Adhesive Contact Between the Surface Wave and a Rigid Strip

1995 ◽  
Vol 62 (2) ◽  
pp. 368-372 ◽  
Author(s):  
O. Y. Zharii
Keyword(s):  
Integral Equation ◽  
Surface Wave ◽  
Stress Distribution ◽  
Contact Area ◽  
Mixed Boundary ◽  
Closed Form Solution ◽  
Adhesive Contact ◽  
Form Solution ◽  
Mixed Boundary Value Problem ◽  
Analysis Of Variation

A problem of adhesive contact between the running surface wave and a rigid strip is investigated. The mixed boundary-value problem of elastodynamics is reduced to a singular integral equation for a complex combination of stresses and an exact closed-form solution of it has been derived. Analysis of variation of contact area dimensions, stress distribution and rotor velocity on the frequency of excitation displayed significant differences between the results corresponding to conditions of adhesion and slipping in contact area. The origin of these differences is discussed.

Two dimensional mixed boundary-value problems in a wedge-shaped region

1968 ◽  
Vol 64 (2) ◽  
pp. 503-505 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Integral Equation ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Two Dimensional ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Mellin Transforms

In a recent paper Srivastav (2) considered the solution of certain two-dimensional mixed boundary-value problems in a wedge-shaped region. The problems were formulated as dual integral equations involving Mellin transforms and were reduced to the solution of a Fredholm integral equation of the second kind. In this paper it will be shown that a closed form solution to the problems treated in (2) may be obtained by elementary means.

Smooth Contact Between the Running Rayleigh Wave and a Rigid Strip

1995 ◽  
Vol 62 (2) ◽  
pp. 362-367 ◽  
Author(s):  
O. Y. Zharii ◽  
A. F. Ulitko
Keyword(s):  
Rayleigh Wave ◽  
Mixed Boundary ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mixed Boundary Value Problem ◽  
Trigonometric Functions ◽  
Frictionless Contact ◽  
Dual Series Equations ◽  
Analytic Expressions ◽  
Smooth Contact

A problem of frictionless contact between the running Rayleigh wave and a rigid strip is investigated. The corresponding mixed boundary value problem of elastodynamics is reduced to a system of dual series equations involving trigonometric functions. On the base of the closed-form solution obtained, explicit analytic expressions for distributions of normal displacements and stresses and of tangential velocities on the surface have been derived.

Frictional Contact Between the Surface Wave and a Rigid Strip

1996 ◽  
Vol 63 (1) ◽  
pp. 15-20 ◽  
Author(s):  
O. Y. Zharii
Keyword(s):  
Surface Wave ◽  
Frictional Contact ◽  
Mixed Boundary ◽  
Closed Form Solution ◽  
Tangential Velocity ◽  
Form Solution ◽  
Load Force ◽  
Tangential Load ◽  
Normal Stress Distribution ◽  
Limiting Case

A problem of frictional contact between a running surface wave and a motionless rigid strip is considered. The corresponding mixed boundary value problem of elastodynamics is reduced to a singular integral equation for the normal stress distribution and a closed-form solution of it has been found. Boundaries of the contact zone are determined from a system of transcendental equations involving trigonometric functions. Also, simple formulae obtained for kinematic characteristics of solution (tangential velocity inside the contact area, velocity and slope of the free surface outside it). The problem considered represents a limiting case of operating ultrasonic motor when it is completely braked by an external tangential load force.

A Mixed Boundary-Value Problem of Elasticity With Parabolic Boundary

1957 ◽  
Vol 24 (1) ◽  
pp. 122-124
Author(s):  
Gunadhar Paria
Keyword(s):  
Stress Distribution ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Complex Variable ◽  
Hilbert Problem ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Two Dimensional ◽  
Mixed Boundary Value Problem ◽  
Parabolic Boundary

Abstract The problem of finding the stress distribution in a two-dimensional elastic body with parabolic boundary, subject to mixed boundary conditions, has been reduced to the solution of the nonhomogeneous Hilbert problem following the method of complex variable. The result has been compared with that for a straight boundary.

Frictionless Contact of Layered Half-Planes, Part I: Analysis

1993 ◽  
Vol 60 (3) ◽  
pp. 633-639 ◽  
Author(s):  
M.-J. Pindera ◽  
M. S. Lane
Keyword(s):  
Integral Equation ◽  
Contact Stress ◽  
Stiffness Matrix ◽  
Fourier Transforms ◽  
Mixed Boundary ◽  
Contact Region ◽  
Closed Form Solution ◽  
Cauchy Type ◽  
Frictionless Contact ◽  
Local Stiffness

