Three-dimensional modeling of the wave dynamics of tensegrity lattices

Composite Structures 173 (2017) 9–16

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Three-dimensional modeling of the wave dynamics of tensegrity lattices F. Fabbrocino a, G. Carpentieri b,⇑ a b

Department of Engineering, Pegaso University, Piazza Trieste e Trento, 48, 80132 Naples, Italy Department of Civil Engineering, University of Salerno, 84084 Fisciano (Salerno), Italy

a r t i c l e

i n f o

Article history: Received 26 March 2017 Accepted 29 March 2017 Available online 1 April 2017 Keywords: Lattice metamaterials Tensegrity structures Wave dynamics Geometric nonlinearities Acoustic lenses

a b s t r a c t This paper develops effective numerical models to study the wave dynamics of highly nonlinear tensegrity metamaterials. Recent studies have highlighted the geometrically nonlinear response of structural lattices based on tensegrity prisms, which may gradually change their elastic response from stiffening to softening through the modification of mechanical, geometrical, and prestress variables. We here study the nonlinear dynamics of columns of tensegrity prisms subject to impulsive compressive loading. An effective nonlinear rigid body dynamics is employed to simulate the dynamic response of such metamaterials. We illustrate how to pass from the matrix to the vector form of the equations of motions, on accounting for a rigid response of the compressive members (bars). Numerical simulations show that the wave dynamics of the examined metamaterials supports compression solitary pulses with profile dependent on the elastic properties of the tensile members (strings), the given impact velocity, and the applied prestress. We conclude that tensegrity columns can be effectively used as tunable acoustic lenses, which are able to generate acoustic solitary waves with adjustable profile in a host medium. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The dynamics of strongly nonlinear metamaterials is receiving increasing attention by the scientific community, (refer, e.g. to review papers [1,2] and references therein). Several studies have shown that elastically hardening discrete systems support compressive solitary waves and the unusual reflection of waves on material interfaces [3–8], while elastically softening systems support the propagation of rarefaction solitary waves under initially compressive impact loading [9,6]. Solitary wave dynamics has been proven to be useful for the construction of a variety of novel acoustic devices. These include: acoustic band gap materials; shock protector devices; acoustic lenses; and energy trapping containers, to name some examples (refer, e.g., to [10] and references therein). It has been found that structural lattices based on tensegrity units (e.g., tensegrity prisms) exhibit a tunable geometrically nonlinear response, which may gradually change from stiffening to softening through the modification of mechanical, geometrical, and prestress variables [11–13,9]. Tensegrity structures are prestressable truss structures, obtained by connecting compressive members (bars or struts) through the use of pre-stretched tensile elements (cables or strings). It is known that tensegrity concepts ⇑ Corresponding author. E-mail addresses: [email protected] (F. Fabbrocino), gerardo_ [email protected] (G. Carpentieri). http://dx.doi.org/10.1016/j.compstruct.2017.03.102 0263-8223/Ó 2017 Elsevier Ltd. All rights reserved.

diffusely appear in nature, such as, e.g., in cells, the structure of the spider silk, and the system of bones and tendons in animals and humans [14]. Attention is increasingly being given to the development of efficient analytical and numerical methods for exploiting tensegrity concepts in engineering design (refer to [14] and references therein). Also the form-finding of tensegrity structures continues to be an active research area, due to both their easy control (geometry, size, topology and prestress control) [15], and the fact that such structures provide minimum mass systems under different loading conditions [16–18,14,19,20]. The use of fractal geometries for the multiscale design of tensegrity systems - diffusely illustrated [20,21] – is of particular interest. The importance of protecting materials and buildings against impacts with external objects is well known (cf., e.g., [4,22]). Equally, there is growing interest in research into noninvasive tools to target defects in materials, and for monitoring structural health in materials and structures [23–26]. In the past, shock protectors and devices used for focusing acoustic waves mainly relied on energy dissipation and the modification of sound propagation through spatially dependent delays. Highly efficient and unconventional mechanisms for protecting materials and focusing mechanical waves through the use of rarefaction and compression solitary waves have recently been discovered by [8,9]. It is worth noting that arrays of tensegrity lattices with elastically hardening response can be employed to fabricate tunable focus acoustic lenses that support extremely compact solitary waves [8,9].

