Electron spin resonance and antiferromagnetic resonance of single crystals of yttrium doped with gadolinium

The nonlinear interaction of oscillation modes is investigated on the basis of Lagrangian formalism. Equations describing the changes of the bound mode amplitudes versus time, are obtained. It is shown that the energy transformation between different modes is of a periodic nature: if in the initial moment of time an appreciable part of the energy is contained, for example, in them-th mode, then after a period of timeTt (called a time of nonlinear interaction) the energy will be transformed to then-th mode. Expressions forT _t for cases with the interaction of two and three modes are obtained. As a particular case the process of nonlinear interaction of the electron “transverse” and “longitudinal” oscillations in the highfrequency hybrid resonance region of a “weakly” inhomogeneous plasma was investigated.     

Linear and nonlinear 2D finite element analysis of sloshing modes and pressures in rectangular tanks subject to horizontal harmonic motions

The influence of nonlinear wave theory on the sloshing natural periods and their modal pressure distributions are investigated for rectangular tanks under the assumption of two-dimensional behavior. Natural periods and mode shapes are computed and compared for both linear wave theory (LWT) and nonlinear wave theory (NLWT) models, using the finite element package ABAQUS. Linear wave theory is implemented in an acoustic model, whereas a plane strain problem with large displacements is used in NLWT. Pressure distributions acting on the tank walls are obtained for the first three sloshing modes using both linear and nonlinear wave theory. It is found that the nonlinearity does not have significant effects on the natural sloshing periods. For the sloshing pressures on the tank walls, different distributions were found using linear and nonlinear wave theory models. However, in all cases studied, the linear wave theory conservatively estimated the magnitude of the pressure distribution, whereas larger pressures resultant heights were obtained when using the nonlinear theory. It is concluded that the nonlinearity of the surface wave does not have major effects in the pressure distribution on the walls for rectangular tanks.     

Comments on the linear theory of the gyrotron

The authors present alternative methods for the calculation of the starting current in a gyrotron, and an analysis of the excitation of parasitic modes in gyrotrons operating in a given mode. Numerical examples are presented which check the validity of some of the assumptions made, and the extension of the formalism to the investigation of the stability of an operating mode when parasitic modes can be excited (mode competition) is also discussed. Results from the problem formulation, which is based on the cold-cathode model of Borie and Jodicke (1987), are compared with both linear and fully nonlinear models. These results illustrate the importance of using a correct model for the field profile and the influence of numerical uncertainties on the results obtained using a linear theory for mode competition. It is noted that the results obtained using the nonlinear theory of mode competition are usually numerically more stable and easier to interpret     

TE modes in a slab waveguide bounded by nonlinear media

A theory is presented of guided TE modes in a one-dimensional slabe waveguide bounded by nonlinear media with intensity dependent refractive indices. It is shown that this is a straightforward generalisation of linear slab waveguide theory with the modes parametrised by the power flow down the guide. Illustrative dispersion curves and mode shapes are discussed.     

Nonlinear gravitational waves on the surface of viscous fluid: Lagrangian approach

The problems of the asymptotic theory of weakly nonlinear surface waves in viscous fluid are discussed. For standing waves on deep water, the solutions obtained in the first- and second-order approximations in a small parameter—wave steepness—are analyzed. The evolution equation for the amplitude of wave packet envelope is obtained where the inverse Reynolds number is equal to the squared steepness. It is shown that this is a nonlinear Schrödinger equation with linear dissipation.     

Leaky mode analysis of circular optical waveguides

A leaky mode is a bound mode below its cut-off frequency. It attenuates in the direction of propagation and approximates the portion of the radiation field within the optical waveguide that is significant far from the source. The characteristics, including attenuation coefficients, of leaky modes on the circular fibre are compared with those of the more familiar slab waveguide. Except for the HE_1m leaky modes, leaky modes attenuate much faster on the slab than on the circular fibre. The greater the /(azimuthal field variation) the smaller the attenuatioN. Thus, theI?1 leaky modes are important even at great distances from the source. An effective cut-off frequency is proposed. Above this frequency the mode is either weakly leaky or trapped. An expression for the number of weakly leaky and trapped modes is given. The power of leaky modes due to illuminating one end of the fibre is determined. The effect of material absorption is considered.     

