Complex Eigenvalue Equations Research Articles

Complex band structure of thermal wave crystals: The plane-wave method

In this paper, we present an extension of the plane-wave method (PWM) to compute the complex band structure of thermal wave crystals (TWCs). The structural periodicity of TWC allows the possibility to manipulate non-Fourier heat via wave interference. While the Cattaneo-Vernotte (CV) heat conduction theory accurately models oscillatory wave-like propagation of heat in TWCs, obtaining an eigenvalue equation for frequency using the CV wave equation is not possible. To overcome this limitation, we propose a novel approach that solves a complex eigenvalue equation for the Bloch wave vectors

An Efficient Method for Predicting the Output Signal of an Eddy Current Probe Around the Edge of a Conductive Cylindrical Rod

We propose a semi-analytical model to predict the output signal of an eddy current probe around the edge of a conductive cylindrical rod. The proposed model is based on the truncated region eigenfunction expansion method and uses the second-order vector potential to formulate the resultant boundary problem. The use of appropriate boundary conditions on the truncation locations leads to less complex eigenvalue equations which remarkably increases the computation speed. The validity and computation efficiency of the proposed model are demonstrated by comparing the simulation results of several case studies with measurements and those obtained using a commercial finite element code.

Pseudospin symmetric solutions of the Dirac equation with the modified Rosen–Morse potential using Nikiforov–Uvarov method and supersymmetric quantum mechanics approach

Employing the Pekeris-type approximation to deal with the pseudo-centrifugal term, we analytically study the pseudospin symmetry of a Dirac nucleon subjected to equal scalar and vector modified Rosen–Morse potential including the spin–orbit coupling term by using the Nikiforov–Uvarov method and supersymmetric quantum mechanics approach. The complex eigenvalue equation and the total normalized wave functions expressed in terms of Jacobi polynomial with arbitrary spin–orbit coupling quantum number k are presented under the condition of pseudospin symmetry. The eigenvalue equations for both methods reproduce the same result to affirm the mathematical accuracy of analytical calculations. The numerical solutions obtained for different adjustable parameters produce degeneracies for some quantum number.

Transverse Vibration Analysis of Axially Moving Trapezoidal Plates

Transverse vibration of axially moving trapezoidal plates is investigated. The differential equation of transverse vibration for a axially moving trapezoidal plate is established by D'Alembert principle. The original trapezoid region can be replaced by regular square region by the medium parameter method for the convenience of calculation. A generalized complex eigenvalue equation is derived by a discrete method (the differential quadrature method). The complex frequency curve of trapezoidal plate is obtained by calculating the eigenvalue equation. The change of the complex frequencies of the axially moving trapezoidal plates with the dimensionless axially moving speed is analyzed. The effects of the aspect ratio and the trapezoidal angle on instability type of the trapezoidal plate are discussed under different boundary conditions. The results of numerical analysis show that there are two main instability types of axially moving trapezoidal plate: divergence and flutter. The modal orders of the two types of instability are also different, which is related to the trapezoidal angle, aspect ratio and boundary condition of the trapezoidal plate.

Analysis of Complex Modal Characteristics of Fractional Derivative Viscoelastic Rotating Beams

For the transverse vibration problem of a fractional derivative viscoelastic rotating beam, the differential equation of the system is obtained based on the Euler–Bernoulli beam theory and Hamilton principle. Then, introducing dimensionless quantities to differential equations and boundary conditions, the generalized complex eigenvalue equations of the system are obtained by the differential quadrature method. The effects of the slenderness ratio, the viscoelastic ratio, the hub radius‐beam length ratio, and dimensionless hub speed and fractional order on the vibration characteristics of fractional derivative viscoelastic rotating beams are discussed by numerical examples. Numerical calculations show that when the dimensionless hub speed is constant, the real part of complex frequency increases with the increase of the fractional order, and the higher‐order growth trend is more obvious. Through the study of displacement response at different points on the beam, it can be seen that the closer to the free end, the larger the response amplitude. And, the amplitude of response has been attenuated, which is also consistent with the vibration law of free vibration considering damping.

