vibration Article
Dynamic Simulation of a Metamaterial Beam Consisting of Tunable Shape Memory Material Absorbers Hua-Liang Hu, Ji-Wei Peng and Chun-Ying Lee * Graduate Institute of Manufacturing Technology, National Taipei University of Technology, Taipei 10608, Taiwan;
[email protected] (H.-L.H.);
[email protected] (J.-W.P.) * Correspondence:
[email protected]; Tel.: +886-2-8773-1614 Received: 21 May 2018; Accepted: 13 July 2018; Published: 18 July 2018
Abstract: Metamaterials are materials with an artificially tailored internal structure and unusual physical and mechanical properties such as a negative refraction coefficient, negative mass inertia, and negative modulus of elasticity, etc. Due to their unique characteristics, metamaterials possess great potential in engineering applications. This study aims to develop new acoustic metamaterials for applications in semi-active vibration isolation. For the proposed state-of-the-art structural configurations in metamaterials, the geometry and mass distribution of the crafted internal structure is employed to induce the local resonance inside the material. Therefore, a stopband in the dispersion curve can be created because of the energy gap. For conventional metamaterials, the stopband is fixed and unable to be adjusted in real-time once the design is completed. Although the metamaterial with distributed resonance characteristics has been proposed in the literature to extend its working stopband, the efficacy is usually compromised. In order to increase its adaptability to time-varying disturbance, several semi-active metamaterials have been proposed. In this study, the incorporation of a tunable shape memory alloy (SMA) into the configuration of metamaterial is proposed. The repeated resonance unit consisting of SMA beams is designed and its theoretical formulation for determining the dynamic characteristics is established. For more general application, the finite element model of this smart metamaterial is also derived and simulated. The stopband of this metamaterial beam with different configurations in the arrangement of the SMA absorbers was investigated. The result shows that the proposed model is able to predict the unique dynamic characteristics of this smart metamaterial beam. Moreover, the tunable stopband of the metamaterial beam with controlling the state of SMA absorbers was also demonstrated. Keywords: metamaterial; tunable vibration absorber; shape memory alloy; stopband; finite element
1. Introduction Metamaterials are artificially fabricated structural materials with special physical properties, such as a negative refraction index of the electromagnetic wave [1], negative effective mass, and negative effective elastic modulus [2], etc. Their potential applications include an invisibility cloak, vibration, acoustic control in structures, and metadevices [3]. Usually, the materials with a periodic distribution of elastic constant or mass density are known as phononic crystals. With analogy to photonic crystal, it is found that when an elastic wave propagates through phononic crystal or an elastic composite with a periodic structure, a similar stopband or bandgap is formed in its transmission [4–6]. Low frequency flexural wave band gaps in Timoshenko beams with locally resonant structures were studied theoretically and experimentally by Yu et al. [7]. They concluded that the existence of low frequency band gaps in the beams provides a method for the flexural vibration control of
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beams. For broadband vibration absorption, Pai [8], by varying the attached absorber’s characteristics along its wave propagating axis, numerically showed the feasibility of the design of a broadband elastic wave absorber in the axial vibration of a longitudinal metamaterial bar. With regards to other structures, the multi-stopband metamaterial beam [9] and plate [10,11] were designed by employing local resonance between the multi-frequency absorbers and external excitation. Moreover, a Helmholtz resonator with an adjustable size [12], a 3-D kagome-sphere lattice with tuned dimensional sizes [13], a chiral elastic lattice with different column materials [14], and a lattice with sinusoidally-shaped ligaments [15] were also employed, respectively, to alter the bandgap of the materials. He and coworkers [16] reported that composite laminate acoustic metamaterials manifested were more effective for broadband vibration absorption with the superior strength to weight ratio of carbon fiber reinforced polymer (CFRP). The design of a layered mechanical metamaterial having implemented negative stiffness inclusions has been presented and its acoustic wave propagation properties have been modeled for beam [17] and plate [18] structures, respectively. A superior performance in a broadband frequency range when compared to a viscoelastic damping constraint layer of equivalent mass was demonstrated using the proposed configurations. Although the above studies proposed the designs of metamaterial which were effective in demonstrating their stopband characteristics in vibration or acoustic control, their designs were unable to be adapted to the change in external excitation source, e.g., machinery which needs to operate at different rotational speeds. Therefore, a variety of metamaterial designs have been incorporated with a tunable capability to be adapted to environmental changes. Wang and co-workers [19] proposed a design of locally tunable resonant acoustic metamaterials composed of easy-to-buckle elastic beams. The controlled static loading induced buckling dramatically alters the stiffness of the beams and consequently the natural frequency of the resonating units, which in turn determines the frequency range of the band gap. The designs of tunable acoustic metamaterials by altering the resonant frequency via piezo shunting [20,21] were also proposed. The performance of this class of design was verified experimentally by Zhu et al. [22]. In addition, Casadei et al. [23] proposed a metamaterial consisting of a slender beam featuring a periodic