + 3(5 − √

5)5

qsqb + 3 − √5

2(qs + qb) = 0.

(4.31)





While a certain range of force densities from equation (4.31) are super-stable,no solutions of equations (4.27)–(4.30) can meet all conditions associated withsuper-stability: The force density matrix from equations (4.27)–(4.29) has atleast five zero-eigenvalues while the minimum eigenvalue of that associated withequation (4.30) is always negative, as shown in figure 4d.

We note that the self-equilibrium conditions of truncated regular cubic,octahedral, dodecahedral and icosahedral tensegrities have been previouslyanalysed by Nishimura (2000) and Murakami & Nishimura (2001, 2003),using icosahedral group graphs and a reduced equilibrium matrix. Theyconsidered structure symmetries to simplify the calculations of nodal forceequilibrium and expressed their final results in three coupled equations. It canbe shown that the solutions of Nishimura (2000) and Murakami & Nishimura(2001, 2003) are fully consistent with those in equations (4.9), (4.15), (4.23)and (4.31).

5. Unified solution for self-equilibrated and super-stable states

Inspired by the solutions derived in §4, a unified and simple solution for theself-equilibrated and super-stable states of all truncated regular polyhedraltensegrities will be conjectured in this section, with coefficients determined fromthree special equilibrated states by considering their geometric features. Theunified solution will be verified by comparison with all solutions in §4.

(a) Conjecture of unified solution

In §4, we have individually derived the self-equilibrated states oftruncated regular tetrahedral, cubic, octahedral, dodecahedral and icosahedraltensegrities by directly solving the self-equilibrium conditions. We have alsoidentified that the solutions possibly satisfying super-stability are given inequations (4.4), (4.9), (4.15), (4.23) and (4.31). Inspired by these results, wefurther attempt to establish a unified solution for the self-equilibrated andsuper-stable states of all truncated regular polyhedral tensegrities.

Conjecture 5.1. For the self-equilibrated and super-stable states of alltruncated regular polyhedral tensegrities, the force densities of elements mustsatisfy the following relation:

(q2s qb + qsq2

b) + A1(q2s + q2

b) + A2qsqb + A3(qs + qb) = 0, (5.1)

where qs and qb are the dimensionless force densities of the remaining-strings andbars, respectively, normalized by the force density of the truncating-strings, andthe coefficients A1, A2 and A3 are constants that depend on the Schläfli symbol{n, m} of the corresponding regular polyhedron, i.e.

A1 = A1(n, m), A2 = A2(n, m) and A3 = A3(n, m). (5.2)

It can be immediately seen that equation (5.1) has a similar form as theindividual solutions given in equations (4.4), (4.9), (4.15), (4.23) and (4.31).





overlapped

overlapped(a) (b)

Figure 5. Un-truncated tetrahedron: (a) its geometry and (b) the corresponding tensegrity. In sucha structure, the two nodes of each truncating-string are merged since the truncating-strings are ofzero length. The bar and the remaining-string in each Z -shaped cell also become overlapped, asshown in the dashed box. (Online version in colour.)

Furthermore, the basic characters of truncated regular polyhedral tensegritiesrequire that the unified solution should meet the following conditions:

— In accordance with the self-equilibrium solutions for all truncated regularpolyhedral tensegrities discussed in the previous section, the unifiedsolution should also be in a polynomial form of the normalized forcedensities of the remaining-strings and bars, i.e. qs and qb.

— In the unified solution, qs and qb should be commutative becauseall truncated polyhedral tensegrity structures are Z -based tensegrities(Li et al. 2010b). In other words, if all remaining-strings and bars in sucha structure are interchanged, the new structure will maintain the sametopology as the original one.

— There should be no constant term in the solution for the following reason.The force density of an element, defined as the ratio of its internal force overlength, is infinite when the element is of zero length. Since the truncatedregular polyhedral tensegrity structures belong to Z -based tensegrities(Li et al. 2010b), they should have a special self-equilibrated state inwhich all truncating-strings have a zero length, corresponding to the un-truncated polyhedra shown in figure 2a. In this case, the force density inthe truncating-strings is infinite and the normalized force densities in thebars and remaining-strings (i.e. qb and qs) are both zero. An un-truncatedtetrahedron is shown as an example in figure 5a, and the correspondingtensegrity with truncating-strings of zero length is illustrated in figure 5b.Clearly, in order for the unified solution to capture the state qs = qb = 0, itcannot have a constant term.

