Time-Periodic Stiffness Modulation in Elastic Metamaterials for Selective Wave Filtering: Theory and Experiment

Abstract

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

Experimental Observation of Nonreciprocal Band Gaps in a Space-Time-Modulated Beam Using a Shunted Piezoelectric Array

“…In this work we experimentally investigate nonreciprocity in a phononic beam, where spatial and temporal modulations are induced upon electric control of equivalent elastic properties. Namely, the spatial modulation is induced by bonding a pattern of piezoelectric elements on a passive substrate, which effectively alter the Young-s modulus of the waveguid through negative capacitance shunts [29], which are manipulated in time through a switching logic. This enables the formation of a traveling stiffness profile, which produces an asymmetric dispersion relation, which is a hallmark of nonreciprocity.As shown in [29], the proposed configuration an effective mean to test non-reciprocity of spatio-temporally modulated media, and may also be adopted as a flexible platform to explore other phenomena associated with temporal and spatio-temporal modulation, among which parametric amplification [29], conversion [12], and topological edge-to-edge pumping [30,31].We consider the electro-mechanical beam illustrated in Fig.…”

“…Namely, the spatial modulation is induced by bonding a pattern of piezoelectric elements on a passive substrate, which effectively alter the Young-s modulus of the waveguid through negative capacitance shunts [29], which are manipulated in time through a switching logic. This enables the formation of a traveling stiffness profile, which produces an asymmetric dispersion relation, which is a hallmark of nonreciprocity.As shown in [29], the proposed configuration an effective mean to test non-reciprocity of spatio-temporally modulated media, and may also be adopted as a flexible platform to explore other phenomena associated with temporal and spatio-temporal modulation, among which parametric amplification [29], conversion [12], and topological edge-to-edge pumping [30,31].We consider the electro-mechanical beam illustrated in Fig. 1, which is made of an aluminum substrate having cross section b × H = 20 mm × 1 mm and total length L = 2400 mm.An array of piezoelectric patches, separated by 2 mm from each other, is placed approximately at the beam's mid-span shifted by l 1 = 690 mm and l 2 = 1134 mm from the left and right boundaries in order to prevent reflections and waves interference during the transient analysis.…”

“…The piezoelectric active domain is characterized by a length l = 576 mm and it consists of 24 pairs of patches (ρ p = 7.9 kg/dm 3 , E p = 62 GPa) of size b×l p ×h p = 20×22×1 mm, bonded on opposite surfaces. Each patch is connected to a shunt circuit emulating a series negative capacitance (NC), for a total of 48 shunts, which provide an effective stiffness reduction to the beam section when the circuit is closed [29]. Finally, a pair of patches bonded to the left end close to the clamp provide excitation.…”

“…In this work we experimentally investigate nonreciprocity in a phononic beam, where spatial and temporal modulations are induced upon electric control of equivalent elastic properties. Namely, the spatial modulation is induced by bonding a pattern of piezoelectric elements on a passive substrate, which effectively alter the Young-s modulus of the waveguid through negative capacitance shunts [29], which are manipulated in time through a switching logic. This enables the formation of a traveling stiffness profile, which produces an asymmetric dispersion relation, which is a hallmark of nonreciprocity.…”

Experimental Observation of Nonreciprocal Waves in a Resonant Metamaterial Beam

“…* Corresponding author: mnouh@buffalo.edu A select number of efforts have conceived ways to realize non-reciprocal media. This includes -but is not limited to -using photo-elastic effects [6], magneto-elastic effects [20][21][22][23], piezoelectric materials [24][25][26], geometric nonlinearity [27,28], or using gyroscopic effects via angular momentum modulations [29]. The band gap tunability observed in granular materials with dielectric properties [30,31] are also promising in providing realizable avenues to break reciprocity.…”