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.     

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.     

The scattering of acoustic waves by a cylinder with a non-uniform elastic coating

The problem of the scattering of a plane acoustic wave by a solid cylinder with a radially non-uniform elastic coating is considered. An analytical expression describing the scattered acoustic field is obtained. The equations of motion of the non-uniform elastic cylindrical layer are reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by the power-series method. The results of calculations of the directional pattern of the scattered field are presented.     

Numerical solution of the propagation problem of longitudinal elastic waves in an orthotropic hollow cylinder

Wave processes in an orthotropic cylinder are analyzed. Harmonic oscillations are considered for which the original problem is reduced to a system of ordinary differential equations. This system is solved by the numerically stable discrete-orthogonalization method. Discrete curves for specific problems are given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 75–78, 1985.     

Saint–Venant torsion of a circular bar with radial cracks incorporating surface elasticity

The Saint–Venant torsion problem of a circular cylinder containing a radial crack with surface elasticity is studied. The surface elasticity is incorporated into the crack faces by using the continuum-based surface/interface model of Gurtin and Murdoch. Both an internal crack and an edge crack are considered. By using the Green’s function method, the boundary value problem is reduced to a Cauchy singular integro-differential equation of the first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials and the collocation method. Due to the incorporation of surface elasticity, the stresses exhibit the logarithmic singularity at the crack tips. The torsion problem of a circular cylinder containing two symmetric collinear radial cracks of equal length with surface elasticity is also solved by using a similar method. The strengths of the logarithmic singularity and the size-dependent torsional rigidity are calculated.     

A numerical solution of magneto-thermoelastic problem in non-homogeneous isotropic cylinder by the finite-difference method

This paper presents the development of the magneto-thermoelastic problem in non-homogeneous isotropic cylinder in a primary magnetic field when the curved surface of the cylinder subject to certain boundary conditions. The governing coupled linear partial differential equations in the hyperbolic-type have been solved numerically using the finite-difference method. Graphical results for the temperature, displacement and components of stresses are illustrated and discussed for copper-like material. The results indicate that the effects of inhomogeneity and magnetic field are very pronounced. Some more interesting particular cases have also been discussed.     

Impact of a box with an elastic bottom on a thin liquid layer

The problem of the impact of a box with an elastic bottom on a thin liquid layer is solved in a planar formulation in the shallow water approximation. Using the method of normal modes, the problem is reduced to a non-linear system of ordinary differential equations, which is solved by the Runge–Kutta method. It is shown that the elasticity of the bottom not only affects all the characteristics quantitatively but also qualitatively.     

Application of the method of straight lines to determining the stressed-strained state and dynamic characteristics of a hollow noncircular cylinder

Using the method of straight lines, problems of elasticity theory for a noncircular hollow cylinder are investigated. By means of separation of variables along the generatrix and finite-difference approximation across the thickness, the input partial differential equations are reduced to a system of ordinary differential equations, which is solvable by the stable numerical method of discrete orthogonalization. Specific features of application of the method proposed to static and dynamic problems of a hollow noncircular cylinder are illustrated by way of examples. Bibliography: 5 titles. Translated from,Obchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 89–87.     

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 analysis of the dispersion of electroelastic waves in a cylinder whose physical properties have a longitudinal axis of symmetry

We develop a numerical method of analyzing the dispersion of electroelastic waves in a hollow circular cylinder. The method is based on the reduction of the wave problem to a spectral problem for a system of ordinary differential equations. This system is solved numerically. The roots of the dispersion equation are determined by the method of bisection. The practical convergence of the algorithm is shown using a specific example. One figure. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 151–156.     

An iterative solver for the Oseen problem and numerical solution of incompressible Navier–Stokes equations

Incompressible unsteady Navier–Stokes equations in pressure–velocity variables are considered. By use of the implicit and semi‐implicit schemes presented the resulting system of linear equations can be solved by a robust and efficient iterative method. This iterative solver is constructed for the system of linearized Navier–Stokes equations. The Schur complement technique is used. We present a new approach of building a non‐symmetric preconditioner to solve a non‐symmetric problem of convection–diffusion and saddle‐point type. It is shown that handling the differential equations properly results in constructing efficient solvers for the corresponding finite linear algebra systems. The method has good performance for various ranges of viscosity and can be used both for 2D and 3D problems. The analysis of the method is still partly heuristic, however, the mathematically rigorous results are proved for certain cases. The proof is based on energy estimates and basic properties of the underlying partial differential equations. Numerical results are provided. Additionally, a multigrid method for the auxiliary convection–diffusion problem is briefly discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Flexure of a micropolar elastic circular cylinder

Flexure problem in the linear theory of micropolar elasticity is solved for a circular cylinder by Iesan’s semi-inverse method. Expressions for the macrodisplacement vector and microrotation vector are obtained.     

