Dynamic Behavior of a Cylindrical Shell with a Liquid under the Action of Nonstationary Pressure Wave

The problem of axisymmetric hydroelastic deformation of a thin cylindrical shell containing a liquid under the action of a moving load is approximately solved. It is reduced to the equation of bending of the shell and the condition of incompressibility of the liquid in the cylinder. The deflections of the shell and the level of lowering of the liquid are unknown. For solution, the Galerkin method is used and the problem is reduced to a system of nonlinear algebraic equations. A simpler solution is considered without taking into account the incompressibility condition. Here, in addition to the deformed state of the shell, the critical speeds of the moving load are determined analytically.

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A Theoretical Series Solution for the Dynamics of Finite-Length Bearings and Squeeze-Film Dampers

Flow-Induced Vibration ◽

10.1115/pvp2003-2094 ◽

2003 ◽

Author(s):

Jose´ Antunes ◽

Miguel Moreira ◽

Philippe Piteau

Keyword(s):

Fourier Series ◽

Galerkin Method ◽

Finite Length ◽

Reynolds Equation ◽

Double Fourier Series ◽

Squeeze Film ◽

Algebraic Equations ◽

Squeeze Film Dampers ◽

Galerkin Solution ◽

The Galerkin Method

In this paper we develop a non-linear dynamical solution for finite length bearings and squeeze-film dampers based on a Spectral-Galerkin method. In this approach the gap-averaged pressure is approximated, in the lubrication Reynolds equation, by a truncated double Fourier series. The Galerkin method, applied over the residuals so obtained, generate a set of simultaneous algebraic equations for the time-dependent coefficients of the double Fourier series for the pressure. In order to assert the validity of our 2D–Spectral-Galerkin solution we present some preliminary comparative numerical simulations, which display satisfactory results up to eccentricities of about 0.9 of the reduced fluid gap H/R. The so-called long and short-bearing dynamical solutions of the Reynolds equation, reformulated in Cartesian coordinates, are also presented and compared with the corresponding classic solutions found on literature.

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Wavelets Galerkin Method for the Fractional Subdiffusion Equation

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4034391 ◽

2016 ◽

Vol 11 (6) ◽

Cited By ~ 3

Author(s):

M. H. Heydari

Keyword(s):

Galerkin Method ◽

Fractional Derivatives ◽

Operational Matrix ◽

Practical Importance ◽

Time Derivative ◽

Algebraic Equations ◽

Legendre Wavelets ◽

Equivalent Problem ◽

Subdiffusion Equation ◽

The Galerkin Method

The time fractional subdiffusion equation (FSDE) as a class of anomalous diffusive systems has obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order 0<α<1. Since analytically solving this problem is often impossible, proposing numerical methods for its solution has practical importance. In this paper, an efficient and accurate Galerkin method based on the Legendre wavelets (LWs) is proposed for solving this equation. The time fractional derivatives are described in the Riemann–Liouville sense. To do this, we first transform the original subdiffusion problem into an equivalent problem with fractional derivatives in the Caputo sense. The LWs and their fractional operational matrix (FOM) of integration together with the Galerkin method are used to transform the problem under consideration into the corresponding linear system of algebraic equations, which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account, automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

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Dynamics of a cylindrical shell with a collapsing elastic base under the action of a pressure wave

INCAS BULLETIN ◽

10.13111/2066-8201.2019.11.s.3 ◽

2019 ◽

Vol 11 (S) ◽

pp. 25-31 ◽

Cited By ~ 10

Author(s):

Boris A. ANTUFEV ◽

Olga V. EGOROVA ◽

Lev N. RABINSKIY

Keyword(s):

Cylindrical Shell ◽

Pressure Wave ◽

Rocket Engine ◽

Elastic Base ◽

The Body ◽

Powder Charge ◽

Solid Propellant Rocket ◽

Solid Propellant Rocket Engine ◽

Deformed State ◽

The Moment

In the dynamic and quasi-static statements, the issue of non-stationary deformation and stability of the solid propellant rocket engine (SPRE) was approximately solved. It is modeled by a thin, smooth cylindrical shell, inside of which, on a part of its length, there is an elastic base corresponding to a gradually burning powder charge. A pressure wave is moving along the outer surface of the body, simulated by the running load. The deformed state of the shell is considered axisymmetric and is determined on the basis of the moment theory of the shells. For diverse variants of mounting the ends of the shell in a closed form, expressions were obtained for the critical velocity of the load. Examples were considered.