Calculation of propagation characteristics of surface plasmons in gold/silver nanowires

We obtain the propagation characteristics of the fundamental plasmon mode guided along Au and Ag nanowires using a simple and accurate method proposed recently by our group for solving the complex eigenvalue equations. Our results are found in excellent agreement with the ones obtained by the finite element method. Furthermore, results obtained using Johnson and Christy’s data [Phys. Rev. B6, 4370 (1972)PLRBAQ0556-2805] for Au and Ag are found in excellent agreement with the recently reported experimental results. On the other hand, Palik’s data [Handbook of Optical Constant of Solids (Academic, 1985)] are found to highly overestimate the losses in Ag nanowires. This study establishes a simple and accurate method for theoretical modeling of surface plasmons guided along Au and Ag nanowires for their applications in various plasmonic devices.

Stability of Axially Moving Piezolaminated Viscoelastic Plate Subjected to Follower Force

The stability of the moving viscoelastic plate with the piezoelectric layer subjected to uniformly distributed tangential follower force is investigated. The force excited by the piezoelectric layer due to external voltage is modeled as the follower tensile force. The differential equation of the axially moving viscoelastic rectangular plate with piezoelectric layer subjected to uniformly distributed tangential follower force is formulated on the basis of the Kirchhoff thin plate theory and the two-dimensional viscoelastic differential constitutive relation. The complex eigenvalue equations are established by the differential quadrature method. Via numerical calculation, the curves of real parts and imaginary parts of dimensionless complex frequencies versus uniformly distributed tangential follower force and dimensionless moving speed are obtained. The effects of nonconservative force, dimensionless axially moving speed, and dimensionless applied voltages on the stability of axially moving nonconservative viscoelastic plate with piezoelectric layer are analyzed.

Dynamic Characteristics and Stability of Axially Moving Viscoelastic Plate with Piezoelectric Layer

The dynamic stability of the moving viscoelastic plate with the piezoelectric layer is studied. On the basis of the thin plate theory and the two-dimensional viscoelastic differential constitutive relation, the differential equation of the axially moving viscoelastic rectangular plate with piezoelectric layer in the Laplace domain is formulated, the equation is suitable for various viscoelastic differential models. Then, the differential equation of motion of the viscoelastic plate with elastic dilatation and Kelvin–Voigt distortion in time domain is derived, with the piezoelectric effect. The complex eigenvalue equations of axially moving viscoelastic plate are established by the differential quadrature method. The generalized eigenvalue equations are solved, and the force excited by the piezoelectric layer due to external voltage is modeled as the follower tensile force; this force is used to improve the stability of the axially moving viscoelastic plate. Via numerical calculation, the results for the instability type and the corresponding critical moving speed of viscoelastic plate are presented to show the variations in these factors with respect to the dimensionless moving speed, the dimensionless delay time and the applied voltages. The dynamic stability of the axially moving viscoelastic plates can be effectively improved by the determination of the optimal location for the piezoelectric layers and the most favorable voltage assignment.

Research on the Modal Damping Ratio for Stay Cable with Viscoselasticity Damper in Cable-Stayed Bridges

Taking the bending stiffness, cable static sag and cable inclined angle into consideration, equations of space free vibration of the cable-damper system are derived in this paper. Joining the variable separation strategy and center difference method, the partial differential equations are discretized in space and a set of complex eigenvalue equations, which are solved by state space method, are derived, and both the maximum modal damping ration and the optimal damper parameter are obtained. Several typical stay cables are investigated for both the in-plane and out-of-plane modes under different cable parameters and damper parameters. The results demonstrate that modal damping ratio for the first in-plane mode is significantly affected by the cable static sag only, but those for the other modes affected by cable sag are slight, and cable static sag do not affect the optimal damper parameter for all modes, however the bending stiffness will changes both the maximum modal damping ratios and the optimal damper parameters. Some valuable suggestions are proposed for the optimal damper design.

Stability Analyses of a Moving Rectangular Plate by the Element Free Galerkin Method

The element-free Galerkin method is proposed to solve the stability of the moving rectangular plates. Utilizing the extended Hamilton’s principle for the elastic dynamics system, the variational expression of the moving thin plate are established. The dimensionless equations of motion of the moving thin plate are obtained by the element-free Galerkin method, and the complex eigenvalue equation is presented. Via numerical calculation, the variation relationship between the first three complex frequencies of the system and the moving speed is obtained. The effects of dimensionless moving speed on the stability and critical load of the thin plates are analyzed.