array of airfoil-shaped masses supported by linear and torsional springs. The resonance characteristics of the airfoils lead to strong attenuation at frequencies defined by the properties of the airfoils and the speed on the incident fluid. There are also designs incorporating the use of tunable smart materials, such as magnetorheological elastomer (MRE) [24], electrorheological fluid (ERF) [25], and shape memory polymer (SMP) [26]. Although the use of smart materials such as MRE and ERF in the design of tunable metamaterials has been proposed, as mentioned previously, the mechanical property change upon activation by the magnetic field or electric field is still limited due to their rubber-like nature. On the other hand, a shape memory alloy (SMA) is metallic in nature and undergoes significant stiffness change upon the thermally activated martensite to austenite transformation [27]. Nevertheless, there is a thermal hysteresis in the phase transformation. Thus, a superlattice material consisting of alternate layers of a shape memory Ni-rich NiAl alloy and NiAl B2 alloy has been proposed to reduce this thermal hysteresis [28]. In general, SMA has been employed in the tunable vibration absorber design for vibration control [29,30]. Therefore, in this study, a theoretical analysis of the feasibility of using SMA cantilevers in the design of a tunable metamaterial beam structure was performed. The geometrical dimensions and configuration of the periodic arrangement on the stopband of vibration transmission were investigated through finite element analysis. 2. Modeling A schematic diagram for the proposed metamaterial beam with shape memory material (SMM) absorbers is shown in Figure 1. Two types of plausible SMM absorber are shown in Figure 1: an absorber with a helical spring and an absorber with cantilevers. Either type can be applied in the construction of the metamaterial beam. l denotes the pitch of the attached absorbers along the axial direction of the beam. In mathematical modeling, each absorber can be represented by a one-degree-of-freedom
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mass-damper-spring system. However, the helical spring can usually be made with lower stiffness Vibration 2018, 2, x FOR PEER REVIEW 3 of 12 stopbands. A suitable canconfigurations be chosen for individual application. By controlling than the cantilever. Thedesign different can be considered for different rangesthe oftemperature stopbands. stopbands. A design can be chosen for individual By controlling the temperature the SMM viasuitable electric current orfor other means, the stiffnessapplication. and subsequently, natural frequency Aof suitable design can be chosen individual application. By controlling thethe temperature of the ofthe the SMM viacurrent electric current or otherthe means, the stiffness and subsequently, thefrequency natural frequency of absorber, can be tuned accordingly. SMM via electric or other means, stiffness and subsequently, the natural of the of the absorber, can be tuned accordingly. absorber, can be tuned accordingly. y y
Pitch of SMM absorbers l Pitch of SMM absorbers l l l
x x
Base structure SMM cantilevers Base structure SMM cantilevers
SMM helical springs SMM helical springs Figure 1. The schematic diagram of the proposed metamaterial beam consisting of different types of Figure 1.memory The schematic of of thethe proposed metamaterial beam consisting of of different types of of Figure 1. The schematic diagram proposed metamaterial beam consisting different types shape materialdiagram absorbers. shape memory material absorbers. shape memory material absorbers.
2.1. Metamaterial Beam with Infinite Length 2.1. Metamaterial Beam with Infinite Length 2.1. Metamaterial Beam with Infinite Length Figure 2 shows a schematic diagram of the repeated unit element for the metamaterial beam. ma, Figure shows schematic diagram of the theand unit the beam. ma, Figure 22 shows a aschematic diagram of repeated unit element for themetamaterial metamaterial beam. ca, and ka denote the equivalent mass, damping, stiffness of element the SMMfor absorber, respectively, while ka denote thethe equivalent mass, damping, andmodeling stiffness of the absorber, respectively, while mlacrepresents ,a,cand ka the denote equivalent mass, damping, and stiffness of SMM the SMM absorber, pitch length of the element. Similar can be found in the work ofrespectively, Pai et al. [9]. a , and l represents length of the element. Similar modeling can be found in the work Pai et al.of[9]. while l represents the pitch length the element. modeling can be found in the work However, in the thispitch study, it should beof noted that ca andSimilar ka can be adjusted by controlling theof temperature However, this study, beitnoted thatbe canoted andtheory, kathat can cbe adjusted bybecontrolling the temperature Pai et al.SMM. [9].in However, in itthis study, should adjusted controlling of the According toshould Euler-Bernoulli’s beam the kdamping for bothby the beam and aifand a can oftemperature the SMM. tothe Euler-Bernoulli’s beamfor theory, if the damping for both the for beam and the of the SMM. According Euler-Bernoulli’s beam theory, if the damping both absorber can beAccording neglected, governingtoequations the beam and absorber can be written as: canabsorber be neglected, governing equations forequations the beamfor andthe absorber can absorber be written as:be theabsorber beam and can bethe neglected, the governing beam and can + EIw(iv ) + EI ( w0′′′+ − w0′′′− ) + ka ( w0 − wa ) δ ( x) = 0 ρ Aw (1) written as: ( iv ) ) + k ( w − w ) δ ( x) = 0 .. w 000w′′′ − 000 ′′′ + ρ A EIw EI w (iv) + (1) ( w0+ 0−+ w0−0− + k a (aw0 −0 wa ) a δ(x) = 0 ρAw + EIw + EI (1) a + k a wa = k a w0 m..a w (2) m am waaw (2) +a k+a wk aa w=a k=a wk0a w0 (2)
l/2 l/2 l/2 l/2 w(x,t) w0 w(x,t) w0 ka ka ma ma
x x
Ca Ca wa wa
Figure2.2.The Theschematic schematicdiagram diagramofofthe therepeated repeatedunit unitelement elementfor forthe themetamaterial metamaterialbeam. beam. Figure Figure 2. The schematic diagram of the repeated unit element for the metamaterial beam.