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The Application of the Galerkin Method to the Solution of the Symmetric and Balanced Counterflow Regenerator Problem

Journal of Heat Transfer ◽

10.1115/1.3247381 ◽

1985 ◽

Vol 107 (1) ◽

pp. 214-221 ◽

Cited By ~ 22

Author(s):

B. S. Baclic

Keyword(s):

Galerkin Method ◽

Space Variable ◽

Analytical Formula ◽

Algebraic Equations ◽

Higher Order Terms ◽

The Matrix ◽

Expansion Coefficients ◽

The Galerkin Method ◽

Analytical Expressions

The Galerkin method is applied to solve the symmetric and balanced counterflow thermal regenerator problem. In this approach, the integral equation, expressing the reversal condition in periodic equilibrium of regenerator matrix, is transformed into a set of algebraic equations for the determination of the expansion coefficients associated with the representation of the matrix temperature distribution at the start of cold period in a power series in the space variable. The method is easy and straightforward to apply and leads to the explict analytical expressions for expansion coefficients. As explicit analytical formula for regenerator effectiveness is derived and the corresponding numerical values are computed. An excellent agreement is found between the present results and those reported in the literature by different numerical methods. The convergence towards the exact results by carrying out the computations to higher order terms, as well as the extension of this method to the more general counterflow regenerator problem is discussed.

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The Gegenbauer Wavelets-Based Computational Methods for the Coupled System of Burgers’ Equations with Time-Fractional Derivative

Mathematics ◽

10.3390/math7060486 ◽

2019 ◽

Vol 7 (6) ◽

pp. 486 ◽

Cited By ~ 1

Author(s):

Neslihan Ozdemir ◽

Aydin Secer ◽

Mustafa Bayram

Keyword(s):

Fractional Derivative ◽

Galerkin Method ◽

Collocation Method ◽

Fractional Integration ◽

Operational Matrix ◽

Coupled System ◽

Algebraic Equations ◽

Burgers Equations ◽

Time Fractional Derivative ◽

The Galerkin Method

In this study, Gegenbauer wavelets are used to present two numerical methods for solving the coupled system of Burgers’ equations with a time-fractional derivative. In the presented methods, we combined the operational matrix of fractional integration with the Galerkin method and the collocation method to obtain a numerical solution of the coupled system of Burgers’ equations with a time-fractional derivative. The properties of Gegenbauer wavelets were used to transform this system to a system of nonlinear algebraic equations in the unknown expansion coefficients. The Galerkin method and collocation method were used to find these coefficients. The main aim of this study was to indicate that the Gegenbauer wavelets-based methods is suitable and efficient for the coupled system of Burgers’ equations with time-fractional derivative. The obtained results show the applicability and efficiency of the presented Gegenbaur wavelets-based methods.

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A New Wavelet Method for Solving a Class of Nonlinear Volterra-Fredholm Integral Equations

Abstract and Applied Analysis ◽

10.1155/2014/975985 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Xiaomin Wang

Keyword(s):

Galerkin Method ◽

Integral Equations ◽

Approximation Scheme ◽

Nonlinear Term ◽

Fredholm Integral Equations ◽

Algebraic Equations ◽

New Approach ◽

Wavelet Approximation ◽

Connection Coefficients ◽

Nonlinear Algebraic Equations

A new approach, Coiflet-type wavelet Galerkin method, is proposed for numerically solving the Volterra-Fredholm integral equations. Based on the Coiflet-type wavelet approximation scheme, arbitrary nonlinear term of the unknown function in an equation can be explicitly expressed. By incorporating such a modified wavelet approximation scheme into the conventional Galerkin method, the nonsingular property of the connection coefficients significantly reduces the computational complexity and achieves high precision in a very simple way. Thus, one can obtain a stable, highly accurate, and efficient numerical method without calculating the connection coefficients in traditional Galerkin method for solving the nonlinear algebraic equations. At last, numerical simulations are performed to show the efficiency of the method proposed.