Study of the behaviour of the fibre Bragg grating fabricated with cladding mode suppression fibre

One effective method for suppression of coupling from guided optical modes into cladding modes in an optical fibre Bragg grating is to use cladding mode suppression fibre. In this specially engineered fibre, the grating is written into the core and the inner cladding, both of which are photosensitive. A theory is presented in this paper to analyse the spectral characteristics of this kind of gratings. A fibre Bragg grating was experimentally fabricated with this kind of photosensitive fibre (PS-RMS-50 from StockerYale). It is shown both theoretically and experimentally that such gratings exhibit strong suppression of the coupling between core mode and cladding mode. The experimental result is in good agreement with the theoretical analysis.     

Analytic theory of a tapered gyrotron resonator

An analytic theory has been derived for determining the eigenfrequencies, RF-field distribution and Q of the TE_mpq modes of a gyrotron resonator consisting of a circular cylinder joined to a slowly tapered section. Explicit results are obtained for a linear taper. The cavity modes are found to have an RF-field distribution which is useful for prebunching the electron beam and enhancing efficiency. For high Q cavities, the cavity Q depends on axial mode number q as q^–2, which is important for mode discrimination. Proper selection of taper length is found to reduce the Q of high q modes, also aiding in mode discrimination. The present approach may be applied to other forms of weakly irregular cavities, such as cavities with nonlinear tapers.Work supported by U.S.D.O.E. Contract DE-AC02-78ET-51013Supported by U.S. Department of Energy     

分层半空间表面非线性瑞利波的激发*

针对非线性瑞利波在均匀分层半空间结构中的激发和传播规律进行研究。根据摄动理论和模态分解将分层半空间结构中瑞利波的二次谐波声场表示为二倍频瑞利波模式的线性组合,经由互易关系得到各模式的展开系数表达式。对不同分层半空间结构中瑞利波二次谐波的激发和传播特性进行讨论,结果表明相速度匹配的瑞利波模式其二次谐波分量随传播距离线性增长,非匹配模式的二次谐波分量则沿传播方向周期震荡传播。此外,文中定义非线性参数表征瑞利波模式产生的非线性程度,这有利于选择出具有明显非线性效应的匹配点,为后续检测工作提供理论依据,具有指导意义。     

Resonant mode analysis in toroidal cavities with elliptical crosssections

Resonant cavity modes in a torus with elliptical cross section are studied by means of a direct variational method. The nonlinear effects of toroidicity and ellipticity on the frequency of the basic mode are analyzed simply and systematically without the restriction of linear theory. It is shown that the toroidicity effect on the m=0 transverse magnetic mode is ⩽11%. The frequency of the mode shifts ~11-29% when the elongation of the cross section changes from 1 to 2. The effects of toroidicity and ellipticity differ for each resonant mode     

Experimental analysis of the linear and nonlinear behaviour of composites with delaminations

An analysis of the linear and nonlinear vibration responses of composites with delaminations is presented. The effect of delamination size on the linear and nonlinear vibration response is studied. The composite material used in this paper is a glass fibre reinforced plastic (GFRP) having delaminations at the plies interfaces. The experimental procedure consists in inducing the specimen on its resonance flexural modes with different excitation levels (amplitudes) for six bending modes and for each delamination length. The presence of the nonlinearity introduced by the delamination was clearly identified by the variation of natural frequencies for increasing excitation levels. Then, nonlinear elastic parameters for progressive delamination length were determined and discussed for the first six bending modes. The linear and the nonlinear elastic parameters were compared in their sensitive modes.     