In Equation (1), ρ, A, I, and E are the mass density, cross-sectional area, second area moment of InIn Equation (1), A,A,I,I,and EEare density, cross-sectional area, second area area moment moment of Equation (1),ρ,ρ, andof arethe themass mass density,δ(x) cross-sectional area, second inertia, and Young’s modulus the beam structure. is the Kroneckler delta function. For a of inertia, inertia, and and Young’s Young’s modulus of the beam structure. δ(x) is the Kroneckler delta function. For a a modulus of the beam structure. δ(x) is the Kroneckler delta function. For metamaterial beam of an infinite length, the representative repeat unit should satisfy the conditions metamaterial beam of an an infinite length, thethe representative repeat unitunit should satisfy the conditions of metamaterial beam infinite length, representative repeat should satisfy the conditions of periodicity at bothofends. Therefore, under harmonic excitation, the lateral displacement functions periodicity at both ends. Therefore, under harmonic excitation, the lateral displacement functions of ofthe periodicity at bothw(x, ends. under harmonic excitation, therespectively, lateral displacement functions of beam structure t) Therefore, and the absorber wa(t) can be expressed, as: theofbeam structure w(x, t) and the absorber w (t) can be expressed, respectively, as: a the beam structure w(x, t) and the absorber wa(t) can be expressed, respectively, as: w ( x, t ) = w0 ei ( β x −Ωt ) = w0 eiβ x (3) i ( βx − Ωt ) t) iβxi β x i(β x −Ω w( w x, (t)x,=t )w=0 ew = w e (3) 0 e = w e (3) 0 0 − iΩt wa (t ) = wa−eiΩt (4) wa (wt) (= w a e −iΩt (4) (4) a t ) = wa e In the above equations, β and Ω are the wave number and frequency of the propagating waves, In the above β and Ω areEquation the wave and frequency of the propagating waves, = −1 . Substituting (3)number into Equation (1) and performing the integration respectively, and iequations,
i =l, we −1can . Substituting Equation (3) into Equation (1) and performing the integration respectively, over the pitchand length have: over the pitch length l, we can have:
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In the above equations, β and Ω are the wave number and frequency of the propagating waves, √ respectively, and i = −1. Substituting Equation (3) into Equation (1) and performing the integration over the pitch length l, we can have: ..
mw0 + kw0 + k a (w0 − wa ) = 0 where 2ρA sin m=
β
(5)
βl 2
, k = 2EIβ3 sin
βl 2
(6)
By further substituting Equations (3) and (4) into Equations (5) and (2), and solving for the nontrivial solution of the resulting homogeneous equations, an equation to determine the wave number β can be obtained: k + k − mΩ2 −k a a (7) =0 2 −k a k a − mΩ Given a vibration frequency Ω, the stopband of this metamaterial beam lies within the range when no real β solution exists in Equation (7). 2.2. Finite Element Formulation of the Metamaterial Beam The equations established in the previous section are based on a metamaterial beam of an infinite length. For practical applications, the beam should be of a finite length and different boundary conditions. Moreover, if multiple stopbands are to be implemented in this metamaterial beam, the absorbers should be tunable in their parameters. The resort to finite element analysis has its necessity. In this study, the structural beam was discretized as an Euler-Bernoulli beam element and the base of each 1-DOF absorber was attached to its node. The corresponding mass matrix and stiffness matrices of the beam element can be found in the standard textbook of the finite element method, e.g., Bathe [31]. As for the damping matrix of the beam element, a proportional damping approach was assumed if considered. The viscous damping of the absorber was appended in the system damping matrix accordingly. By following the standard procedure in finite element analysis, the system matrices and equations of motion could be assembled as in the following equation, where { F } and { X } denote the nodal force vector and nodal displacement vector, respectively. ..
.
[ M]{ X } + [C ]{ X } + [K ]{ X } = { F }
(8)
A sweep sine analysis was performed in Equation (8). The frequency response function of the metamaterial beam over the required frequency span was obtained. For simulating the time response of the metamaterial beam, the Newmark method [31] was adopted in this study: n . ot+∆t n . ot n . ot n .. ot+∆t X = X + (1 − δ ) X + δ X ∆t
{X}
n o n . ot n .. ot+∆t .. t 1 = { X } + X ∆t + −α X +α X (∆t)2 2 n .. ot+∆t n . ot+∆t + [C ] X + [K ]{ X }t+∆t = { F }t+∆t [ M] X
t+∆t
t
(9)
(10) (11)
In the above equations, α and δ are the parameters for adjusting the accuracy and stability of the integration, respectively. The trapezoidal rule of α = 1/4 and δ = 1/2 was adopted in this study. A time incremental step of ∆t was employed in the calculation. Table 1 presents the dimensions and material properties of the structural beam and the parameters of the absorbers used in the numerical simulation. The structural beam was made of aluminum and
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the absorber was designed to damp out the vibration at 2500 Hz, which was within an acoustic range. If it is designed found thattothe mass ofthe thevibration absorber at was about tenth a beam thecalculated, absorber was damp out 2500 Hz, one which wasofwithin anelement. acoustic range.
If calculated, it is found that the mass of the absorber was about one tenth of a beam element.
Table 1. The dimensions and material properties of the structural beam and the parameters of the absorbers used in the simulation of the metamaterial beam. Table 1. The dimensions and material properties of the structural beam and the parameters of the absorbers used in the simulation of the metamaterial beam. Young’s Poisson’s Mass No. of Length Width Height Modulus, E Ratio, ν Density, Elements No. of Young’s Poisson’s Mass ρ Beam Length Width Height Modulus, E Ratio, ν Density, Elements Beam 32 1m 5 mm 3 mm 72.4 GPa 0.33 2780 kg/mρ3 32
Absorber
Absorber
1m 5 mm 3 mm 72.4 GPa ma ca ka ma c a k a 0.1 N·s/m 1.3031 −4 × 10−4 kg 3.21534 × 104 N/m 1.3031 × 10 kg 0.1 N·s/m 3.2153 × 10 N/m
0.33
2780 kg/m3 ωa ωa 2500 Hz 2500 Hz
As for the possible possible experimental implementation, Figure 3 shows an SMA cantilevered beam configuration. Two Two parallel parallel SMA SMA wires wires were were clamped clamped symmetrically symmetrically at at their half lengths by an insulating fixture pad. The wires were separated along their lengths, except except at at the the two two extreme extreme ends. ends. The ends were electrically electricallyconnected connectedto toform formaacircuit circuitloop loopwhere wherethe the control current was brought control current was brought in in through the tabs in the fixture pad. The direction of current flow through this absorber was also through the tabs in the fixture pad. The direction of current flow through this absorber schematically shown in the figure. The control current provided resistance heating to the SMA wires and tuned their phase transformation from martensite to to austenite austenite through through temperature temperature manipulation. manipulation. Therefore, the stiffness of the absorber could be changed from the lower martensitic phase to the higher austenitic phase. Under the assumption assumption of Euler beam theory, the fundamental frequency of the cantilever beam can be calculated as [32]: s 2 1.875 EI 1.8752 EI (12) ωωaa == (12) 2π ρAL 2π ρ AL44 For instance, if Nitinol wire with a diameter of 0.8 mm and length of 10 mm is used, based on its mass density of 6450 kg/m andYoung’s Young’s moduli moduli for for martensite martensite and and austenite austenite phases of 75 GPa and kg/m33and 28 GPa, frequencies of the cantilevered beam are 2332 Hz and GPa, respectively, respectively,the thecalculated calculatedfundamental fundamental frequencies of the cantilevered beam are 2332 Hz 3816 Hz before and and afterafter the martensite-austenite transformation. TheThe estimated working frequency of and 3816 Hz before the martensite-austenite transformation. estimated working frequency this proposed absorber configuration seems to fit parameter presented in Table 1. 1. of this proposed absorber configuration seems to the fit the parameter presented in Table Tabs for heating circuit Face attached to
Nitinol wire: φd Conductive pad
base structure
Direction of current flow
Fixture pad
Length: l
Figure 3. An exemplary implementation of the proposed absorber with SMA wires clamped in Figure 3. An exemplary implementation of the proposed absorber with SMA wires clamped in double-cantilevered configuration. double-cantilevered configuration.
3. Results and Discussion 3. Results and Discussion 3.1. The Stopband of the Metamaterial Beam with Infinite Length 3.1. The Stopband of the Metamaterial Beam with Infinite Length Figure 4 presents the frequency dispersion curves of the metamaterial beam of an infinite length. Figure seen 4 presents the frequency dispersion curves the metamaterial of an infinite length. It is clearly that around the natural frequency, theof absorber existed asbeam a stopband. No vibration It is clearly seen that around the natural frequency, the absorber existed as a stopband. No vibration wave in this frequency band can propagate through this metamaterial beam. When the mass ratio wave this frequency band can propagate through metamaterial beam. the mass of of the in absorber to the beam element increased, the this stopband widened. On When the other hand,ratio as the the absorber to the beam element increased, the stopband widened. On the other hand, as the natural natural frequency of the absorber changed, the stopband of the frequency ratio remained unchanged. frequency of the absorber changed, the stopband of the frequency ratio remained unchanged. However, the stopband of the actual frequency of vibration Ω shifted accordingly with ωa. In other words, the mass ratio of the absorber changed the bandwidth of the stopband, while the natural frequency of
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However, the stopband of the actual frequency of vibration Ω shifted accordingly with ω a . In other Vibration 2018, 2, x FOR PEER REVIEW 6 of 12 words, the mass ratio of the absorber changed the bandwidth of the stopband, while the natural frequency of theshifted absorber the of frequency of thewithout stopband withoutofalteration of itsratio. frequency the absorber the shifted frequency the stopband alteration its frequency The ratio. The simulation of changing the natural frequency of the absorber was conducted to examine the simulation of changing the natural frequency of the absorber was conducted to examine the possible possible influence of switching SMA from martensite to austenite, which generates a change in the influence of switching SMA from martensite to austenite, which generates a change in the Young’s Young’s modulus of folds three[23]. folds [23]. modulus of three 0.20
2.0
ωa=2500/SQRT(3) Hz
ma/m1= 0.1 ma/m1= 0.3
1.0
0.10
0.05
0.5
0.0 0.00
ωa=2500*SQRT(3) Hz
0.15 L/λ (=βL/2π)
Ω/ωa
1.5
ωa=2500 Hz
ma/m1= 0.5
0.00 0.05
0.10 L/λ (=βL/2π)
(a)
0.15
0.20
0.0
0.5
1.0
1.5
2.0
Ω/ωa
(b)
Figure 4. The frequency dispersion curves of metamaterial beams: (a) with the appended absorbers Figure 4. The frequency dispersion curves of metamaterial beams: (a) with the appended absorbers of of different mass ratios; (b) with the appended absorbers of different natural frequencies. different mass ratios; (b) with the appended absorbers of different natural frequencies.