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An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation

Advances in Difference Equations ◽

10.1186/s13662-019-2297-8 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 1

Author(s):

Aydin Secer ◽

Neslihan Ozdemir

Keyword(s):

Galerkin Method ◽

Operational Matrix ◽

Computational Method ◽

Test Problems ◽

Algebraic Equations ◽

Dispersion Parameters ◽

Wavelet Galerkin Method ◽

Operational Matrix Of Integration ◽

Nonlinear Physical Phenomena ◽

The Galerkin Method

Abstract In this paper, our purpose is to present a wavelet Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation, which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters. The presented computational method in this paper is based on Gegenbauer wavelets. Gegenbauer wavelets have useful properties. Gegenbauer wavelets and the operational matrix of integration, together with the Galerkin method, were used to transform the time-fractional KBK equation into the corresponding nonlinear system of algebraic equations, which can be solved numerically with Newton’s method. Our aim is to show that the Gegenbauer wavelets-based method is efficient and powerful tool for solving the KBK equation with time-fractional derivative. In order to compare the obtained numerical results of the wavelet Galerkin method with exact solutions, two test problems were chosen. The obtained results prove the performance and efficiency of the presented method.

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The Galerkin Method for Solving Radiation Transfer in Plane-Parallel Participating Media

Journal of Heat Transfer ◽

10.1115/1.3245095 ◽

1982 ◽

Vol 104 (2) ◽

pp. 351-354 ◽

Cited By ~ 45

Author(s):

M. N. O¨zis¸ik ◽

Y. Yener

Keyword(s):

Galerkin Method ◽

Integral Form ◽

Computer Time ◽

Incident Radiation ◽

Algebraic Equations ◽

Participating Media ◽

Plane Parallel ◽

Reflecting Boundaries ◽

Expansion Coefficients ◽

The Galerkin Method

The Galerkin method is applied to solve radiative heat transfer in an isotropically scattering, absorbing, and emitting plane-parallel medium with diffusely reflecting boundaries. In this approach, the integral form of the equation of radiative transfer is transformed into a set of algebraic equations for the determination of the expansion coefficients associated with the representation of the incident radiation in a power series in the space variable. The method is easy and straightforward to apply and requires relatively little computer time for the computations, since explicit analytical expressions are obtainable for the expansion coefficients.

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Ritz Legendre Multiwavelet Method for the Damped Generalized Regularized Long-Wave Equation

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4004121 ◽

2011 ◽

Vol 7 (1) ◽

Author(s):

S. A. Yousefi ◽

Z. Barikbin

Keyword(s):

Wave Equation ◽

Galerkin Method ◽

Numerical Method ◽

Variable Coefficient ◽

Algebraic Equations ◽

Long Wave ◽

Regularized Long Wave Equation ◽

The Galerkin Method

In this paper, a numerical method is proposed to approximate the solution of the nonlinear damped generalized regularized long-wave (DGRLW) equation with a variable coefficient. The method is based upon Ritz Legendre multiwavelet approximations. The properties of Legendre multiwavelet are first presented. These properties together with the Galerkin method are then utilized to reduce the nonlinear DGRLW equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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An Analytical Buckling Load Formula for a Cylindrical Shell Subjected to Local Axial Compression

Journal of Pressure Vessel Technology ◽

10.1115/1.4052976 ◽

2021 ◽

Author(s):

Licai Yang ◽

Yuguang Li ◽

Tian Qiu ◽

Yuanyuan Dong ◽

Shanglin Zhang

Keyword(s):

Cylindrical Shell ◽

Galerkin Method ◽

Axial Compression ◽

Analytical Formula ◽

Local Buckling ◽

Pressure Vessels ◽

Buckling Load ◽

Perfect Agreement ◽

Buckling Modes ◽

The Galerkin Method

Abstract This paper proposes an analytical buckling load formula for a cylindrical shell subjected to local axial compression for the first time, which is achieved by a carefully constructed load description and perturbation procedure. Local axial load is described by introducing an arctangent function firstly. Then, the analytical solutions of local buckling load coefficients and buckling modes for a locally compressed shell are derived after solving governing differential equations by the perturbation method. For validation, using the presented analytical buckling modes, the Galerkin method is applied to obtain numerical result, which is an infinite order determinant about local buckling load coefficient. Comparative calculation results show that local buckling load coefficients by the analytical formula are in perfect agreement with numerical ones by the Galerkin method and known results in literature. Therefore, the validity and accuracy of the presented formula are verified. Engineering application of the analytical formula is also discussed to evaluate local buckling loads of thin-walled cylindrical shell structures such as silos, pressure vessels, large storage tanks and so on.

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