Parametric Analysis of Tensegrity Plate-Like Structures: Part 1—Qualitative Analysis

applied sciences

Article Parametric Analysis of Tensegrity Plate-Like Structures: Part 1—Qualitative Analysis

Paulina Obara and Justyna Tomasik *

Faculty of Civil Engineering and Architecture, Kielce University of Technology, 25-314 Kielce, Poland; [email protected] * Correspondence: [email protected]; Tel.: +41-34-24-805

Received: 16 September 2020; Accepted: 1 October 2020; Published: 11 October 2020

Featured Application: A qualitative analysis of tensegrity structures is the first step to understand their unique properties, which allow for the control of static and dynamic parameters. The obtained results can be used in the design process of standard tensegrity building structures, i.e., plates and non-standard applications. The tensegrity concept can be used in the design of complex and intelligent structures with self-control, self-diagnosis, self-repair and active control. New and future potential applications will involve tensegrity-inspired metamaterials with exceptional mechanical properties.

Abstract: The study includes parametric analysis of special spatial rod grids called tensegrity plate-like structures. Tensegrity structures consist of only compression and tension components arranged in a system, whose unique mechanical and mathematical properties distinguish them from conventional cable–strut frameworks. Complete analysis of tensegrity structures is a two-stage process. The first stage includes the identification of self-stress states and infinitesimal mechanisms (qualitative analysis). The second stage focuses on the behaviour of tensegrities under external loads (quantitative analysis). In the paper, a qualitative analysis of tensegrity plate-like structures built with modified Quartex modules was conducted. Starting from a single-module structure, more complex cases were sequentially analysed. The different ways of plate support were considered. To carry out a qualitative assessment, a spectral analysis of the truss matrices and singular value decomposition of the compatibility matrix were used. The characteristic features of tensegrity structures were identified. On this basis, the plates were classified into one of the four groups defined in the paper, i.e., ideal tensegrity, “pure” tensegrity and structures with tensegrity features of class 1 or class 2. This classification is important due to different behaviours of the structure under external actions. The qualitative analysis carried out in the paper is the basis for a quantitative analysis.

Keywords: tensegrity; singular value decomposition (SVD); spectral analysis; self-stress state; inﬁnitesimal mechanism; modiﬁed Quartex

1. Introduction Tensegrity is a term derived from the English language as a compound of two words: “tension”—stretching, and “integrity”—stability. The term describes structures composed only of compressed (struts, rods) and tensed elements (cables). Although tensegrities are rod-like structures, some speciﬁc mechanical and mathematical properties distinguish them from conventional systems. The components are in a self-equilibrium system of internal forces called the self-stress state. In the absence of self-stress, the tensegrity structures are unstable, in other words, geometrically variable. Stabilisation occurs only after introducing initial stresses. Their modiﬁcation allows for the control of the static parameters of the structure, among others [1–5].

Appl. Sci. 2020, 10, 7042; doi:10.3390/app10207042 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, x 2 of 18

Appl. Sci.One2020 ,of10 ,the 7042 multitudes of applications of the tensegrity principle in civil engineering2 of 19are double-layer grids. Generally, the elements of a double-layer grid are organised into two parallel planes, which are connected by vertical and diagonal elements. In the horizontal projection, the One of the multitudes of applications of the tensegrity principle in civil engineering are double-layer elements are arranged in a regular pattern. Double-layer tensegrity grids (also called tensegrity grids. Generally, the elements of a double-layer grid are organised into two parallel planes, which are plate-like structures or tensegrity plates) are usually built from basic tensegrity modules such as connected by vertical and diagonal elements. In the horizontal projection, the elements are arranged in Simplex [6–13] or Quartex [6,14–18]. Adjacent modules in tensegrity plates can be connected in a a regular pattern. Double-layer tensegrity grids (also called tensegrity plate-like structures or tensegrity contiguous configuration (struts are connected to each other) or a non-contiguous configuration plates) are usually built from basic tensegrity modules such as Simplex [6–13] or Quartex [6,14–18]. (maintaining a discontinuous arrangement of compressed elements). Modules can be connected Adjacent modules in tensegrity plates can be connected in a contiguous configuration (struts are edge-to-edge, node–node or strut–cable. connected to each other) or a non-contiguous configuration (maintaining a discontinuous arrangement The first two-layer tensegrity grids were constructed by David Emmerich. In the patent [6], of compressed elements). Modules can be connected edge-to-edge, node–node or strut–cable. Emmerich proposed a structure consisting of modified Simplex modules with node–node The first two-layer tensegrity grids were constructed by David Emmerich. In the patent [6], connections and two structures built with Quartex modules linked edge-to-edge. A similar design Emmerich proposed a structure consisting of modified Simplex modules with node–node connections was found by Fuller [7] and Snelson (Snelson’s patent was rejected) [8] (Figure 1). Both of the and two structures built with Quartex modules linked edge-to-edge. A similar design was found abovementioned modules, i.e., Simplex and Quartex, are most often used to build tensegrity by Fuller [7] and Snelson (Snelson’s patent was rejected) [8] (Figure1). Both of the abovementioned plate-like structures. modules, i.e., Simplex and Quartex, are most often used to build tensegrity plate-like structures.

(a) (b) (c)

FigureFigure 1. 1.Patents Patents of of Simplex Simplex modules: modules: (a ()a Emmerich’s) Emmerich’s [6 [6],], (b ()b Fuller’s) Fuller’s [7 [7],], (c ()c Snelson’s) Snelson’s [8 [8].].