Solutions of the Saint Venant problem for a cylinder with helical anisotropy

Solutions of the Saint-Venant problem for a cylinder with helical anisotropy are presented in the form of a linear combination of elementary homogeneous solutions, the construction of which is reduced to boundary-value problems for ordinary differential equations with variable coefficients. These problems are integrated using analytical and numerical methods, and the elements of the stiffness matrix are investigated over a wide range of parameters. It is established that when the cylinder is stretched the sign and value of the torsional deformation depends considerably on the value of the relative angle of twist of the helices.     

采用新方法研究非局部理论中Ⅰ-型裂纹的断裂问题

采用新的方法研究非局部理论中Ⅰ_型裂纹的断裂问题,进而确定裂纹尖端的应力状态,这种方法就是Schmidt方法·　所得结果比艾林根研究同样问题的结果准确和更加合理,克服了艾林根研究同样问题时遇到的数学困难·　与经典弹性解相比,裂纹尖端不再出现物理意义上不合理的应力奇异性,并能够解释宏观裂纹与微观裂纹的力学问题·     

Effect of overload on the integral parameters of a concentric annular die

The non-axisymmetric contact problem in the theory of elasticity is solved for a smooth concentric annular die on a uniform elastic half-space when an overload normal to the boundary of the half-space acts outside the die. A Hankel-Fourier integral transform and triple integral equations are used to reduce the problem to a quasi-regular system of algebraic equations which is solved by a perturbation method and, in general, by a reduction method. The effect of a lumped force on the integral parameters of the die is examined. Graphs are shown which characterize the degree to which the parameters of the problem influence the total force and moment, displacement, and inclination of the die. Kharkov State Economics University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 111–117, 1999.     

The transverse curvature effect on the incompressible laminar boundary layer for longitudinal flow over a cylinder

A solution to the partial differential equations governing incompressible laminar flow in the axial direction over a circular cylinder is presented. A method is employed which reduces the partial differential equations to a set of ordinary differential equations which are then solved consecutively. The solution is initiated at a leading edge and proceeds far downstream. Velocity profiles and values for the coefficient of skin friction, displacement thickness and momentum thickness are obtained. The results are compared to previously obtained solutions valid near the leading edge and asymptotic solutions valid far downstream.     

Methods of analysing ropes. The extension–torsion method

Two new approaches are used for calculating the stress–strain state of a rope and its stiffnesses. The first approach relies on the theory of fibrous composites and Saint-Venant's solution for a cylinder with helical anisotropy. The second approach is based on the solution by the finite element method of the three-dimensional problem of elasticity theory for a solid inhomogeneous cylinder formed by a finite number of elastic fibres arranged in helical lines and connected by a weak filler (in the sense that its Young's modulus is several orders of magnitude less than the Young's modulus of the fibre). The behaviour of the stiffness when the modulus of elasticity of the filler tends to zero is analysed, and the results of the limiting transition are discussed. The numerical results obtained are compared with calculations by other well-known applied theories.     

Stability criterion for a class of nonlinear fractional differential systems

Stability analysis of nonlinear fractional differential systems has been an open problem since the 1990s of the last century. Apparently, Lyapunov’s second method seems to be invalid for nonlinear fractional differential systems (equations). In this paper, we are concerned with this open problem and have solved it partly. Based on Lyapunov’s second method, a novel stability criterion for a class of nonlinear fractional differential system is derived. Our result is simple, global and theoretically rigorous. The conditions to guarantee the stability of the nonlinear fractional differential system are convenient for testing. Compared with the stability criteria in the literature, our criterion is straightforward and suitable for application. Several examples are provided to illustrate the applications of our result.     

The non-linear Saint-Venant problem of the torsion,stretching and bending of a naturally twisted rod

Problems on large stretching, torsional and bending deformations of a naturally twisted rod, loaded with end forces and moments, are considered from the point of view of the non-linear three-dimensional theory of elasticity. Particular solutions of the equations of elastostatics are found, which are two-parameter families of finite deformations and which possess the property that, for these deformations, the initial system of three-dimensional non-linear equations reduces to a system of equations with two independent variables. The use of these equations enables one to reduce certain Saint-Venant problems for a naturally twisted rod to two-dimensional non-linear boundary-value problems for a planar domain in the form of the cross-section of a rod. Different formulations of the two-dimensional boundary-value problem for the cross-section are proposed, which differ in the choice of the unknown functions. A non-linear problem of the torsion and stretching of a circular cylinder with helical anisotropy, which is reduced to ordinary differential equations, is considered as a special case.     

A superconvergent linear FE approximation for the solution of an elliptic system of PDEs

The aim of this work is to study a nonstandard piecewise linear finite element method for elliptic systems of partial differential equations. This nonstandard method was considered by the authors for scalar elliptic equations and for a planar elasticity problem. The method enables us to compute a superconvergent numerical approximation to the solution of the system of partial differential equations.