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 (potentially nonuniform, 1D or 2D) locally resonant metastructures 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 the target frequency range, i.e. respective modal neighborhood. Furthermore, it is observed that there is an optimal, finite number of resonators which gives a bandgap that is wider than the infinite-absorber 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-absorber bandgap. Additionally, 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.

We report on broadband-to-narrowband elastic wave filtering resulting from time-periodic modulation of the stiffness of a one-dimensional elastic waveguide. Time modulation produces flat dispersion bands at frequencies that are multiple integers of half the modulation frequency. These flat bands lead to the selective reflection of a broadband incident wave at the interface between a non-modulated medium, and one with time-modulated stiffness properties. This results from the vanishing group velocity at the flat band frequencies, which prevents their propagation into the modulated domain. Thus, the considered modulated waveguide is understood as a single port system, in which a broadband incident wave (input) results in a narrowband reflected wave (output) at a frequency defined by modulation. The appearance of the flat bands for a time-modulated waveguide is here illustrated analytically and through numerical simulations. The filtering characteristics of a non-modulate/modulated interface are observed experimentally by implementing a square-wave modulation scheme that employs an array of piezoelectric patches bonded to an elastic waveguide subject to transverse motion. The patches are shunted through a negative electrical capaticance that, when connected, implements a stiffness reduction for the resulting electromechanical waveguide. Switching the capacitance on and off effectively modulates the stiffness of the waveguide, and illustrates the filtering characteristics associated with time-modulation of the equivalent elastic properties. We envision that a similar approach could be extended to investigate other properties of time-modulated elastic metamaterials, such as non-reciprocity and one-way filtering of elastic waves.Time-dependent material properties have been the object of considerable attention over the years. Parametric effects in time-modulated media have long been used for amplification of electromagnetic waves 1,2 and surface acoustic waves 3,4 . Non-reciprocal elements based on both up and down converter amplifiers have been introduced in the early 1960s 5 . The interest in time-modulated media, motivated by their application for parametric amplification in electromagnetic waveguides and for signal processing applications, has led to numerous studies of both periodic 6-11 and non-periodic modulation schemes 12 . Recently, time-modulation of relevant physical properties, imposed in a traveling-wave form, has been explored to achieve non-reciprocal behavior not only in optics, but also in acoustics and mechanics [13][14][15] . For example, magnetless, efficient and compact radiofrequency communication systems are designed with spatiotemporally modulated gratings to be shielded from

We investigate the dynamic behavior and topology of quasiperiodic resonant metastructures. We show that the quasiperiodic arrangement of resonators introduces frequency bandgaps in addition to the locally resonant bandgap defined by the natural frequency of the resonators. The concept is illustrated on a beam with an array of mechanical resonators. Numerical evaluation of the spectrum as a function of the quasiperiodic arrangement of resonators reveals a structure reminiscent of a Hofstadter butterfly and allows the study of key topological properties. Results illustrate the occurrence of additional bandgaps that are topologically non-trivial and that host edge localized modes in finite structures. The occurrence of these gaps and of the associated edge states is demonstrated experimentally by measuring the frequency response of the beam and by evaluating the spatial distribution of selected operational deflection shapes. The results unveil the potential of deterministic quasiperiodic structural designs to induce wave localization and attenuation over multiple frequency bands, which may find applications in vibration isolation and energy harvesting, among others.

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