3.2. The Response of the Metamaterial Beam with Finite Length 3.2. The Response of the Metamaterial Beam with Finite Length For the beam with a finite length, the boundary conditions should influence its vibration For the beamBefore with considering a finite length, the boundary conditions should itsthe vibration characteristics. the effect of adding SMM absorbers, the influence reliability of finite characteristics. Before considering the should effect ofbeadding SMM reliability of the finite element model for the pristine beam verified. Tableabsorbers, 2 presentsthe some calculated natural element modelfrom for the finite pristine beammodel should be with verified. 2 presents calculated frequencies element along their Table theoretical results some obtained from thenatural Euler beam theory [32]. the element beam in three different boundary onlyobtained the natural frequencies frequencies from theFor finite model along with their configurations, theoretical results from the Euler from the first modes were calculated. However, forconfigurations, the beam in simply-supported boundary beam theory [32].three For the beam in three different boundary only the natural frequencies conditions, its 19th and 20th frequencies close for to the frequency range of boundary 2500 Hz from the first three modes werenatural calculated. However, theinterested beam in simply-supported were also calculated due to its simpler effort in calculation. The calculation for the higher modes of conditions, its 19th and 20th natural frequencies close to the interested frequency range of 2500 Hz were the beam in other configurations was not attempted because the theoretical results were not readily also calculated due to its simpler effort in calculation. The calculation for the higher modes of the beam available from the vibration textbook. It is seen that thethe accuracy of theresults finite element model in calculating in other configurations was not attempted because theoretical were not readily available the natural frequencies of the structural beam under different boundary conditions was reasonably from the vibration textbook. It is seen that the accuracy of the finite element model in calculating the good, even for the higher frequency modes within the interested frequency range of this study. natural frequencies of the structural beam under different boundary conditions was reasonably good, even for the higher frequency modes within the interested frequency range of this study. Table 2. The calculated natural frequencies of the structural beam from the finite element model along with their theoretical results obtained from the Euler beam theory. Table 2. The calculated natural frequencies of the structural beam from the finite element model along Boundary Method Mode 2 Mode 3 Mode 19 Mode 20 with their theoretical results obtained from the1 Euler Mode beam theory. Theory 2.476 Hz 15.76 Hz 43.40 Hz Cantilever FEM 2.4701Hz 15.50 2Hz 43.40 Hz - 19 - 20 Boundary Method Mode Mode Mode 3 Mode Mode Error −0.242% −1.65% 0.00% Theory 2.476 Hz 15.76 Hz 43.40 Hz Theory 6.942 Hz 27.78 Hz 62.53 Hz 2506 Hz 2777 Hz FEM 2.470 Hz 15.50 Hz 43.40 Hz Cantilever Simply-supported FEM 6.942 Hz 27.77 Hz 62.48 Hz 2525 Hz 2803 Hz Error −0.242% −1.65% 0.00% Error 0.00% −0.04% −0.08% +0.76% +0.94% Theory HzHz 27.78 62.53 2506- Hz 2777- Hz Theory 6.942 15.76 43.40Hz Hz 85.11Hz Hz Simply-supported FEM 6.942 HzHz 27.77 62.48 2525- Hz 2803- Hz Clamped-clamped FEM 15.74 43.38Hz Hz 85.04Hz Hz Error 0.00% − 0.04% − 0.08% +0.76% +0.94% Error −0.13% −0.05% −0.08% -
Theory 15.76 Hz 43.40 Hz 85.11 Hz Clamped-clamped FEM function 15.74 43.38 Hz beam 85.04 - was The frequency response of Hz the metamaterial in Hz cantilevered- configuration Error (8) by− 0.13%sweep − 0.05% −0.08% - 32 equally spaced calculated first from Equation using sine technique. The beam with
absorbers was discretized into 32 elements in the finite element model. With the linear nature of the governing equation, the steady state response was calculated under a sinusoidal input force of unit
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The frequency response function of the metamaterial beam in cantilevered configuration was Vibration 2018, 2, from x FOR Equation PEER REVIEW 7 of 12 calculated first (8) by using sweep sine technique. The beam with 32 equally spaced absorbers was discretized into 32 elements in the finite element model. With the linear nature of the magnitude at x = 0.03125 m. The excitation near to the fixeda end was applied to leave more governing equation, the steady stateinput response was calculated under sinusoidal input force of unit room for observing the vibration propagation toward to the free end. The frequency response spectra magnitude at x = 0.03125 m. The input excitation near to the fixed end was applied to leave more of this beam are toward shown to in the Figure It isThe noted that there was aspectra discrete room for cantilevered observing themetamaterial vibration propagation free 5. end. frequency response stopband and the response of the beam reduced drastically within the stopband. The stopband was of this cantilevered metamaterial beam are shown in Figure 5. It is noted that there was a discrete around 2500 Hz, which was the designed working frequency of this metamaterial beam presented stopband and the response of the beam reduced drastically within the stopband. The stopband wasin Table 1. Moreover, because thedesigned attached working absorbersfrequency to the beam there werebeam several resonance around 2500 Hz, which wasof the ofstructure, this metamaterial presented peaks around the stopband. For the frequency response of the beam away from the stopband, the in Table 1. Moreover, because of the attached absorbers to the beam structure, there were several displacement lower For at location x = 0.0625 m, which nearer cantilevered end resonance peaksamplitude around thewas stopband. the frequency response of thewas beam awaythe from the stopband, than at the free end (x = 1.0 m). This fits the characteristics of a cantilevered beam in usual vibrations. the displacement amplitude was lower at location x = 0.0625 m, which was nearer the cantilevered However, the the response at the freeofend was lower than that the end than at at thefrequency free end near (x = the 1.0 stopband, m). This fits characteristics a cantilevered beam innear usual cantilevered end. The vibration reduction through thefree action absorbers effectively vibrations. However, ataccumulated frequency near the stopband, theeffect response at the endofwas lower than that reduced the vibration of the beam. Therefore, it is clearly seen that the stopband was present more near the cantilevered end. The accumulated vibration reduction effect through the action of absorbers obviously at the free end. As regards to the vibration of the absorber, the displacement amplitude effectively reduced the vibration of the beam. Therefore, it is clearly seen that the stopband was present wasobviously always higher its beam counterpart. Nonetheless, the frequency followed closely more at thethan free end. As regards to the vibration of the absorber, the response displacement amplitude with that of its beam counterpart. was always higher than its beam counterpart. Nonetheless, the frequency response followed closely with that of its beam counterpart.
Amplitude of Frequency Response (m /N)
1e-2 1e-3 1e-4 1e-5 1e-6 1e-7 1e-8 1e-9 1e-10 2000
Beam, x=0.0625 m Absorbers, x=0.0625 m Beam, x=1.0 m Absorber, x=1.0 m
2500
3000
Frequency (Hz)
Figure The frequency response spectranear nearthe the fixed end 0.0625m)m)and andatat the free end Figure 5. 5. The frequency response spectra fixed end (x(x = =0.0625 the free end ofof aa cantilevered metamaterial beam with the SMA spring in martensite phase. cantilevered metamaterial beam with the SMA spring in martensite phase.
