Contact Stresses Between Layered Elastic Cylinders

1984 ◽  
Vol 106 (3) ◽  
pp. 396-404 ◽  
Author(s):  
R. Solecki ◽  
Y. Ohgushi
Keyword(s):  
Integral Equation ◽  
Pressure Distribution ◽  
Elastic Cylinder ◽  
Algebraic Equations ◽  
Contact Pressure Distribution ◽  
Rigid Cylinder ◽  
Linear Algebraic Equations ◽  
Static Contact ◽  
Contact Width ◽  
Elastic Cylinders

Frictionless static contact between two layered elastic cylinders is examined. Local curvature of the the contacting surfaces is taken into account. Green’s function is derived and utilized to reduce the contact problem to Fredholm integral equation of the first kind with unknown contact width, unknown contact pressure distribution and unknown normal approach. Solution of this integral equation is obtained in the form of an infinite system of linear algebraic equations with the aid of Fourier transformation. The resulting equations are solved to give the pressure distribution and the normal approach for assumed contact width. Numerical results of two examples are presented: one in which an elastic layer bonded to a rigid cylinder is indented by another rigid cylinder, and, a second in which a layered elastic cylinder is indented by another layered elastic cylinder. The results obtained in the first example are compared with analytical solutions and with experimental data known from the literature.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

scholarly journals Numerical solution of a weakly singular integral equation by the method of orthogonal polynomials

2021 ◽  
pp. 98-103
Author(s):  
Sergei M. Sheshko
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Orthogonal Polynomials ◽  
Numerical Solution ◽  
Singular Integral ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
Analytical Expressions

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

The Problem of an Elastic Stiffener Bonded to a Half Plane

1971 ◽  
Vol 38 (4) ◽  
pp. 937-941 ◽  
Author(s):  
F. Erdogan ◽  
G. D. Gupta
Keyword(s):  
Mechanical Properties ◽  
Integral Equation ◽  
Contact Problem ◽  
Half Plane ◽  
Elastic Constants ◽  
Infinite System ◽  
Stress Singularity ◽  
Algebraic Equations ◽  
Numerical Example ◽  
Linear Algebraic Equations

The contact problem of an elastic stiffener bonded to an elastic half plane with different mechanical properties is considered. The governing integral equation is reduced to an infinite system of linear algebraic equations. It is shown that, depending on the value of a parameter which is a function of the elastic constants and the thickness of the stiffener, the system is either regular or quasi-regular. A complete numerical example is given for which the strength of the stress singularity and the contact stresses are tabulated.

Etude numérique de la fonction de Green scalaire d'une cavité à l'aide d'une nouvelle équation intégrale

1979 ◽  
Vol 57 (2) ◽  
pp. 190-207
Author(s):  
Jacques A. Imbeau ◽  
Byron T. Darling
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Double Precision ◽  
Integration Formula ◽  
Algebraic Equations ◽  
Equation Intégrale ◽  
Exterior Region ◽  
Von Neumann ◽  
Linear Algebraic Equations

We study numerically, with the aid of an IBM-370 computer, the Green's functions of a cavity afforded by the solutions of a new integral equation (B. T. Darling and J. A. Imbeau. Can. J. Phys. 56, 387 (1978)). A number of prolate spheroidal cavities whose eccentricities cover the complete range zero to one are employed, and the solutions are subject to the Dirichlet and von Neumann conditions at the surface. We use the Gauss–Legendre integration formula to replace the integral equation by a set of linear algebraic equations. The Green's function is evaluated by substituting the solution of this set in the formula of Helmholtz, using the same integration formula. Criteria for the optimization of this procedure also are developed and employed. The Green's function can be determined to high precision except in the immediate vicinity of the surface of the cavity where it suffers a well-known discontinuity. We also explore the use of the Helmholtz formula itself in the exterior region as an integral equation to obtain the Green's function of the cavity. We find that although the precision of the solution is much less than that afforded by the precedingly mentioned integral equation the precision is still within the range of practical application. All calculations used double precision arithmetic (16 significant digits on the IBM-370).

