Finite Integral Transformations Research Articles

Non-stationary Flexural Fluctuations of a Round Flat Bimorph Plate with Graded-varying Thickness

The subject matter of the article is a non-stationary axially symmetric goal for a round flat bimorph plate affecting end faces of electric potential which in itself is an arbitrary time function. The designed model allows to take into account the peculiarity of the electric-field pattern based on same inner electrode, which appears to be a metal support plate. The authors apply the method of finite integral transformation (based on Timoshenko's theory) to develop a new closed solution for the electro-elastic system of graded-varying stiffness and thickness under consideration. The achieved ratio design allows to further analyze frequency and stress-strain behaviour of resonant flexural piezo-ceramic converters.

NON-STATIONARY AXISYMMETRIC PROBLEM OF INVERSE PIEZOEFFECT FOR CIRCULAR BIMORPHOUS PLATE OF STEPPED VARIABLE THICKNESS AND RIGIDITY

Non-stationary axisymmetric problem for a thin circular bimorphous plate under the action on the end surfaces of electric potential, which is an arbitrary function of time is viewed. On the basis of the theory of Tymoshenko by method of finite integral transformation a new closed solution for the viewed electricity and elastic system of stepped variable rigidity and thickness is built. The obtained calculated correlations allow to explore the frequency characteristics and stress-deformed state of bimorphous elements.

Free Vibration Analysis of Rectangular Cantilever Plates by Finite Integral Transform Method

Exact solutions of eigenfrequencies and vibration modes of rectangular cantilever thin plates are derived by using the double finite integral transform method. Since only the basic elasticity equations of the plates are used, the analytical method utilized in this paper eliminates the need to predetermine the deformation function arbitrarily and is hence more reasonable than conventional semi-inverse methods. Numerical results presented in the paper demonstrate the validity of the approach.

Forced nonstationary axisymmetric vibrations of a piezoceramic thin bimorph plate

We consider a nonstationary axisymmetric problem for a thin axially polarized bimorph plate whose end surfaces are under the action of an electric potential, which is an arbitrary function of the radial coordinate and time. On the basis of Timoshenko theory, the finite integral transformation method is used to construct a new closed solution. The obtained computational relations allow one to study the stress-strain state of piezoceramic elements with continuous and split circular electrodes.

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ ТЕПЛОВИХ ПРОЦЕСІВ ПРОСТОРОВИХ КОНСТРУКЦІЙ ЕНЕРГЕТИЧНИХ ПРИСТРОЇВ

The article represents the effectiveness of the S-codes in the regional-analytical structures of thermal conductivity problems decision by giving the example of calculation of stationary thermal field of the magnetic conductor of a transformer. The finite integral transformation, regional-structural and projection methods are applied for jointly decision. Decision structures exactly satisfy the thermal conductivity equation in the regions and the junction conditions of the contact boundaries, and include S-codes, accurately accounting the geometry of the studied area. Regionalanalytical structures are universal regarding the model parameters. Thus, it is possible to use them for multivariate diagnosis and forecasting of thermal conditions in three-dimensional structures. Results of a decision can be represented in an approximate analytical form, so that a compact database of thermal behavior of structures can be created. Regional-analytical decision structures in boundary-value problems have more natural approximation apparatus than analytical structures have in a single expression, which makes them a useful tool for modeling of compound objects.

ВЛИЯНИЕ ХАРАКТЕРИСТИК ВНЕШНЕЙ ЦЕПИ НА ФОРМУ ЭЛЕКТРИЧЕСКОГО ИМПУЛЬСА ПРИ ОПРЕДЕЛЕНИИ ПРОЧНОСТИ БЕТОНА УДАРНО-АКУСТИЧЕСКИМ МЕТОДОМ

