Infinite System Of Algebraic Equations Research Articles

Contact with break-off under bending of an elastic strip with a rigid disk

We consider the problem of the theory of elasticity of the contact interaction of a rigid circular disk and an elastic strip, which rests upon two supports with disturbance of contact in the middle part of the contact region. On the basis of the Wiener–Hopf method, an integral equation of the problem is reduced to an infinite system of algebraic equations. The size of the zone of break-off of the boundary of the strip from the disk and the distribution of contact stresses are determined.

Continued fractions in time varying circuits analysis

Analysis of linear radio circuits with time varying parameters results in infinite system of algebraic equations. Theory of such equations systems with necessary for practice fullness has not been developed. Therefore special case of time varying circuit is of great interest when its analysis is simpler and is possible to be reduced to continued fractions, in spite of all this fundamental characteristics of time varying circuit are preserved.

The contact problem for a circular plate with a stress-free end face

An axisymmetric contact problem for a circular elastic plate with a stress-free end face into which two symmetrically arranged punches are imbedded, is considered. The problem is solved using a method developed earlier for bodies of finite sizes, which is based on the generalized orthogonality of homogeneous solutions. The problem reduces to a Fredholm integral equation of the first kind in a function describing the displacement of the surface of the plate outside the punch. These functions are sought in the form of a sum of a Schlömilch series and a power function with a root singularity. The ill-posed infinite system of algebraic equations obtained as a result is regularized by the introduction of a small positive parameter. Since the matrix elements of the system, as well as the contact stresses, are defined by poorly converging numerical and functional series, the technique of summation of the residues of these series is used. The distribution of the contact pressure and the dimensionless imbedding force are found. Examples of the calculation of a plane punch are given.

Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides

The plane contact problem of the indentation of a rigid punch into a base-sucured elastic rectangle with stress-free sides is considered. The problem is solved by a method tested earlier and reduces to a system of two integral equations in functions describing the displacement of the surface of the rectangle outside the punch and the normal or shear stress on its base. These functions are sought in the form of the sum of trigonometric series and an exponential function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result of this are regularized by introducing small positive parameters. Because the matrix elements of the systems, and also the contact stresses, are defined by poorly converging numerical and functional series, the previously developed method of summation of these series is used. The contact pressure distribution and the dimensionless indenting force are found. Examples of a plane punch calculation are given.

Bending analysis of secured rectangular panels with one free edge

A computational model of the flat walls of various tanks, retaining walls, deck and structural ceilings with one free edge is presented as a rectangular plate, three edges of which are secured, and the fourth free (Fig. 1). The transverse load on the elements is frequently uniform or hydrostatic (in the latter case, it can be represented as the sum of a uniform load and inversely symmetric load). This problem has been examined in [1, 2], and its has no exact solution in closed form. Timoshenko [1] reduces the solution to an infinite system of algebraic equations in terms of the unknown coefficients of trigonometric series with respect to two coordinates. Smotrov [2] used the finite-difference method. Numerical results presented in [1] were obtained for the most part by this approximate method. The purpose of this paper is to acquire more reliable numerical results for the bending analysis of the indicated components of water-development works. The method of superposition of correcting functions will be used to solve the problem in question [3]. The transverse load q is assumed constant. The problem is resolved in dimenionless quantities. The deflection function sought should satisfy the differential bending equation

Solving the n-term time fractional diffusion-wave problem via a method of approximation of tempered convolution

We present a method of approximation of tempered convolution via Laguerre polynomials and apply it in solving the n-term time fractional diffusion-wave problem in one spatial variable. In particular, we give a solution to the two-term diffusion-wave equation as an example of how the method works. The method is applicable in solving different fractional equations with entire and fractional derivatives in 2D. By a fractional operator, we transform them into convolution form and apply the approximation of tempered convolution by Laguerre polynomials. This leads to an infinite system of algebraic equations through coefficients whose solution gives the coefficients in the Laguerre series expansion of the solution to the corresponding equation.

