Formulation and validation of a Ritz-based analytical model of high-frequency periodically layered isolators in compression

Periodically layered isolators exhibit transmissibility “stop bands” or frequency ranges in which there is very low transmissibility. A two-dimensional axisymmetric model was developed to accurately predict the location of these stop bands for isolators in compression. A Ritz approximation method was used to model the axisymmetric elastic behavior of layered cylindrical isolators. A modal analysis was performed for a single elastomer and metal layer combination or cell. A modal synthesis approach was then used to obtain a model of an n-celled isolator, from which overall isolator modal properties are determined. This model of the dynamic behavior of layered isolators was validated with experiments. Analytical and experimental transmissibilities are compared for test specimens having identical elastomer components, but different geometries and different numbers of cells. In all cases, experimental and analytical transmissibilities are in close agreement at frequencies ranging from zero to those associated with the initial roll-off of the stop bands. For three and four cell cases, minimum stop band analytical transmissibilities lie below the minimum experimental measurements, although an experimental noise floor imposed a minimum transmissibility measurement of approximately 1.4×10⁻⁴. Experiment suggests a practical isolator design could limit the minimum number of cells to three or four to ensure a pronounced stop band attenuation effect. In addition, analytical and experimental transmissibilities are compared for geometrically similar test specimens with differing elastomeric damping properties. The analytical and experimental results show that stop band effectiveness is not appreciably affected by the addition of modest damping.     

Flapwise dynamic response of a wind turbine blade in super-harmonic resonance

The flapwise dynamic response of a rotating wind turbine blade in super-harmonic resonance is studied in this paper, while the blade is subjected to unsteady aerodynamic loads. Coupled extensional–bending vibrations of the blade are considered; the governing equations which are coupled through linear and quadratic terms arising from rotating and geometric effects respectively are obtained by applying the Hamiltonian principle. The lth flapwise linear frequency and the rotational frequency are assumed to be in an almost 3:1 ratio, so super-harmonic resonance occurs when this linear frequency is close to the associated nonlinear frequency. By using the first n, no less than l, linear undamped modal functions as a functional basis and applying the Galerkin procedure, a 2n-degree-of-freedom discrete model with quadratic and cubic terms owing to geometric effect is derived. The generalized displacements corresponding to the discrete system are disintegrated into static and dynamic displacements. Perturbation method is adopted to get analytical solutions of the discrete dynamic system for positive aerodynamic dampings. The coning angle and the inflow ratio are chosen as two control parameters to analyze aeroelastic behaviors of the blade. By assuming that the static and dynamic displacements are of the same order in resonance region, and there is no other resonance except the super-harmonic resonance, the multiple-scales method is employed to obtain a set of amplitude modulation equations whose coefficients depend on two control parameters. The frequency-response equation is derived from the amplitude modulation equations. A method to estimate the functional dependence of the detuning parameter on two control parameters is introduced. The amplitude of the harmonic response is derived from the frequency-response equation after knowing the detuning parameter. Then the stability of the steady motion with respect to control parameters can be determined. The evolution of the dynamic response of the resonance mode with decreasing aerodynamic damping is discussed by means of both perturbation and numerical methods.     

Maximum likelihood estimator of operational modal analysis for linear time-varying structures in time–frequency domain

This paper investigates the problem of modal parameter estimation of time-varying structures under unknown excitation. A time–frequency-domain maximum likelihood estimator of modal parameters for linear time-varying structures is presented by adapting the frequency-domain maximum likelihood estimator to the time–frequency domain. The proposed estimator is parametric, that is, the linear time-varying structures are represented by a time-dependent common-denominator model. To adapt the existing frequency-domain estimator for time-invariant structures to the time–frequency methods for time-varying cases, an orthogonal polynomial and z-domain mapping hybrid basis function is presented, which has the advantageous numerical condition and with which it is convenient to calculate the modal parameters. A series of numerical examples have evaluated and illustrated the performance of the proposed maximum likelihood estimator, and a group of laboratory experiments has further validated the proposed estimator.     

非线性薄膜波导TE模色散特性的多层分割法计算

对于芯区为非线性介质、衬底及包层为线性介质的平板波导、提出用多层分割法分析芯区的模场,采用递推公式（等效于推广的传递矩阵法）求解传播常数与光功率间的依赖关系,该法适用地克尔型或非克尔型介质及芯区折射率非均匀的一般情形,实例计算结果与已有的精确数值计算结果十分符合。     

