Vibration of Elastic Structures With Cracks

1993 ◽  
Vol 60 (2) ◽  
pp. 414-421 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
The Other ◽  
Internal Crack ◽  
Two Dimensional ◽  
Antiplane Strain ◽  
Elastic Solids ◽  
Helmholtz Operator ◽  
Crack Configuration ◽  
Analytical Algorithm

An analytical algorithm is proposed to represent eigensolutions [λm2, ψm(r)]m=1∞ of an imperfect structure C containing cracks in terms of crack configuration σc and eigensolutions [ωn2, φn(r)]n=1∞ of a perfect structured without P the cracks. To illustrate this algorithm on mechanical systems governed by the two-dimensional Helmholtz operator, the Green’s identity and Green’sfunction of P are used to represent ψm(r) in terms of an infinite series of φn(r). Substitution of the ψn(r) representation into the Kamke quotient of C and stationarity of the quotient result in a matrix Fredholm integral equation. The eigensolutions of the Fredholm integral equation then predict λm2 and ψm(r) of C. Finally, eigensolutions of two rectangular elastic solids under antiplane strain vibration, one with a boundary crack and the other with an oblique internal crack, are calculated numerically.

Perturbation Eigensolutions of Elastic Structures With Cracks

1993 ◽  
Vol 60 (2) ◽  
pp. 438-442 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Integral Equation ◽  
Crack Opening ◽  
Mechanical Systems ◽  
Rayleigh Quotient ◽  
Internal Crack ◽  
Perturbation Approach ◽  
Two Dimensional ◽  
Antiplane Strain ◽  
Elastic Solids ◽  
Integral Equation Approach

The purpose of this paper is to determine approximate eigensolutions of a class of cracked mechanical systems governed by the two-dimensional Helmholtz equation through a perturbation approach. Shen (1993) shows that exact eigenvalues λm2, and their corresponding crack-opening shapes ΔΨm of such mechanical systems satisfy a Fredholm integral equation A(λm2)ΔΨm = 0. Following the integral equation approach, the approximation in this paper consists of formulating the Rayleigh quotient of the Fredholm operator A(λ2) and estimating eigenvalues μ(λ2) of the operator A(λ2) through perturbation and stationarity of the Rayleigh quotient. The zeros of μ(λ2) then approximate eigenvalues λm2 of the cracked systems. Finally, approximate λm2 are calculated for two-dimensional elastic solids under antiplane-strain vibration with an oblique internal crack and a boundary crack.

Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet

2016 ◽  
Vol 14 (01) ◽  
pp. 1650004 ◽  
Author(s):  
M. Tahami ◽  
A. Askari Hemmat ◽  
S. A. Yousefi
Keyword(s):  
Integral Equation ◽  
Tensor Product ◽  
Numerical Solution ◽  
Fredholm Integral Equation ◽  
Fredholm Integral Equations ◽  
Wavelet Basis ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
One Dimensional

In one-dimensional problems, the Legendre wavelets are good candidates for approximation. In this paper, we present a numerical method for solving two-dimensional first kind Fredholm integral equation. The method is based upon two-dimensional linear Legendre wavelet basis approximation. By applying tensor product of one-dimensional linear Legendre wavelet we construct a two-dimensional wavelet. Finally, we give some numerical examples.

A METHOD FOR MODELING THE RESISTIVITY AND IP RESPONSE OF TWO‐DIMENSIONAL BODIES

Geophysics ◽  
1976 ◽  
Vol 41 (5) ◽  
pp. 997-1015 ◽  
Author(s):  
Donald D. Snyder
Keyword(s):  
Integral Equation ◽  
Fourier Transform ◽  
Boundary Value Problem ◽  
Method Of Moments ◽  
Fredholm Integral Equation ◽  
Electric Potential ◽  
Inverse Fourier Transform ◽  
Two Dimensional ◽  
Numerical Techniques ◽  
Point Of Interest

A method has been developed for the solution of the resistivity and IP modeling problem for one or more two‐dimensional inhomogeneities buried in a space for which the Dirichlet Green’s function is known. The boundary‐value problem reduces to a Fredholm integral equation of the second kind which is parametrically a function of a spatial wavenumber. Using the method of moments, the integral equation is solved for a number of values of the wavenumber. An inverse Fourier transform is then performed in order to obtain the electric potential at any point of interest. The method agrees well with both experimental results and other numerical techniques.