To determine the expressions of A1, A2 and A3 for a truncated regularpolyhedron with given Schläfli index {n, m}, we consider three special self-equilibrated states of truncated regular polyhedral tensegrities, corresponding toone asymptotic line (qs → ∞) and two special points, I (qs = 0) and II (qs = qb),on the curves from equations (4.4), (4.9), (4.15), (4.23) and (4.31), as shownin figure 6.





–3

–2

–1

0

1

2

3

curve 2

curve 3

curve 3

curve 1

curve 2

curve 1

curve 3

curve 2

curve 1

curve 3

forc

e de

nsity

of

bars

, qb

curve 3

curve 2

special point II

special point II special point II

special point II

curve 1 curve 1

asymptotic lineasymptotic line

asymptotic line asymptotic line

special point I special point I

special point Ispecial point I

special point IIspecial point I

asymptotic line

3(qs + qb) = 0

2+3qsqb +

(qsqb + qsqb) + (qs + qb)2 2 2 2

83

23

(qsqb + qsqb) + (qs + qb)2 2 22

–3

–2

–1

0

1

2

3

–3

–2

–1

0

1

2

3

forc

e de

nsity

of

bars

, qb

–3

–2

–1

0

1

2

3

–3 –2 –1 0 21 3

force density of remaining-strings, qs

–3 –2 –1 0 21 3


–3

–2

–1

0

1

2

3

forc

e de

nsity

of

bars

, qb

–3 –2 –1 0 21 3


+ qsqb + (qs + qb) = 0

(qs + qb) = 032+2qsqb +

23

(qsqb + qsqb) + (qs + qb)2 2 22

(qsqb + qsqb)

(qs + qb)

qs qb

2

2 2

(qs + qb) = 0

2

curve 2

+ 5– 55

+ 15– 55

+ 3(5– 5)10

(qsqb + qsqb)

(qs + qb)

2

2 2

(qs + qb) = 0

qs qb

2

+ 5– 55

+ 3– 52

+ 3(5– 5)5

(a) (b)

(c) (d)

(e)

Figure 6. Self-equilibrium and possibly super-stable solutions for truncated regular (a) tetrahedral,(b) cubic, (c) octahedral, (d) dodecahedral and (e) icosahedral tensegrities. The force densitymatrix corresponding to curves 1 and 2 (dashed lines) has negative eigenvalues, and that associatedwith the dash-dotted part of curve 3 does not meet the condition of qb < 0 and qs > 0, hence notmeeting the super-stability conditions. Only the states corresponding to the solid part on curve 3are both self-equilibrated and super-stable. (Online version in colour.)





O

z

yx

15(6)3

2

4

R

Ox y

z 6

5

3

41

2

RO

6

5

3

4

1

2

x y

z

R

twooverlapped

nodes

(a) (b) (c)

Figure 7. Critical truncated tensegrity structures associated with (a) tetrahedron, (b) cube andoctahedron, and (c) dodecahedron and icosahedron. For clarity, we have drawn all truncating-strings (magenta) but only two bars (blue) and six nodes that are connected a specified node R(highlighted as a green solid ball).

(b) Determination of coefficients in conjecture 5.1

(i) Coefficient A1

First, the critical truncated polyhedral tensegrity in which all remaining-strings have a zero length is considered to determine the coefficient A1. Thisspecial self-equilibrated state corresponds to the critical truncated polyhedron indefinition 2.3, as shown in figure 2c, in which case the force density of remaining-strings approaches infinity, corresponding to the asymptotic line of qs → ∞ in theqb − qs curve.

Rewriting equation (5.1) as(qb + q2

b

qs

)+ A1

(1 + q2

b

q2s

)+ A2

qb

qs+ A3

(1qs

+ qb

q2s

)= 0, (5.3)

and then letting qs → ∞ yield

A1 = −q̃b, (5.4)

where q̃b denotes the force density of bars in the critical truncated polyhedraltensegrity as qs → ∞. The problem is now reduced to calculating q̃b.

There are three types of critical truncated polyhedral tensegrities, as shownin figure 7. Consider the elements connecting to a specific node, say node Rhighlighted by a green solid ball in figure 7, where all strings (magenta) are drawnbut only those bars (blue) connected to node R are shown. Here, all strings aretruncating-ones and have force density equal to unity. The force density in thebars can be calculated from force equilibrium. As shown in figure 7, the fournodes connecting to R via strings are numbered from 1 to 4, and the two nodesjoining with R via bars are numbered 5 and 6. Clearly, force equilibrium in thex-direction at node R requires

4∑i=1

(xi − xR) + q̃b

6∑i=5

(xi − xR) = 0, (5.5)

where xR and xi are the x-coordinates of R and connecting node i, respectively.