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Three-dimensional finite element (FE) models of lattice structures usually make use of tetrahedral elements with a large number of degrees of freedom [27]. Such models are hardly applicable to dynamic simulations, even for lattices constituted by a small number of cells. A key goal of the present work is to develop efficient and accurate models of tensegrity lattices that make use of 3D assemblies of one-dimensional models for bars and strings. By describing the bars as rigid members and the cables as elastically deformable elements, we first develop the dynamics of an arbitrary tensegrity network in matrix form, and next we show how to switch such a formulation to vector form (Section 2). The latter proves to be useful in order to coupling the proposed model with standard FE models that may interact with tensegrity networks. The time-integration of the equations of motion is conducted through a Runge–Kutta algorithm that accounts for a rigidity constraint of the bars [28]. In Section 3, we apply the proposed numerical model to investigate the nonlinear wave dynamics of tensegrity columns. We study the dynamic response of such systems to impact loading, by establishing comparisons with the alternative model proposed in Ref. [29], which assumes rigid response of the terminal bases of each prism [30]. We show that our 3D modeling of tensegrity columns allows us to detect different strain wave profiles, as a function of the applied prestress, and a rigidity parameter describing the kinematics of the terminal bases (Section 4). Such tunable response can be profitably used to build adaptive arrays of tensegrity columns, which can subjected to different levels of prestress, so as to generate solitary waves with different phases that coalesce at a focal point in an adjacent host medium [23,24]. We draw the main conclusion of the present study and future research lines in Section 5. 2. A numerical model for the dynamics of tensegrity networks The dynamic problem of a general three-dimensional lattice is hereafter presented via a suitable, vector form reformulation of the tensegrity dynamics presented in [31]. We first introduce some basic notation (Section 2.1). Next we summarize the matrix-form of the tensegrity dynamics diffusely illustrated in [31] (Section 2.2), and then pass to develop the vector form of the equations of motions employed in the present work (Section 2.3). 2.1. Basic notation 2.1.1. Matrices and vectors Throughout the paper, we indicate matrices with bold capital letters (ie X), vectors with bold lower case letters (i.e. x), and scalars with italic letters (ie x). For later use, we introduce the following operators: ^ ¼ diag ðx1 ; x2 ; . . . ; xn Þis an operator that produces a diagonal –x matrix with the components x1 ; x2 ; . . . ; xn of the vector x; – bXc is an operator that keeps only the diagonal terms of the square matrix X and set to zero all the off-diagonal terms; – vecðXÞ indicates the vectorizing operator that stack up all columns of matrix X; – the Kronecker product between two matrices A 2 Rmn and B 2 Rpq , through the equation:

2

6 AB¼6 4

a11 B .. .

 .. .

3

a1n B .. 7 mpnq 7 . 52R am1 B    amn B

where aij is the ith;jth element of the matrix A.

ð1Þ

2.1.2. Tensegrity networks Let us consider a tensegrity network made up of nn nodes (or joints), nb bars and ns cables (Fig. 1). The joints are frictionless hinges, and each member carries only axial forces. The bars (i.e., the compressed members) are assumed to behave as straight rigid bodies (rods) with uniform mass density, constant cross-section, and negligible rotational inertia about the longitudinal axis. The cables are instead modeled as straight elastic springs that can carry only tensile forces. The generic node i, with i 2 ½1; . . . ; nn , is located by the vector ni 2 R3 in the three-dimensional Euclidean space, and is loaded with an external force vector wi 2 R3 . By suitably collecting the vectors ni and wi , we introduce the following nodal and force matrices:

N ¼ ½ n1

. . . ni

n2

W ¼ ½ w1

. . . nnn  2 R3nn

. . . wi

w2

. . . wnn  2 R3nn

ð2Þ ð3Þ

The kth bar (or cable) k of the network, with k 2 ½1; . . . ; nb  (or k 2 ½1; . . . ; ns ), is located by the vector bk 2 R3 (or sk 2 R3 ). For example, if the kth bar connects nodes i and j, then bk ¼ nj  ni . By stacking up the bar and string vectors, we obtain the following matrices describing the geometry of all bars and cables:

 B ¼ b1

b2

S ¼½ s1

s2

. . . bk . . . sk

 . . . bnb 2 R3nb ;

. . . sns  2 R3ns

ð4Þ ð5Þ

The center of mass of the kth bar between nodes i and j is   located by the vector rk ¼ ni þ nj =2. Collecting all the rk vectors, we get the matrix:

 R ¼ r1

r2

. . . rk

 . . . rnb 2 R3nb

ð6Þ

It is useful to rewrite the above matrices as follows:

B ¼ NCTB ;

S ¼ NCTS ;

R ¼ NCTR

ð7Þ

where CB 2 Rnb nn and CS 2 Rns nn are connectivity matrices of bars and cables, respectively. The general ith row of CB (or CS ) corresponds to the ith bar (or cable), and the element CB ij (or CSij ) is equal to: 1 if vector bi (or si ) is directed away from node jth;1 if vector bi (or si ) is directed toward node jth, and 0 if vector bi (or si ) does not touch node j. Similarly, the ith row of CR 2 Rnb nn corresponds to the bar bi , and the element CR ij is equal to: 1 if vector bi is touching node j, or 0 if vector bi does not touch node j. Following Ref. [14], we say that a tensegrity network is of class n, if the maximum number of bars concurring in each node is equal to n. 2.1.3. Cable forces Let us consider now the generic cable (say the kth one) with Young modulus of the material Esk , cross-section area Ask , rest length Lk , and stretched length sk (i.e. sk ¼ ksk k, and sk P Lk ). We define the stiffness ksk and the prestrain pk through the following equations:

Esk Ask ; Lk sk  L k pk ¼ Lk

ksk ¼

ð8Þ ð9Þ

The force density carried by the current cable is given by the following (unilateral) constitutive equation (elastic, no-compression response):





ck ¼ max ksk 1  ck ¼0; if : sk < Lk

 Lk ;0 ; sk

if : sk P Lk ;

ð10Þ ð11Þ

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Fig. 1. General tensegrity network: the red thin lines are cables, the black thick lines are bars.

As we shall see in the sequel, it is convenient to collect all the quantities ck into the diagonal matrix:   c^ ¼ diag c1 c2 . . . cns 2 Rns ns . 2.2. Matrix form of the tensegrity dynamics On adopting the matrix from of the tensegrity dynamics presented in Ref. [31], we write the equations of motion of a class 1 tensegrity network as follows:

€ þ NK ¼ W NM

ð12Þ

where:

1 ^ R 2 Rnn nn þ CTR mC 12 ^CS  CTB ^kCB 2 Rnn nn K ¼ CTS c

^ B M ¼ CTB mC

e1 ¼ ½ 1 0    0 T 2 Rnn T

e2 ¼ ½ 0 1    0  2 R

ð14Þ

ð22Þ

enn ¼ ½ 0 0    1 T 2 Rnn

ð23Þ

we can stack up each column of the matrices appearing in Eq. (12) in order to rewrite Eq. (19) as follows:

2 € NMe1 6 € 6 NMe2 6 .. 6 4 . € NMe n

3

3 2 3 2 NKe1 We1 7 6 NKe 7 6 We 7 2 7 2 7 7 6 6 7 þ 6 . 7 ¼ 6 . 7 2 R3nn 7 6 . 7 6 . 7 5 4 . 5 4 . 5 NKenn

ð15Þ

Wenn

3 eT1 Mei 7 6 T nn 6 e2 Mei 7 X

 7 6 € j eTj Mei € nn 6 n  n 7¼ .. 7 6 . 5 j¼1 4 T en Mei 3 2 Tn e1 Kei 7 6 T nn 6 e2 Kei 7 X

 7 6    n nn 6 . 7 ¼ nj eTj Kei 6 .. 7 j¼1 5 4 eTnn Kei 2

€ ðMei Þ ¼ ½ n €1 N

€2 n

NðKei Þ ¼ ½ n1

n2

ð16Þ ð17Þ

The generalization of the above equations to the case of a class k system is straightforward, by making recourse to the Lagrange multipliers technique illustrated in [32].

