Hindawi Advances in Civil Engineering Volume 2018, Article ID 3081783, 15 pages https://doi.org/10.1155/2018/3081783

Research Article Effective Negative Mass Nonlinear Acoustic Metamaterial with Pure Cubic Oscillator

Ming Gao ,1,2 Zhiqiang Wu ,1 and Zhijie Wen 2

1Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin, China 2College of Mining and Safety Engineering, Shandong University of Science and Technology, Qingdao, China

Correspondence should be addressed to Zhiqiang Wu; [email protected] and Zhijie Wen; [email protected]

Received 4 May 2018; Accepted 11 July 2018; Published 30 September 2018

Academic Editor: Fengqiang Gong

Copyright © 2018 Ming Gao et al. .is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Acoustic metamaterial, which can prohibit effectively the elastic wave propagation in the bandgap frequency range, has broad application prospects in the vibration and noise reduction areas. .e Lindstedt–Poincar´emethod was utilized to analyze the dispersion curves of nonlinear metamaterial with a pure Duffing oscillator. .e first-order perturbation solutions of acoustic and optical branches were obtained. Both the starting and cutoff frequencies of the bandgap are determined consequently. It was found that the soft/hard characteristics of pure Duffing oscillators could lead to the lower/upper movement of the starting and cutoff frequencies of the bandgap. By further researching the degraded linear system, the conclusion that actual nonlinear metamaterial bandgap region is wider than effective negative mass region is drawn and that both mass and stiffness ratio effect on the starting frequency is obtained. Effective positive mass can also lead to the vibration attenuation in bandgap. For nonlinear metamaterial, the translation effect of the external excitation amplitude on the bandgap range and the zero mass at the nonlinear bandgap cutoff frequency were discussed, and all above conclusion are identified by numerical analysis.

1. Introduction waves and studied the attenuation effect of seismic waves by using finite periodic lattices of mass-in-mass barriers [17]. In the last ten years, a significant attention is directed toward Effective negative mass metamaterials come from mass so called metamaterial, a kind of artificial structures, which is effective ratio, an artificial parameter, which is calcu- composed of small substructure that behaves like a continu- lated analogically as the parameter of electromagnetic ous material. .e most remarkable property of metamaterials metamaterials [18, 19] and has not only positive but also is the bandgap in which the acoustic/elastic waves propagate is negative values. Based on the idea of locally resonant mi- prohibited [1]. With this unique characteristic, metamaterials crostructure, Liu et al. [20] designed the first effective can be applied to the wave filtering [2–4], vibration attenu- negative mass metamaterial in certain frequency region. ation [5], acoustic isolation [6, 7], and sonic transmission Subsequently, Liu et al. [21], Milton and Willis [22], Willis [8, 9]. In civil engineering, acoustic metamaterials with the [23], and Huang et al. [24, 25] proposed various analytic band characteristics were also created to replace the tradi- models to analyze the related effective negative mass tional seismic designs for isolating seismic waves from mechanisms. Yao et al. [26] realized effective negative mass buildings. Shi and his coworkers [10–14] proposed a novel and zero-mass phenomena in a 1D mass-spring system concept of foundation, periodic foundation, to isolate civil experimentally. Other different types of acoustic meta- structures from seismic wave with frequencies that might materials with negative effective mass have been proposed resonate with the structures, which has been investigated in [27–32]. From the existing literatures, the starting frequency a number of comprehensive theoretical, numerical, and ex- bandgap of metamaterials is defined by local resonance perimental studies. In particular, Bruˆle´ et al. [15] carried out frequency where the effective negative mass just began. the large-scale test of seismic metamaterials [15] such as However, the bandgap starting frequency will change with seismic metawedge capable [16] capable of creating seismic outer spring stiffness when the local resonance frequency 2 Advances in Civil Engineering remains unchanged [26], and actually the starting frequency Mu€j,1 + K�2uj,1 − uj−1,1 − uj+1,1� − 2k� uj,2 − uj,1 � is lower than resonance frequency in experimental research. ( ) 3 1 Nonlinear acoustic metamaterials also deserve special − 2εΓ�uj,2 − uj,1 � � 0, attention. Vakakis and his coauthors [33, 34] utilized the multiple scales perturbation method to analyze nonlinear where overdots denote derivative with respect to time t. chains subjected to external forcing and ground springs and According to the load on the oscillator in period j, its developed nonlinear dispersion relationships that exhibit differential equation of motion is amplitude dependence. Chakraborty and Mallik [35] studied 3 the cubic chain and the effect of nonlinearities on the mu€j,2 � −2k� uj,2 − uj,1 � − 2εΓ�uj,2 − uj,1 � . (2) propagation constant and natural frequencies. .en Lazarov and Jensen [36] considered a linear chain with attached Rewrite Equations (1) and (2) into a matrix nonlinear damped oscillators by balance approach. Marathe M u€ (K + k) k u 0 ⎣ j,1 ⎦ 2 −2 ⎣ j,1 ⎦ and Chatterjee [37] looked at a damped nonlinearity and � �⎡ ⎤ +� �⎡ ⎤ 0 m u€ −2k 2k u used harmonic balance and multiple scales to uncover the j,2 j,2 ( ) decay rate in the propagation zone. Narisetti et al. [38] 3 3 K� uj , + uj+ , � ⎢ −�uj,2 − uj,1 � ⎥ ⎡⎣ −1 1 1 1 ⎤⎦ + ⎣⎢⎡ ⎦⎥⎤ � . developed a Lindstedt–Poincare´ perturbation technique and − 2εΓ 3 0 analyzed monoatomic cubic chains to capture dispersion 0 �uj,2 − uj,1 � and bandgap shifts. However, their models are not based on effective negative mass, and Duffing oscillator is often Define the external/internal spring rigidity ratio as a damped one. .e damp may have great influence on the α � K/k, the shell/oscillator mass ratio as β � M/√m�����, and the starting and cutoff frequencies of bandgap [39]. As a result, linear natural frequency of oscillator as ωn � 2k/m by in order to investigate how effect of the nonlinearity on the introducing dimensionless time τ � ωt and dimensionless 2 effective negative mass nonlinear metamaterial dispersion nonlinear coefficient Γ � Γ/mωn. .is way, we can non- curves, the pure Duffing oscillator, which is no damped pure dimensionalize Equation (3): Duffing oscillator is adaptive 2 d uj,1 It is worth mentioning that, there is no literature dis- ⎢⎡ ⎥⎤ u β 0 ⎢ d 2 ⎥ 1 + α −1 j,1 cussion whether positive effective mass phenomenon exists 2⎣⎢⎡ ⎦⎥⎤⎢ τ ⎥ ⎣⎢⎡ ⎦⎥⎤⎢⎡ ⎥⎤ ω ⎢ ⎥ + ⎣ ⎦ in the nonlinear metamaterial bandgap based on effective ⎢ 2 ⎥ 0 1 ⎣⎢ d uj, ⎦⎥ −1 1 uj,2 negative mass. 2 d 2 (4) In this paper, effective negative mass nonlinear metamaterial τ mathematical model is given firstly, and Lindstedt–Poincare´ u + u 3 α j−1,1 j+1,1 ⎡⎢ −�uj,2 − uj,1 � ⎤⎥ perturbation method is utilized to calculate metamaterial dis- − ⎣⎢⎡ ⎦⎥⎤ + 2εΓ⎣⎢ ⎦⎥ � 0, 2 3 persion curves. .en, the precise expression of both starting and 0 �uj,2 − uj,1 � cutoff frequencies is deduced consequently. After that, the result is obtained that positive effective mass phenomenon exists in the here ω � ω/ωn is a dimensionless frequency. Now we are nonlinear bandgap. Finally, the numerical calculation verifica- going to find the first-order perturbation solution of ω by tion results agree with our theoretical analysis. Lindstedt–Poincare´ method. Using the asymptotic expansion below:

(0) (1) 2 2. The First-Order Solution for the Dispersive ⎧⎨ uj � uj + εuj + oε �, ⎩ (5) Curve of Acoustic Metamaterial of Effective ω � ω + εω + oε2 �, Negative Mass with Pure Cubic Oscillator 0 0 where the superscript and subscript 0 and 1 are the linear Consider an acoustic metamaterial containing a pure Duffing and first-order asymptotic expansion alternatively. oscillator, illustrated as Figure 1. .e shell mass is M, the Substitute Equation (5) into Equation (4): rigidity coefficient of the spring connecting the two shells is K, 2 (0) 2 (1) d uj, d uj, and the mass of the internal oscillator is m. .e internal os- ⎢⎡ 1 + 1 ⎥⎤ ⎢ 2 ε 2 ⎥ + cillator is a pure Duffing oscillator whose force is proportional 2 β 0 ⎢ dτ dτ ⎥ 1 α −1 ω + εω � ⎡⎣ ⎤⎦⎢ ⎥ +⎡⎣ ⎤⎦ 0 1 ⎢ ( ) ( ) ⎥ to displacement and the cubic of displacement, and the linear 0 1 ⎣⎢ d2u 0 d2u 1 ⎦⎥ −1 1 rigidity coefficient of the spring connecting the oscillator and j,2 + j,2 2 ε 2 the shell is k; the coefficient of the cubic term is Γ. dτ dτ u u (0) (1) (0) (0) (1) (1) In the case of period j, for instance, j,1 and j,2 are the uj, + εuj, u + u + �u + u � ⎢⎡ 1 1 ⎥⎤ α ⎣⎡⎢ j−1,1 j+1,1 ε j−1,1 j+1,1 ⎦⎤⎥ displacements of the shell and the oscillator in period j; uj−1,1 · ⎣⎢ ⎦⎥ − (0) (1) 2 and uj−1,2 are the displacements of the shell and the oscillator uj,2 + εuj,2 0 in period j−1; and uj+ , and uj+ , are the displacements of 1 1 1 2 (0) (0) 3 the shell and the oscillator in period j + 1. Next, we are going −�uj, − uj, � + 2εΓ⎣⎢⎡ 2 1 ⎦⎥⎤o�ε2 � � 0. to discuss kinematic modeling of this system. (0) (0) 3 �uj, − uj, � From the load on the shell in period j, its differential 2 1 equation of motion is (6) Advances in Civil Engineering 3

uj–1,1 uj–1,2 uj,1 uj,2 uj+1,1 uj+1,2 M M M kk kk kk … m m m … K KKK d d d Figure 1: Metamaterial eŽective negative mass with the pure Dung oscillator.

