G. L. Huang 1 Department of Systems Engineering, University of Arkansas at Little Rock, Little Rock, AR 72204 e-mail: glhuang@ualr.edu C. T. Sun School of Aeronautics and Astronautics, Purdue University, W. Lafayette, IN 47907 Band Gaps in a Multiresonator Acoustic Metamaterial In this study, we investigated dispersion curves and the band gap structure of a multi- resonator mass-in-mass lattice system. The unit cell of the lattice system consists of three separate masses connected by linear springs. It was demonstrated that the band gaps can be shifted by varying the spring constant and the magnitude of the internal masses. By using the conventional monatomic (single mass) lattice model as an equivalent system, the effective mass was found to become negative for frequencies in the band gaps. An attempt was made to represent the two-resonator mass-in-mass lattice with a microstruc- ture continuum model. It was found that the microstructure continuum model can capture the dispersive behavior and band gap structure of the original two-resonator mass-in- mass system. DOI: 10.1115/1.4000784 Keywords: acoustic metamaterials, band gap, dispersion curves, negative effective mass, microstructure continuum theory 1 Introduction Recently, many researchers were engaged in exploring electro- magnetic metamaterials 1–3 with unusual properties such as a negative refractive index for various novel applications 4–6. The photonic band gap, a range of frequency in which electromagnetic waves cannot propagate, is of interest to engineers in designing photonic devices 7. Because of the similarity between stress wave propagation in phononic crystals and electromagnetic waves in photonic crystals 8–10, acoustic metamaterials that contain manmade periodic microstructures also attracted much attention recently. Practical applications of these systems include mechani- cal filters, sound and vibration isolators, and acoustic waveguides 11,12. One of the most attractive features in acoustic metamate- rials is the possibility to tailor the desired band gap for stopping propagation of waves of certain frequencies, especially for the construction of the low frequency band gap. Many researches about band gaps of acoustic and elastic wave propagation in phononic crystals have been conducted, both theo- retically and experimentally. One of the first calculations of acous- tic band gaps in a simple periodic composite was performed by Kushwaha et al. 13. However, the calculation in Ref. 13 was carried out only for the case of antiplane shear. By using mechani- cal lattice structures, complete acoustic band gaps were demon- strated by Martinsson and Movchan 14, who provided a method that can quickly determine band gaps. Using the plane-wave ex- pansion method, the propagation of elastic waves through two- dimensional 2D periodic composites, which exhibit full band gaps, was investigated 15. However, it should be mentioned that the above studies about the acoustic band gaps are of the Bragg type, which appears at about an angular frequency  of the order of v / a v is the wave velocity and a is the periodic constant. Recently, studies of the acoustic metamaterials showed that the resonance phononic crystals can be designed to display some novel sonic properties such as refractionlike beam bending or re- focusing 16–18. Different from the Bragg type, the resonance phononic crystals acoustic metamaterials show that the size of the periodic constant could be much smaller than the wavelength of the wave at the low frequency band gap 19,20. The negative refractive behavior for the acoustic wave in metamaterials can be described by introducing a negative effective mass density and/or modulus 21,22. The negative effective mass density arises from the negative momentum of the unit cell with positive velocity fields due to local resonance, which was confirmed both experi- mentally 23. It was demonstrated that the effective mass density becomes negative, owing to the local resonance of the internal masses. The physical mechanism of the negative effect mass density can be also well understood with the help of a simple mass-spring structure. Recently, a one-resonator mass-in-mass lattice model was studied by Huang et al. 24 to understand the physical mean- ing of negative effective mass density and the corresponding stop- ping bands. The model is similar to the one originally introduced by Vincent 25 and more recently by Lazarov and Janson 26, to be used as a possible mechanical filter. However, the aforemen- tioned systems, in general, have only two complete band gaps and may not be suitable for some device applications, which need multiple band gaps. In order to design multiple band gaps at the desired frequency range, a multiresonator mass-in-mass system is needed. In this study, we demonstrated that, with multiresonator mass- in-mass lattice systems, there are several band gaps. Furthermore, these band gaps may be varied by changing the resonance fre- quencies of the resonators. It was also found that if the multireso- nator system is represented by a monatomic single mass lattice system, then the effective lattice mass has to be frequency- dependent and, in the band gap, it becomes negative. The relation- ship between the negative effective mass and wave propagation in the lattice system was discussed. Finally, we employed a micro- structure continuum model 27,28 to represent the multiresonator lattice system, and found that the microstructure continuum model is suitable for describing the wave motion and band structure of multiresonator lattice systems without the need of using negative masses. 2 Band Gaps in Multiresonator Acoustic Metamaterials 2.1 Two-Resonator Mass-in-Mass System. Consider an infi- nitely long one-dimensional lattice consisting of mass-in-mass units, as shown in Fig. 1. The unit-cells are placed periodically at a spacing of L. The three rigid masses are denoted as m 1 , m 2 , and m 3 , from outside to inside of the unit cell, respectively. The three spring constants k 1 , k 2 , and k 3 are assumed to represent the respec- tive interactions among the three masses. Note that, in each unit 1 Corresponding author. Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 11, 2009; final manuscript received October 8, 2009; published online April 14, 2010. Assoc. Editor: Stephen A. Hambric. Journal of Vibration and Acoustics JUNE 2010, Vol. 132 / 031003-1 Copyright © 2010 by ASME Downloaded 23 Jun 2010 to 139.78.10.157. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm