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 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 Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

2021 ◽  
Vol 17 (1) ◽  
pp. 33
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Triangular System ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Linear Algebraic Equations

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0,T). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of  linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is  O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

Solutions of a singular integro-differential equation related to the adhesive contact problems of elasticity theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nugzar Shavlakadze ◽  
Otar Jokhadze
Keyword(s):  
Differential Equation ◽  
Analytic Functions ◽  
Contact Problems ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Algebraic Equations ◽  
Adhesive Interaction ◽  
Normal Contact ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations

Abstract Exact and approximate solutions of a some type singular integro-differential equation related to problems of adhesive interaction between elastic thin half-infinite or finite homogeneous patch and elastic plate are investigated. For the patch loaded with vertical forces, there holds a standard model in which vertical elastic displacements are assumed to be constant. Using the theory of analytic functions, integral transforms and orthogonal polynomials, the singular integro-differential equation is reduced to a different boundary value problem of the theory of analytic functions or to an infinite system of linear algebraic equations. Exact or approximate solutions of such problems and asymptotic estimates of normal contact stresses are obtained.

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.

scholarly journals A MUNTZ-LEGENDRE APPROACH TO OBTAIN SOLUTIONS OF SINGULAR PERTURBED PROBLEMS

2020 ◽  
Vol 20 (3) ◽  
pp. 537-544
Author(s):  
SUAYIP YUZBASI ◽  
EMRAH GOK ◽  
MEHMET SEZER
Keyword(s):  
Error Function ◽  
Approximate Solutions ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Algebraic Equations ◽  
Residual Function ◽  
Collocation Points ◽  
Linear Algebraic Equations ◽  
Collocation Approach ◽  
Estimated Error

Singularly perturbed differential equations are encountered in mathematical modelling of processes in physics and engineering. Aim of this study is to give a collocation approach for solutions of singularly perturbed two-point boundary value problems. The method provides obtaining the approximate solutions in the form of Müntz-Legendre polynomials by using collocation points and matrix relations. Singularly perturbed problem is transformed into a system of linear algebraic equations. By solving this system, the approximate solution is computed. Also, an error estimation is done using the residual function and the approximate solutions are improved by means of the estimated error function. Two numerical examples are given to show the applicability of the method.

scholarly journals An Efficient Algorithm for Solving Integro-Differential Equation

2019 ◽  
Vol 54 (6) ◽  
Author(s):  
Hassan Hamad AL-Nasrawy ◽  
Abdul Khaleq O. Al-Jubory ◽  
Kasim Abbas Hussaina
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Difference Equations ◽  
Matrix Equation ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Approximate Methods ◽  
Linear Algebraic Equations ◽  
Linear Delay

In this paper, we study and modify an approximate method as well as a new collocation method, which is based on orthonormal Bernstein polynomials to find approximate solutions of mixed linear delay Fredholm integro-differential-difference equations under the mixed conditions. The main purpose of this paper is to study and develop some approximate methods to solve the mixed linear delay Fredholm integro-differential-difference equations. We employ a new algorithm to find approximate solution via perpendicular Bernstein polynomials on the interval [0,1], and we construct a new matrix of derivatives that will be used to find an approximate solution of matrix equation, that will reduce it to the systems of linear algebraic equations. We study the convergence approximate solutions to the exact solutions. Finally, two examples are given and their results are shown in figures to illustrate the efficiency and accuracy of this method. All the computations are implemented using Math14.

Approximate Solutions for Solving Fractional-order Painlevé Equations

2019 ◽  
Vol 1 (1) ◽  
pp. 12-24 ◽  
Author(s):  
Mohammad Izadi
Keyword(s):  
Approximate Solutions ◽  
Painlevé Equations ◽  
Algebraic Equations ◽  
Initial Value ◽  
Painleve Equations ◽  
Matrix Operations ◽  
Linear Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
The Matrix ◽  
Collocation Scheme

In this work, Chebyshev orthogonal polynomials are employed as basis functions in the collocation scheme to solve the nonlinear Painlevé initial value problems known as the first and second Painlevé equations. Using the collocation points, representing the solution and its fractional derivative (in the Caputo sense) in matrix forms, and the matrix operations, the proposed technique transforms a solution of the initial-value problem for the Painlevé equations into a system of nonlinear algebraic equations. To get ride of nonlinearlity, the technique of quasi-linearization is also applied, which converts the equations into a sequence of linear algebraic equations. The accuracy and efficiency of the presented methods are investigated by some test examples and a comparison has been made with some existing available numerical schemes.

Sign in / Sign up

Export Citation Format

Share Document