scholarly journals Vibrations of a Cantilevered Thick Plate

2021 ◽  
pp. 106-117
Author(s):  
S. O Papkov
Keyword(s):  
Boundary Conditions ◽  
Variational Approach ◽  
Thick Plate ◽  
Infinite System ◽  
Trigonometric System ◽  
Superposition Method ◽  
Algebraic Equations ◽  
Exact Methods ◽  
Particular Solutions ◽  
Linear Algebraic Equations

It has been for the first time that an analytical solution to the problem of free vibrations of a cantilevered thick orthotropic plate is presented. This problem is quite cumbersome for using the exact methods of the theory of elasticity; therefore, methods based on the variational approach were developed to solve it. The paper suggests using the superposition method to construct a general solution of the vibration equations of a plate in the series form of particular solutions obtained with the help of a variables separation. The particular solutions of one of the coordinates are built in the form of trigonometric functions of a special type (modified trigonometric system). The constructed solution, in contrast to the solutions known in the literature on the basis of the variational approach, accurately satisfies the equations of vibrations. The use of a modified trigonometric system of functions makes it possible to obtain uniform formulas for even and odd vibration shapes and to reduce the quantity of boundary conditions on the plate sides from twelve to nine ones, while five of the nine boundary conditions are also accurately satisfied. The structure of the presented solution on the plate boundary is such that, each of the kinematic or force characteristics of the plate is represented as a sum of two series, i.e. a trigonometric series and a series in hyperbolic functions. Remaining boundary conditions make it possible to obtain an infinite system of linear algebraic equations with respect to the unknown coefficients of the series representing the solution. The convergence of the solution by the reduction method of the infinite system is investigated numerically. Examples of the numerical implementation are given; numerical studies of the spectrum of natural frequencies of the cantilevered thick plate were carried out based on the obtained solution, both with varying elastic characteristics of the material and with varying geometric parameters.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

Cosserat Spectrum of an Axisymmetric Elasticity Problem for a Finite-Length Solid Cylinder

2018 ◽  
Vol 35 (3) ◽  
pp. 343-349
Author(s):  
Yu. V. Tokovyy
Keyword(s):  
Asymptotic Behavior ◽  
Finite Length ◽  
Solution Method ◽  
Superposition Method ◽  
Algebraic Equations ◽  
Solid Cylinder ◽  
Cosserat Spectrum ◽  
Linear Algebraic Equations ◽  
Key Features ◽  
Axisymmetric Elasticity

ABSTRACTAn algorithm for the computation and analysis of the Cosserat spectrum for an axisymmetric elasticity boundary-value problem in a finite-length solid cylinder with boundary conditions in terms of stresses is proposed. By making use of the cross-wise superposition method, the spectral problem is reduced to systems of linear algebraic equations. A solution method for the mentioned systems is presented and the asymptotic behavior of the Cosserat eigenvalues is established. On this basis, the key features of the Cosserat spectrum for the mentioned problem are analyzed with special attention given to the effect of the cylinder aspect ratio.

LEAST-SQUARES MESHFREE COLLOCATION METHOD

2012 ◽  
Vol 09 (02) ◽  
pp. 1240031 ◽  
Author(s):  
BO-NAN JIANG
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Least Squares ◽  
Collocation Method ◽  
Present Method ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
Dual Variables

A least-squares meshfree collocation method is presented. The method is based on the first-order differential equations in order to result in a better conditioned linear algebraic equations, and to obtain the primary variables (displacements) and the dual variables (stresses) simultaneously with the same accuracy. The moving least-squares approximation is employed to construct the shape functions. The sum of squared residuals of both differential equations and boundary conditions at nodal points is minimized. The present method does not require any background mesh and additional evaluation points, and thus is a truly meshfree method. Unlike other collocation methods, the present method does not involve derivative boundary conditions, therefore no stabilization terms are needed, and the resulting stiffness matrix is symmetric positive definite. Numerical examples show that the proposed method possesses an optimal rate of convergence for both primary and dual variables, if the nodes are uniformly distributed. However, the present method is sensitive to the choice of the influence length. Numerical examples include one-dimensional diffusion and convection-diffusion problems, two-dimensional Poisson equation and linear elasticity problems.

scholarly journals Two Contact Problems for Annular Rigid Stamp on an Elastic Half Space

2018 ◽  
Vol 17 (6) ◽  
pp. 458-464
Author(s):  
S. V. Bosakov
Keyword(s):  
Functional Equation ◽  
Half Space ◽  
Bessel Functions ◽  
Infinite System ◽  
Annular Plate ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The paper presents solutions of two contact problems for the annular plate die on an elastic half-space under the action of axisymmetrically applied force and moment. Such problems usually arise in the calculation of rigid foundations with the sole of the annular shape in chimneys, cooling towers, water towers and other high-rise buildings on the wind load and the load from its own weight. Both problems are formulated in the form of triple integral equations, which are reduced to one integral equation by the method of substitution. In the case of the axisymmetric problem, the kernel of the integral equation depends on the product of three Bessel functions. Using the formula to represent two Bessel functions in the form of a double row on the works of hypergeometric functions Bessel function, the problem reduces to a functional equation that connects the movement of the stamp with the unknown coefficients of the distribution of contact stresses. The resulting functional equation is reduced to an infinite system of linear algebraic equations, which is solved by truncation. Under the action of a moment on the annular plate  die, the distribution of contact stresses is searched as a series by the products of the Legendre attached functions with a weight corresponding to the features in the contact stresses at the die edges. Using the spectral G. Ya. Popov ratio for the ring plate, the problem is again reduced to an infinite system of linear algebraic equations, which is also solved by the truncation method. Two examples of calculations for an annular plate die on an elastic half-space on the action of axisymmetrically applied force and moment are given. A comparison of the results of calculations on the proposed approach with the results for the round stamp and for the annular  stamp with the solutions of other authors is made.

