Mixed Antiplane Problem of Electroelasticity for a Piezoceramic Half-Space with a Tunnel Hole

Harmonic vibrations of a piezoceramic half-space weakened by a tunnel hole with a system of active surface electrodes are investigated. Based on integral representations for solutions to two types of conditions on the boundary of the half-space, the boundary-value problem of electroelasticity is reduced to a system of singular integrodifferential equations of the second kind. The results of parametric investigations characterizing the behavior of the electroelastic field components on the boundary and in the region of a piecewise-homogeneous half-space are presented.     

Solution of an axisymmetric problem of free vibrations of piezoceramic hollow cylinders of finite length by the spline collocation method

An axisymmetric problem of longitudinal free vibrations of piezoceramic hollow cylinders for some types of boundary conditions on the end faces is considered. The lateral surfaces of the cylinder are covered with thin short-circuited electrodes. The polarization of piezoceramics is directed along the axis of the cylinder. Using the method of separation of variables and method of spline collocation along the length of the cylinder, we reduce the problem to a system of ordinary differential equations, which is solved by the method of step-by-step search. We present results of calculations for a PZT 4 piezoceramic cylinder.     

Torsional vibrations of piezoceramic hollow cylinders with circular polarization

A problem of natural torsional vibrations of hollow piezoceramic cylinders is considered. The piezoceramic is polarized in the circular direction. The lateral surfaces of the cylinder are covered with shortcircuited thin electrodes. The method of solution of the problem is based on the combination of the spline collocation methods along the longitudinal coordinate and the stepwise search method along the radial coordinate. We present results of a numerical analysis of a cylinder made from PZT 4 ceramic in a wide range of change in the geometrical characteristics of the cylinder.     

The antiplane problem of electroelasticity for a cylinder with a tunnel crack excited by an arbitrary system of surface electrodes

The antiplane mixed boundary-value problem of electroelasticity of the oscillations of an infinite piezoceramic cylinder, weakened by a curvilinear tunnel crack, is considered. Using special integral representations of the solution, the boundary-value problem is reduced to a system of singular integro-differential equations of the second kind with discontinuous kernels. The results of a numerical realization of the algorithm, characterizing the amplitude-frequency characteristics of a piecewise-uniform cylinder and the behaviour of the components of the electroelastic field in the region and on the boundary of the cylinder under conditions of the inverse piezoelectric effect, are presented.     

Homogeneous solutions of the problem of steady vibrations of a piezoceramic cylinder

Steady vibrations of a hollow piezoceramic cylinder with radial polarization are considered. An analytical and numerical analysis is performed of the homogeneous solutions, and the behaviour of the dispersion curves of the real and complex modes is investigated as a function of the geometrical parameters.     

Excitation of Shear Waves in a Piezoceramic Medium with Two Tunnel Holes by a System of Surface Electrodes

An antiplane mixed dynamic problem of electroelasticity for an infinite piezoceramic medium weakened by two tunnel holes is considered. The conjugated fields are excited by the difference of electric potentials applied to a system of electrodes located on the surfaces of the holes. The boundary-value problem is reduced to a system of singular integrodifferential equations with discontinuous kernels, which is solved numerically by quadratures. The results of parametric investigations, characterizing the behavior of the electric field components on the boundary and in the region of the piecewise-homogeneous space, are presented.     

Structure of the frequency spectrum of nonaxisymmetric vibrations of a transversally isotropic hollow ball

We consider nonaxisymmetric vibrations of a hollow sphere made of a spherically Isotropic material. The original boundary-value problem is reduced to a system of ordinary differential equations by introducing new unknown junctions and expanding the sought solution in a series in spherical functions. The nonaxisymmetric problem of vibrations of a hollow ball decomposes into two independent problems that describe modes of first and second kind. The frequency spectrum for modes of first and second kind is analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 32–35, 1986.     

Boundary Control of Physical Fields in a Piezoceramic Layer Having a Tunnel Hole with a System of Electrodes

An approach to the optimal (from the viewpoint of minimum energy consumption) control of the characteristics of coupled fields is illustrated with the example of an antiplane dynamic problem of electroelasticity for a piezoceramic layer with a tunnel hole. The vibrations in the layer are excited by time-dependent harmonic loads of two types: by the difference of electric potentials applied to a pair of infinitely long electrodes located on the surface of the hole and by shear forces or electric charges continuously distributed on the given sections of the layer base, which are assumed as control actions. The inverse problem is solved by using the solution of the corresponding direct mixed antiplane problem of electroelasticity, with the help of which the optimization problem is reduced to the -problem of moments. The functions of optimal control allowing one, for example, to disconnect the external circuit of the generator of electric voltage are given.     

Diffraction of a plane acoustic wave by a non-uniform thermoelastic cylindrical layer bounded by inviscid heat-conducting fluids

The diffraction of sound by a radially laminated isotropic thermoelastic cylindrical shell is considered. The system of equations for small perturbations of a hollow thermoelastic cylinder is reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by the spline-collocation method. Expressions are obtained describing the wave field outside the cylindrical layer. Results of calculations of polar radiation patterns of the amplitude of the scattered acoustic wave in the far zone are presented.     

Numerical solution of the problem of the stress-strain state in hollow cylinders using spline approximations

The three-dimensional theory of elasticity is used for a study of the stress-strain state in a hollow cylinder with varying stiffness. The corresponding problem is solved by a method that is partly analytical and partly numerical in nature: Spline approximations and collocation are used to reduce the partial differential equations of elasticity to a boundary-value problem for a system of ordinary differential equations of higher order for the radial coordinate, which is then solved using the method of stable discrete orthogonalization. Results for an inhomogeneous cylinder for various types of stiffness are presented.     

