Vibration of a Cracked Cylindrical Shell of Rectangular Planform

1985 ◽  
Vol 52 (4) ◽  
pp. 927-932
Author(s):  
R. Solecki ◽  
F. Forouhar
Keyword(s):  
Cylindrical Shell ◽  
Fourier Transformation ◽  
Circular Cylindrical Shell ◽  
Homogeneous System ◽  
Algebraic Equations ◽  
Discontinuous Functions ◽  
Harmonic Vibrations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Gauss Theorem

Harmonic vibrations of a circular, cylindrical shell of rectangular planform and with an arbitrarily located crack, are investigated. The problem is described by Donnell’s equations and solved using triple finite Fourier transformation of discontinuous functions. The unknowns of the problem are the discontinuities of the slope and of three displacement components across the crack. These last quantities are replaced, using constitutive equations, by curvatures and strain in order to improve convergence and to represent explicitly the singularities at the tips. The formulas for differentiation of discontinuous functions are derived using Green-Gauss theorem. Application of the boundary conditions at the crack leads to a homogeneous system of linear algebraic equations. The frequencies are obtained from the characteristic equation resulting from this system. Numerical results for special cases are provided.

Stress Concentration in a Stretched Cylindrical Shell With Two Equal Circular Holes

1975 ◽  
Vol 42 (1) ◽  
pp. 105-109 ◽  
Author(s):  
P. Seide ◽  
A. S. Hafiz
Keyword(s):  
Cylindrical Shell ◽  
Stress Concentration ◽  
Stress Distribution ◽  
Circular Cylindrical Shell ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Wide Range ◽  
Circular Holes ◽  
Limiting Case ◽  
Infinite Set

In this investigation, the stress distribution due to uniaxial tension of an infinitely long, thin, circular cylindrical shell with two equal small circular holes located along a generator is obtained. The problem is solved by the superposition of solutions previously obtained for a cylinder with a single circular hole. The satisfaction of boundary conditions on the free surfaces of the holes, together with uniqueness and overall equilibrium conditions, yields an infinite set of linear algebraic equations involving Hankel and Bessel functions of complex argument. The stress distribution along the boundaries of the holes and the interior of the shell is investigated. In particular, the value of the maximum stress is calculated for a wide range of parameters, including the limiting case in which the holes almost touch and the limiting case in which the radius of the cylinder becomes very large. As is the case for a flat plate, the stress-concentration factor is reduced by the presence of another hole.

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.

On the Interaction at Anti-Flat Deformation of Stress Concentrators of the Type of Cracks and Stringers with Regard to the Layer Manufactured from Miscellaneous Materials

2019 ◽  
Vol 828 ◽  
pp. 81-88
Author(s):  
Nune Grigoryan ◽  
Mher Mkrtchyan
Keyword(s):  
Integral Transform ◽  
Composite Layer ◽  
Singular Integral Equations ◽  
Original System ◽  
Fourier Integral ◽  
Quadrature Formulas ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Tangential Forces

In this paper, we consider the problem of determining the basic characteristics of the stress state of a composite in the form of a piecewise homogeneous elastic layer reinforced along its extreme edges by stringers of finite lengths and containing a collinear system of an arbitrary number of cracks at the junction line of heterogeneous materials. It is assumed that stringers along their longitudinal edges are loaded with tangential forces, and along their vertical edges - with horizontal concentrated forces. In addition, the cracks are laden with distributed tangential forces of different intensities. The case is also considered when the lower edge of the composite layer is free from the stringer and rigidly clamped. It is believed that under the action of these loads, the composite layer in the direction of one of the coordinate axes is in conditions of anti-flat deformation (longitudinal shift). Using the Fourier integral transform, the solution of the problem is reduced to solving a system of singular integral equations (SIE) of three equations. The solution of this system is obtained by a well-known numerical-analytical method for solving the SIE using Gauss quadrature formulas by the use of the Chebyshev nodes. As a result, the solution of the original system of SIE is reduced to the solution of the system of systems of linear algebraic equations (SLAE). Various special cases are considered, when the defining SIE and the SLAE of the task are greatly simplified, which will make it possible to carry out a detailed numerical analysis and identify patterns of change in the characteristics of the tasks.

scholarly journals On the Solubility of Linear Algebraic Equations (Contd.)