The axis-symmetric non-stationary problem of electric elasticity of a piezoceramic cylinder with axial polarization of the material, acting as a converter of energy in the course of the application of the shock-acoustic method of the quality control of concrete structures is considered in the article. The new closed solution represents a consecutive application of methods of finite integral transformations including the Fourier transformation based on the axial coordinate and a generalized algorithm based on the radial variable. Standardization and transformation of boundary conditions into homogeneous ones are performed at each stage. Measurements of electric impulses are taken by a voltmeter that has an electrode coating, and operates as an ideal conductor with negligible mass. The voltmeter is attached to the surface of the cylinder. Simplified electric boundary conditions are used to describe the electric elastic state of the sample under research. More accurate correlations designated to describe the application of a high conductivity voltmeter are considered in this paper to identify the difference of potentials between the electrode planes. Numerous analyses of the research results have proven that the conductivity of the measuring device produces a significant influence on the shape and the maximal intensity of the electric impulse. However, the stress-strained state of the cylinder is changed insignificantly. Besides, dependence of oscillations and shapes of the solid piezoceramic cylinder on electric boundary conditions has been analyzed. The influence of the geometric dimensions of the cylinder on the values of the electric field in the piezoceramic material has been the subject of research.

Exact Solution of Fractional Diffusion Model with Source Term used in Study of Concentration of Fission Product in Uranium Dioxide Particle

The exact solution of fractional diffusion model with a location-independent source term used in the study of the concentration of fission product in spherical uranium dioxide (UO2) particle is built. The adsorption effect of the fission product on the surface of the UO2 particle and the delayed decay effect are also considered. The solution is given in terms of Mittag—Leffler function with finite Hankel integral transformation and Laplace transformation. At last, the reduced forms of the solution under some special physical conditions, which is used in nuclear engineering, are obtained and corresponding remarks are given to provide significant exact results to the concentration analysis of nuclear fission products in nuclear reactor.

Free Vibration of Orthotropic Plate on Foundation with Four Edges Free by Finite Integral Transform Method

The double finite cosine integral transform method is exploited to obtain explicit solutions for the natural frequencies and mode shapes of the orthotropic rectangular plate on foundation with four edges free. In the analysis procedure, the classical Kirchhohh orthotropic rectangular plate is considered and the Winkler elastic foundation is utilized to represent the elastic foundation. Because only are the basic dynamic elasticity equations of the orthotropic thin plate on elastic foundation adopted, it is not need prior to select the deformation function arbitrarily. Therefore, the solution developed by present paper is reasonable and theoretical. In order to illuminate the correction of formulations, the numerical results are also presented to comparing with that of the other references.

Solving the heat conduction problem for an annular cylinder with finite sizes containing internal heat sources

The method of finite integral transformations is used to solve the unsteady heat conductance problem for a finite sizes annular cylinder containing continuously acting internal heat sources, the power of which is a function of coordinates and time, with boundary conditions of the third kind at four boundaries. The cylinder is placed in medium, the temperature of which varies with time. The cylinder’s initial temperature is a function of coordinates.

Deflection of a circular plate by heat sources distributed along a curve

We propose a new approach based on the method of finite integral transformations and the theory of generalized functions to the solution of the problem of thermoelasticity for a circular plate heated by heat sources distributed along a curve. We also present the analysis of numerical results.

Dynamic Thermoelastic Analysis of a Slab Using Finite Integral Transformation Method

In this work the dynamic thermoelastic response of a slab with finite thickness subjected to aerodynamic heating is investigated. The system of generalized governing equations with generalized initial and boundary conditions is solved by the method of finite integral transformation, and the analytic expressions of the transient temperature and dynamical stresses in the slab are obtained. The finite integral transformation method shows its advantage and convenience to deal with the complex boundary conditions of a dynamic thermoelastic problem. Moreover, by introducing the aerodynamic heating and taking into account a typical Mach number curve for hypersonic flight, the analytical solution is obtained to simulate the dynamic thermoelastic response of the slab in hypersonic flight environment. Calculations are carried out for zirconium diboride to obtain transient temperature and dynamical stresses induced by aerodynamic heating.