Stability of a Cracked Viscoelastic Plate of Varying Thickness Subjected to Follower Force

The stability of the cracked visoelastic rectangular plate of varying thickness subjected to uniformly distributed tangential follower force is investigated. Two models describing thickness variation are parabolic and linear variations along one edge of the plate. This is done in order to show how critical loads for cracked visoelastic thin plate subjected to follower force can be obtained using the two-dimensional viscoelastic differential constitutive relation and the thin-plate theory. The differential equations in the Laplace domain are established. The complex eigen-value equations are derived by the differential quadrature method. The generalized eigen-value under different boundary conditions is calculated. The effects of the plate parameters, the crack parameters and the dimensionless delay time on the stability of the viscoelastic plates are analyzed. The crack is found to change the type of instability and reduce the stability of varying thickness viscoelastic plate.

Investigation of Terahertz Wave Propagation Along Shielded Dielectric Multiple-Slot Waveguide

In this paper, a shielded dielectric multiple-slot waveguide is presented and demonstrated by theoretical calculation at terahertz frequencies. Some electromagnetic analysis of waveguide currently employed simplified intrinsic frequency dispersion models for the bulk conductivity of normal metals used in terahertz wave structures. This paper has compared various conductivity models for gold between 0.1-3 THz. The dielectric loss, conductor loss, total propagation loss with different conductivity models and E-field distribution has been obtained by numerically solving the complex eigenvalue equation for the propagation constant. The analysis results are in agreement with the assumption of R. Sun et al. and the experimental results obtained by H. Sun et al. The propagation loss deviation between various models is less than 3% at 0.2-0.5 THz. Comparisons with other slot waveguides are also given. The analysis results show that the proposed line has lower propagation attenuation than other slot waveguides.

Investigation of Terahertz Wave Propagation Along Parallel-Plate Dielectric Waveguide using Various Metal Conductivity Models

In this paper, an investigation of parallel-plate dielectric waveguide is proposed by theoretical calculation at terahertz frequencies. In the previous study, the electromagnetic analysis of waveguide employed simplified intrinsic frequency dispersion models — classical skin-effect model for the bulk conductivity of normal metals used in terahertz wave structures. To analyze the deviation between various metal conductivity models, the accurate classical relaxation-effect frequency dispersion model and the traditional and much simpler classical skin-effect model are used to analyze the propagation characteristic of the proposed waveguide. The propagation loss with different conductivity models has been obtained by numerically solving the complex eigenvalue equation for the propagation constant. The energy coupling characteristic of this waveguide is also proposed. The conductor loss deviation between the classical relaxation-effect model and the classical skin-effect model varies from 1.1% to 6.4%, and the total loss varies from 1.1% to 5.4% at 0.1–1 THz. The analysis results are very useful for the development of terahertz power propagation, spectroscopy and detectors.

Dynamic stability of linearly varying thickness viscoelastic rectangular plate with crack and subjected to tangential follower force

Based on the two-dimensional viscoelastic differential constitutive relation and the thin plate theory, the differential equations of linearly varying thickness viscoelastic plate with crack and subjected to uniformly distributed tangential follower force in the Laplace domain are established, and the expression of the additional rotation induced by the crack is given. The complex eigenvalue equations of linearly varying thickness viscoelastic plate constituted by elastic behavior in dilatation and the Kelvin–Voigt laws for distortion with crack and under the action of uniformly distributed tangential follower force are obtained by the differential quadrature method. The generalized eigenvalue under different boundary conditions is calculated, and the curves of real parts and imaginary parts of the first three order dimensionless complex frequencies versus uniformly distributed tangential follower force are obtained. The effects of the aspect ratio, the thickness ratio, the crack parameters and the dimensionless delay time on the dynamic stability of the viscoelastic plates are analyzed.