scholarly journals Comparison of Linear Vortex Panel Method and Finite Volume Method for Calculation of Generated Lift in Potential Flow Over Two-Dimensional Car Bodies

2020 ◽  
Author(s):  
Saeid Moammaei ◽  
Mehran Khaki Jamei ◽  
Morteza Abbasi
Keyword(s):  
Pressure Distribution ◽  
Lift Coefficient ◽  
Control Point ◽  
Panel Method ◽  
The Body ◽  
Aerodynamic Load ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
The Impact

Abstract This paper describes one of the aspects of the panel method to analyze the aerodynamic characteristics of a sedan. The linear vortex panel method has been developed to simulate the ideal flow over a two-dimensional arbitrary car and, it also calculates the aerodynamic load on the body. By satisfying the boundary conditions on each control point, our linear algebraic equations are obtained. The results are sensitive to the distribution of the panels over the body thus the body is broken up equally into very small panels. After solving the set of equations, the vortices strength is obtained and the pressure distribution for the upper and the lower surface of the body is calculated. The impact of the angle of attack on the aerodynamic behavior of the intended car is investigated and it is found that the lift coefficient increases with the free stream angle from -4 to 4. The accuracy of the results has been determined by checking them against the standard CFD data. The pressure distribution trend is found very much in confirmation with the CFD results, however, a discrepancy at the rear end is observed. Therefore, it can be concluded that this method does not seem practical for geometries with steep slopes in the rear part of the car. Finally, both methods are applied to the other modified geometries with lower slopes at the rear section and the results compare well with the fluent.

scholarly journals The method of secondary sources in electrical engineering science and ill-conditioned matrices.

2020 ◽  
Vol 4 ◽  
pp. 82-94
Author(s):  
V.P. Voloboev ◽  
◽  
V.P. Klymenko ◽  
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Discrete Model ◽  
Three Dimensional ◽  
Electric Circuit ◽  
Electrical Circuit ◽  
Algebraic Equations ◽  
New Approach ◽  
Secondary Sources ◽  
Linear Algebraic Equations

A new approach to solving the problem of instability of a system of linear algebraic equations (SLAE) with an ill-conditioned matrix describing a discrete model of the Fredholm integral equation of the sec-ond kind, which reduces the calculation by the method of secondary sources of three-dimensional static and quasi-stationary electromagnetic fields of any geometry in inhomogeneous and nonlinear media, is considered. The essence of the new approach is all about. There is a method for correctly compiling a description of an electrical circuit. In this method, for the first time, when describing an electrical cir-cuit, the parameters of a specific task are taken into account, but they are not taking into account in other methods. As a result, the solution to the problem is stable even in the case of a SLAE with an ill-conditioned matrix. The disadvantage of this method is the description of the electrical circuit in the form of a graph. The description of the discrete model of the integral equation is proposed to be trans-formed to a form of representation that satisfies the method of describing the electric circuit. To achieve this goal, the following tasks have been completed. The requirements of the method of correct compila-tion of the description, which the form of the description of the discrete model of the integral equation must satisfy, are formulated. The analysis of the linear discrete model of the integral equation is carried out, the graph of the discrete model is constructed, and the requirements for the method of transform-ing this graph to the graph that meets the requirements of the method are formulated. A technique for transforming a graph of a discrete model into a graph that meets the requirements of the method has been developed. Final result: a description of a discrete model of the Fredholm integral equation of the second kind, compiled by the method of secondary sources in the form of a graph, satisfying the re-quirements of the method is presented.