By using the Newmark method mentioned previously, the time responses of a cantilevered By using the Newmark method mentioned previously, the time responses of a cantilevered metamaterial beam at different downstream locations were simulated and shown in Figure 6. For the metamaterial beam at different downstream locations were simulated and shown in Figure 6. For the beam excited with a sinusoidal disturbance at x = 31.25 mm near the fixed end, the response component beam excited with a sinusoidal disturbance at x = 31.25 mm near the fixed end, the response with the same frequency of excitation showed a quick decay at farer downstream locations. Nevertheless, component with the same frequency of excitation showed a quick decay at farer downstream locations. the vibration of lower frequency was still observed. This result reveals that the vibration components Nevertheless, the vibration of lower frequency was still observed. This result reveals that the vibration in the stopband can be effectively diminished after a certain distance away from the excitation. This components in the stopband can be effectively diminished after a certain distance away from the was consistent with the lower magnitude in frequency response spectra at a further downstream excitation. This was consistent with the lower magnitude in frequency response spectra at a further location from the disturbance presented in Figure 5. downstream location from the disturbance presented in Figure 5. Figure 7 presents the variations of displacement amplitudes along the downstream direction for Figure 7 presents the variations of displacement amplitudes along the downstream direction for the metamaterial beams having 16 and 32 absorbers, respectively. For the metamaterial beam with 32 the metamaterial beams having 16 and 32 absorbers, respectively. For the metamaterial beam with absorbers equally-spaced along its full length, it is seen that both the displacement amplitudes of the 32 absorbers equally-spaced along its full length, it is seen that both the displacement amplitudes of beam and absorber decayed drastically away from the location of external disturbance. Because the the beam and absorber decayed drastically away from the location of external disturbance. Because the absorber took over the vibration energy from the beam, it had a larger displacement amplitude than the base beam at all locations. For the metamaterial beam with only 16 absorbers equally-spaced in the first half of its length, the decay in displacement amplitude leveled after the last absorber. Even under this circumstance, the attenuation in vibration still reached an applicable level for engineering
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absorber took over the vibration energy from the beam, it had a larger displacement amplitude than the Vibration 2018, 2, x FOR PEER REVIEW 8 of 12 base beam2018, at all the metamaterial beam with only 16 absorbers equally-spaced in the first Vibration 2, xlocations. FOR PEER For REVIEW 8 of 12 half of its length, the decay in displacement amplitude leveled after the last absorber. Even under this application. Therefore, the configuration design of this metamaterial beam should be an interesting application. the Therefore, the configuration design of this metamaterial beam should be an interesting circumstance, attenuation topic to be further explored.in vibration still reached an applicable level for engineering application. topic to be further explored.design of this metamaterial beam should be an interesting topic to be Therefore, the configuration further explored. 0.00030 0.00030
x=0.5 m x=0.5 mm x=0.25 x=0.25 m m x=0.0625 x=0.0625 m
Displacement Response Displacement Response (m)(m)
0.00025 0.00025 0.00020 0.00020
F=100 sin(Ωt), x0=31.25 mm, Ω=2500 Hz F=100 sin(Ωt), x0=31.25 mm, Ω=2500 Hz
0.00015 0.00015 0.00010 0.00010 0.00005 0.00005 0.00000 0.00000 -0.00005 -0.00005 -0.00010 -0.000100.000 0.000
0.005 0.005
0.010 0.010 Time (s) Time (s)
0.015 0.015
0.020 0.020
Displacement Amplitude Displacement Amplitude (m)(m)
Figure 6. Time responses of a cantilevered metamaterial beam at different downstream locations. Figure 6. Time responses of a cantilevered metamaterial beam at different downstream locations. Figure 6. Time responses of a cantilevered metamaterial beam at different downstream locations. 1e-4 1e-4 1e-5 1e-5 1e-6 1e-6 1e-7 1e-7 1e-8 1e-8 1e-9 1e-9 1e-10 1e-10 1e-11 1e-11 1e-12 1e-12 1e-13 1e-13 1e-14 1e-14 1e-15 1e-15 1e-16 1e-16 1e-17 1e-17 1e-18 1e-18 1e-19 1e-19 1e-20 1e-20 1e-21 1e-21 1e-22 1e-22 1e-23 1e-230.0 0.0
Beam (32 absorbers) Beam (32 absorbers) Absorbers (32 absorbers) Absorbers (32 absorbers) Beam (16 absorbers) Beam (16(16 absorbers) Absorber absorbers) Absorber (16 absorbers)
Ω=2500 Hz Ω=2500 Hz
0.2 0.2
0.4 0.4
0.6 0.6
X Coordinate (m) X Coordinate (m)
0.8 0.8
1.0 1.0
Figure Thevariations variationsinindisplacement displacementamplitudes amplitudes along thelength lengthofofthe the cantileveredmetamaterial metamaterial Figure 7.7. along the Figure 7.The The variations in displacement amplitudes along the length of thecantilevered cantilevered metamaterial beams having different numbers absorbers. beams having different numbers ofof absorbers. beams having different numbers of absorbers.