Dynamic Stability of Cracked Viscoelastic Rectangular Plate Subjected to Tangential Follower Force

Based on the thin plate theory and the two-dimensional viscoelastic differential type constitutive relation, the differential equation of motion of a viscoelastic plate containing an all-over part-through crack and subjected to uniformly distributed tangential follower force is established in Laplace domain. Then, by performing the Laplace inverse transformation, the differential equation of motion of the plate in the time domain is obtained. The expression of the additional rotation induced by the crack is given. The complex eigenvalue equations of the cracked viscoelastic plate subjected to uniformly distributed tangential follower force are obtained by the differential quadrature method, and the δ method is adopted at the crack continuity conditions. The general eigenvalue equations of the cracked viscoelastic plate subjected to uniformly distributed tangential follower force under the different boundary conditions are calculated. The transverse vibration characteristics, type of instability, and corresponding critical loads of the cracked viscoelastic plate subjected to uniformly distributed tangential follower force are analyzed.

Transverse vibration of viscoelastic rectangular plate with linearly varying thickness and multiple cracks

Based on the thin-plate theory and the two-dimensional viscoelastic differential constitutive relation, the differential equations of motion of the viscoelastic plate with linearly varying thickness and an arbitrary number of all-over part-through cracks are established, and the expressions of the additional rotation angle induced by the cracks are deduced. We assume that it is elastic in dilatation, but postulate the Kelvin–Voigt laws for distortion, the complex eigenvalue equations of the viscoelastic plate with linearly varying thickness and multiple cracks are derived by the differential quadrature method. The general eigenvalue equations of the viscoelastic plate with multiple cracks under different boundary conditions are calculated. The effects of various geometric parameters, dimensionless delay time and dimensionless crack parameters on the transverse vibration characteristics of a viscoelastic plate containing multiple all-over part-through cracks are analyzed.

Natural vibration of two-dimensional slender structure–water interaction systems subject to Sommerfeld radiation condition

The dynamic behaviour of two-dimensional flexible slender structure–water interaction systems subject to a Sommerfeld radiation condition at the infinity boundary of the water domain is investigated. A new parameter, the speed of radiation wave, is introduced into the Sommerfeld radiation condition to consider the influences of both of the pressure wave and the free surface wave of the water, which is an extension of the original Sommerfeld condition. The governing equations describing the dynamic behaviour of the system are analysed and solved using a separation of variables method. It is demonstrated that the natural vibration of the two-dimensional slender structure–water interaction system subject to a Sommerfeld radiation condition is governed by a complex eigenvalue equation which has only pairs of complex conjugate eigenvalues. The number of the pairs of complex conjugate natural frequencies equals the number of the natural modes of the corresponding dry structure and is independent of the continuous fluid domain, which has infinite degrees of freedom. The examples, including four cases of shallow water, deep water, no free surface wave and incompressible water, demonstrate and illustrate the developed theoretical and numerical method.

Simulation of the Propagation Property of Metal WiresTerahertz Waveguides

The propagation property of metal wires terahertz waveguides is studiedand simulated under the framework of the Sommerfeld model. The groupvelocity dispersion, attenuation amplitude, transverse magnetic mode andpropagating energy have been obtained by numerically solving the complexeigenvalue equation for the propagation constant. It is found that thegroup velocity dispersion and attenuation amplitude decrease with theincreasing radii of metal wires. The energy power within the dielectriclayer increase with the increase of radiation frequency.

Resonance states of two-electron quantum dots

Autoionizing resonance states of two-electron quantum dots are studied by using the effective-mass complex-eigenvalue equation approach. It is shown that two-electron quantum dots may have very rich spectra of resonance states. The number of them and their lifetimes depend strongly on the dot size. It is observed that a resonance state may change its character from Feshbach to shape type with the dot size decreasing. Since the resonance states may have very long lifetimes they should not be neglected in the description of transport processes in quantum dots.

Theoretical study of polymeric metal clad optical waveguide polarizer

Planar optical waveguides consisting of thin dielectric films and buffer layers with metal cladding have been investigated theoretically. A computer program was written to calculate the exact zeroes of complex eigenvalue equation for TE and TM modes in multilayer metal clad waveguide polarizer. Numerical results and illustrations are given for Polycarbonate waveguide with other polymers as buffer and Al, Ag and Au as cladding metals at 0.6238 μ m . It is also shown that, using thin (finite) films of metal produce more efficient polarizers as compared to semi-infinite metal films. Effect of low index buffer layer on attenuation of TM/TE modes is also investigated.