Nonlinear surface polaritons in 2D electron system in high magnetic field

The paper deals with the theoretical investigation of nonlinear surface polaritons (NSP) in isolated two-dimensional electron system (2DES) arranged at the interface between linear and nonlinear media and placed into the external quantizing magnetic field directed perpendicularly to 2DES. We consider that nonlinear medium dielectric permeability depends upon the tangential component of electric field only. It is shown that under the integer quantum Hall effect conditions all NSP characteristics are represented by the quantized values. It is found that the NSP spectrum contains two NSP modes - high-frequency and low-frequency ones. It is shown that the NSP can exist only in the case where the value of tangential component of electric field at the interface is less than a certain critical value. It is found that the resonant interaction between the NSP high-frequency mode and surface polariton mode occurs in the vicinity of the cyclotron resonance subharmonic. Received 23 September 2001 / Received in final form 31 January 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: bludov@ire.kharkov.ua     

Linear theory of an electron cyclotron maser operating at the fundamental

The linear theory of an electron cyclotron maser (ECM) operating at the fundamental is developed. A set of analytic expressions, valid for all TE cavity modes, is derived for the starting current and frequency detuning using the Vlasov-Maxwell equations in the weakly relativistic limit. These results are applicable for an arbitrary electron velocity distribution as well as any longitudinal distribution of the RF field. It is shown that the starting current can be expressed in a simple form which contains the Fourier trans-form of the longitudinal field distribution. Analytic results are presented for specific longitudinal field variations, including uniform, sinusoidal, and Gaussian. It is found that the starting characteristics of an ECM are strongly influenced by the axial dependence of the RF field, but weakly affected by the velocity spread of the electron beam. The problem of multimode oscillation is treated in the linear theory by using a Slater expansion of the cavity field. The complete formulation for mode competition based on this expansion is presented and preliminary results are derived. This comprehensive analysis of ECM linear theory should be useful as a diagnostic of ECM performance and should facilitate comparison between theory and experiment.Work supported by U.S.D.O.E. Contract DE-AC-02-80ER52059Supported by U.S. Department of EnergySupported by National Science Foundation     

Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification

We consider a linear cantilever beam attached to ground through a strongly nonlinear stiffness at its free boundary, and study its dynamics computationally by the assumed-modes method. The nonlinear stiffness of this system has no linear component, so it is essentially nonlinear and nonlinearizable. We find that the strong nonlinearity mostly affects the lower-frequency bending modes and gives rise to strongly nonlinear beat phenomena. Analysis of these beats proves that they are caused by internal resonance interactions of nonlinear normal modes (NNMs) of the system. These internal resonances are not of the classical type since they occur between bending modes whose linearized natural frequencies are not necessarily related by rational ratios; rather, they are due to the strong energy-dependence of the frequency of oscillation of the corresponding NNMs of the beam (arising from the strong local stiffness nonlinearity) and occur at energy ranges where the frequencies of these NNMs are rationally related. Nonlinear effects start at a different energy level for each mode. Lower modes are influenced at lower energies due to larger modal displacements than higher modes and thus, at certain energy levels, the NNMs become rationally related, which results in internal resonance. The internal resonances of NNMs are studied using a reduced order model of the beam system. Then, a nonlinear system identification method is developed, capable of identifying this type of strongly nonlinear modal interactions. It is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, which makes it valid for multi-frequency beating signals. Our work extends an earlier nonlinear system identification approach developed for nearly mono-frequency (monochromatic) signals. The extended system identification method is applied to the identification of the strongly nonlinear dynamics of the considered cantilever beam with the local strong nonlinear stiffness at its free end.     

Field deformation and polarization change in a step-index optical fibre due to bending

The field distribution and polarization state of waveguide modes propagating around a curved dielectric optical fibre are investigated analytically by using a first-order perturbation theory. It is shown that the polarization direction of all modes except the HE_1m modes changes due to bending. Field distributions are also deformed and the shift in the field maximum, which takes place in the plane of curvature, is clarified for the dominant HE₁₁ mode.     