scholarly journals Diffraction of the field of vertical electric dipole on the spiral conductive sphere in the presence of a cone

2018 ◽  
Keyword(s):  
Hilbert Space ◽  
Electric Dipole ◽  
Infinite System ◽  
Matrix Operator ◽  
Algebraic Equations ◽  
Problem Statement ◽  
Vertical Electric Dipole ◽  
Linear Algebraic Equations ◽  
Method Of Regularization ◽  
The Matrix

The problem of diffraction of a vertical electric dipole field on a spiral conductive sphere and a cone has been solved. By the method of regularization of the matrix operator of the problem, an infinite system of linear algebraic equations of the second kind with a compact matrix operator in Hilbert space $\ell_2$ is obtained. Some limiting variants of the problem statement are considered.

Influence Functions in the Numerical Analysis of Bending of Thin Elastic Plates

1984 ◽  
Vol 8 (2) ◽  
pp. 103-114 ◽  
Author(s):  
Mohammed F.N. Mohsen ◽  
Ali A. Al-Gadhib ◽  
Mohammed H. Baluch
Keyword(s):  
Boundary Conditions ◽  
Influence Function ◽  
Infinite Plate ◽  
Thin Plates ◽  
Influence Functions ◽  
Algebraic Equations ◽  
Elastic Plates ◽  
The Real ◽  
Linear Algebraic Equations ◽  
The Moment

A numerical method for the linear analysis of thin plates of arbitrary plan form and subjected to arbitrary loading and boundary conditions is presented in this paper. This method is an extension of the Wu-Altiero method [1] where use has been made of the force influence function for an infinite plate, whereas the work contained in this paper is based on the use of the moment influence function of an infinite plate. The technique basically involves embedding the real plate into a fictitious infinite plate for which the moment influence function is known. N points are prescribed at the plate boundary at which the boundary conditions for the original problem are collocated by means of 2N fictitious moments placed around contours outside the domain of the real plate. A system of 2N linear algebraic equations in the unknown moments is obtained. The solution of the system yields the unknown moments. These may in turn be used to compute deflection, moments or shear at any point in the thin plate. Finally, the method is extended to include influence functions of both concentrated forces and concentrated moments. This is obtained by applying concentrated moments and forces simultaneously on the contours located outside the domain of the plate.

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.

scholarly journals Analysis of Directional Properties of Angular Horn Antenna

2021 ◽  
Vol 4 (3) ◽  
Author(s):  
Kateryna Andriivma Shyshkova
Keyword(s):  
Helmholtz Equation ◽  
Infinite System ◽  
Horn Antenna ◽  
Acoustic Properties ◽  
Sound Waves ◽  
Algebraic Equations ◽  
Design Elements ◽  
Linear Algebraic Equations ◽  
Electroacoustic Transducer ◽  
Wide Range

In this paper, horn antennas are considered to belong to the class of aperture antennas which usually include a sound wave reflector and an electroacoustic transducer. For the variant of technical implementation of the electroacoustic transducer in the form of a corner antenna, the problem of sound emission by such an antenna is solved . taking into account the repeated reflection of emitted sound waves from the antenna design elements. The study of the acoustic properties of such an antenna was carried out taking into account a number of assumptions. 'what material, the thickness of the walls of the mouthpiece is infinitesimal. These conditions are supplemented by the known conditions of radiation at infinity. All the above assumptions make it possible to greatly simplify the solution of the problem of sound radiation by an angular horn antenna. To do this, the Helmholtz equation under boundary conditions was solved by the method of connected fields in multiconnected domains, corresponding to the physical model of the antenna. The radiation field of such an antenna is presented in the form of three partial regions, which in turn, according to the method of partial regions - in the form of Fourier series expansions, the coefficients of which are determined by solving differential equations describing piezoceramic transducer oscillations and wave processes. in acoustic environments in contact with it. The solution of the Helmholtz equation is reduced to the solution of an infinite system of linear algebraic equations taking into account the above assumptions, as well as the conditions of field continuity at the boundary of partial domains, we obtain an infinite system of linear algebraic equations. Based on the system, an approximate expression for the normalized radiation pattern is obtained. Studying the features of the directional properties of the angular antenna by direct analysis of the expression is not possible. Therefore, such a study was carried out on the basis of calculated directivity diagrams obtained using a computer for a wide range of wave sizes and geometrical characteristics of the angular antenna. cylindrical electroacoustic transducers. In all calculations, a uniform distribution of the oscillating velocity on the surface of the emitter was chosen Analyzing the obtained data, there is a pronounced dependence of the shape of the pattern on the magnitude of the wave size of the speaker. This is manifested in the fact that the main petal splits into two or even three petals, as well as in increasing the overall sharpness of the pattern.  

Diffraction by a set of three parallel impedance half-planes with the one amidst located in the opposite direction

2002 ◽  
Vol 80 (8) ◽  
pp. 893-909 ◽  
Author(s):  
G Çinar ◽  
A Büyükaksoy
Keyword(s):  
Fourier Transform ◽  
Opposite Direction ◽  
Infinite System ◽  
Plane Waves ◽  
Mode Matching ◽  
Algebraic Equations ◽  
Matching Method ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
The One

The problem of diffraction of plane waves by a set of three parallel half-planes with different surface impedances on upper and lower faces where the one in the middle is placed in the opposite direction, is solved by the mode-matching method where available, and by Fourier-transform technique elsewhere. The solution includes two independent Wiener–Hopf equations each involving an infinite number of expansion coefficients that satisfy an infinite system of linear algebraic equations. PACS No.: 41.20J

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