ð24Þ

Consider now that the vectors appearing on left-hand side of Eq. (24) can be expressed as:

and:

  ^ ¼ diag m1 ; m2 ; . . . ; mnb 2 Rnb nb m j k k 1 1 j T ^CS ÞCTB ‘^2 2 Rnb nb ^ ‘^2 ^k ¼ B_ T B_ m þ B ðW  Sc 12 2

 ‘^2 ¼ diag kb1 k2 ; kb2 k2 ; . . . ; kbnb k2 2 Rnb nb

ð21Þ

...

n

ð13Þ

ð20Þ

nn

Substituting Eq. (26) into Eq. (24) we have:

We show in this section how to convert the matrix form (12) of the equations of motion into the following vector form:

€ þ Kn n ¼ w 2 R3nn Mn n

ð18Þ

where: n ¼ v ecðNÞ, Mn ¼ M  I3 , and Kn ¼ K  I3 . On applying the vectorizing operator to (12), we obtain:

 € v ec NM þ v ecðNKÞ ¼ v ecðWÞ 2 R3nn

Upon introducing the following base vectors:

ð19Þ

2 6 6 6 þ6 6 4

ð26Þ

3 € 1 þ eT2 Me1 n € 2 þ    þ eTn Me1 n € nn eT1 Me1 n n 6 T € 1 þ eT2 Me2 n € 2 þ    þ eTn Me2 n € nn 7 7 6 e1 Me2 n n 7 6 7þð27Þ 6 .. 7 6 . 5 4 € 1 þ eT2 Menn n € 2 þ    þ eTn Menn n € nn eT1 Menn n n 3 2 3 eT1 Ke1 n1 þ eT2 Ke1 n2 þ    þ eTnn Ke1 nnn We1 7 eT1 Ke2 n1 þ eT2 Ke2 n2 þ    þ eTnn Ke2 nnn 7 6 We2 7 7 7 6 6 . 7 2 R3nn ð28Þ ¼ 7 6 .. 7 4 .. 7 5 . 5 2

2.3. Vector form of the equations of motions

ð25Þ

eT1 Kenn n1 þ eT2 Kenn n2 þ    þ eTnn Kenn nnn

Wenn

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which implies, after some calculations:



 € þ KT  I3 n ¼ w 2 R3nn MT  I 3 n

ð29Þ

Since M and K are symmetric, upon defining Mn ¼ M  I3 and Kn ¼ K  I3 , we can finally reduce Eq. (29) to the vector form (18). We employ the Runge-Kutta integration algorithm described in Ref. [28] to perform the time-integration of Eq. (18). Such an algorithm prevents numerical violations of the rigidity constraint of the bars, ensuring that the bar vectors bk remain constant at each time step. 3. Tensegrity columns A tensegrity column is a three-dimensional lattice obtained by stacking up an arbitrary number np of tensegrity prisms one over the other [14]. We here consider columns of T3 tensegrity prisms (Fig. 2), which are made of 3 bars (members 1–4, 2–5, and 3–6), 3 bottom horizontal cables (1–2-3), 3 top horizontal cables (4–56), and 3 cross cables (1–6, 2–4, and 3–5). In particular, we consider T3 prisms that are: regular since the top and bottom bases are equilateral triangles, and minimal in the sense that the structure has the minimum possible number of cables required for stability. Let us define: L the length of the bars, a the radius of the circle that inscribes the top or bottom triangles, h the height of the prisms, u the ”twist” angle between bottom and top faces, k the stretched length of the vertical cables, ‘ the stretched length of the top or bottom horizontal cables. The feasible configurations are obtained in the range of u varying between p=3 (strings touching each other) and p (bars touching each other). As already mentioned, we model the bars as rigid bodies and the strings as linear elastic cables (cf. Section 2.1.3). A feasible prestressable configuration (reference configuration with nonzero internal forces under zero external forces) is obtained for / ¼ 5=6p. Note that each prism can be either right-handed or lefthanded, depending whether the upper base is clockwise or counter-clockwise twisted with respect to the lower base, respectively. The obtainable columns can stack blocks with the same orientation, or blocks with different orientations. In either cases, the   column is made of nn ¼ 3 np þ 1 nodes, nb ¼ 3np bars, and   ns ¼ 3 2np þ 1 cables. The static and quasi-static equilibrium problems of T3 prisms have been extensively studied in [11–13,30] using analytical,

