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In this work, we obtain closed-form expressions for the transfer matrix of free oscillations in finite periodic Timoshenko–Ehrenfest beams with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for free oscillations of a beam composed of N cells, we obtain a fourth-order recursive relation for the matrix coefficients, which defines the so-called Tetranacci Polynomials. Such recursive relation provides an algorithm to compute the Nth power of the transfer matrix, avoiding the matrix product of the N matrices. Furthermore, in the symmetric case of free oscillations in finite periodic Timoshenko-Ehrenfest beams, closed-form expressions for the solutions to the recursive relation have recently been derived, which we use to write the transfer matrix of a finite beam composed of N cells in a closed form. We find a good agreement between the natural frequencies calculated with the obtained expressions and finite element simulations. These expressions are very useful for studying the interaction of evanescent oscillations and find application, for instance, in the development of phononic topological insulators. Our formalism can be applied to waves propagating in finite periodic layers described by a 4 × 4-transfer matrix, such as electromagnetic waves in anisotropic optical media.

Most wastewater treatment plants use activated sludge-based biological systems for this purpose. The latter must effectively remove organic matter and, at the same time, show good sedimentability. However, sometimes there is an excessive proliferation of certain bacteria, giving rise to filamentary swelling, compromising the excellent sedimentability of the sludge. In this sense, the study’s objective was to evaluate the effect of applying two different technologies, the application of low-frequency ultrasound and UV radiation. Some bench-scale experiments were performed using the bulked sludge from the secondary clarifier of a wastewater treatment facility in an industrial park (CIVAC) in Morelos, Mexico, affected by filamentous organisms. Results showed that for the UV application for two, four, and 6 min, the settleability of the mixed liquor suspended solids was not improved; on the other hand, the cavitation effect caused by the ultrasound application demonstrated effective action against the destruction of filamentous organisms. The 10 min condition showed a significant decrease in the filament integrity of the microorganisms and a significant improvement of sedimentation velocity and sludge volume index (SVI) values and settleability of the sludge, but not enough to satisfy national discharge regulations related to total suspended solids in the treated effluent. Molecular identification indicates the presence of the genera Thauera and Brevundimonas in the sludge.

In this work, we obtain closed expressions for the transfer matrix and the transmittance of electromagnetic waves propagating in finite one-dimensional anisotropic periodic stratified media with an arbitrary number of cells. By invoking the Cayley–Hamilton theorem on the transfer matrix for the electromagnetic field in a periodic stratified media formed by N cells, we obtain a fourth-order recursive relation for the matrix coefficients that defines the so-called Tetranacci polynomials (TPs). In the symmetric case, corresponding to a unit-cell transfer matrix with a characteristic polynomial where the coefficients of the linear and cubic terms are equal, closed expressions for the solutions to the recursive relation, known as symmetric TPs, have recently been derived, allowing us to write the transfer matrix and transmittance in a closed form. We show as sufficient conditions that the 4 × 4 differential propagation matrix of each layer in the binary unit cell, Δ , a) has eigenvalues of the form ± p 1 , ± p 2 , with p 1 ≠ p 2 , and b) its off-diagonal 2 × 2 block matrices possess the same symmetric structure in both layers. Otherwise, the recursive relations are still solvable for any 4 × 4 matrix and provide an algorithm to compute the N th power of the transfer matrix without carrying out explicitly the matrix multiplication of N matrices. We obtain analytical expressions for the dispersion relation and transmittance, in closed form, for two finite periodic systems: the first one consists of two birefringent uniaxial media with their optical axis perpendicular to the z -axis, and the second consists of two isotropic media subject to an external magnetic field oriented along the z -axis and exhibiting the Faraday effect. Our formalism applies also to lossy media, magnetic anisotropy or optical activity. [ABSTRACT FROM AUTHOR]

