Exact Eigenvalue Equation Research Articles

Exact closed-form solution for free vibration of Euler-Bernoulli and Timoshenko beams with intermediate elastic supports

This paper presents exact closed-form solutions for free vibration of discretely supported Euler-Bernoulli (DEB) and Timoshenko beams (DTB) in the presence of an arbitrary number of intermediate elastic constraints. The exact eigenvalue equations and mode shapes of the DEB and DTB are derived using the generalized function method for the first time, which expands the general solution of mode shapes as a combination of the standard trigonometric/hyperbolic functions with integration constants extended to generalized functions. The second-order eigenvalue equation is formulated from the perspective of the entire domain of the beam without enforcement of any continuity conditions. Several typical constraints of interest in practical engineering are investigated and other cases can be achieved by following similar methodologies elucidated in this paper. The accuracy of present solutions is validated by numerical results obtained from the finite element method (FEM) and previous literature adopting the transfer matrix method (TMM) and Green's function method (GFM). The exiting frequency equation and mode shape of a beam with elastic end constraints are proved to be the degenerate solutions of current study. Courant's maximum-minimum principle is reproduced with detailed discussions on the support stiffnesses and positions, bimodal behavior, and sensitivity to geometrical variations. As an extended application, a new method based on rigorous mathematical deduction is proposed to efficiently determine the critical stiffness leading to the maximum fundamental frequency of a beam with multiple intermediate supports. This study is of significance to the support optimization and dynamic analysis of discretely supported beam-like structures.

Weak Guidance, Multi-index and Exact Mode Eigenvalue Calculation for Mode Intensity Profiles in Tapers

The fundamental mode intensity profile is calculated for different points along the transition of an optical fibre taper from standard size (125 μm external cladding diameter and 8 μm core diameter) down to micro (1 μm external fibre diameter) and submicro size (down to 440 nm fibre diameter) at 1550 nm operational wavelength. The first section of the taper was evaluated using weak guidance approximation, the second section was treated as a three-index layer structure (double-clad) and evaluated with eigenvalue equations for three refractive indices. An external medium being a liquid with a refractive index close to silica is being considered. The third and thinnest section of the taper was evaluated with the exact mode eigenvalue equation. The results show that the fundamental mode intensity for the third section show a discontinuity just at the fibre edge with a peak amplitude larger than the optical field at the centre of the fibre. The taper shape that complies with the adiabaticity criterion and how the mode intensity profile and the spot size (Petermann 1 definition) of the fundamental mode change along their position on the taper were determined.

Fundamental mode intensity evolution in tapered optical fibres

The mode field intensity, spot size, central peak intensity evolution and adiabaticity are calculated for different points along the transition of an optical fibre taper that adiabatically tapers from the standard 125 nm down to 1 µm and then to 440 nm diameter for low loss operation at 1550 nm wavelength. The first section of the taper is evaluated using a weak guidance approximation. The second section is treated as a three-index layer structure (double-clad) and evaluated with eigenvalue equations for three refractive indices. The third and thinnest section of the taper is studied using an exact mode eigenvalue equation. The results show that the fundamental mode for the third section has a discontinuity at the fibre edge with a peak intensity larger than the intensity at the centre of the fibre. Since the guiding by the core disappears in the first section of the taper, the mode field does not simply reduce monotonously along the taper with the outer diameter of the fibre. By this novel approach, and for the first time, to the best of our knowledge, the taper shape that complies with the adiabaticity criterion, the mode intensity profile and the spot size (first Petermann definition) of the fundamental mode evolution, along their position on the taper are determined.

Calculation of Magnetization and Permeability Tensor of a Partially Magnetized Cylindrical Ferrite Resonator

A accurate method of calculating a magnetization and permeability tensor of ferrites is proposed in this paper. We introduce a cylindrical ferrite resonator that is located between parallel metal plates. We theoretically derive exact eigenvalue equations for the mode-splitting phenomenon of the HE 111 mode. We also show that the mode-splitting behavior depends on a single magnetization value of the ferrite. The magnetization and permeability tensor components are computed from the measured resonant frequencies of the resonator. We present an analysis of cylindrical ferrite resonator and its experimental results.