1913 ◽  
Vol 12 ◽  
pp. 137-138
Author(s):  
John Dougall
Keyword(s):  
Linear Equations ◽  
Homogeneous System ◽  
Algebraic Equations ◽  
Similar Reasoning ◽  
Null Solution ◽  
Linear Algebraic Equations ◽  
The Given ◽  
Homogeneous Linear

A system of n non-homogeneous linear equations in n variables has one and only one solution if the homogeneous system obtained from the given system by putting all the constant terms equal to zero has no solution except the null solution.This may be proved independently by similar reasoning to that given for Theorem I., or it may be deduced from that theorem. We follow the latter method.

scholarly journals Markov jump transport of trapped charge in thin dielectric layers

2020 ◽  
Author(s):  
Andrey Pil'nik ◽  
Andrey Chernov ◽  
Damir Islamov
Keyword(s):  
Charge Transport ◽  
Theoretical Models ◽  
Thin Layers ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Dielectric Layers ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

Abstract In 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 1D transport through dielectric layers, might incorrectly describe the charge flow through the ultra-thin layers with a countable number of traps, taking into account 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 dynamics of system evolution for special cases. These solutions can be used to test the code and for studying of charge transport properties of thin dielectric films.

scholarly journals On the Solubility of Linear Algebraic Equations.—

1912 ◽  
Vol 11 ◽  
pp. 125-129
Author(s):  
John Dougall
Keyword(s):  
Unique Solution ◽  
Homogeneous System ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Image Position

(a) It is proved in treatises on Algebra that the equations (in three variables for brevity)have a unique solution given byprovided the determinantdoes not vanish.(b) It is also proved, from (2), that if the “degenerate” homogeneous systemhas a non-null solution (i.e. a solution in which the variables are not all zero), then Δ = 0.

scholarly journals Stability of rectangular cantilever plates with high elasticity

2021 ◽  
Vol 244 ◽  
pp. 04004
Author(s):  
Mikhail Sukhoterin ◽  
Sergey Baryshnikov ◽  
Tatiana Knysh ◽  
Elena Rasputina
Keyword(s):  
Boundary Problem ◽  
Critical Loads ◽  
Compressive Load ◽  
Longitudinal Direction ◽  
Approximate Solutions ◽  
Homogeneous System ◽  
Critical Force ◽  
Cantilever Plate ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The problem of cantilever plate stability has been little studied due to the difficulty of solving the corresponding boundary problem. The known approximate solutions mainly concern only the first critical load. In this paper, stability of an elastic rectangular cantilever plate under the action of uniform pressure applied to its edge opposite to the clamped edge is investigated. Under such conditions, thin canopies of buildings made of new materials can be found at sharp gusts of wind in longitudinal direction. At present, cantilever nanoplates are widely used as key components of sensors to create nanoscale transistors where they are exposed to magnetic fields in the plate plane. The aim of the study is to obtain the critical force spectrum and corresponding forms of supercritical equilibrium. The deflection function is selected as a sum of two hyperbolic trigonometric series with adding special compensating summands to the main symmetric solution for the free terms of the decomposition of the functions in the Fourier series by cosines. The fulfillment of all conditions of the boundary problem leads to an infinite homogeneous system of linear algebraic equations with regard to unknown series coefficients. The task of the study is to create a numerical algorithm that allows finding eigenvalues of the resolving system with high accuracy. The search for critical loads (eigenvalues) giving a nontrivial solution of this system is carried out by brute force search of compressive load value in combination with the method of sequential approximations. For the plates with different side ratios, the spectrum of the first three critical loads is obtained, at which new forms of equilibrium emerge. An antisymmetric solution is obtained and studied. 3D images of the corresponding forms are presented.

scholarly journals Buckling of stabilizers deep-sea vehicles

2020 ◽  
Author(s):  
С.О. Барышников ◽  
М.В. Сухотерин ◽  
Т.П. Кныш ◽  
Н.Ф. Пижурина
Keyword(s):  
Deep Sea ◽  
Critical Loads ◽  
Water Pressure ◽  
High Water ◽  
Homogeneous System ◽  
Order Partial Differential Equation ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Compressive Loads ◽  
The Stability