3.3. The Stopband Tunability of the Metamaterial Beam 3.3. of of thethe Metamaterial Beam 3.3.The TheStopband StopbandTunability Tunability Metamaterial Beam Figure 8 presents the calculated frequency response spectra of a cantilevered metamaterial beam Figure 8 presents the calculated frequency response spectra ofof a acantilevered metamaterial beam Figure 8 presents the calculated frequency response spectra cantilevered metamaterial beam with the SMA spring in martensite and austenite phases, respectively. As mentioned previously, the with and austenite austenite phases, phases,respectively. respectively.AsAs mentioned previously, withthe theSMA SMAspring springin in martensite martensite and mentioned previously, the Young’s modulus of the Nitinol changed from 28 GPa of martensite phase to 75 GPa of austenite the Young’smodulus modulusofofthe theNitinol Nitinolchanged changed from from 28 28 GPa Young’s GPa of of martensite martensite phase phasetoto75 75GPa GPaofofaustenite austenite phase upon complete transformation in heating. The corresponding change in the stiffness of the phase phaseupon uponcomplete completetransformation transformationininheating. heating.The Thecorresponding correspondingchange changeininthe thestiffness stiffnessofofthe the absorber as given in Table 1 was adjusted accordingly. The finite element analysis clearly showed absorber asas given in Table 1 was adjusted accordingly. The finite element analysis clearlyclearly showed that absorber given in Table 1 was adjusted accordingly. The finite element analysis showed that the stopband was tuned to a higher frequency when the absorbers were heated into austenite the stopband was tuned a higher when the absorbers were heated austenite phase. that the stopband was to tuned to a frequency higher frequency when the absorbers wereinto heated into austenite phase. Theoretically, if the SMA was heated to a temperature wherein only partial austenite Theoretically, if the SMA was heated to a temperature wherein only partial austenite transformation phase. Theoretically, if the SMA was heated to a temperature wherein only partial austenite transformation was completed, the stopband could be tuned between the two extremes of simple transformation was completed, the stopband could be tuned between the two extremes of simple martensite and simple austenite. Another possibility would be that the absorbers were tuned in a martensite and simple austenite. Another possibility would be that the absorbers were tuned in a distributed fashion, i.e., the absorbers were divided into several groups and the SMA in each group distributed fashion, i.e., the absorbers were divided into several groups and the SMA in each group had different degrees of austenite transformation. In this case, a metamaterial beam with very wide had different degrees of austenite transformation. In this case, a metamaterial beam with very wide stopband could probably be made. stopband could probably be made.
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was completed, the stopband could be tuned between the two extremes of simple martensite and simple austenite. Another possibility would be that the absorbers were tuned in a distributed fashion, i.e., the absorbers were divided into several groups and the SMA in each group had different degrees Vibration 2018, 2, x FOR PEER REVIEW 9 of 12 of austenite transformation. In this case, a metamaterial beam with very wide stopband could probably be made.
Amplitude of Frequency Response (m / N)
1e-2 Martensite Austenite
1e-3
1e-4
1e-5
1e-6
1e-7 Cantilevered ζa =0.001 ma /m1=0.1
1e-8
1e-9 2000
2500
3000
3500
4000
4500
5000
Frequency (Hz)
Figure8.8.The Thefrequency frequencyresponse responsespectra spectraofof a cantilevered metamaterial beam with SMA spring Figure a cantilevered metamaterial beam with thethe SMA spring in in martensite and austenite phases, respectively. martensite and austenite phases, respectively.
Amplitude of Frequency Response (m / N)
For the metamaterial beams in simply-supported configuration, Figure 9 shows their frequency For the metamaterial beams in simply-supported configuration, Figure 9 shows their frequency responses when the absorbers were arranged in different spacing. These frequency responses were responses when the absorbers were arranged in different spacing. These frequency responses were calculated from the sinusoidal input at x = 0.03125 m and steady state output at x = 0.96875 m. For calculated from the sinusoidal input at x = 0.03125 m and steady state output at x = 0.96875 m. For these these four metamaterial beams, the stopband near the designed frequency of 2500 Hz occurred. However, four metamaterial beams, the stopband near the designed frequency of 2500 Hz occurred. However, only the metamaterial beam with absorber spacing of 0.03125 m showed the complete stopband only the metamaterial beam with absorber spacing of 0.03125 m showed the complete stopband around around 2500 Hz. The other metamaterial beams with larger absorber spacing ≥0.0625 m either presented 2500 Hz. The other metamaterial beams with larger absorber spacing ≥0.0625 m either presented a a smaller or offset stopband. As seen from Table 2, the mode shape of the simply-supported beam smaller or offset stopband. As seen from Table 2, the mode shape of the simply-supported beam with with frequency close to 2500 Hz was the 19th mode, which had a wave length of about 0.05 m. Therefore, frequency close to 2500 Hz was the 19th mode, which had a wave length of about 0.05 m. Therefore, it is suspected that the spacing of the absorbers in a metamaterial beam should be less than the wave it is suspected that the spacing of the absorbers in a metamaterial beam should be less than the wave length of the corresponding mode shape at the designed working frequency. More detailed study is length of the corresponding mode shape at the designed working frequency. More detailed study is needed in this respect. needed in this respect. Figure 10 presents the frequency response spectra of the simply-supported metamaterial beam having the absorbers activated in different states. For the absorbers in an inactivated state, i.e., 1e-2 stopband around 2500 Hz was seen, as expected from a previous example. martensitic phase, the Absorber spacing= 0.03125 m Absorber spacing= 0.0625 m As all the absorber springs were activated or heated to fully austenitic phase, the Young’s modulus 1e-3 Absorber spacing= 0.09375 m of the Nitinol was raised to 75 GPa and the spring constants of the absorber increased accordingly. Absorber spacing= 0.125 m 1e-4 The stopband of this simply-supported beam having all absorbers activated shifted to a higher 1e-5 frequency around 4200 Hz, similar to that of the cantilevered beam shown in Figure 6. In order to explore the effect of1e-6 the tunable capability of the SMA absorber, only alternating absorbers were activated along the beam. The frequency response shown in Figure 10 for this beam with alternatingly 1e-7 activated absorbers demonstrated two discrete stopbands, around 2500 and 4200 Hz, respectively. 1e-8 Although their bandwidths were somewhat compromised, these two stopbands were very distinctive. Simply-supported 1e-9that distributed It is therefore believed activation of the absorbers along the beam’s axis could present more capabilities of 1e-10 the metamaterial beam in active broadband control. 1e-11 1e-12 2000