Determination of the stress state in a half-space in the vicinity of cylindrical defects appearing on the surface under torsional vibrations

We solve the problem of the determination of the stress state under torsional vibrations of a half-space with a cylindrical defect (a crack or a thin rigid inclusion) that crosses its surface. The method of solution is based on the use of discontinuous solutions of the equations of torsional vibrations and consists in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of an angular displacement or a tangential stress.     

Solution of problems of free torsional vibrations of thick-walled orthotropic inhomogeneous cylinders

With the use of the 3D theory of elasticity, we investigate the problem of free torsional vibrations of an anisotropic hollow cylinder with different boundary conditions at its end faces. We have proposed a numerical-analytic approach for the solution of this problem. The original partial differential equations of the theory of elasticity with the use of spline approximation and collocation are reduced to an eigenvalue problem for a system of ordinary differential equations of high order in the radial coordinate. This system is solved by the stable numerical method of discrete orthogonalization together with the method of step-by-step search. We also present numerical results for the case of orthotropic and inhomogeneous material of the cylinder for some kinds of boundary conditions.     

Dynamics of a Sea Platform Interacting with the Seabed and the Water Medium

Transient vibrations of the system {structure–seabed–water layer} are investigated. The schematic representation of the structure consists of a rigid circular foundation slab and a superstructure in the form of a vertical viscoelastic cylinder with lumped masses. The interaction of the foundation with a viscoelastic half-space (the seabed) is taken into account by using the method of orthogonal polynomials to solve the dynamical contact boundary-value problem. A procedure is developed for including in the analysis the interaction of a cylindrical foundation-supported structure with the water layer (to simulate the dynamics of an oil-drilling platform). The assumed load is seismic action specified by an analog accelerogram of the local area. A computational system is developed for numerical simulation of the vibrations and for determining the forces, displacements, accelerations, and contact stresses.     

Diffraction of water waves by a slotted dual cylindrical caisson breakwater

The linearized theory of water waves is used to examine the diffraction of incident regular waves by a slotted dual cylindrical caisson breakwater. The breakwater consists of a vertically slotted outer cylinder circumscribing an impermeable inner cylinder. Under the assumption that the wavelength is much greater than its thickness, each panel is replaced by a thin structure and the permeability is modeled by suitable boundary conditions applied on its surface. The mixed boundary-value problem for the outer cylinder is transformed to dual series relations and the least-square method is applied to get the forces on the structure and associated diffraction field. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The coupled problem of thermoelastic vibrations of a cylinder with cavities

Using a method based on the machinery of analytic functions of complex variables, we construct the solution of the problem of steady planar strain of an isotropic cylinder with curvilinear cavities caused by uneven heating of the surfaces of the cylinder and the cavities. It is an exact solution of the equations and an approximate solution of the boundary conditions. The results of numerical studies are given for a hollow cylinder undergoing thermoelastic radially symmetric vibrations. Two figures. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 82–86.     

On the triviality condition for the kernel of the linearized first boundary-value problem for a system of equations of magnetic hydrodynamics

The first boundary-value problem for a system of equations of magnetic hydrodynamics in a cylinder of infinite length with ideally conducting surface is considered under conditions in the case where it has a two-dimensional solution. Sufficient conditions are obtained for the problem linearized in the neighborhood of this solution to have the trivial solution.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 168–173, February, 1995.     

An asymptotic calculation of the vibrations of a viscoelastic cylinder with a slowly varying internal boundary

The vibrational problem is considered for a long hollow viscoelastic cylinder with a variable internal boundary, this being enclosed in an elastic shell. An asymptotic method based on averaging is used to find the resonant vibrations in response to a small uniformly distributed internal pressure that varies periodically with time. The calculation is carried through to quadrature working formulas.Moscow Institute of Electric Machine Construction. Translated from Mekhanika Polimerov, No. 5, pp. 935–939, September–October, 1973.     

Harmonic and pulse excitations of multiply connected cylindrical bodies

The three-dimensional problem of the theory of elasticity of the harmonic oscillations of cylindrical bodies (a layer with several tunnel cavities on a cylinder of finite length) is considered for uniform mixed boundary conditions on its bases. Using the Φ-solutions constructed, the boundary-value problems are reduced to a system of well-known one-dimensional singular integral equations. The solution of the problem of the pulse excitation of a layer on the surface of a cavity is “assembled” from a packet of corresponding harmonic oscillations using an integral Fourier transformation with respect to time. The results of calculations of the dynamic stress concentration in a layer (a plate) weakened by one and two openings of different configuration are given, as well as the amplitude-frequency characteristics for a cylinder of finite length with a transverse cross section in the form of a square with rounded corners, and data of calculations for a trapeziform pulse, acting on the surface of a circular cavity, are presented.     

The method of homogeneous solutions in an axisymmetric problem of statics for a finite cylinder with homogeneous conditions on the ends

Using the property of Papkovich generalized orthogonality of eigenfunctions, we develop a method of satisfying the boundary conditions on the lateral surface of a cylinder. The stresses and displacements in a finite cylinder with homogeneous conditions on the ends are represented in terms of the axial displacement. The solution is constructed as an expansion in a series of eigenfunctions of the corresponding homogeneous boundary-value problem. We find a class of boundary conditions that admits a solution of the problem without reduction to an infinite system of algebraic equations. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 135–139.     

Spectral problem for an anisotropic elastic waveguide

We consider the homogeneous boundary-value problem of propagation of elastic waves in a hollow anisotropic cylinder with free walls. The problem is solved by a method based on series expansion of traveling-wave amplitudes in powers of the radial coordinate. A dispersion relationship is obtained. The different types of wave motion are analyzed as a function of the degree of anisotropy of the material.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 76–80, 1985.