Two-dimensional and layered structures in the discrete φ4 model

The properties of phase transitions in two-dimensional and layered systems are investigated on the basis of a discrete φ⁴ model by numerical and analytical methods. The only parameter a of the discrete φ⁴ model determines the behavior of the system and makes it possible to investigate phase transitions ranging from transitions of the displacement type (a → +0) to order-disorder type (a → +∞). The behavior of a two-dimensional system is investigated in a wide range of values of the parameter a. The temperature dependences of the squared order parameter η²(T) and the phase transition temperature T _c as a function of the thickness N of the system are obtained for three characteristic values of the parameter a using the Monte Carlo method. The properties of phase transitions in the discrete φ⁴ model are investigated on the basis of the mean-field approximation and the independent-mode approximation. The results obtained in the numerical experiments are compared with the analytical approximations. It is shown that the mean-field approximation qualitatively describes the behavior of the phase-transition temperature T _c as a function of the thickness N of the system for a wide range of values of the parameter a, and the independent-mode approximation describes quantitatively, to within 5%, the results of the numerical simulation for small values of a.     

Excitation of nuclear collective states by heavy ions within the model of semimicroscopic optical potential

The (semi)microscopic double-folding nucleus-nucleus optical potentials are suggested for consideration of inelastic scattering with excitation of collective nuclear states by using the adiabatic approach and the elastic scattering amplitude in the high-energy approximation. The analytical expression for inelastic scattering amplitude is obtained keeping the first-order terms in the deformation parameter of a potential. Calculations of inelastic cross sections for the ¹⁷O heavy ions scattered on different nuclei at about hundred MeV/nucleon are made, and the acceptable qualitative agreement with the experimental data is obtained without introducing free parameters. The prospect of the method for further applications is discussed. The text was submitted by the authors in English.     

Resonant modal conversion in a two-mode waveguide

We characterize a system consisting of a two-mode waveguide coupled to a single-mode microring resonator possibly presenting a nonlinear response of Kerr type. By using the scattering parameter formalism extended to the multimode domain, we show that in the linear regime and for an ideally transparent medium, each resonance of the system can be exploited to perform complete even-to-odd (respectively, odd-to-even) modal conversion. Moreover, when the Kerr nonlinearity is effective, the microring enables a power-dependent modal switching mediated by phase bistability. Thanks to its mode-processing capabilities, this configuration is suitable to find application as a functional building-block in mode-division multiplexing (MDM) photonic integrated circuits.     

New nonlinear intensity Kerr effect in the polar geometry

The problem of the electromagnetic wave reflection on the second optical harmonic from a semi-infinite optically isotropic magnetic medium is considered for the uniform magnetization direction corresponding to the polar Kerr effect. In the first approximation in magnetization, the method of Green’s tensor functions is used to derive expressions for the complex amplitudes of the wave fields for the incidence of s-and p-polarized waves (as well as of their superposition) on the medium. It is shown that, in the latter case, the nonlinear polar Kerr effect is of the intensity type. The dependences of the intensity effect on the angle of incidence of the inducing wave and on the angle of its polarization, which were obtained in numerical experiments, are presented. A comparative analysis of the linear and nonlinear intensity Kerr effects is carried out.     

Cutoff frequencies in optical fibres with central refractive-index depression

Exact analytical solutions are given for modal cutoff frequencies in round optical fibres with a central refractive-index depression represented by a radial index distribution Rp in the core. The cutoff frequencies Vc for modes with azimuthal order m are the solutions of Jq−1(2Vc/{2+p})=0, where q = 2m/(2 + p). An accurate analytical approximation formula for the cutoff of the LP11 mode is developed, which even circumvents the calculation of real-order Bessel function zeros. For the LP02 mode, as well as for all other circularly symmetric modes, the exact cutoff frequency is merely a linear function of the profile parameter p. This revised version was published online in November 2006 with corrections to the Cover Date.     

Experimental identification of viscous damping in linear vibration

This paper is concerned with the experimental evaluation of the performance of viscous damping identification methods in linear vibration theory. Both existing and some new methods proposed by the present authors [A.S. Phani, J. Woodhouse, Viscous damping identification in linear vibration, Journal of Sound and Vibration 303 (3–5) (2007) 475–500] are applied to experimental data measured on two test structures: a coupled three cantilever beam with moderate modal overlap and a free–free beam with low modal overlap. The performance of each method is quantified and compared based on three norms and the best methods are identified. The role of complex modes in damping identification from vibration measurements is critically assessed.     

A new semi-analytical method for the analysis of tapered optical waveguides

Tapered waveguide are used in number of integrated optic devices such as directional couplers, modulators, switches mode converters, etc. Most of the methods analyzing tapered waveguide are numerical in nature. In this paper we present a simple, fast and accurate semi analytical method for z-varying waveguide. However, very few idealized structures can be analyzed directly. The present method consists of separating transverse and longitudinal variation in the wave equation, leading to a differential equation with z-varying coefficients for the field variation along z-axis. For the transverse variation local normal theory is applied. Now this equation is applied to specific taper geometries like linear down taper. Computational are done assuming typical values. We observe variation of power mode profile and mode width. Waveguide loss is also including in the analysis. Finally the analytical simulation results have been verified by the commercial Opti-BPM software.     