Elasticity Solution of Deep Beam Under Settlement of Supports

1976 ◽  
Vol 43 (4) ◽  
pp. 613-618 ◽  
Author(s):  
S. N. Prasad ◽  
P. K. Chiu
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
The Other ◽  
Tangential Displacement ◽  
Deep Beam ◽  
Two Dimensional ◽  
Singular Behavior ◽  
Elasticity Solution ◽  
Collocation Technique

The present study investigates the two-dimensional stresses and displacements in a finite rectangle whose one set of parallel edges is given a relative tangential displacement by means of rigidly attached planes. The other set of parallel edges is free from tractions. The problem is formulated in terms of a singular integral equation of the first kind, which yields the correct singular behavior of stresses at the corners. The integral equation is solved numerically by employing Gauss-Jacobi quadrature in conjunction with certain collocation technique. Numerical results of quantities of practical interests are shown graphically and also compared with the classical bending analysis.

scholarly journals Total Ponderomotive Force on an Extended Test Body

2009 ◽  
Vol 2009 ◽  
pp. 1-12
Author(s):  
D. Langemann
Keyword(s):  
Integral Equation ◽  
Series Expansion ◽  
Fredholm Integral Equation ◽  
Ponderomotive Force ◽  
Test Body ◽  
Total Force ◽  
Two Dimensional ◽  
Insulating Material ◽  
Fourier Techniques ◽  
Numerical Effort

Droplets on insulating material suffer a nonvanishing total ponderomotive force because of the inhomogeneity of the surrounding electric field. A series expansion of this total force is proven in a two-dimensional setting by determining the line charge density at the boundary of the test body via a Fredholm integral equation, which is solved by Fourier techniques. The influence of electric charges in the neighborhood of the test body can be estimated as well as the convergence speed of the series expansion. In all realistic applications the series converges very fast. The numerical effort in the simulation of the motion of rainwater droplets on outdoor insulators reduces considerably.

scholarly journals A Triangle Algorithm of Padé-Type Approximant for Two-Dimensional Fredholm Integral Equations of the Second Kind

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Meilan Sun ◽  
Chuanqing Gu
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Recursive Algorithm ◽  
High Order ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Initial Value ◽  
Numerical Examples ◽  
Type Approximation

The function-valued Padé-type approximation (2DFPTA) is used to solve two-dimensional Fredholm integral equation of the second kind. In order to compute 2DFPTA, a triangle recursive algorithm based on Sylvester identity is proposed. The advantage of this algorithm is that, in the process of calculating 2DFPTA to avoid the calculation of the determinant, it can start from the initial value, from low to high order, and gradually proceeds. Compared with the original two methods, the numerical examples show that the algorithm is effective.

scholarly journals Solving fuzzy volterra-fredholm integral equation by fuzzy artificial neural network

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Seiyed Hadi Abtahi ◽  
Hamidreza Rahimi ◽  
Maryam Mosleh
Keyword(s):  
Neural Network ◽  
Integral Equation ◽  
Artificial Neural Network ◽  
Fredholm Integral Equation ◽  
The Other ◽  
Programming Algorithm ◽  
Effective Solution ◽  
Equation Modelling ◽  
Engineering Problems ◽  
Artificial Neural

<p style='text-indent:20px;'>The volterra-fredholm integral equation in all forms are arose from physics, biology and engineering problems which is derived from differential equation modelling. On the other hand, the trained programming algorithm by the fuzzy artificial neural networks has effective solution to find the best answer. In this article we try to estimate the equation and its answer by developed fuzzy artificial neural network to fuzzy volterra-fredholme integral. Our attempts would lead to benchmark other extended forms of this type of equation.</p>

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