2200
2400
2600
2800
3000
Frequency (Hz)
Figure 9. The frequency responses of the simply-supported metamaterial beams having absorbers in
a smaller or offset stopband. As seen from Table 2, the mode shape of the simply-supported beam with frequency close to 2500 Hz was the 19th mode, which had a wave length of about 0.05 m. Therefore, it is suspected that the spacing of the absorbers in a metamaterial beam should be less than the wave length of the corresponding mode shape at the designed working frequency. More detailed study is needed in this respect. Vibration 2018, 1, 7
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Amplitude of Frequency Response (m / N)
1e-2
Absorber spacing= Absorber spacing= Absorber spacing= Absorber spacing=
1e-3
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1e-5 the frequency response spectra of the simply-supported metamaterial beam Figure 10 presents having the absorbers 1e-6 activated in different states. For the absorbers in an inactivated state, i.e., martensitic phase, the stopband around 2500 Hz was seen, as expected from a previous example. As 1e-7 all the absorber springs were activated or heated to fully austenitic phase, the Young’s modulus of 1e-8 the Nitinol was raised to 75 GPa and the spring constants of the absorber increased accordingly. The Simply-supported stopband of this simply-supported beam having all absorbers activated shifted to a higher frequency 1e-9 around 4200 Hz, similar to that of the cantilevered beam shown in Figure 6. In order to explore the 1e-10 effect of the tunable capability of the SMA absorber, only alternating absorbers were activated along 1e-11 the beam. The frequency response shown in Figure 10 for this beam with alternatingly activated 1e-12 two discrete stopbands, around 2500 and 4200 Hz, respectively. Although absorbers demonstrated 2000 2200 2400 2800 their bandwidths were somewhat compromised, these2600 two stopbands were3000 very distinctive. It is (Hz) therefore believed that distributed activationFrequency of the absorbers along the beam’s axis could present moreFigure capabilities of the metamaterial in active broadband control.beams having absorbers in 9. The frequency responses ofbeam the simply-supported metamaterial
Figure 9. The frequency responses of the simply-supported metamaterial beams having absorbers in different spacing. different spacing.
Amplitude of Frequency Response (m / N)
1e-2 All martensite All austenite Alternating
1e-3
1e-4
1e-5
1e-6
1e-7 Simply-supported 1e-8
1e-9 2000
2500
3000
3500
4000
4500
5000
Frequency (Hz)
Figure The frequency responses simply-supported metamaterial beam having absorbers Figure 10.10. The frequency responses of of thethe simply-supported metamaterial beam having thethe absorbers activated in different states. activated in different states.
Conclusions 4. 4. Conclusions The finite element formulation a metamaterial beam which features distributed SMM absorbers The finite element formulation onon a metamaterial beam which features distributed SMM absorbers was established. Based theoretical analyses, tuning absorber mass was more effective was established. Based onon thethe theoretical analyses, thethe tuning in in absorber mass was more effective in in increasing the bandwidth of the stopband in vibration transmission, while both mass and stiffness increasing the bandwidth of the stopband in vibration transmission, while both mass and stiffness tuning had capability shift stopband.The Thesimulation simulationresults resultsofof both cantilevered and tuning had thethe capability to to shift thethe stopband. both cantilevered and simply-supported metamaterial beams showed the shift of the stopband to a higher frequency by simply-supported metamaterial beams showed the shift of the stopband to a higher frequencythe phasephase transformation fromfrom martensite to austenite of the SMA absorber spring. This bycontrolled the controlled transformation martensite to austenite of the SMA absorber spring. proposed design could acousticmetamaterial metamaterialstructure. structure.Moreover, Moreover, This proposed design couldconstitute constituteanother anotherclass class of of smart smart acoustic installation of the absorbers along a partial segment was able to dampen out the harmonic installation of the absorbers along a partial segment was able to dampen out the harmonic disturbance disturbance of target frequency with an acceptable performance. Thus, the configuration design of of target frequency with an acceptable performance. Thus, the configuration design of this metamaterial this can metamaterial beam can beto anbe interesting topic to be further explored. beam be an interesting topic further explored. Author Contributions: Conceptualization, C.-Y.L.; Methodology, H.-L.H.; Programming, J.-W.P.; WritingOriginal Draft Preparation, H.-L.H.; Writing-Review & Editing, C.-Y.L. Funding: This research was funded by Ministry of Science and Technology, Taiwan, under Grant No. MOST 106-2221-E-027-054.
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Author Contributions: Conceptualization, C.-Y.L.; Methodology, H.-L.H.; Programming, J.-W.P.; Writing-Original Draft Preparation, H.-L.H.; Writing-Review & Editing, C.-Y.L. Funding: This research was funded by Ministry of Science and Technology, Taiwan, under Grant No. MOST 106-2221-E-027-054. Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature A [C ] ca E { F} I [K ] ka k l [ M] ma m w(x, t) w0 wa (t) wa {X} α β δ(x) δ ρ Ω ωa
cross-sectional area of the beam structure damping matrix of the metabeam in finite element formulation the equivalent stiffness of the SMM absorber Young’s modulus of the beam structure external force vector of the metabeam in finite element formulation second area moment of inertia of the beam structure stiffness matrix of the metabeam in finite element formulation the equivalent stiffness of the SMM absorber equivalent spring constant defined in Equation (6) pitch length between the absorbers mass matrix of the metabeam in finite element formulation the equivalent mass of the SMM absorber equivalent mass defined in Equation (6) the lateral displacement function of the beam structure amplitude of the lateral displacement of the beam structure the displacement function of the absorber amplitude of the lateral displacement of the absorber nodal displacement vector of the metabeam in finite element formulation the parameter used in Newmark method wave number of the propagating waves is the Kroneckler delta function the parameter used in Newmark method the mass density of the beam structure frequency of the propagating waves natural frequency of the absorber
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