Nonlinear intensity-related magneto-optical Kerr effects in the planar geometry

The reflection of an electromagnetic wave at the second-harmonic frequency from a semi-infinite optically isotropic magnetic medium for the directions of uniform magnetization corresponding to the meridional and equatorial Kerr effects is considered. Using the Green’s tensor function technique, in the first approximation in magnetization, the expressions for complex amplitudes of the field are obtained for the s and p polarizations of the incident beam and their superpositions. It is shown that in the latter case, the meridional effect becomes intensity-related. Dependences of the intensity-related meridional and equatorial Kerr effects on the angle characterizing polarization of the pump wave are obtained by numerical calculations. A comparative analysis of the linear and nonlinear Kerr effects is made.     

Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space–time

Gravitational radiation of binary systems can be studied by using the adiabatic approximation in General Relativity. In this approach a small astrophysical object follows a trajectory consisting of a chained series of bounded geodesics (orbits) in the outer region of a Kerr Black Hole, representing the space time created by a bigger object. In our paper, we study the entire class of orbits, both of constant radius (spherical orbits), as well as non-null eccentricity orbits, showing a number of properties on the physical parameters and trajectories. The main result is the determination of the geometrical locus of all the orbits in the space of physical parameters in Kerr space–time. This becomes a powerful tool to know if different orbits can be connected by a continuous change of their physical parameters. A discussion on the influence of different values of the angular momentum of the hole is given. Main results have been obtained by analytical methods.     

Study of the P1 approximation in an inverse scattering problem

We examine the validity of the P₁ approximation in determining the albedo and asymmetry factor of a scattering medium from the hemispherical transmittance data. For this purpose, an analytical expression of the hemispherical transmittance is derived with the P₁ approximation and then fitted to hemispherical transmittance data of a scattering medium using three methods. These include the asymptotic expansion method, the unconstrained least mean-squares method and the constrained least mean-squares method. We find that estimation of radiative properties with the P₁ approximation is effective only for high albedos. Only the constrained least mean-squares method yields physically resonable values for the radiative properties in all cases.     

Study on analytical solutions and their simplification for one-component multi-Yukawa fluids and test by Monte-Carlo simulation

Using the mean spherical approximation, an explicit analytical equation of state with non-dimensional variables for one-component multi-Yukawa fluids is established based on the work of Blum and Ubriaco. Two simplified calculation methods (one is the linear summation method and the other is the negligible entropy method) are proposed. The compressibility factors for one-component 1–4 Yukawa fluids are simulated with the Monte-Carlo method. Comparisons between various methods under different range parameters are listed and discussed for 1–4 Yukawa fluids. These indicate that our methods are reliable for a wide range of parameters and can be used conveniently.     

Demkov-kunike model for cold atom association: Weak interaction regime

We study the nonlinear mean-field dynamics of molecule formation at coherent photo- and magneto-association of an atomic Bose-Einstein condensate for the case when the external field configuration is defined by the quasi-linear level crossing Demkov-Kunike model, characterized by a bell-shaped pulse and finite variation of the detuning. We present a general approach to construct an approximation describing the temporal dynamics of the molecule formation in the weak interaction regime and apply the developed method to the nonlinear Demkov-Kunike problem. The presented approximation, written as a scaled solution to the linear problem associated to the nonlinear one we treat, contains fitting parameters which are determined through a variational procedure. Assuming that the parameters involved in the solution of the linear problem are not modified, we suggest an analytical expression for the scaling parameter.     

A new method for calculation of the magnetic properties in one- and two-dimensional spin systems with anisotropic Heisenberg exchange

A new approximation method is proposed for the calculation of the magnetic susceptibility of one-dimensional assembly of spins and the critical temperature of two-dimensional one both with the anisotropic Heisenberg exchange. In a linear chain system, every spins are grouped into pairs of adjacent spins (pair-approximation) or clusters of adjacent three spins ((q+1)-approximation), and the partition function of the total spin system is approximated as a sum of products of the partitions functions for the pairs or the clusters. Then the partition function of the anisotropic Heisenberg spin system is shown to reduce into a form of the Ising spin system with modified coupling constants. The exact result for the Ising chain system enables us to obtain an analytical expression for the magnetic susceptibility of anisotropic Heisenberg chain system. The same approximations are also applied to two-dimensional lattices, and the critical temperatures of the square, triangular, and honeycomb lattices with anisotropic Heisenberg exchange are calculated as a function of anisotropy parameter. The results are compared with those of the existing theories and shown to be quite excellent.