Cosserat Spectrum of an Axisymmetric Elasticity Problem for a Finite-Length Solid Cylinder

2018 ◽  
Vol 35 (3) ◽  
pp. 343-349
Author(s):  
Yu. V. Tokovyy
Keyword(s):  
Asymptotic Behavior ◽  
Finite Length ◽  
Solution Method ◽  
Superposition Method ◽  
Algebraic Equations ◽  
Solid Cylinder ◽  
Cosserat Spectrum ◽  
Linear Algebraic Equations ◽  
Key Features ◽  
Axisymmetric Elasticity

ABSTRACTAn algorithm for the computation and analysis of the Cosserat spectrum for an axisymmetric elasticity boundary-value problem in a finite-length solid cylinder with boundary conditions in terms of stresses is proposed. By making use of the cross-wise superposition method, the spectral problem is reduced to systems of linear algebraic equations. A solution method for the mentioned systems is presented and the asymptotic behavior of the Cosserat eigenvalues is established. On this basis, the key features of the Cosserat spectrum for the mentioned problem are analyzed with special attention given to the effect of the cylinder aspect ratio.

Multi-Level Residue Harmonic Balance Method for the Transmission Loss of a Nonlinearly Vibrating Perforated Panel

2016 ◽  
Vol 16 (02) ◽  
pp. 1450100 ◽  
Author(s):  
Y. Y. Lee
Keyword(s):  
Differential Equations ◽  
Harmonic Balance ◽  
Solution Method ◽  
Transmission Loss ◽  
Mass Movement ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Multi Level

This paper investigates the transmission loss of a nonlinearly vibrating perforated panel using the multi-level residue harmonic balance method. The coupled governing differential equations which represent the air mass movement at each hole and the nonlinear panel vibration are developed. The proposed analytical solution method, which is revised from a previous harmonic balance method for single mode problems, is newly applied for solving the coupled differential equations. The main advantage of this solution method is that only one set of nonlinear algebraic equations is generated in the zero level solution procedure while the higher level solutions to any desired accuracy can be obtained by solving a set of linear algebraic equations. The results obtained from the multi-level residue harmonic balance method agree reasonably with those obtained from a numerical integration method. In the parametric study, the velocity amplitude convergences have been checked. The effects of excitation level, perforation ratio, diameter of hole, and panel thickness are examined.

scholarly journals Vibrations of a Cantilevered Thick Plate

2021 ◽  
pp. 106-117
Author(s):  
S. O Papkov
Keyword(s):  
Boundary Conditions ◽  
Variational Approach ◽  
Thick Plate ◽  
Infinite System ◽  
Trigonometric System ◽  
Superposition Method ◽  
Algebraic Equations ◽  
Exact Methods ◽  
Particular Solutions ◽  
Linear Algebraic Equations

It has been for the first time that an analytical solution to the problem of free vibrations of a cantilevered thick orthotropic plate is presented. This problem is quite cumbersome for using the exact methods of the theory of elasticity; therefore, methods based on the variational approach were developed to solve it. The paper suggests using the superposition method to construct a general solution of the vibration equations of a plate in the series form of particular solutions obtained with the help of a variables separation. The particular solutions of one of the coordinates are built in the form of trigonometric functions of a special type (modified trigonometric system). The constructed solution, in contrast to the solutions known in the literature on the basis of the variational approach, accurately satisfies the equations of vibrations. The use of a modified trigonometric system of functions makes it possible to obtain uniform formulas for even and odd vibration shapes and to reduce the quantity of boundary conditions on the plate sides from twelve to nine ones, while five of the nine boundary conditions are also accurately satisfied. The structure of the presented solution on the plate boundary is such that, each of the kinematic or force characteristics of the plate is represented as a sum of two series, i.e. a trigonometric series and a series in hyperbolic functions. Remaining boundary conditions make it possible to obtain an infinite system of linear algebraic equations with respect to the unknown coefficients of the series representing the solution. The convergence of the solution by the reduction method of the infinite system is investigated numerically. Examples of the numerical implementation are given; numerical studies of the spectrum of natural frequencies of the cantilevered thick plate were carried out based on the obtained solution, both with varying elastic characteristics of the material and with varying geometric parameters.

scholarly journals Analysis of a Coaxial Waveguide with Finite-Length Impedance Loadings in the Inner and Outer Conductors

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Feray Hacıvelioğlu ◽  
Alinur Büyükaksoy
Keyword(s):  
Reflection Coefficient ◽  
Boundary Value Problem ◽  
Finite Length ◽  
Formal Solution ◽  
Coaxial Waveguide ◽  
Algebraic Equations ◽  
Coaxial Cylinders ◽  
Infinite Systems ◽  
Linear Algebraic Equations ◽  
Band Stop Filter

A rigorous Wiener-Hopf approach is used to investigate the band stop filter characteristics of a coaxial waveguide with finite-length impedance loading. The representation of the solution to the boundary-value problem in terms of Fourier integrals leads to two simultaneous modified Wiener-Hopf equations whose formal solution is obtained by using the factorization and decomposition procedures. The solution involves 16 infinite sets of unknown coefficients satisfying 16 infinite systems of linear algebraic equations. These systems are solved numerically and some graphical results showing the influence of the spacing between the coaxial cylinders, the surface impedances, and the length of the impedance loadings on the reflection coefficient are presented.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

scholarly journals Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrey A. Pil’nik ◽  
Andrey A. Chernov ◽  
Damir R. Islamov
Keyword(s):  
Charge Transport ◽  
Thin Layers ◽  
Jump Processes ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Statistical Solution ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

AbstractIn this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

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