Se estudió numéricamente la radiación electromagnética de nanoemisores colocados en una microesfera multicapas con metamateriales con dispersión (materiales izquierdos LH). Dicha investigación no puede hacerse analíticamente, sino que requiere intensivas simulaciones numéricas con el uso de la técnica tensor de Green y algoritmos específicos para las funciones esféricas. Todo esto lleva a un intenso consumo de recursos de computo. Como resultado de los experimentos numericos se encuentra que en el rango de frecuencia donde las capas de LH tienen un índice de refracción negativo el espectro de frecuencia del campo consiste en una serie de resonancias estrechas y bien separadas. En la banda de estos picos, gran parte de la energía del campo se encuentra en una capa de LH y prácticamente no sale de la microesfera. Estas conclusiones se obtuvieron como resultado no sólo de simulaciones, sino también de una interacción persona-ordenador. Se discuten los aspectos avanzados de la Interacción Persona-Ordenador con la interfaz gráfica y usabilidad que se han desarrollado en nuestro estudio y que nos ha permitido explorar a fondo un nanosistemas complejos.

Estudiamos numéricamente el control del espectro de frecuencias de nano-emisores puestos en una micro-esfera recubiertas con una pila cuasi periódica. Se investiga la evolución espectral de transmisión con el cambio de espesor de bloques de dos capas, construidos siguiendo la secuencia de Fibonacci. Encontrado básicamente un comportamiento no lineal en dicho sistema: cuando el número de capas (el orden de Fibonacci) aumenta, la estructura del espectro de transmisión adquiere una forma fractal. Nuestros cálculos muestran el confinamiento y una mejora gigantesca del campo, cuando la relación entre el ancho de las capas en la pila (parámetro de control) está cerca del valor de la regla de oro. Se discute también la necesidad de un código de programación estructura orientada a objetos, para el estudio numérico de tal sistema no lineal.

In a previous work an elastic bar with a groove or notch that presents a doorway state was studied when the system was excited with 20 cycles of harmonic signals. The strength function had a Lorentzian width Γd = 1/πτd, where τd is the decay time of the prompt response. In the present paper, the doorway-state phenomenon is analyzed again for the same harmonic signals but for a very large number of cycles. The strength-function phenomenon is once more obtained, but now with a Lorentzian width Γ' which is larger than Γd. A qualitative and numerical explanation of this fact is given, leading therefore to further understanding of doorway states in elastic systems. The numerical results show a very good agreement with the values measured in the laboratory. [ABSTRACT FROM AUTHOR]

Three classical systems formed by the pairs of coupled resonators are analyzed, including a system of elastically coupled masses, a system of rigid rods separated by a notch, and an optical system made by a pair of dielectric films separated by a thin metallic layer. We show numerically that these systems are analogous to each other. For the latter two systems, the theoretical results are confirmed experimentally. This study allows us to discuss in classical systems properties that typically are only described quantum mechanically, such as the avoided crossing theorem, the strength function phenomenon, and the effect of the coupling element. [ABSTRACT FROM AUTHOR]

We systematically studied the optical properties (narrows peaks position) of the transmission spectra for microspheres coated by a multilayered stack. Three different sequences of spherical stack-periodic, quasiperiodic, and disordered-are studied by the transfer matrix approach. Dependence of the number of resonances in the transmission spectrum as function of number of layers in the stack is numerically investigated with details. It is shown that characteristic shape of the recursive return map forms well-defined ordered spectrum in the state space. The shape of such structures is different for different frequency ranges and various spherical quantum numbers. For quasiperiodic case, the latter leads to a specific signature of studied sequences and generates the self-similarity in the transmittance spectra. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Two elastic systems are considered in this work: A special linear chain of harmonic oscillators and a quasi one-dimensional vibrating rod. Starting in both cases with a locally periodic system formed by unit cells with a single element, these cells are converted into binary cells. The acoustic and optical bands then appear. For the vibrating rod experimental values are compared with theoretical results; in particular, the normal-mode amplitudes are obtained and the agreement is excellent. [ABSTRACT FROM AUTHOR]

We studied numerically the frequency spectrum of photons in a multilayered microsphere coated by a quasiperiodic (Pascal) dielectric stack. It is found that the transmittancy spectrum of such a stack consists of quasiband gaps and narrow resonances caused by re-relection of optical waves. When the number (Pascal order) of layers increases, the band gaps and resonances split, and the structure of the frequency spectrum acquires a fractal form. Some parts of the spectrum show the self-similarity in the diferent frequency scales. [ABSTRACT FROM AUTHOR]