0 1 Separate the ε and ε coecients and order them to be 0 0 uj,( 1) A( ) zero. en, we get, 1 1 iqaj iτ 0 e e c.c., ε coecient: 0  0 2 + ( ) ( ) ( ) 0 uj,2 A2 d2u( ) j,1     9 0     β 0 2 1 α 1 uj,( )  0    dτ 1 u( ) 0 2 + j 1,1 A1( ) 1 iqa j 1 iτ ω0 e ( )e c.c., + 0 0 0 01 2 0  11 u( ) u( )  A( ) 2 + ( )  d uj,( 2)  − j,2 j±1,2 2 ±       d 2     ( ) 10   τ     where a is the periodic±   unit size, qaj is the j periodic phase   −     0 0 0 0 7 factor, and A1( )A2( ) are the steady-state amplitude of shell α uj( )1,1 uj( )1,1 + + 0. and oscillator, respectively. c.c. stands for the conjugation of 2  − 0 the equations above. Substitute Equations (9) and (10) into Equation (7): − ε1 coecient:  qa ω2β 1 2α sin2 1 0 2 1 0 A1( ) d uj,( 1) + + 2 1 iqaj iτ 1 e e . . . ( ) 1 1 c c 0 β 0 dτ2 1 α 1 uj,1 u( ) u( ) 0 2 +  2 + α j 1,1 j 1,1 − − A( ) ω + +  2 2 0  11ω0  1 + 1 2 −    01 d2u( )  11 u( ) 0     j,2  − j,2          ( )   d 2     −   − −   τ         − In order for Equation (11) to have nonzero solutions   0 0 11   2 0 A( )A( ) d uj,( 1) 1 2 , the determinant of its coecient matrix must be 0 0 3 zero. Hence, β 0 dτ2 uj,( 2) uj,( 1) 2ω0ω1 2 . 1 qa  3 2 2 01 2 0  0 0 ω0 β 2 2α sin ,  d uj,( )  − uj,( ) −uj,( ) +  ( )  2  2 1 1 ω0 2 −   Γ    dτ2  −       0 0 12   − and A1( ) and A2( ) have the following relation:    −  ( ) 0 2 0 A( ) 1 ω A( ). 1  0 2 ( ) 8 2 2.1. Linear Dispersive Curve Equation. e steady-state so- By †nding ω0 in Equation− (12), we get the dispersive13 lution of Equation (7) is expressed as curve equation of the two frequency branches:

√2 ββ 1 2α sin2 qa/2 β 1 2 4α sin2 qa/2 α sin2 qa/2 β 1 + + ( ) ( + ) + ( ) ( ) + ωaco , ( ) 0  2β  − −     14 √2 ββ 1 2α sin2 qa/2 β 1 2 4α sin2 qa/2 α sin2 qa/2 β 1 + + ( )+ ( + ) + ( ) ( ) + ωopt , ( ) 0  2β  −     15

aco opt where ω0 and ω0 stand for the acoustic and optical branch is a high-frequency dispersive curve. On this frequency branches. An acoustic frequency branch is a low- basis, we can obtain linear dispersive curve Equations (14) frequency dispersive curve while an optical frequency and (15). 4 Advances in Civil Engineering

2.2. Nonlinear Dispersive Curve Equation. Next, let us We get consider the first-order perturbation solution which, for (0) 2ΓN ω0 Equation (8), is expressed as ω1 � . (23) A(0) + A(0)β1 − ω2 � u(1) (1) 2 1 0 ⎡⎢ j,1 ⎤⎥ A1 ⎢ ⎥ ⎢⎡ ⎥⎤ 1 iqaj iτ ⎣⎢ ⎦⎥ �⎣⎢ ⎦⎥ e e + c.c., (16) So, the perturbed solution is expressed as (1) (1) 2 (0) uj,2 A2 2ΓN ω ω � ω + εω � ω + ε 0 + o�ε2 �. ( ) 0 1 0 (0) (0) 2 24 A2 + A1 β1 − ω0 � u(1) (1) ⎡⎢ j±1,1 ⎤⎥ A1 1 iqa(j±1) iτ ( ) ( ) ⎣⎢ ⎦⎥ �⎡⎣ ⎤⎦ e e + c.c. (17) Considering the relation between the A 0 and A 0 u(1) A(1) 2 1 2 j±1,2 2 shown in Equation (13), Equation (23) can be written as � �2 � (0)� Substitute Equations (16) and (17) into Equation (8): 3Γω07�A2 � ω � ω + ε + o�ε2 �. (25) qa 0 2 2 2 + + 2� � (1) 2� 1 + β1 − ω � � ⎢⎡ −ω0β 1 2α sin −1 ⎥⎤ A 0 ⎢ 2 ⎥⎡⎢ 1 ⎤⎥ 1 iqaj i ⎢ ⎥⎢ ⎥ e e τ + . . ⎢ ⎥⎣ ⎦ c c opt aco ⎣⎢ ⎦ (1) 2 By substituting ω0 and ω 0 shown in Equations (14) 2 A2 −1 1 − ω0 and (15) into the equation above, we can obtain the dispersive curve equations of the first-order asymp- (0) (0) totically expanded acoustic and optical frequency −2ω0ω1A + 4ΓN ⎢⎡ 1 ⎥⎤ 1 iqaj iτ �⎣⎢ ⎦⎥ e e branches: (0) (0) 2 7� �2 −2ω0ω1A2 − 4ΓN opt � (0)� 3Γ�ω0 � �A2 � ω � ωopt + ε , ( ) opt 0 2 (26) A 0 opt 2 ⎢⎡ 1 ⎥⎤ 3iqaj 3iτ 2� 1 + β�1 −�ω � � � +⎣⎢ ⎦⎥Ce e + c.c., 0 A(0) 2 � � 7� (0)�2 (18) 3Γωaco � �A � � aco + 0 2 , ( ) ( ) ωaco ω0 ε (27) N(0) � ( )(A 0 A 0 )(A(0) A(0))2 aco 2 2 where 3/4 2 − 1 2 − 1 . 2� 1 + β�1 −ω0 � � � To remove the secular term, considering only the iqaj iτ equality between the e e coefficients on both sides of .ese are the dispersive curve equations of the Equation (18). .en we have acoustic and optical frequency branches that include qa for nonlinear factors. We can see that these dispersive 2 + + 2� � (1) ⎢⎡ −ω0β 1 2α sin −1 ⎥⎤ A ⎢ 2 ⎥⎢⎡ 1 ⎥⎤ curve equations are relevant to nonlinear small pertur- ⎢ ⎥⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ bation ε, coefficient before nonlinear term Γ, and steady- A(1) −1 1 − ω2 2 state vibration amplitude of the pure Duffing oscillator 0 (19) (0) |A2 |. (0) At α � 5, β � 3, ε � 0.0135, the influence of coefficient −2ω ω A + 4ΓN(0) ⎢⎡ 0 1 1 ⎥⎤ �⎣⎢ ⎦⎥. before the nonlinear term Γ, on the dispersive curve is ( ) analyzed as presented in Figure 2. Here, Γ � 1 represents −2ω ω A 0 − 4ΓN(0) 0 1 2 a hard Duffing oscillator, Γ � −1 represents a soft Duffing 2 2 oscillator, and Γ � 0 represents linearity. From this dia- Order C1 � −ω0β + 1 + 2α sin (qa/2), and consider Equation (12). .en, we have gram, when the local resonance oscillator is a hard Duffing 1 oscillator, its two dispersive curves will translate toward C1 � 2. (20) the high band relative to the linear oscillator; when the 1 − ω0 local resonance oscillator is a soft Duffing oscillator, on .rough linear transform of the augmented matrix of the contrary, its two dispersive curves will translate to- Equation (19), we get ward the low band relative to the linear oscillator. Also, under the same conditions, nonlinear factors make greater (0) (0) C1 −1 2ω0ω1A1 + 4ΓN differences to the dispersive equation of the optical fre- ⎛⎝ ⎞⎠. 0 0� 2ω ω A(0) − 4ΓN(0)�C + 2ω ω βA(0) + 4ΓN(0) quency branch than to that of the acoustic frequency 0 1 2 1 0 1 1 branch. (21) In order for Equation (19) to have nonzero solutions (1) (1) 2.3. Rediscussion on the Starting and Cutoff Frequencies of A1 A2 , the rank of its coefficient matrix must be the same as that of its augmented rank. Hence, a Corresponding Linear System. For acoustic frequency branch ωacoin Equation (14), when qa � π, the bandgap (0) (0) (0) (0) 0 �2ω0ω1A2 − 4ΓN �C1 + 2ω0ω1βA1 + 4ΓN . (22) starting frequency is Advances in Civil Engineering 5