Substituting the nodal coordinates of critical truncated polyhedra intoequation (5.5) leads to

q̃b = − 23 + 2 cos(2p/p)

. (5.6)

It follows from equation (5.4) that the coefficient A1 is

A1 = 23 + 2 cos(2p/p)

, (5.7)

where p = max(n, m) is the maximum total number of edges of truncatingpolygons in the critical truncated polyhedron, {n, m} being the Schläfli symbol ofthe corresponding regular polyhedron. It can be shown that p = 3 and A1 = 1 forthe truncated tetrahedral tensegrities, p = 4 and A1 = 2/3 for the truncated cubicand octahedral tensegrities, and p = 5 and A1 = (5 − √

5)/5 for the truncateddodecahedral and icosahedral tensegrities. These results are tabulated in table 1.Note that A1 has the same value for dual tensegrities such as the truncateddodecahedron and icosahedron (see definition 2.2).

(ii) Coefficient A3

Next we proceed to determine coefficient A3. Consider the truncated polyhedraltensegrity in which all remaining-strings are of zero force density and can thus beeliminated without affecting self-equilibrium (Li et al. 2010a). Once all remaining-strings are removed, the resulting structure corresponds to the special point Iof qs = 0 in the qb − qs curve. As an example, a self-equilibrated configurationof truncated tetrahedral tensegrity with all remaining-strings removed is shownin figure 8a.

Substituting qs = 0 into equation (5.1) gives

A3 = −A1q̄b, (5.8)

where q̄b denotes the force density of bars in the truncated polyhedral tensegritywith qs = 0. The problem is now reduced to finding q̄b.

In the special state with all remaining-strings removed, there are onlytwo truncating-strings and one bar connected to each node. In this case,force equilibrium requires that the three elements must lie in the same plane(Motro 2003), as shown in figure 8a. Because of the structural symmetry, the solidcentre of the structure, the centres of all truncating polygons and the midpointsof all bars must be overlapped. Therefore, a specific node (e.g. node R denoted asa solid ball in figure 8b) and its three connecting nodes must lie on a circle withcentre located at the midpoint of a bar and diameter equal to the length of thebar, as shown in figure 8b. Based on these considerations, the nodal coordinatesand force density of bars can be determined.

As shown in figure 8b, nodes 1 and 2 are connected to the specified node R viatwo strings, and node 3 to R via one bar. In the local Cartesian coordinate system(x-O-y), where the origin O is located at the centre of the circle, the x-axis andy-axis are along and perpendicular to the bar, respectively, the force equilibrium





Tab

le1.

Uni

fied

solu

tion

for

the

self-

equi

libri

umof

trun

cate

dre

gula

rpo

lyhe

dral

tens

egri

tyst

ruct

ures

.

(q2 sq b

+q s

q2 b)+

A1(

q2 s+

q2 b)+

A2q

sqb

+A

3(q s

+q b

)=0

trun

cate

dpo

lyhe

dral

tens

egri

ties

nm

p=

max

(n,m

)A

1=

23

+2

cos(

2p/p)

A2=

6+

4[cos

(2p/p)

−co

s(2p

/m

)]3

+2

cos(

2p/p)

A3=

2[1−

cos(

2p/m

)]3

+2

cos(

2p/p)

tetr

ahed

ral

33

31

33/

2cu

bic

43

42/

38/

31

octa

hedr

al3

44

2/3

22/

3do

deca

hedr

al5

35

(5−

√ 5)/5

(15

−√ 5)

/5

3(5

−√ 5)

/10

icos

ahed

ral

35

5(5

−√ 5)

/5

3(5

−√ 5)

/5

(3−

√ 5)/2





2 mπ π

−

O

x

1 2

3

y

R

(a) (b)

Figure 8. A special self-equilibrated state of truncated tetrahedral tensegrity with zero forcedensities in the remaining-strings. (a) The structure with all remaining-strings removed; (b) inequilibrium, the two truncating-strings and one bar connected to a generic node R (solid ball)should be in co-planar. (Online version in colour.)

