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Effective Negative Mass Nonlinear Acoustic Metamaterial with Pure Cubic Oscillator 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 4 j;1 5 2 −2 4 j;1 5 and Chatterjee [37] looked at a damped nonlinearity and "#2 3 +" #2 3 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+ ; � 6 −�uj;2 − uj;1 � 7 24 −1 1 1 1 35 + 462 573 � : 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/pm�����, 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- 62 73 u β 0 6 d 2 7 1 + α −1 j;1 cussion whether positive effective mass phenomenon exists 2462 5736 τ 7 462 57362 73 ω 6 7 + 4 5 in the nonlinear metamaterial bandgap based on effective 6 2 7 0 1 46 d uj; 57 −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 26 −�uj;2 − uj;1 � 37 perturbation method is utilized to calculate metamaterial dis- − 462 573 + 2εΓ46 57 � 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 8< 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- 62 1 + 1 73 6 2 ε 2 7 + cillator is a pure Duffing oscillator whose force is proportional 2 β 0 6 dτ dτ 7 1 α −1 ω + εω � 24 356 7 +24 35 0 1 6 ( ) ( ) 7 to displacement and the cubic of displacement, and the linear 0 1 46 d2u 0 d2u 1 57 −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 � 62 1 1 73 α 426 j−1;1 j+1;1 ε j−1;1 j+1;1 537 displacements of the shell and the oscillator in period j; uj−1;1 · 46 57 − (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; � + 2εΓ462 2 1 573o�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 eective negative mass with the pure Dung oscillator.
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