1 2 ω 2ββ 1 2α 4α2 4αβ 4α β2 2β 1 , 01  2β + + + + + +  − − 1.5 1   β 1 2α 4α2 4αβ 4α β2 2β 1 ,  2β + + + + + +  – − − ω 1 1   β 1 2α β 1 2α 2 4β ,  2β + + ( + + ) +   0.5 − − 1   2β β 1 2α β 1 2α 2 4β ,  2β +( + + ) ( + + ) +  − − − –π –(3π/4) –(π/2) –(π/4) 0 π/4 π/2 3π/4 π   qa β 1 2α β 1 2α 2 4β ( + + ) ( + + ) + , 1 Γ = 1  + 2β − − − Γ = 0 Γ = –1 2 1 , Figure 2: e in˜uence of nonlinear factors on the dispersive  2 β 1 2α β 1 2α 4β curve. ( + + )+ ( + + ) + −  − − ( ) It is easy to prove that, in Equation (27), β 1 2α + + 28+ β 1 2α 2 4β 0. Hence, the bandgap dimensionless ( + + ) + starting frequency is smaller than 1. Namely,− the bandgap  starting− frequency is smaller> than the resonance frequency ω0. Furthermore, ω 1 is the start area of eŽective negative 01  mass. In the literature [26], ω01 1 is used to de†ne the  0.9 bandgap starting frequency. Comparing with the ω01 value, we can see that it is higher than the actual starting frequency. 0.8 It is relatively conservative for bandgap design, but does not 0.7 re˜ect the in˜uence of spring stiŽness of the shell. ω 0.6 01 Figure 3 shows how the starting frequency changes when 0.5 α and β are de†ned diŽerently. From Figure 3, the variation 0.4 0.3 of the nondimensionalized starting frequency ω01 is not only relevant to the shell/oscillator rigidity ratio α, but also to 0.2 their mass ratio β. Hence it is inappropriate to simply de†ne the bandgap starting frequency as a resonance frequency. 1 1 Literature [26] oŽers a good example supporting our ob- 3 3 servation with resonance frequency of 6.35 Hz and the β 5 5 starting frequency at 5.8 Hz. α For ωopt in Equation (15), when qa 0, the bandgap 7 7 0  cutoŽ frequency is Figure 1 3: e in˜uence of α and β on the starting frequency. ω 1 . 02  + β ( ) mass in bandgap and the phenomenon that cutoŽ frequency From Equation (28), the bandgap  cutoŽ frequency29 is is eŽective zero mass should also exist in the nonlinear only relevant to the shell/oscillator mass ratio β and de- system However, none of the above has been reported. en, creases when the mass ratio increases. And the cutoŽ fre- verify them by numerical methods as mentioned in the quency is corresponding to the eŽective zero mass. following. Considering the dispersive curve equations of the †rst- order asymptotically expanded acoustic and optical fre- 3. Numerical Simulation and Validation quency branches shown in Equations (25) and (26), for nonlinear acoustic metamaterial with weak nonlinearity (ε is e bandgap characteristics of in†nite acoustic metamaterial small), nonlinear system characteristics depend on the can be assumed by the †nite one (the periodic number is less corresponding degraded linear systems. As a result, in linear than 5) [40]. In order to validate our theoretical analysis system the phenomenon that both eŽective and negative above, we are going to build a model of an eŽective negative 6 Advances in Civil Engineering

