A general theory for bandgap estimation in locally resonant metastructures

Abstract

Locally resonant metamaterials are characterized by bandgaps at wavelengths that are much larger than the lattice size, enabling low-frequency vibration attenuation. Typically, bandgap analyses and predictions rely on the assumption of traveling waves in an infinite medium, and do not take advantage of modal representations typically used for the analysis of the dynamic behavior of finite structures. Recently, we developed a method for understanding the locally resonant bandgap in uniform finite metamaterial beams using modal analysis. Here we extend that framework to general locally resonant 1D and 2D metastructures (i.e. locally resonant metamaterial-based finite structures) with specified boundary conditions using a general operator formulation. Using this approach, along with the assumption of an infinite number of resonators tuned to the same frequency, the frequency range of the locally resonant bandgap is easily derived in closed form. Furthermore, the bandgap expression is shown to be the same regardless of the type of vibration problem under consideration, depending only on the added mass ratio and target frequency. For practical designs with a finite number of resonators, it is shown that the number of resonators required for the bandgap to appear increases with increased target frequency, i.e. more resonators are required for higher vibration modes. Additionally, it is observed that there is an optimal, finite number of resonators which gives a bandgap that is wider than the infinite-resonator bandgap, and that the optimal number of resonators increases with target frequency and added mass ratio. As the number of resonators becomes sufficiently large, the bandgap converges to the derived infinite-resonator bandgap. Furthermore, the derived bandgap edge frequencies are shown to agree with results from dispersion analysis using the plane wave expansion method. The model is validated experimentally for a locally resonant cantilever beam under base excitation. Numerical and experimental investigations are performed regarding the effects of mass ratio, non-uniform spacing of resonators, and parameter variations among the resonators.

Introduction

Inspired by photonic crystals in electromagnetism, researchers have long investigated phononic crystals for their potential to filter or redirect elastic waves [1]. Phononic crystals exhibit bandgaps (i.e. frequency ranges where elastic or acoustic waves cannot propagate) produced by Bragg scattering [2], [3], [4], which occurs when the wavelength of the incident wave is on the order of the lattice constant of the crystal [5], [6]. Therefore, a fundamental limitation of Bragg-based phononic crystals is that it is only possible to create low-frequency bandgaps using very large structures. In their seminal work, Liu et al. [7] showed the potential for locally resonant metamaterials to create bandgaps at wavelengths much larger than the lattice size, enabling the creation of low-frequency bandgaps in relatively small structures. Locally resonant metamaterials contain resonating elements, whether mechanical [7], [8] or electromechanical [9], [10], [11], which are capable of storing and transferring energy. A significant body of research has examined locally resonant elastic/acoustic metamaterials of various types. Ho et al. [8] examined a similar system to Liu et al. [7] using a rigid frame with rubber-coated metal spheres as resonators. For that same type of system, Liu et al. [12] found analytic expressions for the effective mass densities of 3D and 2D locally resonant metamaterials, showing that the effective mass becomes negative near the resonant frequency. Simplifying the analysis, others have used lumped-mass models to obtain the locally resonant bandgap [13], [14]. Other researchers have studied different implementations of resonators for different types of elastic waves [15], [16], [17], [18], [19], [20], [21], and two-degree-of-freedom resonators [22]. Moving towards analytical predictions for the bandgap edge frequencies, Xiao et al. [23] used the plane wave expansion method to study flexural waves in a plate with periodically attached resonators, giving a method to predict the edges of the bandgap. Peng and Pai [24] also studied a locally resonant metamaterial plate, finding an explicit expression for the bandgap edge frequencies.

Much of the research on locally resonant metamaterials has relied on unit-cell based dispersion analysis, using techniques such as the plane wave expansion method to obtain the band structure of the metamaterial. This type of analysis lacks the information of modal behavior and cannot readily answer questions such as the dependence of the bandgap width on the number and spatial distribution of attachments in a finite structure. To this end we recently presented [25] a modal analysis approach to bridge the gap between the lattice-based dispersion characteristics of locally resonant metamaterials and the modal interaction between a primary structure and its resonators. This paper extends the framework presented for beams in bending in [25] to general (potentially non-uniform) 1D and 2D linear vibrating metastructures, i.e. locally resonant metamaterial-based finite structures (Fig. 1). A general form for the governing equations of the system is assumed using differential operator notation, and a modal expansion using the mode shapes of the structure without resonators provides significant simplification. Applying the assumption of an infinite number of resonators placed on the structure, the locally resonant bandgap edge frequencies are derived in closed form. This expression for the bandgap edge frequencies depends only on the resonant frequency of the resonators and the ratio of resonator mass to plain structure mass, and can be used for any typical vibrating structure (strings, rods, shafts, beams, membranes, or plates). To tie this work to other research in this field, the bandgap expression is validated numerically with the band structure obtained from the plane wave expansion method. To demonstrate that the derived bandgap expression is useful for design, we validate the bandgap expression for a finite number of uniformly distributed resonators. Additionally, we discuss the performance of metastructures with non-uniform distributions of resonators, as well as the effect of parameter variation in the resonant frequencies of the resonators. Finally, experimental validations are presented.