Research on adaptive temperature control in sound field induced by self-focused concave spherical transducer

Temperature control of hyperthermia treatments is generally implemented with multipoint feedback system comprised of phased-array transducer, which is complicated and high cost. Our simulations to the acoustic field induced by a self-focused concave spherical transducer (0.5 MHz, 9 cm aperture width, 8.0 cm focal length) show that the distribution of temperature can keep the same “cigar shape” in the focal region during ultrasound insonation. Based on the characteristic of the temperature change, a two-dimensional model of a “cigar shape” tumor is designed and tested through numerical simulation. One single-point on the border of the “cigar shape” tumor is selected as the control target and is controlled at the temperature of 43 °C by using a self-tuning regulator (STR). Considering the nonlinear effects of biological medium, an accurate state-space model obtained via the finite Fourier integral transformation to the bioheat equation is presented and used for calculating temperature. Computer simulations were performed with the perfusion rates of 2.0 kg/(m 3 s) and 4.5 kg/(m 3 s) to the different targets, it was found that the temperatures on the border of the “cigar shape” tumor can achieve the desired temperature of 43 °C by control of one single-point. A larger perfusion rate requires a higher power output to obtain the same temperature elevation under the same insonation time and needs a higher cost for compensating the energy loss carried away by blood flow after steady state. The power output increases with the controlled region while achieving the same temperature at the same time. Especially, there is no overshoot during temperature elevation and no oscillation after steady state. The simulation results demonstrate that the proposed approach may offers a way for obtaining a single-point, low-cost hyperthermia system.

Temperature field of a screened wall with a thermoactive lining on exposure to an axisymmetric thermal effect

A mathematical model of the process of formation of a temperature field in the plane wall — thermoactive lining — heatproof coating system, the outer surface of which is under the influence of an axisymmetric heat flux with Gaussian-type intensity, is suggested as the base for developing a hierarchy of simplified analogs. To determine the temperature field studied, a finite integral transformation for a three-layer region with specific conjugation conditions has been developed.

Critical velocity of a load moving on a beam with a sudden change of foundation stiffness: Applications to high-speed trains

The transient dynamic response of a beam supported on a foundation with sudden stiffness change and subjected to a force moving with constant velocity is analysed. The abrupt change is located at the mid-section of the beam of finite length. Two analytical approaches are implemented. In the first one, the response is obtained by finite integral transformations incorporating global modes of vibration, while in the second the analytical responses of each half of the beam are linked by continuity conditions. The values obtained are used to study the influence of the abrupt change on the critical velocities. The analyses carried out enable to reach results and draw conclusions directly related to the knowledge of ground vibrations induced by high-speed trains.

Nonlinear interaction between a moving vehicle and a plate elastically mounted on a tunnel

The paper presents a theoretical study of the interaction between a nonlinear model of a moving vehicle (velocity v ) and a plate elastically mounted in a tunnel. An efficient approach to the solution of the problem of vehicle– slab-track– tunnel–soil interaction is developed on the basis of a coupling of the Finite Element and Integral Transform methods (FEM and ITM). According to this approach the tunnel which may have an arbitrary shape and a portion of the surrounding soil is modelled by Finite Elements while the soil (half-space) is described by the ITM. The corresponding solution is found using the solutions for the uniform half-space and for the continuum with a cylindrical cavity. To exploit the invariance of the structure in longitudinal direction x, for this direction additionally to the time–frequency transform ( t ▪ ω ) a space–wavenumber transform ( x ▪ k x ) is used. The case of a half-space is analyzed using a Fourier transform also in the second horizontal direction ( y ▪ k y ) and an analytical solution for the z-direction on the basis of exponential functions. In the case of the infinite continuum with the cavity a Fourier series (for the circumferential direction) and a series of cylindrical functions (for the radial direction) are used for the solution regarding the cross-sectional coordinates ( y , z ) . The tunnel structure, that may have an arbitrary shape, and a portion of the surrounding soil will be modelled by Finite Elements in a ( x ▪ k x , t ▪ ω ) transformed domain. In order to observe the boundary conditions at the surface of the half-space as well as at the surface of the cavity, the superposition of the two solutions has to be performed after an inverse Fourier transform (IFT) in the ( k x , y , z , ω ) -domain. The solution for the complete system floating slab-track– tunnel– half-space is obtained in the ( k x , y , z , ω ) -domain. Once the system transfer function H ( ω ′ ) (with ω ′ = ω + vk x ) for this complete coupled system is found, the displacements of the plate can be calculated in time domain by the IFT of the product p ( ω ′ ) H ( ω ′ ) , i.e. the convolution of the loading p ( t ) with the impulse response function h ( t ) , which completely represents the behavior of the coupled system. In the context of the coupling of systems in a relative movement the problems of differential algebraic equations (DAE) have to be observed.