F sin ωt M M M M M kk kk kk kk kk m m m m m K KKKK d d d d d

Figure 4: EŽective negative mass system models of the Dung oscillator of †ve diŽerent periods. mass system with pure Dung oscillator in †ve periods as starting frequency gradually translates toward the high- shown in Figure 4. By adding a simple harmonic exciting frequency region. When n is negative, the corresponding force F sin ωt on the shell of period 1 and picking up the internal Dung oscillator is a softened Dung oscillator. response in period 5, the steady-state displacements of the Hence, as n decreases, the bandgap starting frequency shell and the oscillator are calculated at each of the excitation gradually translates toward the low-frequency region. frequencies. is way, we get the vibration transmission e same method also applies to the in˜uence of non- characteristics of this †nite periodic structure to analyze the linear factor n on the bandgap cutoŽ frequency. As shown in bandgap and vibration transmission pattern of this struc- Figure 6(b), when n is positive, the corresponding internal ture. When the coecient before the nonlinear term is 0, Dung oscillator is a hardened Dung oscillator. Hence, we  it represents the corresponding linear system. can observe that, as n increases, the bandgap cutoŽ fre- Γ quency gradually translates toward the high-frequency re- gion. When n is negative, the corresponding internal Dung 3.1. Inuence of Nonlinear Terms on the Bandgap Starting and oscillator is a softened Dung oscillator. As n decreases, the Cuto­ Frequencies. To validate the theoretical analysis result bandgap cutoŽ frequency gradually translates toward the above, we may compare it eŽectively against experimental low-frequency region. parameters provided in the literature [26] and establish As discussed above, existence of nonlinear factor aŽects a periodic structure vibration system of †ve periods with the bandgap starting and cutoŽ frequencies of the structure. shell mass M 0.1011 kg, internal oscillator mass  e smaller the n value is, the smaller the corresponding m 0.04647 kg, internal nonlinear spring linear rigidity  starting or cutoŽ frequency is; the larger the n value is, the k 37 N/m, external linear spring damping coecient  larger the starting or cutoŽ frequency is. Under the same c 0.05 NS/m, and rigidity coecient of spring linking the  conditions, nonlinear factor makes greater diŽerence to external oscillators K 117 N/m. If the nonlinear pertur-  cutoŽ frequency than it does to starting frequency. bation term is taken as ε 0.01, to facilitate expression, we  de†ne nk. 3.2. Inuence of the Shell Rigidity on the Bandgap Frequency.  ( ) As analyzed above, existence of a nonlinear term makes n When is positive, DungΓ is hard rigid; when n30is a little diŽerence to the bandgap starting frequency of the negative, Dung is soft rigid; when n is zero, the system has corresponding linear system; starting frequency is relevant been degraded into a linear system. e vibration trans- to shell/oscillator rigidity ratio α and mass ratio β, and cutoŽ mission characteristic curves of the corresponding system frequency is only relevant to mass ratio β. As α decreases, the are calculated at n 0, 1, 1, 3, 3, 10, and 10 as presented starting frequency decreases, the cutoŽ frequency remains  in Figure 5. e vibration transmission can be con†rmed by the same, and the bandgap width increases. Now, let us draw the ratio of the †fth shell− steady-state− amplitude− to the †rst the vibration transmission pro†le of the corresponding shell steady-state amplitude. system by only changing the rigidity coecient of the spring As illustrated in Figure 5, existence of a nonlinear term between the oscillator and the shell, K, from K 117 N/m to  does not make much diŽerence to the bandgap, mainly 37 N/m and taking the n value as n 0, 1, 1, 3, 3, 10, and  because the Dung oscillator we selected is weakly non- 10 as presented in Figure 7. linear. By enlarging the points of the bandgap starting and When the shell connecting rigidity K 117 N/m is −  − cutoŽ frequencies in the chart, we will see how a nonlinearity changed− to 37 N/m, the masses of the Dung oscillator and factor aŽects the bandgap. the shell remain the same, and according to analysis results When the longitudinal coordinate (i.e., transmission in Equation (27) and Figure 3, after the shell rigidity is rate) of the transmission characteristic curve turns from changed, the bandgap cutoŽ frequency will remain the same, positive to negative, the vibration in period 1 cannot be but the starting frequency will decrease. at is, the band- transmitted to the shell of period 5. en, a bandgap will width of the bandgap will increase, and the bandgap will appear. Hence, we have to †nd the point where the trans- expand toward the low-frequency region. As can be ob- mission curve runs across the horizontal axis, i.e., the served from Figure 5, the bandgap cutoŽ frequency remains bandgap starting frequency, as presented in Figure 6. the same at 7.76 Hz while the bandgap starting frequency has From Figure 6(a), when n is positive, the corresponding decreased to 3.76 Hz, proving our analysis result. Hence, if internal Dung oscillator is a hardened Dung oscillator. we want a wider bandgap width, when other conditions Hence, we can observe that, as n increases, the bandgap remain unchanged, the rigidity coecient of the linear Advances in Civil Engineering 7

40

20

0

–20

–40 Transmission (dB) Transmission

–60

–80 0 246810 f (Hz) n = –10 n =0 n =3 n = –3 n =1 n = 10 n = –1 Figure 5: Transmission characteristics with diŽerent n (n 117 N/m). 