numerical and experimental approaches. The highly nonlinear dynamics of tensegrity metamaterials alternating tensegrity prisms and lumped masses has been investigated in [8,9]. The latter feature frictionless contact between the T3 prisms and the lumped masses, so that the masses can move only in the longitudinal direction. The studies conducted in [8,9] have shown that elastically hardening tensegrity metamaterials support compressive solitary waves and the unusual reflection of waves on material interfaces, while elastically softening systems support the propagation of rarefaction solitary waves under initially compressive impact loading. A mechanical modeling of the wave dynamics of tensegrity columns has been recently presented in [29], with reference to the case with prisms featuring rigid response of the terminal bases. The study presented in [29] accounts for the coupling between longitudinal and twisting modes of tensegrity columns, and highlights that also such systems, as well as the tensegrity metamaterials analyzed in Refs. [8,9], support the propagation of compressive solitary waves under impact loading. The next section presents the application of the numerical model presented in Section 2 to the study of the impact dynamics of right-handed tensegrity columns, which may feature either flexible (Fig. 3(a-b)) or rigid bases (Fig. 3(c-d)) in each T3 prism. We examine columns that are constrained at one end and free at the other end, under impulsive loading generated by given initial velocities of the nodes of the free end. The limiting case with rigid bases of the T3 prisms is modeled by replacing the horizontal cables with bars allowed to work in tension (Fig. 3(c-d)).

4. Numerical results Let us examine the wave dynamics of columns equipped with np ¼ 50 right-handed prisms and cables featuring Young modulus Es ¼ 5:48  106 N=m2 , and cross-section radius rs ¼ 0:14 mm. The lattice constant a of the analyzed columns is set equal to 5 mm, while the reference height of each prism is equal to h0 ¼ 6:2 mm, giving a total height of the column of 0.31 m. The bases of the prisms forming the column are endowed with lumped masses and the total mass of each unit is equal to 0.0249 kg. We characterize the state of prestrain/prestress of the column through the cross-string prestrain p defined according the notation introduced in Section 2.1.3. We refer to the quantity  ¼ ðh  h0 Þ=h as

Fig. 2. Geometry of a T3 tensegrity prism.

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Fig. 3. Top views and front views of tensegrity columns of right handed prisms with flexible bases (a-b: FB columns), and rigid bases (c-d: RB colums).

the axial strain of the prism (positive when the prism is stretched from the reference configuration), h denoting the current (deformed) height of the generic prism. The wave dynamics of the analyzed systems is studied through the numerical model given in Section 2, by applying different initial velocities v 0 to the nodes of the free end (right-end), which are directed along the axis of the column, so as to generate a compressive impulsive loading (impact velocities). Two different sets of simulations are presented, on considering systems subject to zero (Section 4.1) and nonzero (Section 4.2) prestress, respectively. In both cases, we compare the dynamics of columns composed of prisms with flexible bases (hereafter referred to as FB-columns) with that of columns composed of prisms equipped with rigid bases (RB-columns). The analyzed columns have the same geometric and mass data of those analyzed in [29]. 4.1. Wave dynamics of columns under zero prestress We first examine the wave dynamics of FB columns (Fig. 3(a-b)) under zero prestress. Fig. 4 shows the axial strain wave profiles generated in such columns on applying three different impact velocities: (a) v 0 ¼ 0:001 m=s, (b) v 0 ¼ 0:002 m=s, (c) v 0 ¼ 0:005 m=s. For each examined value of v 0 , we show in Fig. 4 time-shots of the traveling axial strain wave. We observe the formation of a leading compression pulse with significantly large amplitude followed by a dispersive, oscillatory tail. The leading compression pulses span approximatively 6 prisms t ¼ 0:5, for each of the analyzed cases. We numerically estimates the following amplitudes of the leading pulses at time t ¼ 0:5 s :  ¼ 7:19  104