AnAn analysis analysis of of plates plates composed composed of of Simplex Simplex modules modules was was carried carried out, out, among among others, others, by by Kono Kono andand Kunieda Kunieda [ 19[19],], Gomez-Jauregui Gomez-Jauregui et et al. al. [20 [20]] and and Olejnikova Olejnikova [21 [21].]. Kono Kono and and Kunieda Kunieda created created the the firstfirst experimental experimental model model of of the the tensegrity tensegrity plate. plate. It It was was built built with with thirty-three thirty-three modified modified Simplex Simplex modules.modules. The The span span of of the the structure structure was was 9 9 m, m, and and its its area area was was 80 80 m m2.2 Skelton. Skelton and and Oliveira Oliveira compared compared thethe properties properties of of tensegrity tensegrity structures structures created created from from the the same same modules, modules, but but connected connected in in di differentfferent ways.ways. The The Gomez-Jauregui Gomez-Jauregui team team proposed proposed a a method method of of obtaining obtaining tensegrity tensegrity panels panels based based on on the the geometrygeometry of of traditional traditional double-layer double-layer trusses, trusses, while while Olejnikova Olejnikova studied studied constructions constructions with with single single and and doubledouble curvatures. curvatures. TensegrityTensegrity plates plates composed composed of of Quartex Quartex modules modules were were considered, considered, among among others, others, by by Wang Wang and and XuXu [22 [22],], Faroughi Faroughi and and Lee Lee [23 [23]] and and Sulaiman Sulaiman et et al. al. [24 [24]] (Figure (Figure2). 2). Wang Wang and and Xu Xu used used semidefinite semidefinite programmingprogramming (SDP) (SDP) to determineto determine the optimal the optimal topology topo oflogy the tensegrity of the tensegrity plate-like structureplate-like consisting structure ofconsisting nine modules. of nine Faroughi modules. and Faro Leeughi used and a genetic Lee used algorithm a genetic to optimise algorithm the to cross-sections optimise the ofcross-sections cables and strutsof cables of a structureand struts composed of a structure of twenty composed modules, of tw whereasenty modules, Sulaiman whereas et al. considered Sulaiman et the al. application considered ofthe double-layered application of tensegrity double-layered grids consisting tensegrity of grids four consisting or eight modules of four or as eight roofing. modules as roofing. Appl.Appl. Sci. Sci. 20202020, ,1010, ,x 7042 33 of of 18 19

(a) (b) (c)

FigureFigure 2. 2. Double-layeredDouble-layered tensegrity gridsgrids builtbuilt with with modiﬁed modified Quartex: Quartex: (a )(a model) model of Wangof Wang and and Xu [Xu22], [22],(b) model (b) model of Faroughi of Faroughi and Leeand [Lee23], [23], (c) model (c) model of the of Sulaimanthe Sulaiman et al. et [al.24 ].[24].

AA lot lot of of papers papers about about tensegrity tensegrity plates plates are are authored authored by by Wang Wang [15,25,26]. [15,25,26]. Wang Wang proposed proposed many many tensegritytensegrity plate-like plate-like structures structures composed composed of of Qu Quartexartex modules modules connected connected together together in in various various ways. ways. WangWang used other tensegrity modules to build platesplates as well. He He presented presented numerous numerous interesting interesting examplesexamples of of tensegrity tensegrity plate-like plate-like structures structures bu builtilt from from regular regular or or modified modified octahedron octahedron modules. modules. WangWang alsoalso proposedproposed a temporarya temporary tensegrity tensegrity plate-like plate-like structure, structure, which which could becould folded be andfolded unfolded and unfoldedif necessary. if necessary. IsraeliIsraeli researcher researcher Ariel Ariel Hanaor Hanaor had had a a great great impact impact on on the the development development of of research research on on tensegrity tensegrity platesplates [[27–31].27–31]. HanaorHanaor tested tested how how to combineto combine basic basi tensegrityc tensegrity modules modules to obtain to systemsobtain systems with suffi withcient sufficientrigidity and rigidity optimal and features optimal in terms features of weight. in term Hes presentedof weight. a geometricallyHe presented rigid a geometrically two-layer tensegrity rigid two-layergrid and, liketensegrity Wang, proposedgrid and, the like use Wang, of tensegrity proposed plates the as use temporary of tensegrity structures. plates as temporary structures.Two-layer tensegrity grids were also widely investigated by Motro and Averseng with their teams.Two-layer The aim tensegrity of the French grids scientists’ were also research widely was investigated to confirm by the Motro possibility and Averseng of the application with their of teams.tensegrity The structuresaim of the in French engineering scientists’ structures research [14 ,was32–37 to]. confirm An experimental the possibility structure of the model application consisting of tensegrityof nine modified structures Quartex in engineering modules was structures built. As part[14,32–37]. of the TensarchAn experimental project, which structure was the model main consistingpart of Raducan’s of nine dissertation,modified Quartex an experimental modules was model built. of aAs tensegrity part of the board Tensarch consisting project, of 2V which expander was themodules main waspart createdof Raducan’s [14,33, 34disser]. Aversengtation, an also experimental considered similarlymodel of structured a tensegrity plates. board Averseng, consisting Jamin of 2Vand expander Quirant proposedmodules was the usecreated of tensegrity [14,33,34]. plates Averse consistingng also ofconsidered 2V expander simi moduleslarly structured as a temporary plates. Averseng,structure forJamin people and with Quirant reduced proposed mobility the [35 –use37]. of tensegrity plates consisting of 2V expander modulesOther as interestinga temporary concepts structure of two-layerfor people tensegrity with reduced grids mobility have been [35–37]. presented by Papantoniou [38]. The modelsOther interesting were built withconcepts modified of two-layer Quartex tensegri modulesty and grids described have been on complicated presented by geometric Papantoniou solids. [38]. ApartThe models from standard were built applications with modified of tensegrity Quartex plates, non-standardmodules and ones described can be found.on complicated Interesting geometricproposals solids. of using double-layered tensegrity grids were presented by Al Sabouni-Zawadzka andApart Gilewski from [39 standard–41] (Figure applications3). In the of paper tensegrity [ 39], plates, the researchers non-standard presented ones can the conceptbe found. of Interestingmodeling tensegrityproposals plates of asusing a continuous double-layered model. tensegrity This approach grids was were also usedpresented by Obara by [4Al,5]. Sabouni-ZawadzkaAl Sabouni-Zawadzka and and Gilewski Gilewski [39–41] additionally (Figure 3). proposed In the paper the use [39], of tensegritythe researchers plate-like presented structures the conceptbuilt with of modeling Simplex tensegrity and Quartex plates modules as a contin asuous intelligent model. This constructions approach was [40] also and used structures by Obara of [4,5].metamaterials Al Sabouni-Zawadzka [41]. As in the worksand Gilewski of Al Sabouni-Zawadzka additionally proposed and Gilewski, the use the of use tensegrity of tensegrity plate-like in the structurescontext of built intelligent with Simplex construction and Quartex was dealt modules with by as Fest, intelligent Shea and constructions Smith [42,43 [40]]. They and proposedstructures theof metamaterialsconcept of a plate [41]. built As in with the threeworks or of five Al adjustableSabouni-Zawadzka tensegrity and star Gilewski, modules. the use of tensegrity in the context of intelligent construction was dealt with by Fest, Shea and Smith [42,43]. They proposed the concept of a plate built with three or five adjustable tensegrity star modules. Appl. Sci. 2020, 10, 7042 4 of 19 Appl. Sci. 2020, 10, x 4 of 18 Appl. Sci. 2020, 10, x 4 of 18