condition of node R in the x-direction is

(x1 − xR) + (x2 − xR) + q̄b(x3 − xR) = 0. (5.9)

In writing equation (5.9), we have used the fact that all truncating-strings haveforce density equal to unity. Taking the radius of the circle as unit of length, wefind xR = 1, x1 = − cos(2p/m), x2 = − cos(2p/m) and x3 = −1, where m as one ofthe Schläfli parameters in {n, m} is the total number of edges in each truncatingpolygon. Substituting these nodal coordinates into equation (5.9) leads to

q̄b = −(

1 − cos2p

m

). (5.10)

Combining equations (5.7) and (5.10) with (5.8) results in

A3 = 2[1 − cos(2p/m)]3 + 2 cos(2p/p)

, (5.11)

where m = 3 for the truncated tetrahedral, cubic and dodecahedral tensegrities,m = 4 for the truncated octahedral tensegrities, and m = 5 for the truncatedicosahedral tensegrities. The values of A3 calculated from equation (5.11) aretabulated in table 1.

(iii) Coefficient A2

Since the truncated regular polyhedral tensegrity structures belong to Z -basedtensegrities (Li et al. 2010b), we take hyper-truncated polyhedral tensegrities, inwhich the four elements of each Z -shaped elementary cell are in a line, as thethird special self-equilibrated state to determine the coefficient A2. This specialstate corresponds to the hyper-truncated polyhedron in definition 2.4, as shown infigure 2d. For example, figure 9 shows the hyper-truncated tetrahedron and itscorresponding hyper-truncated tetrahedral tensegrity.

In a hyper-truncated polyhedral tensegrity structure, the force densities in allremaining-strings (q̂s) and bars (q̂b) are identical and force equilibrium along the





(a) (b) overlapped

overlapped

Figure 9. Hyper-truncated tetrahedron: (a) its geometry and (b) the corresponding tensegrity. Inthis structure, nodes that are not connected by any element in a Z -shaped elementary cell becomeoverlapped, as shown in the dashed box. (Online version in colour.)

element direction showsq̂s = q̂b = −1. (5.12)

Note that the strings are allowed to bear compressive forces in the hyper-truncated polyhedral tensegrity only for the sake of calculations. In thisspecial case, the force densities correspond to the special point II in theqb − qs curve, which is at the same location (−1, −1) for all truncated regularpolyhedral tensegrities.

Substituting equation (5.12) into (5.1) leads toA2 = 2 − 2A1 + 2A3. (5.13)

It follows from equations (5.7) and (5.11) that

A2 = 6 + 4 [cos(2p/p) − cos(2p/m)]3 + 2 cos(2p/p)

. (5.14)

(c) Unified solution

With the determination of coefficients A1, A2 and A3 in equation (5.1), wecan establish the following necessary condition for the self-equilibrated andsuper-stable states of all truncated regular polyhedral tensegrities:

Theorem 5.2. For the self-equilibrated and super-stable states of all truncatedregular polyhedral tensegrities, the dimensionless force densities of the remaining-strings (qs) and bars (qb) normalized by the force density of the truncating-stringsmust satisfy the following relation:

(q2s qb + qsq2

b) + A1(q2s + q2

b) + A2qsqb + A3(qs + qb) = 0, (5.15)

where

A1 = 23 + 2 cos(2p/p)

, A2 = 6 + 4 [cos(2p/p) − cos(2p/m)]3 + 2 cos(2p/p)

and

A3 = 2 [1 − cos(2p/m)]3 + 2 cos(2p/p)

, (5.16)

with parameters m and p = max(n, m) following the definition of Schläfli symbol{n, m} for the corresponding regular polyhedron.





Table 1 lists the self-equilibrium solutions of truncated regular tetrahedral,cubic, octahedral, dodecahedral and icosahedral tensegrities given by equations(5.15) and (5.16). It can be verified that the earlier-mentioned unified solutioncovers all super-stable solutions derived in §4. It is a necessary condition for theself-equilibrium and super-stability of all truncated regular polyhedral tensegritystructures. The sufficient condition will be determined in §6.

6. Super-stability analysis

In order to determine the necessary and sufficient condition for the self-equilibrated and super-stable states, we must find the super-stable solutionsamong the self-equilibrated ones defined by the unified solution in §5.

