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 Stability of orthotropic plates

2020 ◽  
Vol 166 ◽  
pp. 06004
Author(s):  
Mykola Surianinov ◽  
Dina Lazarieva ◽  
Iryna Kurhan
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Boundary Conditions ◽  
Critical Loads ◽  
Orthotropic Plate ◽  
Algebraic Equations ◽  
Stability Equation ◽  
Element Analysis ◽  
Linear Algebraic Equations ◽  
The Stability

The solution to the problem of the stability of a rectangular orthotropic plate is described by the numerical-analytical method of boundary elements. As is known, the basis of this method is the analytical construction of the fundamental system of solutions and Green’s functions for the differential equation (or their system) for the problem under consideration. To account for certain boundary conditions, or contact conditions between the individual elements of the system, a small system of linear algebraic equations is compiled, which is then solved numerically. It is shown that four combinations of the roots of the characteristic equation corresponding to the differential equation of the problem are possible, which leads to the need to determine sixty-four analytical expressions of fundamental functions. The matrix of fundamental functions, which is the basis of the transcendental stability equation, is very sparse, which significantly improves the stability of numerical operations and ensures high accuracy of the results. An analysis of the numerical results obtained by the author’s method shows very good convergence with the results of finite element analysis. For both variants of the boundary conditions, the discrepancy for the corresponding critical loads is almost the same, and increases slightly with increasing critical load. Moreover, this discrepancy does not exceed one percent. It is noted that under both variants of the boundary conditions, the critical loads calculated by the boundary element method are less than in the finite element calculations. The obtained transcendental stability equation allows to determine critical forces both by the static method and by the dynamic one. From this equation it is possible to obtain a spectrum of critical forces for a fixed number of half-waves in the direction of one of the coordinate axes. The proposed approach allows us to obtain a solution to the stability problem of an orthotropic plate under any homogeneous and inhomogeneous boundary conditions.

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 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.

Unsteady flow in a supersonic cascade with strong in-passage shocks

1977 ◽  
Vol 83 (3) ◽  
pp. 569-604 ◽  
Author(s):  
M. E. Goldstein ◽  
Willis Braun ◽  
J. J. Adamczyk
Keyword(s):  
Unsteady Flow ◽  
Aircraft Engine ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Trailing Edges ◽  
Linearized Theory ◽  
Closed Form Analytical Solution ◽  
Supersonic Region ◽  
Shock Motion ◽  
The Stability

Linearized theory is used to study the unsteady flow in a supersonic cascade with in-passage shock waves. We use the Wiener–Hopf technique to obtain a closed-form analytical solution for the supersonic region. To obtain a solution for the rotational flow in the subsonic region we must solve an infinite set of linear algebraic equations. The analysis shows that it is possible to correlate quantitatively the oscillatory shock motion with the Kutta condition at the trailing edges of the blades. This feature allows us to account for the effect of shock motion on the stability of the cascade.Unlike the theory for a completely supersonic flow, the present study predicts the occurrence of supersonic bending flutter. It therefore provides a possible explanation for the bending flutter that has recently been detected in aircraft-engine compressors at higher blade loadings.

On splitting for cubic spline wavelets with four zero moments on an interval

2021 ◽  
pp. 72-87
Author(s):  
Борис Михайлович Шумилов
Keyword(s):  
Cubic Spline ◽  
Spline Space ◽  
Dirichlet Boundary Conditions ◽  
Second Derivative ◽  
Algebraic Equations ◽  
Splitting Algorithm ◽  
Spline Wavelets ◽  
Wavelet Space ◽  
Linear Algebraic Equations ◽  
The Stability

В пространстве кубических сплайнов построены вейвлеты, удовлетворяющие однородным граничным условиям Дирихле и обнулению первых четырех моментов. Получены неявные соотношения, связывающие сплайн-коэффициенты разложения на начальном уровне со сплайн-коэффициентами и вейвлет-коэффициентами на вложенном уровне ленточной системой линейных алгебраических уравнений с невырожденной матрицей. После расщепления на четные и нечетные уравнения матрица преобразования имеет пять (вместо трех в случае двух нулевых моментов) диагоналей. Доказано наличие строгого диагонального доминирования по столбцам. Для сравнения использованы вейвлеты с двумя нулевыми моментами и интерполяционные кубические сплайновые вейвлеты. Результаты численных экспериментов показывают, что схема с четырьмя нулевыми моментами точнее при аппроксимации функций, но грубее при аппроксимации второй производной. The article examines the problem of constructing a splitting algorithm for cubic spline wavelets. First, a cubic spline space is constructed for splines with homogeneous Dirichlet boundary conditions. Then, using the first four zero moments, the corresponding wavelet space is constructed. The resulting space consists of cubic spline wavelets that satisfy the orthogonality conditions for all thirddegree polynomials. The originality of the research lies in obtaining implicit relations connecting the coefficients of the spline expansion at the initial level with the spline coefficients and wavelet coefficients at the embedded level by a band system of linear algebraic equations with a nondegenerate matrix. Excluding the even rows of the system, the resulting transformation algorithm is obtained as a solution to a sequence of band systems of linear algebraic equations with five (instead of three in the case of two zero moments) diagonals. The presence of strict diagonal dominance over the columns is proved, which confirms the stability of the computational process. For comparison, we adopt the results of calculations using wavelets orthogonal to first-degree polynomials and interpolating cubic spline wavelets with the property of the best mean-square approximation of the second derivative of the function being approximated. The results of numerical experiments show that the scheme with four zero moments is more accurate in the approximation of functions, but becomes inferior in accuracy to the approximation of the second derivative.