×10–3 2 0.01 1 0.005

0 0

–0.005 –1 –0.01 Transmission (dB) Transmission Transmission (dB) Transmission –2 –0.015

–0.02 –3 5.6755 5.676 5.6765 5.677 7.6752 7.6753 7.6754 7.6755 7.6756 7.6757 7.6758 f (Hz) f (Hz) n = –10 n =0 n =3 n = –10 n =0 n =3 n = –3 n =1 n = 10 n = –3 n =1 n = 10 n = –1 n = –1 (a) (b)

Figure 6: Starting and cutoŽ frequencies region (K 117 N/m). (a) Starting frequency region. (b) CutoŽ frequency region.  spring connecting the shell must be as small as practically positive mass below the linear resonance frequency possible. constituted by the Dung oscillator and the shell in each period, and eŽective negative mass does not appear except above the resonance frequency and below the cutoŽ 3.3. Responses of the Shell and the Oscillator on the Bandgap. frequency. Next, the responses of the shells and the os- As can be summarized from the discussions on bandgap cillators of the system in the eŽective negative mass area starting frequency above, existence of weakly nonlinear and the eŽective positive mass area within the spectrum of factor results in some minor changes near the frequency bandgap are examined with rigidity ratio between springs value of its degraded linear system, and that both the connecting external large oscillators K 37 N/m while  Dung oscillator/shell linear rigidity ratio α and mass keeping all the other parameters unchanged. ratio β make a diŽerence to the starting frequency and that the starting frequency is lower than the resonance fre- quency of a linear oscillator In other words, generation of 3.3.1. E­ective Negative Mass Phenomenon in Bandgap. bandgap is not always resulted from eŽective negative From Figure 7, with bandgap between 3.7 Hz and 7.67 Hz and mass. Within the spectrum of bandgap, it must be eŽective resonance frequency of linear oscillator at 6.35 Hz, analysis is 8 Advances in Civil Engineering

40

20

0

–20

–40 Transmission (dB) Transmission

–60

–80 0 246810 f (Hz) n = –10 n =0 n =3 n = –3 n =1 n = 10 n = –1 Figure 7: Transmission characteristics with diŽerent n (K 37 N/m). 

0.01 0.3

0.005 0.2 0.1

0 0

–0.1 Velocity (m/s) Velocity Displacement (m) Displacement –0.005 –0.2

–0.3 –0.01 1499.7 1499.75 1499.8 1499.85 1499.9 1499.95 1500 –0.01 –0.005 0 0.005 0.01 t (s) Displacement (m) Period 1 Period 4 Period 1 Period 4 Period 2 Period 5 Period 2 Period 5 Period 3 Period 3 (a) (b)

Figure 8: Five periodic shells’ (a) displacement time history curves and (b) phase diagrams. carried out under eŽective negative mass taking the external loaded on the shell in period 1 cannot be propagated in this excitation frequency 7 Hz. Figure 8 shows the vibration periodic structure but is suppressed, which concurs with the  time history curves and phase diagrams of the shells in †ve characteristics of bandgap. periods. Figure 9 showsω the oscillators in †ve periods. Figure 10 shows the vibration time histories of the shell From Figures 8 and 9, some time later, the vibration and the Dung oscillator in period 1. Here, the dotted lines amplitude of each shell and oscillator no longer attenuates are the time histories of the shell, and the solid lines are the but has stabilized; the displacement-velocity phase diagrams time histories of the Dung oscillator. As can be observed, of the shell and the oscillator both consist of a circle, and all when the motion displacement of the shell M is positive the shells and oscillators of this periodic structure have amplitude, the motion of the Dung oscillator, m, is achieved steady-state vibration. a negative amplitude; when the motion displacement of M is Now, let us compare the shell vibration in each period. a negative amplitude, the motion displacement of m is also From period 1 through period 5, the shell vibration gradually positive amplitude. is means that M and m always have an decreases. is vibration is almost zero in the last period. e opposite motion displacement, and the vibrations of the internal Dung oscillator shows very similar vibration shell and the oscillator are reverse phased. Here, the black characteristics to the shell. is indicates that vibration curves are the time displacement curves of m vibration of the Advances in Civil Engineering 9

0.15 5 0.1

0.05

0 0 –0.05 Velocity (m/s) Velocity

Displacement (m) Displacement –0.1

–0.15 0 –5 –0.2 1499.7 1499.75 1499.8 1499.85 1499.9 1499.95 1500 –0.1 –0.05 0 0.05 0.1 t (s) Displacement (m) Period 1 Period 4 Period 1 Period 4 Period 2 Period 5 Period 2 Period 5 Period 3 Period 3 (a) (b)

Figure 9: Five periodic oscillators’ (a) displacement time history curves and (b) phase diagrams (eŽective negative mass).