We first solve equation (5.15) for the force density qb of bars as a functionof the force density qs of remaining-strings. Rewriting equation (5.15) in thequadratic form

(qs + A1)q2b + (q2

s + A2qs + A3)qb + (A1q2s + A3qs) = 0 (6.1)

leads to two possible solutions

qb = −(q2s + A2qs + A3) + √

(q2s + A2qs + A3)2 − 4(qs + A1)(A1q2

s + A3qs)2(qs + A1)

(6.2)

and

qb = −(q2s + A2qs + A3) − √

(q2s + A2qs + A3)2 − 4(qs + A1)(A1q2

s + A3qs)2(qs + A1)

. (6.3)

Using parameter values listed in table 1, it can be verified that the followinginequality holds for all truncated regular polyhedral tensegrities:

(q2s + A2qs + A3)2 − 4(qs + A1)(A1q2

s + A3qs) ≥ 0, (6.4)

indicating that the solution (qs, qb) exists in the entire range of −∞ < qs < +∞.Thus all self-equilibrated states, expressed in terms of the force densities (qs,qb), can be obtained from equations (6.2) and (6.3). For each type of truncatedregular polyhedral tensegrities, the self-equilibrated states captured by the unifiedsolution form three qb − qs curves, as shown in figure 6. Curves 1 and 3 are fromequation (6.2), which is singular and discontinuous at qs = −A1, while curve 2from equation (6.3) is continuous throughout −∞ < qs < +∞. Once the self-equilibrated state (qs, qb) has been found, the corresponding force density matrixcan be determined from equation (3.3) or (3.4).

As discussed in §3c, the super-stability of a tensegrity structure first requiresthat the strings have positive (qs > 0) and the bars negative force densities(qb < 0). Also, the force density matrix must be positive semi-definite with exactlyfour zero-eigenvalues. Our calculations show that all force density matricescorresponding to the solid part of curve 3 satisfy the earlier two conditions, whilethe force density matrices associated with curves 1 and 2 (the dashed lines) hasat least one negative eigenvalue and the dash-dotted part of curve 3 are notin the region qs > 0 and qb < 0. In addition, our calculations further show thatall geometry matrices corresponding to the solid part of curve 3 exactly have





type

self-equilibrated configurations of truncated regular polyhedral tensegrities

tetrahedral octahedralcubic dodecahedral icosahedral

(a)super-stable

(b) notsuper-stable

Figure 10. Example self-equilibrated configurations of truncated regular polyhedral tensegritieswhich are (a) super-stable and (b) not super-stable. Here, the force densities of both truncating-strings and remaining-strings are set to be unity, while the force densities of bars in (a) and (b)are determined from equations (6.2) and (6.3), respectively.

a rank of 6, satisfying the super-stability condition (iv) in §3c. Therefore, allsuper-stable states of truncated regular polyhedral tensegrities will be locatedin the solid part of curve 3, and other states in the three curves are not super-stable. Some representative self-equilibrated configurations of truncated regularpolyhedral tensegrities determined by the unified solution as qs = 1 are plottedin figure 10. The configurations in the first column (figure 10a) are super-stable,with qb solved from equation (6.2), while those in the second column (figure 10b)from equation (6.3) are not super-stable. For clearer observation of the three-dimensional self-equilibrated and super-stable configurations in figure 10a, thereader may refer to the website (Connelly & Terrell 2008).

Therefore, we have obtained the following necessary and sufficient conditionfor the self-equilibrated and super-stable states of all truncated regularpolyhedral tensegrities:

Theorem 6.1. A truncated regular polyhedral tensegrity is self-equilibrated andsuper-stable if and only if its force densities satisfy the following relation:

qb = −(q2s + A2qs + A3) + √

(q2s + A2qs + A3)2 − 4(qs + A1)(A1q2

s + A3qs)2(qs + A1)

(6.5)

where qs > 0, and A1, A2, A3 have been given in equation (5.16).

7. Conclusions

In this paper, we have performed a detailed theoretical analysis of self-equilibrium and super-stability properties of truncated regular polyhedraltensegrity structures. The most important result of the work is that we havefound the necessary and sufficient condition, expressed in a simple and explicit





form in equation (6.5), for a self-equilibrated and super-stable truncated regularpolyhedral tensegrity structure. We hope the present analysis will stimulatefurther theoretical studies in the community on super-stable tensegrity structures.

Supports from the National Natural Science Foundation of China (grant nos 10972121 and10732050), Tsinghua University (2009THZ02122) and the 973 Program of MOST (2010CB631005)are acknowledged.

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