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.     

Antiplane Problem of Electroelasticity for a Hollow Piezoceramic Cylinder Excited by a System of Surface Electrodes

An antiplane mixed boundary-value problem of electroelasticity is considered for a hollow piezoceramic cylinder with an arbitrary system of active surface electrodes generating its harmonic vibrations. The problem is solved using a method elaborated earlier for investigating vibrations of a solid piezoceramic cylinder with a system of active surface electrodes. The scheme of numerical solution of the obtained singular integro-differential equations of the boundary-value problem is based on the quadrature method. Calculation results are presented that describe the amplitude-frequency characteristics of a hollow cylinder and the behavior of some mechanical and electric quantities both within the cylinder and on its boundary.     

The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel

In this paper, the blood flow problem is considered in a blood vessel, and a coupling system of Navier–Stokes equations and linear elastic equations, Navier–Lame equations, in a cylinder with cylindrical elastic shell is given as the governing equations of the problem. We provide two finite element models to simulating the three-dimensional Navier–Stokes equations in the cylinder while the asymptotic expansion method is used to solving the linearly elastic shell equations. Specifically, in order to discrete the Navier–Stokes equations, the dimensional splitting strategy is constructed under the cylinder coordinate system. The spectral method is adopted along the rotation direction while the finite element method is used along the other directions. By using the above strategy, we get a series of two-dimensional-three-components (2D-3C) fluid problems. By introduce the S-coordinate system in E³ and employ the thickness of blood vessel wall as the expanding parameter, the asymptotic expansion method can be established to approximate the solution of the 3D elastic problem. The interface contact conditions can be treated exactly based on the knowledge of tensor analysis. Finally, numerical test shows that our method is reasonable.     

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.     

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.     

Verification of software codes for simulation of unsteady flows in a gas centrifuge

A simple semi-analytical solution is proposed for the problem of an unsteady gas flow in a gas centrifuge. The circulation in the centrifuge is driven by a source/sink of energy and by an external force (deceleration/acceleration of the gas rotation) acting on the gas at a given frequency. In the semi-analytical solution, the rotor is infinite, while the given forces vary harmonically with a given wave-length along the axial coordinate. As a result, the unsteady flow problem is reduced to a system of ordinary differential equations, which can be quickly solved to any prescribed accuracy. This problem is proposed for verifying numerical codes designed for the simulation of unsteady processes in gas centrifuges. A similar unsteady problem is solved numerically, in which case the cylinder is finite with the rotor length equal to the wavelength of the external force along the axis of rotation. The periodicity of the solution is set at end faces of the cylinder. As an example, the semi-analytical solution is compared with the numerical one obtained with these boundary conditions. The comparison confirms that the problem formulations are equivalent in both cases.     

Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction

We study the asymptotics of the Stokes problem in cylinders becoming unbounded in the direction of their axis. First we assume that the applied forces are independent of the axis coordinate, then we assume that they are periodic along the axis of the cylinder. Finally in Section 4, we make an asymptotic analysis under much more general assumptions on the applied forces.     

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.     

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.     

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.     

Saint-Venant's problem for heterogeneous anisotropic elastic solids

Summary In this paper we give a method to solve Saint-Venant's problem for inhomogeneous and anisotropic elastic cylinders when the elastic coefficients are independent of the axial coordinate. The cross-section of the cylinder is assumed to be occupied by different inhomogeneous and anisotropic elastic materials. Dedicated to Prof.Dario Graffi on his 70th birthday Entrata in Redazione il 15 maggio 1975.     

The construction of a nose shape of minimum drag for specified external dimensions and volume using Euler equations

The problem of constructing the axisymmetric nose shape which gives minimum wave drag for a specified volume and external dimensions is solved by a direct method using Euler equations. As in the Newton's formula approximation, the optimum contours together with the front faces – a segment of the boundary extremum along the longitudinal coordinate and the gently sloping segment of the bilateral extremum – may contain a cylindrical end part with a horizontal segment of the boundary extremum with respect to the maximum admissible radial coordinate. In the direct method, the required parameters (“controls”), which define the shape of the optimum contour, are the radii corresponding to the points of the segment of a bilateral extremum, including the radius of the face for fixed abscissas. For each aspect ratio (the ratio of the length to the radius of the base), when a certain value of the volume coefficient (the ratio of the volume to the volume of a cylinder of maximum external dimensions) is exceeded, the optimum nose shape is completed by a rear cylindrical part. The optimum nose shape, which begins from a certain initial contour, that satisfies the limitations of the problem, is constructed after a finite number of cycles. In each cycle, all the controls are corrected, and together with the directions of the change, their increments are found, while the information necessary for this for any number of controls is obtained after three direct calculations. One other advantage of the method is the rapid, close to quadratic, convergence. The nose shapes constructed are compared with the nose shapes that are optimum in the Newton's formula approximation.     