(a)(a) (b) (b) Figure 3. Double-layered tensegrity grids proposed by Al Sabouni-Zawadzka and Gilewski: FigureFigure 3. 3.Double-layered Double-layered tensegrity tensegrity grids grids proposed proposed by by Al Al Sabouni-Zawadzka Sabouni-Zawadzka and and Gilewski: Gilewski: (a) model (a) model from [39], (b) model from [41]. from(a) model [39], ( bfrom) model [39], from (b) model [41]. from [41]. The abovementioned examples are theoretical or experimental works. However, there are also The abovementioned examples are theoretical or experimental works. However, there are practicalThe abovementioned applications of examplestensegrity areplates theoretical in civil engineering.or experimental The mostworks. famous However, and the there most are also alsopractical practicalspectacular applications applications example ofof a oftensegritytwo-layer tensegrity tensegrity plates plates in grid in civil civil wa sengineering. the engineering. Blur Building The The pavilion most most famous[44–46], created andand thethe as mostmost spectacularspectaculara temporary example example structure of of a a two-layer fortwo-layer Expo 2012 tensegrity tensegrity in Switzerlan grid grid was wad (Figures the the Blur Blur 4a). Building BuildingThe structure pavilion pavilion was [ 44[44–46],created–46], createdfrom created as as aa temporary temporaryoctahedron structure structure modules for withfor Expo Expoadditional 2012 2012 incables in Switzerland Switzerlan connectingd (Figurethe (Figure top 4surface.a). 4a). The The The structure structuretensegrity was structurewas created created was from from octahedronoctahedronalso used modules modules as the roofing with with additional ofadditional a bank annex cables cables patio connecting connect in Athensing the the [47] top top (Figure surface. surface. 4b). The AThe similar tensegrity tensegrity construction structure structure is was was alsoalso usedused used as as the the roofing rooﬁng roofing of of the aof bank exhibitia bank annexon annex pavilion patio patio in in AthensPatras, in Athens Greece [47] (Figure[47] [47] (Figure (Figure4b). A 4b).4c). similar OtherA similar construction suggestions construction for is used is asused thethe rooﬁngas application the roofing of the of exhibitionof tensegrity the exhibiti plates pavilionon inpavilion real in Patras, engineering in Patras, Greece constructions Greece [47] (Figure [47] (Figurecan4c). be Otherfound, 4c). Other suggestions for example, suggestions forin the for [48,49]. applicationthe application of tensegrity of tensegrity plates plates in real in engineering real engineering constructions constructions can be found,can be forfound, example, for example, in [48,49 ].in [48,49].

(a)

(b) (c)

Figure 4. Practical applications of tensegrity in civil engineering: (a) Blur Building [46], (b) a patio roof of a bank annex (Athens, Greece) [47], (c) an exhibition pavilion for Patras Cultural Capital of Europe (Patras, Greece) [47]. (b) (c) Appl. Sci.Appl.2020 Sci., 102020, 7042, 10, x 5 of 185 of 19

Figure 4. Practical applications of tensegrity in civil engineering: (a) Blur Building [46], (b) a patio The reviewroof of aof bank the annex literature (Athens, showed Greece) that[47], the(c) an vast exhibition majority pavilion of works for Patras are Cultural devoted Capital mainly of to the search forEurope geometrical (Patras, Greece) configuration, [47]. control of the shape of tensegrity structures under the influence of external loads and the application of these structures. The parametric analysis, determining the influenceThe of self-stress review of statesthe literature on the showed static properties that the vast of majority these structures, of works are has devoted been developed mainly to slightly.the It shouldsearch be for noted geometrical that complete configuration, parametric control analysis of the of shape tensegrity of tensegrity structures structures is a two-stage under process.the influence of external loads and the application of these structures. The parametric analysis, The first stage includes the identification of self-stress states and infinitesimal mechanisms (qualitative determining the influence of self-stress states on the static properties of these structures, has been analysis),developed whereas slightly. the It second should stage be noted focuses that complete on the behaviour parametric analysis of tensegrities of tensegrity under structures external is a loads (quantitativetwo-stage analysis). process. The first stage includes the identification of self-stress states and infinitesimal Inmechanisms this paper, (qualitative the qualitative analysis), analysis whereas is the conducted. second stage This focuses analysis on the is thebehaviour basis forof tensegrities a quantitative analysis.under Tensegrity external loads plate-like (quantitative structures analysis). built with modified Quartex modules are considered. Starting fromIn this asingle-module paper, the qualitative structure, analysis more is complexconducted. cases This areanalysis sequentially is the basis analysed. for a quantitative The different methodsanalysis. of plates Tensegrity support plate-like are considered. structures To carrybuilt with out amodified qualitative Quartex assessment, modules the are spectral considered. analysis of the trussStarting matrices from (compatibility a single-module matrix structure, and sti ffmoreness matrixcomplex with cases the are eff ectsequentially of self-equilibrated analysed. forces)The is different methods of plates support are considered. To carry out a qualitative assessment, the used, including the singular value decomposition (SVD) of the compatibility matrix. The characteristic spectral analysis of the truss matrices (compatibility matrix and stiffness matrix with the effect of features of tensegrity structures are identified. On this basis, the plates are classified into one of the self-equilibrated forces) is used, including the singular value decomposition (SVD) of the four groupscompatibility defined matrix. in the The paper. characteristic This classification features of te isnsegrity important structures due are to diidentified.fferent behaviours On this basis, of the structurethe plates under are external classified actions. into one of the four groups defined in the paper. This classification is important due to different behaviours of the structure under external actions. 2. Mathematical Description The2. Mathematical space truss description finite element of Young’s modulus Ee, cross-sectional area Ae and length Le (Figure5a)The are space used totruss model finite the elem componentsent of Young’s of the modulus tensegrity , structure.cross-sectional In a area global coordinate and length system e 1 6 (x, y,(Figurez) a finite 5a) elementare used isto describedmodel the bycomponents the compatibility of the tensegrity matrix Bstructure.( R × )In: a global coordinate system (, , ) a finite element is described by the compatibility matrix∈ (∈ℝ×): e h i B = cx cy cz cx cy cz , (1) =−[−c −−c −−c c c c], (1) x x y y z z where= cj−=i =,c =j− i ,c== j− i . where cx Le , cy Le , cz Le .