A Note on the Elastohydrodynamic Lubrication of Soft Contacts

1986 ◽  
Vol 200 (3) ◽  
pp. 189-194 ◽  
Author(s):  
C J Hooke
Keyword(s):  
Pressure Distribution ◽  
Transition Zone ◽  
Contact Pressure ◽  
Elastohydrodynamic Lubrication ◽  
Contact Pressure Distribution ◽  
Rigid Cylinder ◽  
Dry Contact ◽  
Highly Loaded

The clearances in highly loaded non-Hertzian contacts can be calculated directly from the dry contact pressure distribution. This note presents a method of extending the analysis into less highly loaded regions. It is shown that the method accurately predicts the clearance over much of the transition zone for Hertzian contacts and its use in a non-Hertzian situation is illustrated using the contact between a rigid cylinder and an elastomer-lined surface as an example.

A New Prediction Methodology For Dynamic Contact Pressure Distribution In A Disc Brake

2012 ◽  
Author(s):  
Abd Rahim Abu Bakar ◽  
Mohd Kameil Abdul Hamid ◽  
Huajiang Ouyang
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Pressure Distribution ◽  
Surface Topography ◽  
Contact Pressure ◽  
Dynamic Contact ◽  
Element Model ◽  
Disc Brake ◽  
Contact Pressure Distribution ◽  
Static Contact

Taburan tekanan sentuhan dinamik masih lagi tidak dapat diukur secara uji kaji. Ini menjadikan kaedah berangka melalui analisis unsur terhingga merupakan pilihan alternatif yang terbaik bagi tujuan tersebut. Namun begitu, model unsur terhingga yang dibina perlu terlebih dahulu diujisahkan agar hasil ramalan yang diperolehi memuaskan dan realistik. Kertas kerja ini mencadangkan dan menjalankan pengujisahan ke atas model secara tiga peringkat iaitu mengujisahkan aspek kelakuan dinamik pada setiap komponen brek cakera dan juga pemasangan selain daripada pengujisahan tekanan sentuhan statik dengan keputusan daripada pengujian. Model 3-dimensi telah dibina berdasarkan komponen sebenar. Permukaan topografi bahan geseran diambilkira dan dimodelkan dalam model unsur terhingga. Hasil analisis mencatatkan keputusan yang memberangsangkan di mana model menunjukkan persamaan dengan keputusan uji kaji bagi kelakukan dinamik dan juga tekanan sentuhan statik. Setelah model diujisahkan, analisis tekanan sentuhan dinamik dilakukan. Kata kunci: brek cakera, tekanan sentuhan dinamik, topografi permukaan, ujian sentuhan, analisis modal, unsur terhingga The dynamic contact pressure distribution in a disc brake system remains impossible to measure through experimental methods. This makes numerical analysis using the finite element method an indispensable alternative tool to its prediction. However, the finite element model must first be validated through appropriate analyses so that realistic predicted results can be obtained. This paper proposes and carries out a three-stage validation methodology: validating the dynamic aspect of each brake component and the brake assembly using modal testing data and the contact aspect using the experimental results of static contact pressure. A detailed 3-dimensional finite element model of an actual disc brake was developed. Brake pad surface topography is also taken into consideration. Good agreement is achieved between predicted and experimental results both in modal analysis and static contact pressure distributions. Once a validated model was obtained, contact analysis for dynamic condition of the disc brake is performed. Key words: disc brake, dynamic contact pressure, surface topography, contact tests, modal analysis, finite element

APPROXIMATE SOLUTIONS OF NONLINEAR VOLTERRA INTEGRAL EQUATION SYSTEMS

2010 ◽  
Vol 24 (32) ◽  
pp. 6235-6258 ◽  
Author(s):  
SALIH YALÇINBAŞ ◽  
KÜBRA ERDEM
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Nonlinear Volterra Integral Equation ◽  
Solution Function ◽  
Taylor Coefficients ◽  
Linear Algebraic Equations ◽  
New System

The purpose of this study is to implement a new approximate method for solving system of nonlinear Volterra integral equations. The technique is based on, first, differentiating both sides of integral equations n times and then substituting the Taylor series the unknown functions in the resulting equation and later, transforming to a matrix equation. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method.

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