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 On the seven-diagonals splitting for the cubic spline wavelets with six vanishing moments on an interval

2021 ◽  
Vol 2099 (1) ◽  
pp. 012016
Author(s):  
B M Shumilov
Keyword(s):  
Cubic Spline ◽  
Dirichlet Boundary Conditions ◽  
Algebraic Equations ◽  
Splitting Algorithm ◽  
Discrete Function ◽  
Vanishing Moments ◽  
Spline Wavelets ◽  
Linear Algebraic Equations ◽  
Comparative Results ◽  
The Stability

Abstract This study uses a zeroing property of the first six moments for constructing a splitting algorithm for the cubic spline wavelets. First, we construct a system of cubic basic spline-wavelets, realizing orthogonal conditions to all polynomials up to any degree. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the orthogonality to all polynomials up to the fifth degree on the closed interval. The originality of the study consists of obtaining implicit finite relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, the resulting transformation matrix has seven diagonals, instead of five as in the previous case with four zero moments. A modification of the system is performed, which ensures a strict diagonal dominance, and, consequently, the stability of the calculations. The comparative results of numerical experiments on approximating and calculating the derivatives of a discrete function are presented.

Unrestricted Harmonic Balance

1980 ◽  
Vol 35 (10) ◽  
pp. 1054-1061 ◽  
Author(s):  
Friedrich Franz Seelig
Keyword(s):  
Linear Systems ◽  
Harmonic Balance ◽  
Open Systems ◽  
Periodic Structures ◽  
Simulation Method ◽  
Algebraic Equations ◽  
Good Convergence ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
The Stability

Abstract Periodic structures in chemical kinetic systems can be evaluated by an extension of the well-known method of harmonic balance, which yields very simple expressions in the case of linear systems containing only zero and first order reactions. The far more interesting non-linear systems containing e.g. second order reactions which in case of open systems far from thermodynamic equilibrium give rise to non-classical phenomena like oscillations, chemical waves, excitability, hysteresis, multistability, dissipative structures etc. can be treated in a similar way by introducing new pseudo-linear quantities utilizing certain group properties of harmonic expansions. The resulting complicated implicit non-linear algebraic equations are solved by a method developed by Powell and show good convergence. Since this method - in contrast to the conventional method of simulation - is independent from the stability of the periodic structure to be evaluated it can even be applied to unstable cases where the simulation method necessarily fails. An evaluation of the stability is included in the developed computer program.

scholarly journals Common Lyapunov Function Based on Kullback–Leibler Divergence for a Switched Nonlinear System

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Omar M. Abou Al-Ola ◽  
Ken'ichi Fujimoto ◽  
Tetsuya Yoshinaga
Keyword(s):  
Lyapunov Function ◽  
Switched Systems ◽  
Switched System ◽  
Algebraic Equations ◽  
Common Lyapunov Function ◽  
Arbitrary Switching ◽  
Switched Nonlinear System ◽  
Linear Algebraic Equations ◽  
Leibler Divergence ◽  
The Stability

Many problems with control theory have led to investigations into switched systems. One of the most urgent problems related to the analysis of the dynamics of switched systems is the stability problem. The stability of a switched system can be ensured by a common Lyapunov function for all switching modes under an arbitrary switching law. Finding a common Lyapunov function is still an interesting and challenging problem. The purpose of the present paper is to prove the stability of equilibrium in a certain class of nonlinear switched systems by introducing a common Lyapunov function; the Lyapunov function is based on generalized Kullback–Leibler divergence or Csiszár'sI-divergence between the state and equilibrium. The switched system is useful for finding positive solutions to linear algebraic equations, which minimize theI-divergence measure under arbitrary switching. One application of the stability of a given switched system is in developing a new approach to reconstructing tomographic images, but nonetheless, the presented results can be used in numerous other areas.

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