A method is presented for the solution of frictionless contact problems on multilayered half-planes consisting of an arbitrary number of isotropic, orthotropic, or monoclinic layers arranged in any sequence. A displacement formulation is employed and the resulting Navier equations that govern the distribution of displacements in the individual layers are solved using Fourier transforms. A local stiffness matrix in the transform domain is formulated for each layer which is then assembled into a global stiffness matrix for the entire multilayered half-plane by enforcing continuity conditions along the interfaces. Application of the mixed boundary condition on the top surface of the medium subjected to the force of the indenter results in an integral equation for the unknown pressure in the contact region. The integral possesses a divergent kernel which is decomposed into Cauchy type and regular parts using the asymptotic properties of the local stiffness matrix and the ensuing relation between Fourier and finite Hilbert transform of the contact pressure. For homogeneous half-planes, the kernel consists only of the Cauchy-type singularity which results in a closed-form solution for the contact stress. For multilayered half-planes, the solution of the resulting singular integral equation is obtained using a collocation technique based on the properties of orthogonal polynomials. Part I of this paper outlines the analytical development of the technique. In Part II a number of numerical examples is presented addressing the effect of off-axis plies on contact stress distribution and load versus contact length in layered composite half-planes.

scholarly journals Electrostatic potential calculation by application of walk-on-spheres method

2018 ◽  
pp. 152-160
Author(s):  
Liudmila Vladimirova ◽  
Irina Rubtsova ◽  
Nikolai Edamenko
Keyword(s):  
Integral Equation ◽  
Problem Solving ◽  
Boundary Value Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Value Problem ◽  
Equation Solving ◽  
Electrostatic Potential Calculation ◽  
Walk On Spheres Algorithm ◽  
Boundary Form

The paper is devoted to mixed boundary-value problem solving for Laplace equation with the use of walk-on-spheres algorithm. The problem under study is reduced to finding a solution of integral equation with the kernel nonzero only at some sphere in the domain considered. Ulam-Neumann scheme is applied for integral equation solving; the appropriate Markov chain is introduced. The required solution value at a certain point of the domain is approximated by the expected value of special statistics defined on Markov paths. The algorithm presented guarantees the average Markov trajectory length to be finite and allows one to take into account boundary conditions on required solution derivative and to avoid Markov paths ending in the neighborhood of the boundaries where solution values are not given. The method is applied for calculation of electric potential in the injector of linear accelerator. The purpose of the work is to verify the applicability and effectiveness of walk-on-spheres method for mixed boundary-value problem solving with complicated boundary form and thus to demonstrate the suitability of Monte Carlo methods for electromagnetic fields simulation in beam forming systems. The numerical experiments performed confirm the simplicity and convenience of this method application for the problem considered.

scholarly journals An exact solution of linearized flow of an emitting, absorbing and scattering grey gas

1971 ◽  
Vol 45 (4) ◽  
pp. 673-699 ◽  
Author(s):  
Ping Cheng ◽  
A. Leonard
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Transport Theory ◽  
Neutron Transport ◽  
Separation Of Variables ◽  
Closed Form Solution ◽  
Strong Shock ◽  
Form Solution ◽  
Isotropic Scattering ◽  
Differential Approximation

The governing equations for the problem of linearized flow through a normal shock wave in an emitting, absorbing, and scattering grey gas are reduced to two linear coupled integro-differential equations. By separation of variables, these equations are further reduced to an integral equation similar to that which arises in neutron-transport theory. It is shown that this integral equation admits both regular (associated with discrete eigenfunctions) and singular (associated with continuum eigenfunctions) solutions to form a complete set. The exact closed-form solution is obtained by superposition of these eigen-functions. If the gas downstream of a strong shock is absorption–emission dominated, the discrete mode of the solution disappears downstream. The effects of isotropic scattering are discussed. Quantitative comparison between the numerical results based on the exact solution and on the differential approximation are presented.

Vibratory Motion of a Body on an Elastic Half Plane

1968 ◽  
Vol 35 (4) ◽  
pp. 697-705 ◽  
Author(s):  
P. Karasudhi ◽  
L. M. Keer ◽  
S. L. Lee
Keyword(s):  
Integral Equation ◽  
Half Plane ◽  
Good Accuracy ◽  
Coupling Effect ◽  
Mixed Boundary ◽  
Mixed Boundary Value Problem ◽  
Dual Integral Equations ◽  
Entire Surface ◽  
Vibratory Motion ◽  
Rocking Vibration

The vertical, horizontal and rocking vibrations of a body on the surface of an otherwise unloaded half plane are studied. The problems are formulated so that one stress vanishes over the entire surface, and an oscillating displacement is prescribed in the loaded region. The problems are mixed with respect to the prescribed displacement and the remaining stress. Each case leads to a mixed boundary value problem represented by dual integral equations which are reduced to a single Fredholm integral equation. Although numerical methods are used to solve the integral equation, the contact stresses are found to be presentable in closed form to good accuracy. An estimate of the stiffnesses for coupled horizontal and rocking vibration is also suggested and it is found that the coupling effect is significant.

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