В данной работе исследуется устойчивость прямоугольной консольной панели как приближенной расчетной модели стабилизаторов глубоководных аппаратов. Вследствие высокого давления воды сжимающие усилия в плоскости стабилизатора, приложенные к свободным граням, могут быть значительными и приводить к потере устойчивости. Целью настоящей работы является разработка эффективного метода численного моделирования устойчивости стабилизаторов принципиально новых судов и кораблей, в том числе из новых материалов. Задачей исследования является определение спектра критических сжимающих нагрузок, а также соответствующих форм закритического равновесия для этих элементов. Краевая задача устойчивости прямоугольной консольной панели описывается дифференциальным уравнением четвертого порядка в частных производных по двум переменным для искомой функции прогибов и системой граничных условий, содержащих частные производные этой функции до третьего порядка включительно. В качестве параметра основное уравнение изгиба содержит интенсивность равномерно распределенного давления на свободные края панели. Функция прогибов выбирается в виде суммы двух гиперболо-тригонометрических рядов по двум координатам и дополняется затем специальными компенсирующими членами. Проблема сводится к исследованию бесконечной однородной системы линейных алгебраических уравнений относительно неизвестных коэффициентов рядов. Поиск критических нагрузок осуществляется перебором величины давления и анализом бесконечной системы. Получен спектр нескольких первых критических нагрузок, при которых появляется новая форма равновесия. In this paper, we study the stability of a rectangular console panel as an approximate computational model of deep-sea vehicle stabilizers. Due to high water pressure, compressive forces in the stabilizer plane applied to free faces can be significant and lead to loss of stability. The purpose of this work is to develop an effective method for numerical modeling of stability of stabilizers of fundamentally new vessels and ships, including those made of new materials. The aim of the study is to determine the spectrum of critical compressive loads, as well as the corresponding forms of supercritical equilibrium for these elements. The boundary value problem of stability of a rectangular console panel is described by a fourth-order partial differential equation for two variables for the desired deflection function and a system of boundary conditions containing partial derivatives of this function up to and including the third order. As a parameter, the basic bending equation contains the intensity of evenly distributed pressure on the free edges of the panel. The deflection function is selected as the sum of two hyperbolic-trigonometric series over two coordinates and then supplemented with special compensating terms. The problem is reduced to the study of an infinite homogeneous system of linear algebraic equations with respect to unknown series coefficients. The search for critical loads is performed by searching the pressure value and analyzing the infinite system. The spectrum of the first few critical loads at which a new form of equilibrium appears is obtained.

scholarly journals Dynamic Buckling of an Axially Compressed Cylindrical Shell With Discrete Rings and Stringers

1973 ◽  
Vol 40 (3) ◽  
pp. 736-740 ◽  
Author(s):  
C. A. Fisher ◽  
C. W. Bert
Keyword(s):  
Cylindrical Shell ◽  
Circular Cylindrical Shell ◽  
Nonlinear Differential Equations ◽  
Dynamic Buckling ◽  
Algebraic Equations ◽  
Discrete Elements ◽  
Governing Equations ◽  
Aircraft Cabins ◽  
Dirac Delta ◽  
Aircraft Cabin

As an exploratory effort toward improving the crashworthiness of light aircraft cabins, a theoretical analysis was made to predict the dynamic buckling load and buckling time of a stiffened, thin-walled circular cylindrical shell. To provide for the large stiffener spacing in light aircraft, the stiffeners were considered as discrete elements by means of a Dirac delta procedure. The nonlinear governing equations were derived using Hamilton’s principle and the final equations were obtained by means of Galerkin’s method. Solution was carried out by using a Gauss-Jordan technique on the algebraic equations and a Runge-Kutta technique on the nonlinear differential equations. Numerical results are presented for an idealized model of a typical light aircraft cabin.

XII.—Studies in Practical Mathematics. I. The Evaluation, with Applications, of a Certain Triple Product Matrix

1938 ◽  
Vol 57 ◽  
pp. 172-181 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Form ◽  
Matrix Multiplication ◽  
Matrix Product ◽  
Algebraic Equations ◽  
Product Matrix ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
The Matrix ◽  
Practical Mathematics ◽  
General Operation

The solution of simultaneous linear algebraic equations, the evaluation of the adjugate or the reciprocal of a given square matrix, and the evaluation of the bilinear or quadratic form reciprocal to a given form, are all special cases of a certain general operation, namely the evaluation of a matrix product H′A-1K, where A is square and non-singular, that is, the determinant | A | is not zero. (Matrix multiplication is like determinant multiplication, but exclusively row-into-column. The matrix H′ is obtained from H by transposition, that is, by changing rows into columns.) The matrices H′ and K may be rectangular. If A is singular, the reciprocal A-1 does not exist; and in such a case the product H′(adj A) K may be required. Arithmetically, the only difference in the computation of H′A-1K and H(adj A) K is that in the latter case a final division of all elements by | A | is not performed.

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