Impression of a semiinfinite stamp in an elastic strip with regard for friction and adhesion

We consider the problem of contact interaction between a semiinfinite stamp with rectilinear base and an elastic strip with one rigid side. Friction forces in the contact region are taken into account. These forces lead to the division of the contact region into slipping and adhesion zones. With the use of the Wiener-Hopf method, a system of integral equations is reduced to an infinite system of algebraic equations. The computational results of stresses and strains at the boundary and at inner points of the elastic strip are presented.

Diffraction of a plane acoustic wave by a rigid sphere in a cylindrical cavity: An axisymmetric problem

The paper studies the interaction of a rigid spherical body and a cylindrical cavity filled with an ideal compressible fluid in which a plane acoustic wave of unit amplitude propagates. The solution is based on the possibility of transforming partial solutions of the Helmholtz equation between cylindrical and spherical coordinates. Satisfying the interface conditions between the cavity and the acoustic medium and the boundary conditions on the spherical surface yields an infinite system of algebraic equations with indefinite integrals of cylindrical functions as coefficients. This system of equations is solved by reduction. The behavior of the system is studied depending on the frequency of the plane wave

The contact problem for hollow and solid cylinders with stress-free faces

The contact problem for hollow and solid circular cylinders with a symmetrically fitted belt and stress-free faces is considered. Homogeneous solutions corresponding to zero stresses on the cylinder faces are obtained. The generalized orthogonality of homogeneous solutions is used to satisfy the modified boundary conditions. In the final analysis the problem is reduced to a system of integral equations in the functions describing the displacement of the outer and inner surfaces of the cylinders. These functions are sought in the form of the sum of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result, are regularized by introducing small positive parameters [Ref. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978] and, after reduction, have stable regularized solutions. Since the elements of the matrices of the system are given by poorly converging numerical series, an effective method of calculating the residues of these series is developed. Formulae for the distribution function of the contact pressure and the integral characteristic are obtained. Since the first formula contains a third-order derivative of the functional series, a numerical differentiation procedure is employed when using it [Refs. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978; Danilina NI, Dubrovskaya NS, Kvasha OP et al. Numerical Methods. A Student Textbook. Moscow: Vysshaya Shkola; 1976]. Examples of the analysis of a cylindrical belt are given.

PROPAGATION OF WAVES IN A BIFURCATED CYLINDRICAL WAVEGUIDE WITH WALL IMPEDANCE DISCONTINUITY

In the present work the radiation of sound from a bifurcated circular waveguide formed by a semi-infinite rigid duct inserted axially into a larger infinite tube with discontinuous wall impedance is reconsidered through an alternative approach which consists of using the mode matching technique in conjunction with the Wiener-Hopf method. By expressing the total field in the appropriate waveguide region in terms of normal modes and using the Fourier transform technique elsewhere, we end up with a single modified Wiener-Hopf equation whose solution involves an infinite system of algebraic equations. This system is solved numerically and the influence of some parameters on the radiation phenomenon is shown graphically. The equivalence of the direct method described in (1) and the present mixed method are shown numerically.

Eigenvalues of the Laplacian on an elliptic domain

The importance of eigenvalue problems concerning the Laplacian is well documented in classical and modern literature. Finding the eigenvalues for various geometries of the domains has posed many challenges which include infinite systems of algebraic equations, asymptotic methods, integral equations etc. In this paper, we present a comprehensive account of the general solutions to Helmholtz’s equations (defined on simply connected regions) using complex variable techniques. We consider boundaries of the form z z ̄ = f ( z ± z ̄ ) or its inverse z ± z ̄ = g ( z z ̄ ) . To illustrate the theory, we reduce the problem on elliptic domains to equivalent linear infinite algebraic systems, where the coefficients of the infinite matrix are known polynomials of the eigenvalues. We compute truncations of the infinite system for numerical values. These values are compared to approximate values and some inequalities available in literature.