(a) (b)

FigureFigure 5. ( a5.) ( Spacea) Space truss truss ﬁnite finite element, element, ((b) global degrees degrees of of freedom freedom of element of element . e.

Additionally,Additionally, due todue the to existing the existing self-stress self-stress state Sstatee, the standard, the standard truss ﬁnite truss element finite element description × description needs to be complemented with the geometric stiffnesse matrix6 6 (∈ℝ ): needs to be complemented with the geometric stiﬀness matrix KG R × : 100∈ − = ; =000.  (2) − e " # 0001 0 0  e S I I   K = − ; I =  0 0 0 . (2) The qualitative analysis is GprovidedLe forII an n-element space truss ( = 1, 2, … , ) described by the −  0 0 0  elasticity matrix (∈ℝ×): The qualitative analysis is provided for an n-element space truss (e = 1, 2, ... , n) described by the n n elasticity matrix E( R × ): ∈ h 1 1 2 2 n n i E = diag E A E A E A (3) L1 L2 ... Ln Appl. Sci. 2020, 10, 7042 6 of 19

m 1 with m degrees of freedom q( R × ): ∈ T q = [ q1 q2 ... qm ] . (4)

n m m m The compatibility matrix B( R × ) and the geometric stiﬀness matrix KG( R × ) for tensegrity ∈ ∈ structures are determined using the ﬁnite element formalism [50–59]:  B1C1     2 2   B C  B =  , (5)  ...     BnCn 

Xn e T e e KG = (C ) KG C , (6) i=1

e 6 m where C R × is a Boolean matrix. It is assumed that the number of global degrees of freedom ∈ of each element qi (i = 1, 2, ... , 6) corresponds to the number of global nodes of elements n1, n2, as shown in Figure5b. Consequently, the non-zero elements of Ce can be expressed as Ciqi = 1. The non-linearity equations of the equilibrium with the influence of the self-stress state in the structure can be written as: KT1q = P; KT1 = (KL + KG), (7) T where KT1 is the tangent stiffness matrix and KL = B EB is the linear stiffness matrix. The qualitative analysis of tensegrity structures can be done through the singular value decomposition [60–66] of the compatibility matrix B [58,67–71]. The singular value decomposition (SVD) of matrix B is a factorisation in the form:

B = YNXT, (8)

n n m m where Y( R × ) = [ y1 y2 ... yn ] and X( R × ) = [ x1 x2 ... xm ] are orthogonal ∈ n m ∈ matrices and N( R × ) is a rectangular diagonal matrix. The SVD of matrix B was described ∈ in depth in the paper [58], but to clarify, it should be remembered that the orthogonal matrices Y and X, as well as matrix N, are related to eigenvectors and eigenvalues of the following problems: BBT µI y = 0, BTB λI x = 0. (9) − − whose solutions can be expressed as: h i h i H = diag µ1 µ2 . . . µn , L = diag λ1 λ2 . . . λm . (10)

Zero eigenvalues (if they exist) of the matrix (9)1 µi = 0 are related to the non-zero solution of homogeneous equations called self-stress, or more precisely, self-equilibrated normal forces that satisfy homogeneous equations of equilibrium. The self-stress can be considered as an eigenvector related to zero eigenvalue yi = S(µi = 0). However, zero eigenvalues (if any) of the matrix (9)2λi = 0 are related to the finite mechanisms (rigid body motions) or infinitesimal mechanisms (local geometrical variability), which can be considered an eigenvector related to zero eigenvalue xi = q(λi = 0). In order to identify whether the mechanism is infinitesimal or finite, nonlinear analysis (7) with the use of the geometric stiffness matrix (6) should be applied. This is only if the self-stress exists. If the eigenvalues of KT1 are positive numbers, the mechanism is infinitesimal and the structure is stable. Zero eigenvalues are related to finite mechanisms, whereas a negative is responsible for instability of the structure. Appl. Sci. 2020, 10, 7042 7 of 19

The full solution to the eigen problem is provided by the spectral analysis of the tangent stiﬀness matrix KT1, which takes into consideration the inﬂuence of the self-equilibrated forces:

(K σI)z = 0. (11) T1 − Solutions of the eigen problem (11) can be expressed as: h i O = diag σ1 σ2 . . . σn . (12)

