Active control for acoustic wave propagation in nonlinear diatomic acoustic metamaterials
Introduction
Over the past decades, the exploration of new devices and new structures has never been stopped. Metamaterials, with a lot of non-conventional and peculiar properties such as wave-guiding [1], [2], [3], [4], noise reduction [5], [6], [7], cloaking [8], [9] and band gaps [10], [11] always attract considerable academic and research curiosities for their potentials in wide engineering applications. Generally, metamaterials are not new materials invented or created. These are artificially structured composite materials which can be regarded as a smart design of structures. The prefix “meta”, translated from the Greek, means “beyond”, the combination of “meta” and “material” coined by pioneers was to describe the material which is “beyond conventional materials” [12].
Acoustic metamaterials, as one significant branch of metamaterials, are designed to manipulate and control the dispersive properties of vibrating wave propagation. Comparing to phononic crystals which are designed to control wave through Bragg scattering [13], [14], acoustics can also generate local resonance properties to control and guide wave propagation [15]. For local resonance of acoustic metamaterials, the unique properties are not dependent on design periodicity. The resonant frequency of a unit cell usually depends on the inner structure or an attached item (e.g., mass), and the inner restoring force (e.g., that of the linear or nonlinear spring). At resonant frequency of the inner structures, the wave that propagates through the structure can generate enormous deflection, even to a degree several orders of the magnitude of the resonance item. Thus, the local resonance band gap can be produced in the subwavelength region. The wave number expressed as a function of frequency is always utilized to analyze the band properties and dispersion relations of the structures. The generation or control of band gaps is one of the interesting properties of acoustic metamaterials [16], [17], [18], [19], [20].
To address the problem easily, the linear analysis approach is always adopted to capture the main features of structures. However, structural nonlinearity cannot always be neglected. Currently, metamaterial nonlinearity has received considerable attention [21], [22], [23], [24], [25]. For this reason, the Lindestedt–Poincaré (L–P) perturbation method [26], [27] provides a novel approach to solve the weakly nonlinear problems in metamaterials. In 2018, Zhou et al. [28] analytical and numerically illustrated how the wave package propagates through nonlinear acoustic metamaterials. Via the spectro-spatial analysis approach, the novel nonlinear wave mechanism is revealed and the direction-biased waveguide is designed. There are numerous wave solutions available currently with structural nonlinearity depending on various parameters such as wave amplitude, wave interaction, hard or soft nonlinearities, etc.
Once metamaterials are manufactured and ready for applications, it is rather difficult to impose any additional control over the entire system. However, with active or passive control components installed at the outset, the structures can exhibit distinct characteristics. In 2001, Tang and Wang [29] demonstrated and compared active–passive hybrid piezoelectric networks using analytical methods and experiments. They successfully strengthened the coupling coefficients and subsequently developed active–passive hybrid piezoelectric network control. Chen et al. [30] presented a band gap control system with an active control using negative capacitance piezoelectric shunting. In 2018, Wang et al. [31] studied the nonlinear effects on a diatomic lattice chain with an active control using a piezoelectric spring model. It was able to capture different properties such as a new system stop band, soft nonlinear property and hard nonlinear property. One year later, Li et al. [32] investigated acoustic waveguiding which shows negative bulk modules near the resonant frequencies. The effective bulk modulus of an acoustic metamaterial can be controllable, and thus it can exhibit more tunable abilities as compared to the one with a single cavity. All studies show that with an active control, the whole smart metamaterial structures can be more promising and applicable [4], [33], [34], [35], [36], [37].
In this study, the band properties of a nonlinear diatomic mass-in-mass chain with an active control are analyzed theoretically and the results are illustrated in several numerical examples. The Lindestedt–Poincaré (L–P) perturbation method is employed to approximate the nonlinear behaviors of such a system. It is easy to obtain the dispersion relation of the zeroth order perturbation via this promising analysis approach. However, due to the coupling properties of the neighboring cells which form a unit cell, it is not convenient to solve the equation by simply assuming the secular item as zero in the first-order perturbation analysis. Thus, the orthogonality conditions of the eigenvector corresponding to the mass and stiffness matrix, i.e., the Hermitian matrix [38], are utilized to obtain the first-order perturbation relation. Afterwards, different nonlinear parameters are adopted to illustrate the nonlinearity effects on the system band gap properties. It is easily achieved from the results that nonlinearity will influence the third band gap and the fourth branch of the dispersion relation. This band-folding-induced band gap can generate topological interface states [39], thus making it significant to make a more precise prediction for the topological analysis [40], [41], [42]. In real applications, the prediction and control of band gap properties are fundamental to any efficient design of the structure for its functionality and easy tuneability. In what follows, we examine the band gap behaviors by adopting an active piezoelectric spring, the system stiffness can be artificially and readily changed by applying different electric conditions. This provides an approach for an efficient active control of the whole system for its availability for wider real applications. This study concludes that a topological approach for active control can be associated with the designs of nonlinear acoustic metamaterial devices and systems.
Section snippets
Nonlinear diatomic mass-in-mass system with active control
Consider the unit cell of a mass-in-mass chain indicated by a dashed black frame as illustrated in Fig, 1. The unit cell consists of two mass-in-mass cells , and , . For simplicity but without the loss of generality, the masses are taken aswhile masses are . Each unit cell is connected by nonlinear springs and bounded to the ground by linear piezoelectric springs. The entire system is governed by three groups of parameters, i.e., a nonlinear inner spring that
Comparison between linear and nonlinear systems
In the previous section, new dispersion relations for the linear and nonlinear systems have been derived analytically via the Lindestedt–Poincaré (L–P) perturbation method. From Eq. (36), it can be easily noticed that the whole system is governed by several parameters: the generalized passive springs with linear stiffness and , the generalized stiffness of active spring , the generalized nonlinear coefficient and , and the mass ratio .
Firstly, a comparison with reference
Conclusion
The effect of nonlinearity on the band properties of diatomic mass-in-mass chain with an active control is demonstrated. The dispersion relation of linear and nonlinear diatomic mass-in-mass system is obtained via the Lindestedt–Poincaré (L–P) perturbation method. It is found that the fourth branch and the third gap are more sensitive as compared to other branches and gaps. Specially, a piezoelectric spring model is applied to the diatomic mass-in-mass to make the system available for wider
CRediT authorship contribution statement
Zhenyu Chen: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Visualization, Writing - original draft. Weijian Zhou: Conceptualization, Methodology, Validation, Writing - review & editing. C.W. Lim: Funding acquisition, Project administration, Resources, Supervision, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The work described in this paper was supported by General Research Grant from the Research Grants Council of the Hong Kong Special Administrative Region (Project Nos. CityU 11212017, 11216318), and by City University of Hong Kong (Project No. 7005273).
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