Feedback control of flow alignment in sheared liquid crystals

Based on a continuum theory, we investigate the manipulation of the nonequilibrium behavior of a sheared liquid crystal via closed-loop feedback control. Our goal is to stabilize a specific dynamical state, that is, the stationary "flow alignment," under conditions where the uncontrolled system displays oscillatory director dynamics with in-plane symmetry. To this end we employ time-delayed feedback control (TDFC), where the equation of motion for the ith component q(i)(t) of the order parameter tensor is supplemented by a control term involving the difference q(i)(t)-q(i)(t-τ). In this diagonal scheme, τ is the delay time. We demonstrate that the TDFC method successfully stabilizes flow alignment for suitable values of the control strength K and τ; these values are determined by solving an exact eigenvalue equation. Moreover, our results show that only small values of K are needed when the system is sheared from an isotropic equilibrium state, contrary to the case where the equilibrium state is nematic.

Solution methods of exact solutions for free vibration of rectangular orthotropic thin plates with classical boundary conditions

The exact solutions for free vibrations of orthotropic rectangular thin plates are presented in an elegant way using separation of variables method. And the exact solutions of three configurations (G-G-C-C, SS-G-C-C and C-C-C-G) are solved for the first time. In separation of variables method, the general formulation of the natural mode function and the relations among two spatial eigenvalues and a temporal eigenvalue are directly obtained from the characteristic differential equation, and the coefficients of exact mode functions and two exact eigenvalue equations are determined by two pairs of opposite edge conditions. It was regarded before 2009 that there were not exact solutions for rectangular thin plates with at least two adjacent clamped edges, and the others to be arbitrary combinations of clamped (C), simply supported (SS) and guided (G) conditions. For these configurations, the exact eigensolutions are solved successfully and the first ten exact frequencies are tabulated for a few of aspect ratios.

Eigenvalue equation for an asymmetric left-handed metamaterial planar slab optical waveguide

In this paper, we have derived the exact eigenvalue equation by using the transmission line method for double negative metamaterial slab. The derived results have exactly matched with the existing results found in the literature. The unusual behavior of double negative metamaterial is predicted. Dispersion study, modified profile and the power con- finement factor have been predicted for the case of double negative metamaterials.

The eigenvalue equations of equatorial waves on a sphere

Zonally propagating wave solutions of the linearized shallow water equations (LSWE) in a zonal channel on the rotating spherical earth are constructed from numerical solutions of eigenvalue equations that yield the meridional variation of the waves’ amplitudes and the phase speeds of these waves. An approximate Schr¨odinger equation, whose potential depends on one parameter only, is derived, and this equation yields analytic expressions for the dispersion relations and for the meridional structure of the waves’ amplitudes in two asymptotic cases. These analytic solutions validate the accuracy of the numerical solutions of the exact eigenvalue equation. Our results show the existence of Kelvin, Poincar´e and Rossby waves that are harmonic for large radius of deformation. For small radius of deformation, the latter two waves vary as Hermite functions. In addition, our results show that the mixed mode of the planar theory (a meridional wavenumber zero mode that behaves as a Rossby wave for large zonal wavenumbers and as a Poincar´e wave for small ones) does not exist on a sphere; instead, the first Rossby mode and the first westward propagating Poincar´e mode are separated by the anti-Kelvin mode for all values of the zonal wavenumber.

Electrical tuning of photonic crystals infilled with liquid crystals

In this paper we perform a complete study of electrical tuning in liquid crystal-infilled two-dimensional (2D) photonic crystals (PCs). The nematic liquid crystal (NLC) is characterized by a full range of bulk and surface elastic parameters. An essentially DC tuning field is applied in the axial direction. By minimizing the total (elastic plus electromagnetic) free energy, the configuration of the NLC directors, as a function of radial distance, is obtained. Three possible configurations are considered: escaped radial, planar radial, and axial. It is found that, in general, the escaped radial configuration is the preferred one. However, for sufficiently large applied fields, a phase transition occurs to the axial configuration. For example, in the case of the NLC 5CB, this transition is realized at about 14V/μm provided that the cylinder radius is greater than about 50nm. The configuration of the NLC directors determines the dielectric tensor as function of radial distance and this, in turn, leads to the eigenvalue equation for the PC. We present two such equations: one exact and the other approximate. The exact eigenvalue equation is based on the full anisotropy of the dielectric tensor and does not result in the usual separation of normal modes in a 2D PC. The approximate eigenvalue equation is derived from the average (over the cylinder cross-section) dielectric tensor and leads to modes that are polarized in the directions either parallel (E-mode) or perpendicular (H-mode) to the cylinders. Our calculations of the photonic band structure, by both methods, show that the approximate calculation works very well for the 5CB NLC cylinders in a silicon oxide (silica) host. This allows us to introduce the terminology quasi-E and quasi-H polarizations. We show how the partial photonic band gap in the [100] direction for these polarizations can be tuned and even completely closed. This behavior could be applied to the design of versatile, tunable polarization filters.