3. Classification of Tensegrity Structures In terms of structural mechanics, self-stress states and infinitesimal mechanisms stabilised by these states are the two most important features of tensegrity structures. Based on the definitions known from the literature, tensegrity structures demonstrate six characteristic features [58]: T — the structure is a truss, • S — there is a self-stress state, • C — tensile elements are cables and have no rigidity in compression, • M — there is an infinitesimal mechanism stiffened by the self-stress state, • I — the set of struts is contained within the continuous net of tensile elements, • D — compressed elements form a discontinuous set so its extremities do not touch each other. • The structure has to meet the first three obligatory features (T, S, C) and at least one of the others (M, D, I) to be considered a structure with tensegrity features. If a structure meets all the above requirements, it can be classified as a “pure” tensegrity structure [3,45,57,70,71]. That classification was revised by Obara [70] (60–62) and extended to four classes, namely: ideal tensegrity—structures which meet all requirements (T, S, C, M, D, I) and all self-stress states • (including the superposed one) must ensure the stability of the structure, “pure” tensegrity—structures satisfy the first five requirements, that is, T, S, C, M, D, and all • self-stress states ensure the stability of the structure, structures with tensegrity features of class 1—structures meet the conditions of the first four • features (T, S, C, M) and at least one self-stress state that ensures the stability of the structure, structures with tensegrity features of class 2—structures meet only the obligatory requirements (T, • S, C) and additionally either feature I or D. The classification is shown concisely in Table1. Features S and M (the presence of self-stress states and infinitesimal mechanisms) can be determined directly by the spectral analyses of the truss matrices, while the features D, I, and C (struts do not share ends and are located inside the continuous net of tensile elements with zero compression rigidity) are recognised indirectly by the identified self-stress state.

Table 1. Tensegrity classiﬁcation, based on [70].

TSCMID ideal tensegrity + + + + + + “pure” tensegrity + + + + + – structures with tensegrity features of class 1 + + + + –– + – structures with tensegrity features of class 2 + + + – – +

The proposed classification systematises and precisely defines tensegrity structures and ensures the correct use of the term “tensegrity” in all applications. In engineering practice, such systematisation facilitates the analysis and design of tensegrities because it is a consequence of different behaviours of tensegrity structures under the influence of external action. Appl. Sci. 2020, 10, 7042 8 of 19

4. Examples In this paper, the behaviour of structures built with modified Quartex modules is taken into account. Singular value decomposition of compatibility matrix B and spectral analysis of tangent stiffness matrix KT1 are used for the analysis. Twenty-five structures are considered. For each structure, the following matrices are determined:

H—eigenvalues of matrix BBT, • y —eigenvectors of BBT corresponding to the zero eigenvalues µ (if any) in H responsible for the • i i existence of the self-stress state (S), L—eigenvalues of matrix BTB, • x —eigenvectors of BTB corresponding to the zero eigenvalues λ (if any) in L responsible for the • i i existence of the mechanism (M), O—eigenvalues of matrix K . • T1 Firstly, the single module is considered. Because the characteristic features of tensegrity structures do not depend on geometrical and mechanical characteristics, the constant longitudinal stiffness EeAe = 1 and constant lengths a = 1 are assumed. The single modified Quartex module results are presented explicitly. Next, more complex cases are analysed. Modules were joined in a contiguous configuration, i.e., in a way so that the ends of struts do not stay separated from each other. The bottom surfaces are connected edge-to-edge, the top ones connected node-to-node. The considered tensegrity plate-like structures are built with four, sixteen or sixty-four modules. A plate strip model is taken into account as well. The different methods of plates support are considered. For all proposed structures, the characteristic features of tensegrity structures are identified. On this basis, the plates are classified into one of the four groups defined in the paper, i.e., ideal tensegrity, “pure” tensegrity and structures with tensegrity features of class 1 or class 2. In the case of plates, due to the complexity of the results, only the number of self-stress states, the number of mechanisms and the classification of structures are shown in the paper. The calculation module was written in the Mathematica environment, owing to which operations were simplified by using functions and commands implemented there.

4.1. Single Modified Quartex Module The first considered structure is the single modified Quartex module (Figure6). The module consists of sixteen elements (n = 16), i.e., four struts and twelve cables, and eight nodes (w = 8). Its shape is based on a regular prism. The projection of the top surface of the modified Quartex module is inscribed into the bottom one, allowing for the easy connection of single units into multi-module structures [13,23,72–75]. The considered module’s dimensions allow it to fit into a unit cube. The coordinates of the nodes and the numbers of elements are shown, respectively, in Tables2 and3.

Figure 6. Geometry of the single modiﬁed Quartex module.

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Table 2. Coordinates of the nodes of the single modiﬁed Quartex module.

No. of Node x y z 1 a 0 0 2 0 0.5a a 3 a a 0 4 0.5a 0 a 5 0 a 0 6 a 0.5a a 7 0 0 0 8 0.5a a a

Table 3. Numbers of elements of the single modiﬁed Quartex module.

No. of No. of First No. of Second Element Node Node 1 1 2 2 3 4 struts 3 5 6 4 7 8 5 1 3 6 1 7 bottom cables 7 5 7 8 3 5 9 3 6 10 1 4 middle cables 11 5 8 12 2 7 13 2 8 14 2 4 upper cables 15 4 6 16 6 8

The module with sixteen degrees of freedom (m = 16) is analysed. The blocked displacements are q1, q3, q8, q9, q13, q15, q20, q21 (Figure7). As the number of elements and the number of degrees of 16 16 freedom are equal (n = m = 16), the compatibility matrix B( R × ) is square, and, in turn, this means ∈ that the matrices BBT and BTB are equal and their eigenvalues are the same:

H = L = diag[2.72 2.71 2.71 2.7 1.39 1.34 1.34 1.29 1.0 0.96 0.96 0.88 0.23 0.1 0.1 0.0]

Appl. Sci. 2020, 10, 7042 10 of 19

Figure 7. Normalised self-stress state of the single modiﬁed Quartex module.

