scholarly journals Vibrations of dam–plate of a hydro-technical structure under seismic load

2021 ◽  
Vol 264 ◽  
pp. 05057
Author(s):  
A Tukhtaboev ◽  
Sergey Leonov ◽  
Fozil Turaev ◽  
Kudrat Ruzmetov
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Numerical Method ◽  
Quadrature Formulas ◽  
Seismic Load ◽  
Geometrically Nonlinear ◽  
Material Parameters ◽  
Weakly Singular ◽  
Technical Structure ◽  
Nonlinear Statement

In present paper, the problem of the vibration of a viscoelastic dam-plate of a hydro-technical structure is investigated, based on the Kirchhoff-Love hypothesis in the geometrically nonlinear statement. This problem is reduced to a system of nonlinear ordinary integro-differential equations by using the Bubnov-Galerkin method. The resulting system with a weakly-singular Koltunov-Rzhanitsyn kernel is solved using a numerical method based on quadrature formulas. The behavior of the viscoelastic dam-plate of hydro-technical structure is studied for the wide ranges of physical, mechanical, and geometrical material parameters.

scholarly journals NUMERICAL METHOD OF CALCULATION OF ROUND PLATES IN A GEOMETRICALLY NONLINEAR STATEMENT

Vestnik MGSU ◽  
2017 ◽  
pp. 631-635
Author(s):  
Radek Fatykhovich Gabbasov ◽  
Natalia Borisovna Uvarova
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Geometrically Nonlinear ◽  
Method Of Calculation ◽  
First Order ◽  
Nonlinear Statement ◽  
Minimum Number ◽  
Dead Loading ◽  
The Right ◽  
The Given

The article considers the axisymmetric problem about the calculation of round plates with dead loading in a geometrically nonlinear system. To solve the problem some generalized equations of finite difference method (FMD) are needed that allow to solve tasks within intergrable scope taking into account discontinuities of the required function, its first-order derivative and the right-hand side of the primitive differential equation. Resolvent differential equations of the question comprised fractionally the required function of the inflection and stresses are reduced to four differential equations, two of which are linear of the first-order and two are nonlinear of the second order. The obtained system of differential equations is solved numerically. The proposed method is shown with the example of calculation of a round plate; the given data are taken from work [1]. The calculation data with the minimum number of partitions are compared to the known solution of A.S. Vol’mir [1] and they indicate the possibility of using a numerical method for handling the problem in nonlinear statement.

scholarly journals Calculation of plates in a geometrically nonlinear setting with the use of generalized equations of finite difference method

2018 ◽  
Vol 196 ◽  
pp. 01024 ◽  
Author(s):  
Natalia Uvarova ◽  
Radek Gabbasov
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Geometrically Nonlinear ◽  
Generalized Equations ◽  
Nonlinear Formulation ◽  
First Derivative ◽  
Equation Solving ◽  
Flexible Plates ◽  
Minimum Number ◽  
The Right

The article proposes a numerical method and an algorithm for analysis rectangular flexible plates in a geometrically nonlinear formulation. The generalized equations of the method of finite differences (MD) are used to solve the problem within the integrable region taking into account the discontinuities of the desired function, its first derivative and the right part of the original differential equation. Solving differential equations of the problem, composed with respect to the desired functions of deflection and stress are reduced to the 4th differential equations of the second order, which are solved numerically. As an example, a square plate loaded with a uniformly distributed load is considered. The results of the calculation with a minimum number of partitions are compared with the known analytical solution of A. S. Volmir [1] and indicate the possibility of using the numerical method for solving problems in a nonlinear formulation.

Nonlinear Vibrations of Viscoelastic Composite Cylindrical Panels

2006 ◽  
Vol 129 (3) ◽  
pp. 285-296 ◽  
Author(s):  
Bakhtiyor Eshmatov ◽  
Subrata Mukherjee
Keyword(s):  
Numerical Method ◽  
Nonlinear Vibrations ◽  
Rheological Parameters ◽  
Quadrature Formulas ◽  
Relaxation Kernel ◽  
Viscoelastic Composite ◽  
Nonlinear Statement ◽  
Cylindrical Panels ◽  
Singular Kernels ◽  
Waves Propagation