Nonstationary heat transfer in a hollow composite cylinder

Solution of a nonstationary heat-transfer problem for bounded end-conjugated hollow dissimilar cylinders is presented. In the volume of the cylinders, time- and coordinate-dependent heat release of known intensity takes place. The problem is solved using finite integral transformations over two coordinates.

Shallow Prismatic Shell in a Nonuniform Temperature Field

We study the thermal stressed state of a shallow prismatic shell composed of two flat elements. The analytic solutions of the problems of heat conduction and thermoelasticity are obtained by the method of finite integral transformations in the form of double series. The distributions of forces and moments along the coordinate lines are analyzed for different angles of inflection of the shell.

Exact solutions of some mixed problems of uncoupled thermoelasticity for a truncated hollow circular cone with a groove along the generatrix

An elastic body of finite dimensions in the form of a truncated hollow circular cone with a groove along the generatrix is considered. The uncoupled problem of thermoelasticity is formulated for this body for different types of boundary conditions on all the surfaces. These are the conditions for specifying the displacements or sliding clamping on surfaces with fixed angular coordinates and the conditions for specifying the stresses on surfaces with a fixed radial coordinate (shear stresses are assumed to be zero). It is assumed that the temperature is a specified function of all the spherical coordinates. Some auxiliary functions, related to the displacements, are introduced first, and equations for these functions are then derived using Lamé's equations. A finite integral Fourier transformation with respect to one of the angular variables is then employed. After this, by solving certain Sturm-Liouville problems, a new integral transformation is constructed and is applied to the equations with respect to the other angular variable. As a result a one-dimensional system of differential equations is obtained, to solve which an integral Mellin transformation is employed in a special way. Finally, exact solutions of some problems of thermoelasticity are constructed in series for this body.

Exact solutions of some mixed problems of uncoupled thermoelasticity theory for a finite hollow circular cylinder with a groove along the generatrix

Uncoupled problems of thermoelasticity for a finite hollow circular cylinder with a groove along the generatrix for different types of boundary conditions on all the surfaces are considered. These are the conditions for specifying displacements equal to zero or sliding clamping on the cylindrical surfaces and on the edges of the groove and the conditions for specifying stresses on the ends of the cylinder (shear stresses are assumed to be zero). It is assumed that the problem of thermoelasticity has been solved and the temperature is known. Certain auxiliary functions, related to the displacements, are initially introduced, and equations for these functions are derived using Lamé's equations. A finite integral Fourier transformation with respect to the polar angle and a finite Hankel transformation with respect to the radius are then used. As a result a one-dimensional boundary-value vector problem is obtained for a first-order differential equation, which is solved using Green's matrix constructed. Finally, exact solutions of the problems are constructed in series.

Quasistatic thermoelastoplastic problem for a circular eccentric tube

By the methods of small parameter and finite integral transformation, we determine the nonstationary two-dimensional distribution of temperature in a circular eccentric tube subjected to the action of normal compressive stresses applied to its lateral surfaces under the conditions of convective heat exchange with an ambient medium and the quasistatic thermoelastic and thermoelastoplastic states of mis tube.