Rashba quantum wire: exact solution and ballistic transport

The effect of Rashba spin–orbit interaction in quantum wires with hard-wall boundaries isdiscussed. The exact wavefunction and eigenvalue equation are worked out, pointing outthe mixing between the spin and spatial parts. The spectral properties are also studiedwithin perturbation theory with respect to the strength of the spin–orbit interaction anddiagonalization procedure. A comparison is made with the results of a simple model,the two-band model, that takes account only of the first two sub-bands of thewire. Finally, the transport properties within the ballistic regime are analyticallycalculated for the two-band model and through a tight-binding Green function forthe entire system. Single and double interfaces separating regions with differentstrengths of spin–orbit interaction are analysed by injecting carriers into the first andthe second sub-band. It is shown that in the case of a single interface the spinpolarization in the Rashba region is different from zero, and in the case of twointerfaces the spin polarization shows oscillations due to spin-selective bound states.

Eigenvalue and field equations of three-layered uniaxial fibers and their applications to the characteristics of long-period fiber gratings with applied axial strain

By using the Debye potentials, the exact eigenvalue equations and the corresponding field distributions of the core and cladding modes for three-layered, radially stratified, and dielectric uniaxial optical fibers are derived completely; the modes include TE, TM, and hybrid HE/EH modes. The strain characteristics of long-period fiber gratings' (LPFGs') with applied axial strain are investigated theoretically by studying three-layered uniaxial optical fibers. When uniform axial strain is applied to fiber, the core, and inner and outer cladding become uniaxial crystal optically; that is, the optical axes are parallel to the fiber's axis. Analytic expressions of the strain sensitivities of LPFGs are derived. The analysis reveals that the strain sensitivities of LPFGs based on various cladding modes, including the shift value and direction of the resonance wavelength, differ greatly.

Exact eigenvalue equations for weakly guiding optical fibers with arbitrary graded-index profile

A modified analytical transfer matrix method (ATMM) is extended to investigate the LP/sub lm/ modes of weakly guiding optical fibers with arbitrary graded-index profile. Two classes of eigenvalue equation with respect to azimuthal number l = 0 and l /spl ne/ 0 are developed, respectively. It is confirmed that the phase shift at a turning point in optical fibers is equal to /spl pi/ as well as in planar optical waveguides. The existence of the phase shift at the center of the core in optical fibers, which is equal to 2arctan2 for the case of l = 0, is also demonstrated. Moreover, a numerical comparison with the available reference solution shows that the proposed method is exact.

Thermal relaxation effect on magnetohydrodynamic instability in a rotating micropolar fluid layer heated from below

The effect of a vertical magnetic field on the onset of convective instability in a conducting micropolar fluid (Oldroyd fluid) layer heated from below confined between two horizontal planes under the simultaneous action of the rotation of the system and a vertical temperature gradient is considered. Linear stability theory and normal mode analysis are used to derive an eigenvalue system of order twelve, and an exact eigenvalue equation for natural instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. The effects of magnetic field, the relaxation time and micropolar parameters on the critical Rayleigh number and critical wave number are discussed and presented graphically. The analysis presented in this paper is more general than any previous investigation.

Electrohydrodynamic instability of a rotating layer of a viscoelastic fluid heated from below

The stability of a rotating layer of viscoelastic dielectric liquid (Walters’ liquid B’) heated from below is considered. Linear stability theory is used to derive an eigenvalue system of ten orders and exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh heat numbers and wavenumber for the onset of instability are presented graphically as function of the Taylor number for various values of electric Rayleigh number and the elastic parameters.

