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

In this work we experimentally achieve 1 kHz-wide directional band-gaps for elastic waves spanning a frequency range from approximately 8 to 11 kHz. One-way propagation is induced by way of a periodic waveguide consisting in an aluminum beam partially covered by a tightly packed array of piezoelectric patches. The latter are connected to shunt circuits and switches which allow for a periodic modulation in time of the cell properties. A traveling stiffness profile is obtained by opportunely phasing the temporal modulation of each active element, mimicking the propagation of a plane wave along the material, therefore establishing unidirectional wave propagation at bandgap frequencies.Nonreciprocal devices have been pursued in various research domains and physical platforms, including quantum [1], electromagnetic [2,3], acoustic [4][5][6] and elastic [7-9] media. These devices support wave propagation from a point (A) to one other (B), but not vice-versa, opening up new possibilities for the control of energy flow with unprecedented performance in communication systems [10], unidirectional insulators [11] and converters [12,13], among others. Important contributions in the context of one-way phonon transport have been formulated by Fleury et. al. [14,15], demonstrating directional wave manipulation in acoustic cyrculator devices. Also, elastic and acoustic directional waveguides have been conceived and physically realized, in analogy with the Quantum Hall effect (QHE), achieving backscattering immune and one-way topological edge states [16][17][18][19][20]. Other approaches to nonreciprocity leverage nonlinear phenomena [21,22], metastability [23], bifurcation and chaos [24] which are particularly attractive solutions due to the presence of solely passive elements. However, the exploitation of nonlinear dynamics usually requires high wave amplitudes, thus making the physical realization impractical for compact devices. An effective platform to break reciprocity is offered by space-time modulated systems [25,26]. Notable recent examples have employed programmable magnetic lattice elements [27] and magnetic springs [28]. 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], conversi...

No abstract

No abstract

We present an enhanced version of the parametric nonlinear reduced-order model for shape imperfections in structural dynamics we studied in a previous work. In this model, the total displacement is split between the one due to the presence of a shape defect and the one due to the motion of the structure. This allows to expand the two fields independently using different bases. The defected geometry is described by some user-defined displacement fields which can be embedded in the strain formulation. This way, a polynomial function of both the defect field and actual displacement field provides the nonlinear internal elastic forces. The latter can be thus expressed using tensors, and owning the reduction in size of the model given by a Galerkin projection, high simulation speedups can be achieved. We show that the adopted deformation framework, exploiting Neumann expansion in the definition of the strains, leads to better accuracy as compared to the previous work. Two numerical examples of a clamped beam and a MEMS gyroscope finally demonstrate the benefits of the method in terms of speed and increased accuracy.