Three-Dimensional Analytical Solution for an Axisymmetric Biharmonic Problem

In this paper, we propose a method for the solution of the axisymmetric boundary value problem for a finite elastic cylinder with assigned stress and/or displacements acting on the ends and side. The technique utilizes the Love representation, which allows for reduction of the solution of the elastic problem to the search for a biharmonic function on a cylindrical domain. In the solution method suggested here, we write the Love function with a Bessel expansion and analyze in detail the conditions under which it is possible to differentiate the expansion term by term. We show that this is possible only for a restricted class of elastic solutions. In the general case, we introduce two new auxiliary functions of the z-coordinate. In this way, we obtain the general form of the axisymmetric biharmonic function, which is discussed in relation to certain specific boundary conditions applied on the side and ends of the cylinder. We obtain an exact explicit solution of practical interest for a cylinder with free ends and assigned displacements applied to the side.     

Combined effects of axial load and temperature on finite deformation of incompressible thermo-hyperelastic cylinder

A finite deformation problem is examined for a cylinder composed of a class of incompressible thermo-hyperelastic Mooney-Rivlin materials under an equal axial load at its two fixed ends and a temperature field at its lateral boundary. Firstly, a thermomechanical coupling term is taken into account in the strain energy density function, and a governing equation of the problem is obtained. Secondly, an implicit analytical solution is derived by using the incompressibility and the boundary conditions. Significantly, numerical examples show that the middle portion of the cylinder undergoes almost a uniform radial deformation. However, the deformation near the two ends varies remarkably along the axial direction for relatively large axial loads. In addition, the rising temperature can increase the deformation of structures, and its influence is linear approximately. Specially,in the case of tensile load, the jump increase of the axial deformation may occur.     

Free axisymmetric vibrations of solid cylinders: numerical problem solving

The problem of free vibrations of a solid cylinder with different boundary conditions is solved using the three-dimensional theory of elasticity and a numerical analytic approach. The spline-approximation and collocation methods are used to reduce the partial differential equations of elasticity to systems of ordinary differential equations of high order with respect to the radial coordinate. These equations are solved by stable numerical discrete orthogonalization and incremental search. Calculated results are presented for transversely isotropic and inhomogeneous materials of the cylinder and for several types of boundary conditions at its ends     

Forced vibrations of a boxed shell of square cross-section

We construct the solution of the problem on the steady-state vibrations of a finite boxed shell of square cross-section with symmetry conditions at the shell ends. We present the dispersion curves, find the natural frequencies, and study the stress distribution in the shell. We obtain a simple formula for the approximate analysis of the shell in the case of low-frequency vibrations on the basis of the expansion of the solution in two small parameters and on the Lagrange interpolation formula.     

Using spline-approximation to solve problems of axisymmetric free vibration of thick-walled orthotropic cylinders

The three-dimensional theory of elasticity is used to study the free vibrations of an anisotropic hollow cylinder with different boundary conditions at the ends. The relevant problem is solved by a numerical-and-analytic method. Spline approximation and collocation is used to reduce the partial differential equations of elasticity to a boundary-value problem for a system of ordinary differential equations of high order for the radial coordinate, which is solved using the stable discrete-orthogonalization and incremental-search methods. The calculated results for an orthotropic inhomogeneous cylinder with boundary conditions of several types are presented Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 74–85, October 2008.     

Free v/s forced vibrations of a cylinder at low Re

Free vibrations of a circular cylinder of low non-dimensional mass are investigated at low Reynolds numbers. Computations are carried out for 5% blockage. Lock-in is observed for a range of Re and is accompanied with hysteresis at both lower as well as higher Re ends of the synchronisation/lock-in region. It is well known that the lock-in regime for free vibrations depends on the non-dimensional mass of the oscillator. The results from the present computations are compared with the data for forced vibrations from Koopmann (Journal of Fluid Mechanics, 28, 501–512, 1967) on a Y _max/D vs. f* plot, where Y _max is the maximum oscillation amplitude and f* is the ratio of cylinder vibration frequency to the vortex shedding frequency for a stationary cylinder. Good agreement is observed for the critical amplitude needed for onset of synchronisation between the forced and free vibrations. The results from the free vibrations are compared to the predictions from the linear oscillator model by assuming that the forces on the cylinder are unaffected as a result of vibrations. It is found that, for low mass oscillators, the modification of vortex shedding frequency and lift coefficient due to cylinder oscillations leads to the enhancement of the lock-in regime.     

Three-dimensional spectral approximations to Stokes flow between eccentrically rotating cylinders

A three-dimensional spectral algorithm for the solution of Stokes flow between eccentrically rotating cylinders is described. Included in the model are pressure boundary conditions at the two ends of the finite length cylinders and the effect of a fluid line source on the inner cylinder. A comparison of results for the load and couple on the inner cylinder is made with those available from lubrication theory in the absence of a line source. Good agreement is shown for long, short and finite journal bearings when the various geometrical assumptions inherent in the lubrication analysis are satisfied.     

一类可压缩超弹性圆柱体轴线上的空穴现象

对于由一类均匀各向同性可压缩的广义Varga材料组成的实心圆柱体,研究其在给定的外表面拉伸和轴向拉伸或压缩共同作用下的轴对称变形问题.利用能量变分原理得到了问题的控制方程和边界条件,并求得了描述柱体径向对称变形的参数型解析解和描述圆柱体轴线上空穴生成和增长的空穴分岔解.给出了与泊松比和轴向伸长相关的径向临界伸长的表达式和空穴生成后的应力表达式;并通过数值算例讨论了这些参数对圆柱体轴线上空穴生成和增长、圆柱体的径向位移以及应力的集中和突变的影响,同时给出了相应的数值模拟.     

Influence of general rotation on the transfer of torsional vibrations through axisymmetric layers of viscous liquid

A study is made of the motion of a viscous incompressible liquid in a gap between a cylinder of finite length and a jacket that contains it. The motion is due to small torsional vibrations of the jacket, whereas the complete system rotates uniformly around the common symmetry axis. The equation is linearized under the assumption that the Rossby number is small. Three problems are considered: the one-dimensional problem of vibrational motion in the cylindrical gap, the self-similar problem for the flow in the end interdisk gaps, and the two-dimensional problem that describes the flow in the corner region. It is established that the superposition of the general rotation makes the damping properties of the liquid layers between the jacket and the ends of the cylinder much less good. The influence of boundary effects is clarified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 26–31, May–June, 1991.     

Vibrations of a cylinder in a uniform flow in the presence of a no-slip side-wall

A circular cylinder placed in a uniform flow, and that spans the entire length between two side walls, may experience either parallel or oblique vortex shedding depending on the end conditions. It was shown by Mittal and Sidharth (2014) that the spatio-temporal periodicity of the oblique vortex shedding results in constant-in-time force experienced by the cylinder. On the contrary, parallel vortex shedding leads to fluid force that fluctuates with time. The free vibrations of a circular cylinder, in the presence of a wall, are investigated. For comparison, computations with end walls, where a slip condition on velocity is specified, are also carried out. The Reynolds number, based on the diameter of the cylinder and free-stream speed of the flow, is Re=100. The initial condition for the free vibrations is the fully developed unsteady flow past a stationary cylinder with oblique shedding. It is found that as the amplitude of vibration of the cylinder builds up, the vortices shed from the cylinder align with its axis leading to parallel shedding. The response of the cylinder is associated with two branches: initial and lower. On the lower branch, the response of the cylinder is virtually identical from two- and three-dimensional computations. The flow as well as the response is different on the initial branch and outside the synchronization regime. Forced vibrations confirm the phenomena.     

Three‐dimensional computation for flow‐induced vibrations in in‐line and cross‐flow directions of a circular cylinder

We present numerical results for in‐line and cross‐flow vibrations of a circular cylinder, which is immersed in a uniform flow and is elastically supported by damper‐spring systems to compute vibrations of a rigid cylinder. In the case of a circular cylinder with a low Scruton number, it is well‐known that two types of self‐excited vibrations appear in the in‐line direction in the range of low reduced velocities. On the other hand, a cross‐flow vibration of the circular cylinder can be excited in the range of high reduced velocities. Therefore, we compute the flow‐induced vibrations of the circular cylinder in the wide range of the reduced velocities at low and high Scruton numbers and discuss about excitation mechanisms in the in‐line and cross‐flow directions. Copyright © 2011 John Wiley & Sons, Ltd.     

Nonaxisymmetric vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials

The nonaxisymmetric problem of natural vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials is solved. The properties of the material change continuously along a radial coordinate according to an exponential law. The lateral surfaces of the cylinder are free of external tractions and short circuited by electrodes. After separation of variables and representation of the components of the displacement vector in the form of standing circumferential waves, the initially three-dimensional problem is reduced to a two-dimensional partial differential equation problem. By using the method of spline-collocations with respect to a longitudinal coordinate, this two-dimensional problem is reduced to a boundary-value problem for the eigenvalues expressed in terms of ordinary differential equations. This problem is solved by the stable discrete-orthogonalization technique in combination with a step-by-step search method with respect to the radial coordinate. Results were obtained numerically and subsequently analyzed in this paper.     

Direct Numerical Simulation and Experimental Investigation on Suppression of Vortex Induced Vibrations of Circular Cylinders by Radial Water Jets

The present paper deals with suppression of vortex induced vibrations (VIV) by introducing radial water jets from circular openings in the wall of the oscillating cylinder. Overpressure within the water-filled cylinder propels water jets blowing out into the ambient flow as a means to alter the vortex shedding process. This flow will introduce a disturbance that is expected to yield reduced VIV amplitudes. Results are presented from experiments in a towing tank testing a spring-supported cylinder with two straight rows of radial water jets along the the cylinder, located at positions +120° and?120° on the cylinder circumference. A smooth cylinder with no openings is tested for comparison. Direct Numerical Simulations (DNS) have been performed using the Spectral/hp element code Nεκταr. Outflow through openings in the cylinder wall is modeled, and a parameter study is performed where number of jets as well as jet location on the cylinder circumference and jet flow rate are varied.     

Influence of Initial Geometric Imperfections on the Vibrations and Dynamic Stability of Elastic Shells

The results from studies into the vibrations and dynamic stability of thin elastic shells with initial geometric imperfections are analyzed. The corresponding dynamic problems are solved in both linear and nonlinear formulations. The influence of initial axisymmetric and nonaxisymmetric deflections on natural, forced, parametrically excited, and self-excited vibrations (flutter) is studied. The dynamic buckling of imperfect shells under short-term impulsive loading is examined. Some aspects of experimental investigation into the vibrations of shells with small geometric imperfections (deviations from the design shape) are considered     

Forced vibrations of orthotropic shells: nonclassical boundary-value problems

The forced vibrations of a cylindrical orthotropic shell are studied. Two types of boundary conditions on the outer surface are examined considering that the displacement vector prescribed on the inner surface varies harmonically with time. Asymptotic solutions of associated dynamic equations of three-dimensional elasticity are found. Amplitudes of forced vibrations are determined and conditions under which resonance occurs are established. Boundary-layer functions are defined. The rate of their decrease with distance from the ends inside the shell is determined. A procedure of joining solutions for the internal boundary-layer problem is outlined in the case for the, if clamping boundary conditions are prescribed at the ends     

Coupled seismic vibrations of a pipeline in an infinite elastic medium

The exact solution of the problem of coupled seismic vibrations of an underground pipeline and an infinite elasticmediumis given. A method dramatically simplifying the solution of the exterior problem for themedium is proposed on the basis of the established theorem on the separation of the boundary conditions for the wave potentials on the surface of the cylinder. The obtained results permit improving the incorrect consideration of the problem accepted in the literature.     

可压缩流体中封闭薄球壳的自由振动

通过引入位移函数，成功地研究了薄球壳在可压缩流体中的自由振动。发现存在两类自由振动：第一类与外界流体无关；第二类则受到流体性质的影响．证明了第二类振动的频率方程具有多项式形式并只存在复频率（除ｎ＝１时有Ω＝０）．求解ｎ＝０，１和２时的频率方程，并讨论参数的影响及给出根轨迹图。最后就小阻尼系数法作了对比分析。     

Effect of magnetic field and non-homogeneity on the radial vibrations in hollow rotating elastic cylinder

In this paper, the natural frequencies of the radial vibrations of a hollow cylinder with different boundary conditions under influences of magnetic field, rotation and non-homogeneity have been studied. The solution of the problem is obtained by using technique of variables separation. In the present paper three different boundary conditions are considered, namely the free, fixed and mixed boundary conditions. The displacement and stresses components have been obtained in analytical form involving Bessel function of first and second kind and of order n. The determination is concerned with the eigenvalues of the natural frequencies of the radial vibrations for different boundary conditions in the case of harmonic vibrations. Numerical results are given and illustrated graphically for each case considered. Comparisons are made with the results in the absence of magnetic field, rotation and non-homogeneity. The results indicate that the effect of magnetic field, rotation and non-homogeneity are very pronounced.     

Plane nonaxisymmetric deformation of a viscoelastic cylinder with consideration of its physical and geometric nonlinearity

The nonaxisymmetric plane problem of the nonlinear theory of viscoelasticity is solved for a cylinder reinforced by an elastic circular shell. The cylinder has an internal cut resembling a Maltese cross in shape. The identification of the nonlinear endochronous theory of aging viscoelastic materials is conducted by a genetic algorithm method on the basis a nonmonotonic experimental stress-strain dependence. Some numerical results obtained for the stress-strain state of this cylinder under the action of internal pressure are discussed with consideration of the above physical nonlinearity and the finite logarithmic strains.     

Flow control in flow–structure interaction

We employ passive flow control using two-dimensional hydrofoils to reduce vortex-induced vibrations (VIV) and drag on a cylinder of circular cross-section. We test the hypothesis that by using foils to bend the streamlines around the cylinder, and hence forcing the flow to approach potential flow-like patterns VIV and drag will be reduced. A systematic parametric search, first using groups of two and then four foils, shows that it is possible to completely eliminate vibrations and reduce the drag coefficient to about C_d=0.50 at sub-critical Reynolds numbers. This parametric search is conducted in conjunction with force measurement and particle image velocimetry on a fixed towed cylinder. The effectiveness of the foils in regards to VIV was further tested with an apparatus allowing free transverse vibrations of a towed cylinder.