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In search of novel phononic crystals to effectively control the propagation of elastic waves, we propose a new single-material phononic crystal (PnC) with unit cells containing tapered resonators (TRs). The thickness of the circular taper radially decreases outward from the center. The device modulates dispersion of the wave by a local resonance mechanism and by slowly varying the group velocity of elastic waves. The TRs are layered on the top of a conventional PnC slab with a square arrangement of air holes. The band structure of the PnC is theoretically studied and a comparison is drawn between the avoided level crossings and the symmetry-protected ordinary degeneracies. In the absence of a bandgap, the zero group velocity at the band maximum restricts the waves from propagating. Moreover, the design shows anomalous dispersion phenomena such as self-collimation and bi-refringence, which are rare in conventional PnCs. We trace the origins of these phenomena by analyzing equifrequency contours associated with relevant frequencies. We show that the self-collimation effect persists even with a small variation in the angle of incidence and a perturbative hole at the center of each of the TRs. Within the classical limit, the scale invariance of the elastic wave equation makes the device useful in both the low frequency ultrasonic and the high frequency phononic regime.
In search of novel phononic crystals to effectively control the propagation of elastic waves, we propose a new single-material phononic crystal (PnC) with unit cells containing tapered resonators (TRs). The thickness of the circular taper radially decreases outward from the center. The device modulates dispersion of the wave by a local resonance mechanism and by slowly varying the group velocity of elastic waves. The TRs are layered on the top of a conventional PnC slab with a square arrangement of air holes. The band structure of the PnC is theoretically studied and a comparison is drawn between the avoided level crossings and the symmetry-protected ordinary degeneracies. In the absence of a bandgap, the zero group velocity at the band maximum restricts the waves from propagating. Moreover, the design shows anomalous dispersion phenomena such as self-collimation and bi-refringence, which are rare in conventional PnCs. We trace the origins of these phenomena by analyzing equifrequency contours associated with relevant frequencies. We show that the self-collimation effect persists even with a small variation in the angle of incidence and a perturbative hole at the center of each of the TRs. Within the classical limit, the scale invariance of the elastic wave equation makes the device useful in both the low frequency ultrasonic and the high frequency phononic regime.
The Berry-Tabor (BT) conjecture is a famous statistical inference in quantum chaos, which not only establishes the spectral fluctuations of quantum systems whose classical counterparts are integrable but can also be used to describe other wave phenomena. In this paper, the BT conjecture has been extended to Lévy plates. As predicted by BT conjecture, level clustering is present in the spectra of Lévy plates. The consequence of level clustering is studied by introducing the distribution of nearest neighbor frequency level spacing ratios $P\left( {\widetilde r} \right)$, which is calculated through the analytical solution obtained by the Hamiltonian approach. Our work investigates the impact of varying foundation parameters, rotary inertia, and boundary conditions on the frequency spectra, and we find that $P\left( {\widetilde r} \right)$ conforms to a Poisson distribution in all cases. The reason for the occurrence of the Poisson distribution in the Lévy plates is the independence between modal frequencies, which can be understood through mode functions.
In this work, the Poincaré map numerical method was successfully developed to solve the fourth-order differential equation that describes the flexural vibrations of a beam, within the Timoshenko beam theory. The Euler-Bernoulli continuity conditions were considered, which are valid for frequencies smaller than the critical frequency. As an example, this method was used to design a complex elastic structure, characterized by a flexural frequency spectrum with a broad band gap. Such structure consists of two coupled phononic crystals, which were designed with filling factor values in such a way that in their bending frequency spectra, an allowed band of the first part, overlaps with a band gap of the second one and vice versa. The resulting composed system has a much wider effective gap than its original components, between 4 and 10.5 kHz. This system works as an elastic bending wave filter. Finally, these three structured elastic systems were constructed, and characterized by the acoustic resonance spectroscopy technique. The natural flexural frequencies as well as the corresponding wave amplitudes of each structured beam were measured. The experimental measurements show excellent agreement with the numerical simulation.
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