v 0 ¼ 0:001 m=s; ¼ 1:44  103 for v 0 ¼ 0:002 m=s; and  ¼ 3:59  103 for v 0 ¼ 0:005 m=s. In all the eamined cases, we for

compute the mean speed of the leading pulses equal to 0:248m=s. It is worth noting that the amplitude of the leading pulse increases with the impact velocity, while its velocity stays almost constant. We now compare the results in Fig. 4 with similar ones referred to RB columns subject to equal impulsive disturbances v 0 (Fig. 5). In the cases analyzed in Fig. 5, we observe the formation of compact

solitary

pulses

with

amplitudes:

 ¼ 7:33  103

for

v 0 ¼ 0:001 m=s; ¼ 1:25  10 for v 0 ¼ 0:002 m=s; and  ¼ 1:77  102 for v 0 ¼ 0:005 m=s. The mean speeds of the compression solitons are: 0:0248 m=s for v 0 ¼ 0:001 m=s;0:031 m=s 2

Fig. 4. Axial strain wave profiles in a FB chain of 50 prisms under zero prestrain, for various initial velocities: (a) v 0 ¼ 0:001 m=s, (b) v 0 ¼ 0:002 m=s, (c) v 0 ¼ 0:005 m=s.

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Fig. 5. Axial strain wave profiles in a RB chain of 50 prisms under zero prestrain, for various initial velocities: (a) v 0 ¼ 0:001 m=s, (b) v 0 ¼ 0:002 m=s, (c) v 0 ¼ 0:005 m=s.

for v 0 ¼ 0:002 m=s; and 0:0496 m=s for v 0 ¼ 0:005 m=s. The strain waves span about 3 prisms for each of the cases in Fig. 5. It is worth noting that both the amplitude and the speed of compression solitary waves increase with the impact velocity in RB columns. 4.2. Impact dynamics of prestressed columns We now consider tensegrity columns subject to a state of prestress in the reference configuration, which is obtained by applying a prestrain p ¼ 0:002 to the cross cables. The compression wave dynamics of prestressed FB columns is illustrated in Fig. 6, for impact velocities ranging between v 0 ¼ 0:1 m=s, and v 0 ¼ 0:15 m=s. We apply larger impact velocities to the prestressed column, as compared to the column under zero prestress, to account for the prestress-induced increase in the acoustic impedance of the system. Also in prestressed FB columns (Fig. 6), as well as in the cases illustrated in Fig. 4 (zero prestress), we note the propagation of leading compression pulses with oscillatory tails, which now span approximatively 4 prisms at t ¼ 0:5 s, and exhibit speed varying from 0:248 m=s for v 0 ¼ 0:1 m=s to 0:372 m=s for v 0 ¼ 0:15 m=s. The leading strain pulses illustrated in Fig. 6 have

Fig. 6. Axial strain wave profiles in a FB chain of 50 prisms under cross string prestrain p ¼ 0:002, for various initial velocities: (a) v 0 ¼ 0:1 m=s, (b) v 0 ¼ 0:125 m=s, (c) v 0 ¼ 0:15 m=s.

 ¼ 5:4  102 for v 0 ¼ 0:1 m=s; ¼ 6:97  102 for v 0 ¼ 0:125 m=s; and  ¼ 8:5  102 for v 0 ¼ 0:15 m=s.

amplitudes of:

The response of prestressed RB columns to impact velocities ranging between v 0 ¼ 0:1 m=s, and v 0 ¼ 0:15 m=s is illustrated in Fig. 7. One observes the propagation of compact solitary pulses spanning about 3 units and featuring the following amplitudes:

v 0 ¼ 0:1 m=s; ¼ 0:106 for mean speeds of the compression pulses are: 0:186 m=s for v 0 ¼ 0:1 m=s, and 0:248 m=s for v 0 ¼ 0:125  0:15 m=s.  ¼ 8:93  102 for v 0 ¼ 0:125  0:15 m=s. The

5. Concluding remarks We have presented a three-dimensional numerical model for the dynamics of arbitrary tensegrity networks that accounts for a rigidity constraint of the compressed members, elastic response of cable elements, and vector form of the equations of motions. Such a model can be easily coupled with standard FE models of bodies and structures interacting with tensegrity networks, and proves to be useful for studying the highly nonlinear dynamics of

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Fig. 8. Illustration of a tensegrity acoustic lens (1) consisting of an array of chains alternating lumped masses and tensegrity units with strongly nonlinear response (2) subjected to different levels of prestress. An incident signal (3) generates compact compression solitary waves with different phases (4) within the lens, which coalesce at a focal point (5) in the adjacent host medium (6). The focal point is controlled through variable precompression of the individual chains, generating different wave speeds within the lens [33].

Fig. 7. Axial strain wave profiles in a RB chain of 50 prisms under cross string prestrain p ¼ 0:002, for various initial velocities: (a) v 0 ¼ 0:1 m=s, (b) v 0 ¼ 0:125 m=s, (c) v 0 ¼ 0:15 m=s.

tensegrity metamaterials. It has been applied to investigate the wave dynamics of tensegrity columns traversed by propagating compressive strains waves under impulsive impact loading. A recent study on of the nonlinear dynamics of tensegrity columns [29] assumes that the bases of the prisms forming the column behave as rigid bodies during an arbitrary motion of the structure [30]. Such an assumption is not always matched in real-life tensegrity columns, and may significantly affect the nature of the mechanical response of the prism units (hardening/softening response), as shown in a number of theoretical and experimental studies [11–13,9]. The model proposed in this work accounts for the presence of deformable bases in the prisms composing the column, and reduces to the model presented in [29], as a special case, when the horizontal strings are replaced by rigid bars allowed to work in tension. It is also worth noting that the model proposed in this work regards the units of an arbitrary tensegrity network as fully three-dimensional structures. It therefore allows us extends the modeling of tensegrity metamaterials presented in [8,9], which instead describes the units of such systems as (nonlinear) one-dimensional springs connecting lumped masses.

The numerical results presented in Section 4 allow us to conclude that the more rigid is the response of the bases of the units, the more compact is the nature of the compressive solitary pulses that traverse tensegrity columns, under initial compressive disturbances. We are led to conclude that it is possible to exploit the use of highly nonlinear dynamic response in tensegrity units to create novel metamaterials that will enable unconventional wavefocusing methodologies. Arrays of tensegrity columns may indeed be employed to fabricate tunable focus acoustic lenses supporting extremely compact solitary waves. Such lattices can be subjected to different levels of prestress, so as to generate compact solitary waves with different phases within the lens, which will coalesce at a focal point [23,24] in an adjacent host medium (i.e., a material defect to be targeted, cf. Fig. 8). The 3D modeling presented in this work offers a very useful tool to simulate the mechanical response of such spatial arrays of tensegrity columns, and can also be used to deal with their design by computation. As compared to acoustic lenses based on arrays of granular metamaterials [23,24], tensegrity acoustic lenses will profit from the adjustable width of compression solitary waves in such metamaterials (cf. Section 4). While compression solitary waves in uniform granular chains have a constant width, which is independent of the amplitude [6], the width of similar waves in tensegrity metamaterials changes with amplitude and speed, and the solitary wave tends to concentrate on a single lattice spacing in the high energy regime [8,9]. We address specific studies about engineering applications of tensegrity networks to future work. A key goal of such a research will regard the design of 3D innovative devices for monitoring structural health and damage detection in materials and structures. Combined tensegrity actuators and sensors will be tested to detect the mechanical properties and/or the presence of damage in materials and structures through closed-loop identification procedures [33,25,26]. A second goal will regard the design, manufacture and testing of effective impact mitigation systems based on tensegrity metamaterials with softening-type response. Such nonlinear metamaterials will be able to transform compressive disturbances into solitary rarefaction waves with progressively vanishing oscillatory tail, and/or rarefaction shock-like waves [9]. 6. Compliance with ethical standards The authors declare that they have no conflict of interest.

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