There is one zero eigenvalue in both matrices, thus one self-stress state and one mechanism is identiﬁed for the single module. The values of the eigenvector y16 correlate to the zero diagonal value of the matrix H, i.e., the self-stress forces (values are normalised in such a way that the maximum compressed force in struts is equal to 1), are equal to: − y = [ 1.0000 1.0000 1.0000 16 − − − 1.0000 0.3333 0.3333 0.3333 0.3333 − 0.7454 0.7454 0.7455 0.7454 0.4714 0.4714 0.4714 0.4714] and presented graphically in Figure7. The values of the eigenvector x16, correlated to the zero diagonal value of the matrix L, are the components of the displacement vector related to the mechanism and are equal: x16 = [0.0000 0.0000 0.4472 0.2236 0.0000 0.4472 0.0000 0.2236 − 0.0000 0.0000 0.4472 0.2236 0.0000 0.4472 0.0000 0.2236]. − In this case, the all eigenvalues of the tangent stiﬀness matrix KT1 are positive:

O = diag[6.7 108 6.7 108 6.7 108 6.7 108 2.7 108 2.6 108 2.6 108 2.5 108 2.1 108 · · · · · · · · · 2 108 2 108 1.8 108 3.8 107 1.7 107 1.7 107 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.07], · · · · · · which confirms the stability of the module. It means the identified mechanism x16 is infinitesimal. The infinitesimal mechanism of the modified Quartex is realised by the displacements of the top nodes (Figure8 and Video S1). If the displacements of nodes 2, 4, 6, 8 are blocked, the mechanism would 1 not exist.

Appl. Sci. 2020, 10, x 10 of 18

Figure 7. Normalised self-stress state of the single modified Quartex module.

There is one zero eigenvalue in both matrices, thus one self-stress state and one mechanism is identified for the single module. The values of the eigenvector correlate to the zero diagonal value of the matrix , i.e., the self-stress forces (values are normalised in such a way that the maximum compressed force in struts is equal to −1), are equal to:

= [−1.0000 − 1.0000 − 1.0000 − 1.0000 0.3333 0.3333 0.3333 0.3333

0.7454 0.7454 0.7455 0.7454 0.4714 0.4714 0.4714 0.4714] and presented graphically in Figure 7. The values of the eigenvector , correlated to the zero diagonal value of the matrix , are the components of the displacement vector related to the mechanism and are equal:

= [0.0000 0.0000 − 0.4472 0.2236 0.0000 0.4472 0.0000 0.2236

0,0000 0.0000 0.4472 0.2236 0.0000 − 0.4472 0.0000 0.2236].

In this case, the all eigenvalues of the tangent stiffness matrix are positive: = [6.7 ∙ 10 6.7 ∙ 10 6.7 ∙ 10 6.7 ∙ 10 2.7 ∙ 10 2.6 ∙ 10 2.6 ∙ 10 2.5 ∙10 2.1 ∙ 10

2∙10 2 ∙ 10 1.8 ∙ 10 3.8 ∙ 10 1.7 ∙ 10 1.7 ∙ 10 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.07], which confirms the stability of the module. It means the identified mechanism is infinitesimal. The infinitesimal mechanism of the modified Quartex is realised by the displacements of the top nodes (Figure 8 and Video S1). If the displacements of nodes 2, 4, 6, 8 are blocked, the mechanism Appl. Sci. 2020, 10, 7042 11 of 19 would not exist.

FigureFigure 8. 8. InfinitesimalInﬁnitesimal mechanism mechanism of of the the single single modified modiﬁed Quartex Quartex module. module.

TheThe single single modified modified Quartex module meetsmeets allall tensegritytensegrity features, features, i.e., i.e., the the structure structure is is a trussa truss (T ) (withT) with a continuous a continuous net net of tensileof tensile components—cables components—cables (C )—including(C)—including the the discontinuous discontinuous set set (D )(D of) theof thecompressed compressed elements elements (I) and (I) it and features it features the existence the existence of the self-stress of the state self-stress (S) and thestate mechanism (S) and the (M). mechanismIt means that (M it). isIt classifiedmeans that as it the is classified ideal tensegrity. as the ideal tensegrity. 4.2. Four-Module Tensegrity Plate-Like Structures 4.2. Four-Module Tensegrity Plate-Like Structures The structure built with four modified Quartex modules is considered (Figure9a). The plate The structure built with four modified Quartex modules is considered (Figure 9a). The plate consists of 56 elements (n = 56) and 21 nodes (w = 21). Ten models with different methods of support, consists of 56 elements (n = 56) and 21 nodes (w = 21). Ten models with different methods of support, shown in Figure9b, are considered: shown in Figure 9b, are considered: • model P4-1—the truss (T) with 39 d.o.f. (m = 39), • model P4-1—the truss (T) with 39 d.o.f. (m = 39), model P4-2—the truss (T) with 45 d.o.f. (m = 45), • model P4-3—the truss (T) with 45 d.o.f. (m = 45), • model P4-4—the truss (T) with 48 d.o.f. (m = 48), • model P4-5—the truss (T) with 48 d.o.f. (m = 48), • model P4-6—the truss (T) with 39 d.o.f. (m = 39), • model P4-7—the truss (T) with 39 d.o.f. (m = 39), • model P4-8—the truss (T) with 38 d.o.f. (m = 38), • model P4-9—the truss (T) with 38 d.o.f. (m = 38), • model P4-10—the truss (T) with 39 d.o.f. (m = 39). •

Appl. Sci. 2020, 10, 7042 12 of 19

Figure 9. Four-module tensegrity plate-like structure: (a) geometry, (b) models—bottom means that lower nodes 1–4 and 6–9 are supported; top1 means that upper nodes 10–12, 14–16 and 18–21 are supported.