The contact problem for a rectangle with stress-free side faces

The plane contact problem for an elastic rectangle into which two symmetrically positioned punches are impressed is considered. Homogeneous solutions are constructed that leave the side faces of the rectangle stress-free. When the modified boundary conditions using generalized orthogonality of the homogeneous solutions are satisfied, the problem reduces to a Friedholm integral equation of the first kind in the function describing the displacement of the surface of the rectangle outside the contact area. This function is sought in the form of the sum of a trigonometric series and a power function with a root singularity. The ill-posed infinite system of algebraic equations thereby obtained is regularized by introducing a small positive parameter (Ref. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978), and, after reduction, has a stable regularized solution. Since the matrix elements of the system are determined by a poorly converging number series, an effective method was developed for calculating the residues of the series. Formulae are found for the contact pressure distribution function and dimensionless indentation force. Since the first formula contains a third-order derivative of the functional series, when it is used, a numerical differentiation procedure is employed (Refs. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978; Danilina NI, Dubrovskaya NS, Kvasha OP et al. Numerical Methods. Textbook for Special Colleges. Moscow: Vysshaya Shkola; 1976). Examples of a calculation for a plane punch are given.

Dynamic interaction of an oscillating sphere and an elastic cylindrical shell filled with a fluid and immersed in an elastic medium

The paper studies the interaction of a harmonically oscillating spherical body and a thin elastic cylindrical shell filled with a perfect compressible fluid and immersed in an infinite elastic medium. The geometrical center of the sphere is located on the cylinder axis. The acoustic approximation, the theory of thin elastic shells based on the Kirchhoff—Love hypotheses, and the Lame equations are used to model the motion of the fluid, shell, and medium, respectively. The solution method is based on the possibility of representing partial solutions of the Helmholtz equations written in cylindrical coordinates in terms of partial solutions written in spherical coordinates, and vice versa. Satisfying the boundary conditions at the shell—medium and shell—fluid interfaces and at the spherical surface produces an infinite system of algebraic equations with coefficients in the form of improper integrals of cylindrical functions. This system is solved by the reduction method. The behavior of the hydroelastic system is analyzed against the frequency of forced oscillations.

Equilibrium of a Prestressed Strip Reinforced with Elastic Plates

The contact problem for a prestressed elastic strip reinforced with equally spaced elastic plates is considered. The Fourier integral transform is used to construct an influence function of a unit concentrated force acting on the infinite elastic strip with one edge constrained. The transmission of forces from the thin elastic plates to the prestressed strip is analyzed. On the assumption that the beam bending model and the uniaxial stress model are valid for an elastic plate subjected to both vertical and horizontal forces, the problem is mathematically formulated as a system of integro-differential equations for unknown contact stresses. This system is reduced to an infinite system of algebraic equations solved by the reduction method. The effect of the initial stresses on the distribution of contact forces in the strip under tension and compression is studied

Scattering From an Elliptic Crack by an Integral Equation Method: Normal Loading

The scattering of normally incident elastic waves by an embedded elliptic crack in an infinite isotropic elastic medium has been solved using an analytical numerical method. The representation integral expressing the scattered displacement field has been reduced to an integral equation for the unknown crack-opening displacement. This integral equation has been further reduced to an infinite system of Fredholm integral equation of the second kind and the Fourier displacement potentials are expanded in terms of Jacobi’s orthogonal polynomials. Finally, proper use of orthogonality property of Jacobi’s polynomials produces an infinite system of algebraic equations connecting the expansion coefficients to the prescribed dynamic loading. The matrix elements contains singular integrals which are reduced to regular integrals through contour integration. The first term of the first equation of the system yields the low-frequency asymptotic expression for scattering cross section analytically which agrees completely with previous results. In the intermediate and high-frequency scattering regime the system has been truncated properly and solved numerically. Results of quantities of physical interest, such as the dynamic stress intensity factor, crack-opening displacement scattering cross section, and back-scattered displacement amplitude have been given and compared with earlier results.