The effect of thermal relaxation time on a electrohydrodynamic viscoelastic fluid layer heated from below

The problem of the onset of stability in a horizontal layer of a viscoelastic dielectric liquid with one relaxation time heated from below under the action of a vertical ac electric field is analyzed. Linear stability theory is used to derive an eigenvalue system of order eight and an exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. The critical Rayleigh number and wave number for the onset of instability are presented graphically as functions of the Grashof number for various values of the Prandtl number and one relaxation time. A comparison is made with the results obtained in the absence of an electric field. PACS No.: 47.27.T

Electromagneto-hydrodynamic instability in a horizontal viscoelastic fluid layer with one relaxation time

The stability of viscoelastic conducting liquid (Walters's liquid B′) heated from below in the presence of a magnetic field is considered. Linear stability theory is used to derive an eigenvalue system of sixth order, and an exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh number and wavenumber for the onset of instability are presented graphically as functions of the Chandrasekhar number at a Prandtl numberPr=100 and for various values of the one relaxation time and the elastic parameters.

Lowest excitation energy inatoms in the adiabatic approximation related to the single-particle kineticenergy functional

Time-dependent density-functional theory (DFT) leads to a formally exact eigenvalue equation for determining excitation energies. In an adiabatic approximation, we have first calculated the lowest excitation energies for various neutral atoms and positively charged atomic ions, for comparison with experimental data. Then, to gain further insight, the time-dependent theory is reformulated by using the chemical potential equation of time-independent DFT. The central new quantity then appearing is a second functional derivative of the single-particle kinetic energy Ts. If chemical hardness can be treated as a correction to the term involving Ts, then further analytical progress is effected. Good numerical results testify to the usefulness of invoking the time-independent DFT within the present context.

Analysis of optical fibers with graded-index profile by a combination of modified Airy functions and WKB solutions

An almost exact eigenvalue equation for optical fibers with graded-index profile is derived mathematically based on a combination of the modified Airy functions and the Wentzel, Kramers, Brillouin (WKB) trial solution. By applying proper boundary conditions, a phase shift correction term /spl delta/ is found out which improves the inherent error problems of the conventional WKB method. It is shown through computer simulations that results of the derived eigenvalue equation are in excellent agreement with those of the finite-element method.

Exact eigenvalue equations of modes in elliptical fibers of step‐index profile

The exact eigenvalue equations of modes in elliptical fibers of step‐index profile are not available in the literature. By using a rigorous method, an elliptical fiber mode of step‐index profile is expressed in terms of the even and odd Mathieu wave functions of the first kind in its core and in terms of their decaying even and odd modes in its cladding. Then, by matching the tangential components of the total electromagnetic fields across the core‐cladding boundary and using the orthogonalities of the circumferential Mathieu functions, the eigenvalue equations are derived in closed forms. Results are shown for the propagation characteristics of some low orders of the elliptic modes.

Complex permeability of demagnetized microwave ferrites near and above gyromagnetic resonance

A wide variety of microwave ferrite phase-shifting materials have been measured in the demagnetized state. The relative magnetic permeability and loss factor were determined near and above natural gyromagnetic resonance using H/sub 011/ cylindrical dielectric ring resonators. These low-loss dielectric sleeves were dimensioned for accurate magnetic property measurements of single ferrite rod samples at logarithmically sampled resonant frequencies from 2 GHz to 25 GHz. Permeability and magnetic loss factor are computed from the measured resonant frequencies and Q factors of these resonators, with and without the ferrite sample, using exact eigenvalue equations. Generally, the real part of the complex magnetic permeability increases with decreasing saturation magnetization, while the magnetic loss factor increases nonlinearly with increasing saturation. Schloemann's theoretical model for the real part of initial permeability of a cylindrically symmetric domain configuration in the completely demagnetized state shows excellent agreement with measured data when 2/spl pi//spl gamma//spl mu//sub s///spl omega/<0.75. The data allow design optimization of circulators and dual-mode and polarization-insensitive phasers, which are widely used in antenna array systems.