0.15 phenomenon occurs, meanwhile bandgap in 3.7 Hz–6.35 Hz should be the eŽective positive mass region. Analysis is 0.1 carried out under eŽective positive mass taking the external excitation frequency 5 Hz. Figure 11 shows the vibration 0.05  time history curves and phase diagrams of the shells in †ve 0 periods, and Figure 12ω shows the oscillators in †ve periods. From Figures 11 and 12, all the shells and oscillators of –0.05 this periodic structure have achieved steady-state vibration. e vibrations loaded on the shell and oscillators in period 1 Displacement (m) Displacement –0.1 cannot be propagated in this periodic structure which –0.15 concurs with the characteristics of bandgap. Figure 13 shows the vibration time histories of the shell –0.2 and the Dung oscillator in period 1. As can be observed, 1499.4 1499.5 1499.6 1499.7 1499.8 1499.9 1500 when the motion displacement of the shell M is positive t (s) amplitude, the motion of the Dung oscillator, m, is also of M the same positive amplitude. M and m always have the m same motion displacement, and the vibrations of the shell Figure 10: Displacement time history curves of shell and oscillator and the oscillator in the same period are cophasal. Now, the in period 1. system in every period must have a positive momentum, and the system helps control vibration just as eŽective as negative mass systems does. at is, eŽective positive mass internal mass bock. is is because M has a larger mass than phenomenon also occurs in the bandgap of nonlinear m, and under the same external excitation amplitude, objects metamaterial. with a smaller mass will †nd it easier to change their velocity, resulting in a larger displacement. Further analysis revealed that the Dung oscillator has a greater momentum than the 3.3.3. Discussion on E­ective Zero Mass. As analyzed above, shell. Now, the system comprising the shell and the Dung when the eŽective mass gradually changes from negative to oscillator in individual periods has a negative momentum. positive through a frequency point at which the eŽective As the eŽective velocity is positive, the system possesses mass is zero, the place corresponding to zero mass is where negative mass, and consequently the system helps control the bandgap cutoŽ frequency stands. Existence of nonlinear vibration. factor, however, makes a minor diŽerence to the frequency point at zero mass, and it would be possible that the actual cutoŽ frequency of bandgap cannot be found. To solve this 3.3.2. E­ective Positive Mass Phenomenon in Bandgap. problem, we can closely sample the external excitation Based on the above analysis, bandgap in 6.35 Hz–7.67 Hz, as frequency so as to bind the bandgap cutoŽ frequency of the shown in Figure 7, where the eŽective negative mass nonlinear system, i.e., the point of zero-mass frequency. 10 Advances in Civil Engineering

0.1

2 0.05 1

0 0

Velocity (m/s) Velocity –1 Displacement (m) Displacement –0.05 –2

–0.1 1499.5 1499.6 1499.7 1499.8 1499.9 1500 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 t (s) Displacement (m) Period 1 Period 4 Period 1 Period 4 Period 2 Period 5 Period 2 Period 5 Period 3 Period 3 (a) (b)

Figure 11: Five periodic shells’ (a) displacement time history curves and (b) phase diagrams.

0.2 6 0.15 4 0.1

0.05 2

0 0

–0.05 Velocity (m/s) Velocity –2 Displacement (m) Displacement –0.1 –4 –0.15 –6 –0.2 1499.5 1499.6 1499.7 1499.8 1499.9 1500 –0.2 –0.15 –0.1 –0.05 0 0.05 0.1 0.15 0.2 t (s) Displacement (m) Period 1 Period 4 Period 1 Period 4 Period 2 Period 5 Period 2 Period 5 Period 3 Period 3 (a) (b)

Figure 12: Five periodic oscillators’ (a) displacement time history curves and (b) phase diagrams (eŽective positive mass).

Finally, with a constant nonlinear factor n 10, shell mass From these charts in Figures 14 and 15, the shells of all  M 0.1011 kg, internal oscillator mass m 0.04647 kg, in- periodic units move in the same phase and with the same   ternal nonlinear spring linear rigidity k 37 N/m, and ex- displacement pattern, keeping the spring K linking the  ternal linear spring damping coecient c 0.1 NS/m, we periodic units from deformation. If we ignore the motion of  found the frequency point at zero mass to be 8.2205 Hz. the internal Dung oscillators from outside, the entire system Figures 14 and 15 show the time histories of the shells and will appear to be a rigid rod. e entire system maintains an the Dung oscillators using this frequency point. From in-phase translation while all the internal Dung oscillators these curves, the shells and Dung oscillators in all the move in the same phase but in the opposite direction to the periods are moving. From the phase diagrams, both the shells, and there is no phase diŽerence among the internal shells and the Dung oscillators have reached a steady state, Dung oscillators. is is because a zero mass unit signi†es and the shells in all the periods have the same maximum that its inertia force is zero, making it appear that the entire displacements as the Dung oscillators in all the periods. system is composed of massless rigid bodies. It can be proved Advances in Civil Engineering 11

0.2

0.15

0.1

0.05

0

–0.05 Displacement (m) Displacement –0.1

–0.15

–0.2 1499 1499.2 1499.4 1499.6 1499.8 1500 t (s) M m Figure 13: Displacement time history curves of shell and oscillator in period 1.

0.05 2 1.5 1 0.5 0 0 –0.5 Velocity (m/s) Velocity

Displacement (m) Displacement –1 –1.5 –2 –0.05 7999.7 7999.75 7999.8 7999.85 7999.9 7999.95 8000 –0.04 –0.03 –0.02 –0.01 0 0.01 0.02 0.03 0.04 t (s) Displacement (m) Period 1 Period 4 Period 1 Period 4 Period 2 Period 5 Period 2 Period 5 Period 3 Period 3 (a) (b)

Figure 14: Five periodic shells’ (a) displacement time history curves and (b) phase diagrams.