For all analysed four-module models, the self-stress states (S) are identified—the models differ in the number of self-stress states. However, mechanisms (M) are identified only for five models, i.e., P4-1–P4-5. Model P4-1 is characterised by one mechanism, models P4-2, P4-4 and P4-5–by two mechanisms and model P4-3 by three mechanisms. None of the self-stress states correctly identifies the type of elements (that is, what is a strut and what is a cable) so for the last part of the analysis, superposed and normalised self-stress states for the single modified Quartex module is taken into account. All eigenvalues of the tangential stiffness matrix KT1 for models P4-1–P4-5 are positive and that confirms the stability of these plates. It means the identified mechanisms are infinitesimal. Other than the abovementioned characteristic features S and M, all models satisfy the requirements of characteristics T (plates are trusses) and C (tensile elements are cables and have no rigidity in compression). Models P4-1–P4-5 feature the presence of the mechanisms, so they can be classified as structures with tensegrity features of class 1. The feature D cannot be met because of the method of connecting modules. This feature does not occur for any of the structures presented in the paper. Due to lack of mechanisms, the other models, i.e., P4-6–P4-10, are qualified as structures with tensegrity features of class 2.

4.3. Sixteen-Module Tensegrity Plate-Like Structures The plate consisting of sixteen modiﬁed Quartex modules is taken into account (Figure 10a). This structure consists of 212 elements (n = 212) and 69 nodes (w = 69). Six models, shown in Figure 10b, are considered:

model P16-1—the truss (T) with 153 d.o.f. (m = 153), • model P16-2—the truss (T) with 156 d.o.f. (m = 156), • model P16-3—the truss (T) with 105 d.o.f. (m = 105), • model P16-4—the truss (T) with 129 d.o.f. (m = 129), • model P16-5—the truss (T) with 122 d.o.f. (m = 122), • model P16-6—the truss (T) with 146 d.o.f. (m = 146). • Appl. Sci. 2020, 10, x 12 of 18

4.3. Sixteen-Module Tensegrity Plate-Like Structures The plate consisting of sixteen modified Quartex modules is taken into account (Figure 10a). This structure consists of 212 elements (n = 212) and 69 nodes (w = 69). Six models, shown in Figure 10b, are considered: • model P16-1—the truss (T) with 153 d.o.f. (m = 153), • model P16-2—the truss (T) with 156 d.o.f. (m = 156), • model P16-3—the truss (T) with 105 d.o.f. (m = 105), • model P16-4—the truss (T) with 129 d.o.f. (m = 129), • Appl. Sci. 2020model, 10, 7042P16-5—the truss (T) with 122 d.o.f. (m = 122), 13 of 19 • model P16-6—the truss (T) with 146 d.o.f. (m = 146).

Model P16-1 Model P16-2 Model P16-3 Model P16-4 Model P16-5 Model P16-6 (a) (b)

FigureFigure 10. 10.Sixteen-module Sixteen-module tensegrity tensegrity plate-like structure: structure: (a) ( atop) top view, view, (b) (models.b) models.

For allFor models, all models, self-stress self-stress states states (S ()S) exist. exist. OneOne mechanism mechanism (M (M) is) isidentified identified for formodels models P16-1 P16-1 and and P16-6.P16-6. Only Only superposed superposed self-stress self-stress states states correctly correctly identified identified elements. elements. None of the eigenvalues eigenvalues of of the tangentialthe tangential stiffness stiffness matrix matrixKT1 for models for models P16-1 P16-1 andP16-6 and P16-6 is non-positive, is non-positive, so theso the structures structures are are stable and thestable identified and the identified mechanisms mechanisms are infinitesimal. are infinitesi Exceptmal. Except for the for abovementioned the abovementioned features, features, all modelsall are trussesmodels ( Tare) and trusses their (T tensile) and their elements tensile are elements cables withare cables no rigidity with no in rigidity compression in compression (C). Models (C). P16-1 Models P16-1 and P16-6, due to the existence of the mechanism, are classified as structures with and P16-6, due to the existence of the mechanism, are classified as structures with tensegrity features tensegrity features of class 1, while models P16-2–P16-5 satisfy the conditions of structures with of class 1, while models P16-2–P16-5 satisfy the conditions of structures with tensegrity features of tensegrity features of class 2 because of the absence of mechanisms. class 2 because of the absence of mechanisms. 4.4. Sixty-Four-Module Tensegrity Plate-Like Structures 4.4. Sixty-Four-Module Tensegrity Plate-Like Structures Next, the structure built with sixty-four modified Quartex modules is taken into account Next,(Figure the11a). structure The plate builtconsists with of 800 sixty-four elements modified (n = 800) and Quartex 225 nodes modules (w = 225). is taken Three intomodels, account (Figureshown 11a). in Figure The plate 11b, consistsare considered: of 800 elements (n = 800) and 225 nodes (w = 225). Three models, shown• inmodel Figure P64-1—the 11b, are considered: truss (T) with 579 d.o.f. (m = 579), • modelmodel P64-1—the P64-2—the truss truss (T )(T with) with 579 621 d.o.f. d.o.f. ((m = 621),579), • • model P64-3—the truss (T) with 621 d.o.f. (m = 621). model P64-2—the truss (T) with 621 d.o.f. (m = 621), • model P64-3—the truss (T) with 621 d.o.f. (m = 621). • Appl. Sci. 2020, 10, 7042 14 of 19

Figure 11. Sixty-four-module tensegrity plate-like structure: (a) geometry, (b) models. For all three models, self-stress states (S) exist. However, none of the self-stress states other than the superimposed one properly identifies tensile and compressed elements. One mechanism M ( ) is identified for each model. The identified mechanisms are infinitesimal and ensure the stability of the structure—the tangential stiffness matrix KT1 for all modes has no non-positive eigenvalues.

All models also meet the requirements of features T and C—they are trusses with tensile elements that are cables and have no rigidity in compression. Both models satisfy the conditions of structures with tensegrity features of class 1. For the sixty-four-module structure supported on two opposite edges (models P64-2 and P64-3), unlike the case the of the four-module structure (models P4-2 and P4-3), the change in the direction of support has no inﬂuence on the number of self-stress states. Both models feature 180 self-stress states and one mechanism.