We numerically investigate the optical transmission through a compound spherical stack with conventional and metamaterial (MM) layers and also embedded MM defect. A formation of extremely narrow resonant peak with nearly complete transmittance in area of a band gap is found. We demonstrate that photon fields of certain frequencies can be strongly confined by a left-handed (LH) defect. The influence of a random deviation in the width of compound spherical layers as well the transit to the whispering gallery mode (WGM) is also discussed. [ABSTRACT FROM AUTHOR]

Abstract: We studied the frequency spectrum of photons in a multilayered microsphere coated by a quasiperiodic (Fibonacci) dielectric stack numerically. We found that the transmittancy spectrum of such a stack consists of quasiband gaps and narrow resonances caused by re-reflection of optical waves. When the number (Fibonacci order) of layers increases, the band gaps and resonances split, and the structure of the frequency spectrum acquires a fractal form. We found the self-similarity of the frequency spectrum and evaluated the fractal dimension both for lossless and dissipative cases. The influence of a weak random deviation of spherical layers width is also discussed. [Copyright &y& Elsevier]

The doorway-state phenomenon has been observed in many quantum and classical undulatory systems when two oscillating systems are coupled, one that has a high level density and the other one a very low density. Up to now the systems analysed have in common that they are governed by second-order differential equations. In the present work it is shown that the doorway state mechanism also appears when dealing with flexural vibrations of elastic systems, which are governed by fourth-order differential equations. It should be mentioned that this phenomenon also emerges from the two coupled second-order differential Navier equations. [ABSTRACT FROM AUTHOR]

In this work, we demonstrate, analytically and numerically, that degenerate states of a beam with free ends, with anti-symmetrical transverse mode shape and occurring in the first-degeneracy points above the cutoff frequency, tend asymptotically to the thickness-shear mode in the infinitely long beam limit, which has not been previously reported. We compare our numerical results, obtained using a Timoshenko–Ehrenfest beam theory of second order, with those obtained with the Finite Element method and plane stress elastodynamics theory, obtaining an excellent agreement. We show that anti-symmetric states within the vicinity of these first-degeneracy states are very similar and closely resemble the so-called thickness-shear mode but for a finite beam with free ends under flexural vibrations. • Mode shapes of degenerate states are studied in Timoshenko–Ehrenfest beams. • First-degeneracy asymmetric states turn out very similar to the thickness-shear mode. • Instead, it is shown that transverse oscillations of the cutoff states are non-negligible. • Results for a beam with free ends are corroborated with the Finite Element method. • It is demonstrated that analogs to the thickness-shear mode exist for finite beams. [ABSTRACT FROM AUTHOR]

The doorway state phenomenon has been recently analysed in many different systems, both quantum and classical. The systems range from nuclei to sedimentary valleys, therefore covering a range in size of 19 orders of magnitude. It also applies to systems with chaotic spectra as well as to integrable systems. In all these works, the doorway state has been discussed only in the energy or frequency domains. In this letter we present numerical and experimental results for a quasi–one-dimensional elastic system which presents a doorway state and, for the first time, the temporal evolution of the phenomenon is measured directly. [ABSTRACT FROM AUTHOR]

The optical analogues of Bloch oscillations and their associated Wannier-Stark ladders have been recently analyzed. In this Letter we propose an elastic realization of these ladders, employing for this purpose the torsional vibrations of specially designed one-dimensional elastic systems. We have measured, for the first time, the ladder wave amplitudes, which are not directly accessible either in the quantum-mechanical or optical cases. The wave amplitudes are spatially localized and coincide rather well with theoretically predicted amplitudes. The rods we analyze can be used to localize different frequencies in different parts of the elastic systems and vice versa.

The flexural vibrations of a locally periodic rod, which consists of N unit cells, are discussed both from the experimental and theoretical points of view. Timoshenko's beam theory and the transfer matrix method are used to calculate the normal-mode frequencies and amplitudes. The theoretical values are then compared with the experimental ones, which are obtained using an electromagnetic acoustic transducer (EMAT). Good agreement between the numerical results and the experimental measurements is obtained. It is shown that as N grows, a band spectrum emerges.