双层非线性耦合反应扩散系统中复杂Turing斑图

采用双层耦合的Brusselator模型, 研究了两个子系统非线性耦合时Turing 模对斑图的影响, 发现两子系统Turing 模的波数比和耦合系数的大小对斑图的形成起着重要作用. 模拟结果表明: 斑图类型随波数比值的增加, 从简单斑图发展到复杂斑图; 非线性耦合项系数在0–0.1时, 系统1中短波模在系统2失稳模的影响下不仅可形成简单六边形、四边形和条纹斑图, 两模共振耦合还可以形成蜂窝六边形、超六边形和复杂的黑眼斑图等超点阵图形, 首次在一定范围内调整控制参量观察到由简单正四边形向超六边形斑图的转化过程; 耦合系数在0.1–1时, 系统1中短波模与系统2失稳模未发生共振耦合仅观察到与系统2相同形状的简单六边形、四边形和条纹斑图. 关键词： Brusselator模型 非线性耦合 Turing模     

Nonlinear capillary vibration of a charged bubble in an ideal dielectric liquid

The problem of nonlinear radial pulsations and surface vibrations of a charged bubble placed in an ideal incompressible dielectric liquid is asymptotically solved up to the second order of smallness by the method of many scales. It is shown that, in the case of nonlinear vibrations, resonance energy exchange may take place not only between surface modes but also between the radial mode and a surface mode. A new type of instability (other than Rayleigh instability against the self-charge), instability against the excess vapor pressure in the bubble, is discovered. The new type of instability shows up as energy transfer from the centrosymmetric pulsation mode to all initially excited surface vibration modes simultaneously.     

Bilinear modal representations for reduced-order modeling of localized piecewise-linear oscillators

A novel reduced-order modeling method for vibration problems of elastic structures with localized piecewise-linearity is proposed. The focus is placed upon solving nonlinear forced response problems of elastic media with contact nonlinearity, such as cracked structures and delaminated plates. The modeling framework is based on observations of the proper orthogonal modes computed from nonlinear forced responses and their approximation by a truncated set of linear normal modes with special boundary conditions. First, it is shown that a set of proper orthogonal modes can form a good basis for constructing a reduced-order model that can well capture the nonlinear normal modes. Next, it is shown that the subspace spanned by the set of dominant proper orthogonal modes can be well approximated by a slightly larger set of linear normal modes with special boundary conditions. These linear modes are referred to as bi-linear modes, and are selected by an elaborate methodology which utilizes certain similarities between the bi-linear modes and approximations for the dominant proper orthogonal modes. These approximations are obtained using interpolated proper orthogonal modes of smaller dimensional models. The proposed method is compared with traditional reduced-order modeling methods such as component mode synthesis, and its advantages are discussed. Forced response analyses of cracked structures and delaminated plates are provided for demonstrating the accuracy and efficiency of the proposed methodology.     

Sliding window proper orthogonal decomposition: Application to linear and nonlinear modal identification

The Proper Orthogonal Decomposition (POD) method is a tool well adapted to analyze vector signals whereas Continuous Gabor Transform (CGT) is suitable for scalar signal with multi-frequency components. In this paper, a method named Sliding Window Proper Orthogonal Decomposition (SWPOD) combining POD and CGT to analyze Multi-Degrees-Of-Freedom (MDOF) vibration system responses is presented. SWPOD gives access to the evolution of the coherent spacial structures and their frequency components versus time. The method is of principal interest in the case of swept-sine excitation of linear or nonlinear systems to access the resonance frequencies, mode shapes and modal damping ratios. A procedure is proposed to extract the linear and nonlinear normal modes of weakly damped MDOF mechanical systems and illustrated using numerical examples.