4.5. Plate Strips The last considered structures are plate strips. They are modelled like the four-module structures, presented in Figure9a, with blocked displacements in the y direction in nodes 4, 6, 12, 16, 18 and 19. The structures consist of 56 elements (n = 56) and 21 nodes (w = 21). Five models (Figure 12) are taken into consideration: model PS-1—the truss (T) with 39 d.o.f. (m = 39), • model PS-2—the truss (T) with 40 d.o.f. (m = 45), • model PS-3—the truss (T) with 27 d.o.f. (m = 27), • model PS-4—the truss (T) with 33 d.o.f. (m = 33), • model PS-5—the truss (T) with 32 d.o.f. (m = 32). •

Figure 12. Models of the tensegrity plate strips. 2 All models feature self-stress states (S) but only for model PS-1 does a mechanism (M) exist. Only the superposed self-stress state identiﬁes elements properly. The tangential stiﬀness matrix

2 Appl. Sci. 2020, 10, 7042 15 of 19

KT1 for model PS-1 has all positive eigenvalues. It means that the mechanism is infinitesimal and the structure is stable. All considered models satisfy the requirements of the characteristics T and C (structures are trusses and tensile elements are cables with zero rigidity in compression). Due to the occurrence of the mechanism, model PS-1 satisfies the conditions of structures with tensegrity features of class 1. The rest of the considered models, i.e., models PS-2–PS-5, are classified as having only tensegrity features of class 2 because of the lack of mechanisms. Comparing the analyses performed for the sixteen-module plate and plate strip, it can be noticed that the number of mechanisms obtained from the corresponding models (i.e., P16-1 and PS-1, P16-2 and PS-2, etc.) is equal.

5. Conclusions The qualitative evaluation of truss structures presented in this paper was done for the assessment of the tensegrity plate-like structures. The qualitative analysis leads us to define the unique properties of tensegrities, which allow for the control of static and dynamic parameters. With regard to structural mechanics, the uniqueness of tensegrity structures lies in the existence of self-stress states (S) and mechanisms (M). The other features, i.e., the discontinuity of struts (D) and the inclusion of struts by tension elements (I), which are cables with no rigidity in compression (C), are determined indirectly by the identified self-stress state. Based on the presence of the characteristic tensegrity features T, S, C, M, D and I, a four-category classification was proposed. The structures were classified as ideal tensegrities, “pure” tensegrities and structures with tensegrity features of class 1 or 2. Plates built with modified Quartex modules were considered. Starting from a single-module structure, more complex cases were sequentially analysed. The different methods of plates support were considered. The presence of self-stress states (S) and mechanisms (M) was determined by the singular value decomposition of the compatibility matrix. If any mechanism existed, the spectral analysis of the tangent stiffness matrix, which includes the self-stress state, was used to establish whether the mechanism was finite or infinitesimal. The compatibility matrix and the tangent stiffness matrix were determined using the finite element method. The geometrical and mechanical characteristics did not affect the peculiar tensegrity features, so all constants were assumed as a unit. Results for all analysed plates are summarised in Table4. The following conclusions were obtained:

Only the single modified Quartex module is the ideal tensegrity. This module meets • all tensegrity features, i.e., the structure is a truss (T) with a continuous net of tensile components—cables (C)—including a discontinuous set (D) of the compressed elements (I) and it features the existence of one self-stress state (S) and one mechanism (M). The plates built with four, sixteen or sixty-four modules cannot be the ideal tensegrities because • of the way of connecting the modules. These structures do not satisfy the condition for the discontinuity of struts (D). All of the plates are characterised by the existence of more than one self-stress state (S). Due to the • fact that only one self-stress state accurately identifies the elements, they cannot be considered as “pure” tensegrities. Actually, the only one appropriate self-stress state for the all considered plate-like structures is the one that is a superposition of the self-stress state of the single module. Considered plates can be classified as structures with features of class 1, if there is a mechanism, • or class 2, if there is no mechanism (see Table4). Appl. Sci. 2020, 10, 7042 16 of 19

Table 4. Results of the qualitative analysis of the structures built with the modiﬁed Quartex modules.

No. of No. of No. of Degrees of No. of No. of Single Model Nodes Elements Freedom Self-Stress Classiﬁcation Mechanisms Modules (w) (n) (m) States 1 SM 8 16 16 1 1 ideal tensegrity P4-1 39 1 18

P4-2 45 2 13 structures with tensegrity P4-3 45 3 14 features of class 1 4 P4-4, P4-521 56 48 2 10 P4-6, P4-7 39 0 17 structures with tensegrity P4-8, P4-9 38 0 18 features of class 2 P4-10 39 0 17 structures with tensegrity P16-1 153 1 60 features of class 1 P16-2 156 0 56 P16-2 105 0 107 16 69 212 structures with tensegrity P16-3 129 0 83 features of class 2 P16-4 122 0 90 structures with tensegrity P16-6 146 1 67 features of class 1

P64-1 579 1 222 structures with tensegrity 64 225 800 P64-2, P64-3 621 1 180 features of class 1 structures with tensegrity PS-1 39 1 18 features of class 1 PS-2 40 0 16 4 21 56 PS-3 27 0 29 structures with tensegrity PS-4 33 0 23 features of class 2 PS-5 32 0 24

The complete analysis of tensegrity structures is a two-stage process. The qualitative analysis carried out in the paper is the first stage and it is the basis for a quantitative analysis, which is the second stage. A quantitative analysis, which focuses on the behaviour of tensegrities under external loads, will be provided in part 2 of the paper. The proposed classification is important due to different behaviours of the structure under external actions. If a mechanism exists for the structure, that is when the structure is assigned to the one of the first three categories, and it is possible to control static properties, like, for example, stiffness, by the change in the self-stress level. The proposed tensegrity plate-like structures can be used in the design process of standard tensegrity building structures, i.e., plates and non-standard applications. The tensegrity concept can be used in metamaterial unit cells and in deployable or smart engineering structures with monitoring, self-control, self-repair and active control.

Supplementary Materials: The following are available online at http://www.mdpi.com/2076-3417/10/20/7042/s1, Video S1: mechanism. Author Contributions: Introduction was prepared by J.T. Definition of tensegrity and mathematical description were written by P.O. Results were obtained by J.T. The analysis of the results and conclusions were written by P.O. and J.T. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

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