On the solutions of mixed plane problems that are unbounded where the boundary conditions change

In this paper, we apply a modification to the method of the integral equation formulation of mixed plane boundary value problems so that it enables us to obtain the solutions unbounded at the points where the boundary conditions change. Such solutions are of great physical interest. The modification is illustrated by means of a typical problem. As is it the case in the original method proposed by Cherskii [1], the problem is reduced to an infinite system of algebraic equations. The justification of the truncation of such systems has been established.

Modelling and stability of magnetosoft ferromagnetic plates in a magnetic field

The equations of magnetoelasticity of magnetoelastic plates remain three dimensional even after adopting the Kirchhoff hypothesis. In the present work, two-dimensional equations of magnetoelasticity are obtained for a magnetoelastic ferromagnetic plate of finite size. In the general case the components of the induced magnetic field inside and outside the plate and the amplitudes of lateral vibrations are connected through both the equations of motion and the boundary conditions. This connection essentially complicates the problem of lateral vibrations of magnetoelastic plates of finite size. Assuming that the magnetic susceptibility is very large compared with unity, the solution of the magnetostatics problem in the interior and exterior domains is constructed through asymptotic series with respect to the reciprocal of the magnetic susceptibility. As a result, a system of separated problems with corresponding boundary conditions is obtained, connected by recurrent relations. This allows us to obtain an analytical representation for the components of the excited magnetic field in the case of a very thin plate. The problem of determining the lateral vibrations of magnetosoft ferromagnetic plates of finite size in a magnetic field reduces to a twodimensional singular integrodifferential equation. Using the Galerkin method, the determination of the frequencies reduces to an infinite system of algebraic equations with a normal-type determinant.

Directional properties of an array of sound receivers positioned in an impedance screen recess

Expressions for calculating the directional characteristics of an array of sound receivers positioned in a waveguide with impedance walls are obtained from the solution to the problem on the diffraction of a plane sound wave by the waveguide open end with impedance flanges. The waveguide can be of a finite length, and, in this case, it can be considered as an open cavity in an impedance screen. The solution of the integral equation for the sound pressure distribution over the opening area is reduced to the solution of an infinite system of algebraic equations for the coefficients of the field expansion in normal waveguide waves. Examples of calculated directional characteristics are presented for arrays with receivers positioned at different distances from the opening and for different values of the impedances of the waveguide walls and flanges.

Effective electrical characteristics of a two-dimensional three-component doubly-periodic system with circular inclusions

The problem of the electric conductivity of a two-dimensional three-component system containing periodically arranged conducting circular inclusions of two types has been solved. A consistent method for the calculation of the conductivity and other effective electrical characteristics of this model is proposed, which is applicable in the case of arbitrary component concentrations. A complex potential outside the inclusions is expressed in terms of the Weierstrass zeta function and its derivatives. Undetermined coefficients entering into the general expression for the potential are determined from an infinite system of algebraic equations. In the case of a small concentration of inclusions, this system yields a virial expansion for the conductivity. A numerical analysis of this system of equations provides for the principal possibility of investigating various effective characteristics of the model (including the Hall coefficient and thermo emf) within the entire range of the problem parameters.

Diffraction of a sound wave by the open end of a flanged waveguide with impedance walls

Diffraction of a plane sound wave by the open end of an impedance-wall waveguide connected to an opening in an impedance screen is considered. The plane wave is incident on the waveguide from a free half-space. Two versions of the problem are considered: for a semi-infinite waveguide and for a finite-length waveguide with a specified bottom impedance; the impedances of the walls, screen, and waveguide bottom can be different. The finite-length waveguide can be treated as an open cavity in the impedance screen. For the cavity of zero length, the problem is reduced to the diffraction by an impedance insert in the impedance screen. The solution in the external region determines the scattered field; the solution in the internal region allows one to determine the directional pattern of an array of receivers located in the cavity. The problem is solved using the integral Helmholtz equation with a specially selected Green’s function that provides the fulfillment of the boundary conditions. Formally, the problem is reduced to an infinite system of algebraic equations. The computational results obtained for bistatic and monostatic scattering patterns are presented.