that the springs connecting the units will not deform, and the contrary, the system response is usually sensitive to and entire system looks as if one massless rigid rod is translating. can vary with the external excitation amplitude. Hence, we e lattice kymatology of the periodic system can also provide are going to analyze how bandgap changes with the ex- an explanation. is phenomenon similar to zero mass is how ternal excitation amplitude. electromagnetic waves propagate through a matching met- With a constant nonlinear factor n, we change external amaterial with zero refraction index in electromagnetics. is excitation amplitude to examine how it aŽects the periodic material also features zero dielectric constant and zero system again with shell mass M 0.1011 kg, internal os-  magnetic permeability and no phase diŽerence in the elec- cillator mass m 0.04647 kg, internal nonlinear spring  tromagnetic †eld. linear rigidity k 37 N/m, external linear spring damping  coecient c 0.05 NS/m, and rigid coecient between  springs connecting external large oscillators K 117 N/m.  If the nonlinear perturbation term is taken as ε 0.01, 3.4. Inuence of the Excition Amplitude on the Start and Stop  n 1, f is taken as 1, 2, 3, 5, and 10 to calculate the vibration Frequencies of the Bandgap. As is known to all, for a linear  system, the system response is irrelevant to the external transmission characteristics. e result is presented in excitation amplitude. For a nonlinear system, on the Figure 16. When the external excitation amplitude changes, 12 Advances in Civil Engineering

0.1 8 6 0.05 4 2 0 0 –2 Velocity (m/s) Velocity Displacement (m) Displacement –0.05 –4 –6 –8 –0.1 7999.7 7999.75 7999.8 7999.85 7999.9 7999.95 8000 –0.15 –0.1 –0.05 0 0.05 0.1 0.15 t (s) Displacement (m) Period 1 Period 4 Period 1 Period 4 Period 2 Period 5 Period 2 Period 5 Period 3 Period 3 (a) (b)

Figure 15: Five periodic oscillators’ (a) displacement time history curves and (b) phase diagrams (eŽective zero mass).

40 decreases, the bandgap stop frequency moves toward the low band. As the transmission characteristic chart is much more 20 separated than the previous two subsections, changing the f value makes a greater diŽerence to the starting and cutoŽ 0 frequencies. rough comparison above, we can see that external –20 excitation amplitude makes a great diŽerence to bandgap. As the external excitation amplitude increases, so does the –40 bandgap starting frequency of the metamaterial. is will Transmission (dB) Transmission further reduce the bandgap width, which is negative for the –60 application of vibration control. Hence, when a metamaterial of eŽective negative mass with Dung oscillator is used for –80 vibration control, the external excitation amplitude must be 0 246810kept within a reasonable limit. Frequency (Hz) f =1 f =5 4. Conclusions f =2 f = 10 f =3 In this paper, the Lindstedt–Poincare´ method is utilized to Figure 16: External excitation amplitude has changed the bandgap calculate the †rst-order perturbation solution of acoustic width. and optical branches for eŽective negative mass nonlinear acoustic metamaterial with pure Dung oscillator. e starting and cutoŽ frequencies of bandgap are deduced. en, the in˜uence of mass ratio α and stiŽness ratio β and so do the bandgap start and stop frequencies and the nonlinearity in˜uence on starting and cutoŽ frequency of bandgap bandwidth. Figure 17 shows enlarged areas of the bandgap are discussed. e main conclusions can be drawn starting and cutoŽ frequencies. as follows: As can be observed from Figure 17(a), the external excitation amplitude makes quite a big diŽerence to the (1) e nonlinear part of the dispersion curves solution bandgap starting frequency. As the external excitation has relationships with coecient of nonlinear item, amplitude f increases, the bandgap start frequency moves steady-state amplitude of Dung oscillator, and β. toward the high band; when f decreases, the bandgap starting Both soft and hard Dung oscillators have eŽect on frequency moves toward the low band. the starting and cutoŽ frequencies. e soft/hard Figure 17(b) shows an enlarged view near the stop characteristic of Dung oscillator leads to the trend frequency. From this, we can see that, as f increases, the that both starting and cutoŽ frequencies move bandgap stop frequency moves toward the high band; as f down/up relative to the degraded linear system, and Advances in Civil Engineering 13

both eŽective negative and eŽective positive mass occurs in bandgap. 1 (4) EŽective zero-mass phenomenon also exists in this 0.5 nonlinear metamaterial. is is corresponding to the cutoŽ frequency. On this frequency, there is no relative 0 motion for every shell of this nonlinear metamaterial, and every Dung oscillator also has the same char- –0.5 acteristics. e phases of shell and Dung oscillator in Transmission (dB) Transmission every period are opposite, and the aggregate mo- –1 mentum of every periodic unit is close to zero. –1.5 (5) e external excitation amplitude has in˜uence on bandgap. As the amplitude increases, the starting and 5.35 5.4 5.45 5.5 5.55 cutoŽ frequencies shift to the high-frequency band. f (Hz) f =1 f =5 Data Availability f =2 f = 10 f =3 e data used to support the †ndings of this study are (a) available from the corresponding author upon request. 2 Disclosure 1 Gao Ming now works in College of Mechanical and Elec- 0 tronic Engineering, Shandong Agricultural University in Taian, China. –1 Conflicts of Interest

Transmission (dB) Transmission –2 e authors declare that they have no con˜icts of interest. –3 Acknowledgments 7.55 7.6 7.65 7.7 7.75 7.8 7.85 7.9 e authors acknowledge the State Key Research Devel- f (Hz) opment Program of China (Grant no. 2016YFC0600708), f =1 f =5 the National Natural Science Foundation of China (Grant f =2 f = 10 nos. 11172198 and 51605264), the National Basic Research f =3 Program of China (Grant nos. 2013CB035402 and (b) 2014CB046800), the Shandong Province Higher Educational Science and Technology Program (Grant no. J15LH04), the Figure 17: External excitation amplitude has changed the starting Natural Science Foundation of Shandong Province (Grant and cutoŽ frequencies. (a) Starting frequency region. (b) CutoŽ nos. ZR2018 and MEE001), and the State Key Laboratory of frequency region. Open Funds (Grant no. MDPC201601).

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