This paper is devoted to mathematical models of problems of nonlinear vibrations of viscoelastic, orthotropic, and isotropic cylindrical panels. The models are based on Kirchhoff-Love hypothesis and Timoshenko generalized theory (including shear deformation and rotatory inertia) in a geometrically nonlinear statement. A choice of the relaxation kernel with three rheological parameters is justified. A numerical method based on the use of quadrature formulas for solving problems in viscoelastic systems with weakly singular kernels of relaxation is proposed. With the help of the Bubnov-Galerkin method in combination with a numerical method, the problems in nonlinear vibrations of viscoelastic orthotropic and isotropic cylindrical panels are solved using the Kirchhoff-Love and Timoshenko hypothesis. Comparisons of the results obtained by these theories, with and without taking elastic waves propagation into account, are presented. In all problems, the convergence of Bubnov-Galerkin’s method has been investigated. The influences of the viscoelastic and anisotropic properties of a material, on the process of vibration, are discussed in this work.

scholarly journals ON FULLY DISCRETE COLLOCATION METHODS FOR SOLVING WEAKLY SINGULAR INTEGRO‐DIFFERENTIAL EQUATIONS

2010 ◽  
Vol 15 (1) ◽  
pp. 69-82 ◽  
Author(s):  
Raul Kangro ◽  
Enn Tamme
Keyword(s):  
Differential Equations ◽  
Convergence Rates ◽  
Approximate Solutions ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Fully Discrete ◽  
Weakly Singular ◽  
Good Accordance ◽  
Theoretical Results ◽  
Simple Quadrature

In order to find approximate solutions of Volterra and Fredholm integro‐differential equations by collocation methods it is necessary to compute certain integrals that determine the required algebraic systems. Those integrals usually can not be computed exactly and if the kernels of the integral operators are not smooth, simple quadrature formula approximations of the integrals do not preserve the convergence rate of the collocation method. In the present paper fully discrete analogs of collocation methods where non‐smooth integrals are replaced by appropriate quadrature formulas approximations, are considered and corresponding error estimates are derived. Presented numerical examples display that theoretical results are in a good accordance with the actual convergence rates of the proposed algorithms.

scholarly journals Nonlinear oscillations of a viscoelastic cylindrical panel with concentrated masses

2018 ◽  
Vol 245 ◽  
pp. 01001 ◽  
Author(s):  
Dadakhan Khodzhaev D.A ◽  
Rustamkhan Abdikarimov ◽  
Nikolay Vatin
Keyword(s):  
Numerical Method ◽  
Nonlinear Oscillation ◽  
Viscoelastic Properties ◽  
Nonlinear Oscillations ◽  
Nonlinear Problems ◽  
Equation Of Motion ◽  
Cylindrical Panel ◽  
Geometrically Nonlinear ◽  
Nonlinear Statement ◽  
Concentrated Masses

The problems of oscillations of a viscoelastic cylindrical panel with concentrated masses are investigated, based on the Kirchhoff-Love hypothesis in the geometrically nonlinear statement. The effect of the action of concentrated masses is introduced into the equation of motion of the cylindrical panel using the δ function. To solve integro-differential equations of nonlinear problems of the dynamics of viscoelastic systems, a numerical method is suggested. With the Bubnov–Galerkin method, based on a polynomial approximation of the deflection, in combination with the suggested numerical method, the problems of nonlinear oscillation of a viscoelastic cylindrical panel with concentrated masses were solved. Bubnov–Galerkin’s convergence was studied in all problems. The influence of the viscoelastic properties of the material and concentrated masses on the process of oscillations of a cylindrical panel is shown.

Fully Discrete Galerkin Method For Fredholm Integro-Differential Equations With Weakly Singular Kernels

2008 ◽  
Vol 8 (3) ◽  
pp. 207-222 ◽  
Author(s):  
H. BRUNNER
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Optimal Order ◽  
Piecewise Polynomial ◽  
Fully Discrete ◽  
Proportional Delays ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Discrete Galerkin Method ◽  
Polynomial Collocation

AbstractWe analyze the optimal superconvergence properties of piecewise polynomial collocation solutions on uniform meshes for Volterra integral and integrodifferential equations with multiple (vanishing) proportional delays. It is shown that for delay integro-differential equations the recently obtained optimal order is also attainable. For integral equations with multiple vanishing delays this is no longer true.

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