The virtual mass coefficients of a circular cylinder moving in an ideal fluid between parallel walls

A modification of the method of successive approximations to find the dependence of the virtual mass coefficients of a circular cylinder, moving perpendicular to its axis at an arbitrary point between parallel walls, on the dimensionless distances to both walls, is proposed. The velocity field of the corresponding plane problem is modelled by an infinite sequence of dipoles, situated along a line passing through the centre of the cylinder and perpendicular to the walls. The relations obtained are approximated by simple continuous functions. In special cases, the results obtained are compared with well-known solutions of the problem of the motion of the cylinder near one wall and transverse in rotational flow around an array of cylinders.     

Uniqueness of the solution of the first mixed problem for a two-dimensional nonlinear parabolic equation

The first mixed problem for a two-dimensional nonlinear parabolic equation with nonlinear occurrences of the second derivatives of the unknown function is considered. Under the assumption that a solution, possessing continuous second derivatives with respect to the coordinate variables exists in a closed cylinder and under certain constraints on the initial data of the problem, the uniqueness of this solutions is proved by applying the longitudinal version of the method of straight lines. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 60–69. Translated by N. S. Zabavnikova     

Thermal effects in orthotropic porous elastic beams

This paper is concerned with the linear theory of anisotropic porous elastic bodies. The extension and bending of orthotropic porous elastic cylinders subjected to a plane temperature field is investigated. The work is motivated by the recent interest in the using of the orthotropic porous elastic solid as model for bones and various engineering materials. First, the thermoelastic deformation of inhomogeneous beams whose constitutive coefficients are independent of the axial coordinate is studied. Then, the extension and bending effects in orthotropic cylinders reinforced by longitudinal rods are investigated. The three-dimensional problem is reduced to the study of two-dimensional problems. The method is used to solve the problem of an orthotropic porous circular cylinder with a special kind of inhomogeneity.     

The Fourier-finite element method for the Poisson problem on a non-convex polyhedral cylinder

We study the Poisson problem with zero boundary datum in a (finite) polyhedral cylinder with a non-convex edge. Applying the Fourier sine series to the equation along the edge and by a corner singularity expansion for the Poisson problem with parameter, we define the edge flux coefficient and the regular part of the solution on the polyhedral cylinder. We present a numerical method for approximating the edge flux coefficient and the regular part and show the stability. We derive an error estimate and give some numerical experiments.     

Thermal effects in orthotropic porous elastic beams

This paper is concerned with the linear theory of anisotropic porous elastic bodies. The extension and bending of orthotropic porous elastic cylinders subjected to a plane temperature field is investigated. The work is motivated by the recent interest in the using of the orthotropic porous elastic solid as model for bones and various engineering materials. First, the thermoelastic deformation of inhomogeneous beams whose constitutive coefficients are independent of the axial coordinate is studied. Then, the extension and bending effects in orthotropic cylinders reinforced by longitudinal rods are investigated. The three-dimensional problem is reduced to the study of two-dimensional problems. The method is used to solve the problem of an orthotropic porous circular cylinder with a special kind of inhomogeneity.      

Concerning an approach to identifying the Lamé parameters of an elastic functionally graded cylinder

Based on the general linear elasticity relations, an axisymmetric problem on the steady-state oscillations of a functionally graded hollow cylinder is formulated. The Lamé parameters are considered variable in radial coordinate. Oscillations are caused by the distributed load applied to the outer part of the cylinder boundary. Using the variable separation method, the direct problem on determining the radial and longitudinal components of the displacement field is investigated. The influence of the laws of variation for the Lamé parameters on acoustic characteristics is analysed. The inverse coefficient problem on the identification of the variable Lamé parameters from the data on the amplitude-frequency characteristic is stated. Based on the weak formulation of the problem for an elastic inhomogeneous body, a general linearised relation for the desired and given characteristics is obtained. A system of the Fredholm integral equations of the first kind is formulated with respect to two unknown corrections to the restored laws of the Lamé parameters change. The solution is built by means of an iterative process. A reconstruction of various laws of changing the Lamé parameters is carried out. The accuracy of the presented algorithm is estimated, and recommendations for the most efficient implementation of the reconstruction procedure are proposed.     

Nonstationary nonaxisymmetric temperature fields of multilayer orthotropic cylinders

A solution of the three-dimensional nonstationary nonaxisymmetric problem of heat conduction for a preliminarily heated multilayer infinite orthotropic cylinder under the action of heat sources and in the presence of convective heat exchange is obtained. In this case, a Green function of the corresponding problem of heat conduction is used. As an example, we consider the heating of the cylinder by a mobile normally distributed heat flow. The temperature field in a two-layer cylinder caused by a heat flow that moves along the generator and a helical curve is investigated.     

Influence of the mechanical properties of a cylinder on the temperature stresses with allowance for the edge effect

The volume state of stress of a hollow homogeneous cylinder with temperature varying along the radius is analyzed for various values of Poisson's ratio, which in polymers may fluctuate over a wide range. The stresses in a cylinder of polymeric material, calculated with and without allowance for the temperature dependence of the modulus of elasticity for the same loading conditions, are subjected to a comparative analysis. Numerical results are obtained on the basis of a solution of the axisymmetric three-dimensional problem of thermoelasticity for an inhomogeneous cylinder using a collocation method.Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1083–1086, November–December, 1969.