Dynamic stability of cylindrical orthotropic shells with an elastic core under a longitudinal periodic load

The dynamic instability of a cylindrical orthotropic shell with an elastic core subjected to a longitudinal periodic load is considered. Equations are obtained for determining the regions of dynamic instability for different core models.     

Parametric vibrations of cylindrical shells with a viscoelastic core

The method of calculating the axisymmetric and nonaxisymmetric parametric vibrations of a cylindrical shell bonded to an elastic core [2] is extended to the case of hollow and solid viscoelastic cores by substituting for the material moduli in the equations of motion of the core integral operators with kernels in the form of an exponential and a sum of exponentials. Expressions are given for the reaction of the viscoelastic core, together with the equation of the boundaries of the spectrum of principal regions of dynamic instability. The effect of relaxation time and the long-term modulus of elasticity of the core on the shape and location of the regions of dynamic instability is analyzed.     

Parametric vibrations of an orthotropic cylindrical shell with an elastic core. 2. Some numerical results

It is shown that the operator of the problem belongs to the class of oscillation operators. The first five regions of dynamic instability of an elastic orthotropic cylindrical shell with an elastic isotropic core are determined in the first approximation. The effect of the core and transverse shear in the shell on the width and location of these regions of dynamic instability of the system is determined. The effect of transverse shear on the natural frequencies of the empty shell is established.     

Torsional stability of a glass-reinforced plastic cylindrical shell with an elastic core

The problem of the stability of a glass-reinforced plastic cylindrical shell with an elastic core subjected to twisting moments applied to the edges of the shell is considered. As in various other studies [4–6], the glass-reinforced plastic is treated as an elastically orthotropic material. The core is treated as an isotropic elastic cylinder, whose outer surface is bonded to the shell. Expressions for the critical stresses are obtained for an infinitely long shell and a shell of finite length.Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1082–1086, November–December, 1970.     

Axially symmetric temperature stresses in an elastic isotropic cylinder of finite length

We consider an axially symmetric problem of the thermostressed state of a solid cylinder of finite length with a load-free surface. Using the method of superposition, we have constructed the complete analytical solution of this problem, which is reduced to the solution of a system of linear algebraic equations. We have proposed a method for determining the asymptotic behavior of coefficients in these systems, which enables us to develop an efficient algorithm for the calculation of stresses in the cylinder, including regions near its end-face circles. Typical examples are considered.     

Dynamic stability of a viscoelastic orthotropic cylindrical shell

A method based on the use of Laplace transforms has been developed for reducing the system of equations of motion of a viscoelastic orthotropic cylindrical shell to a single integro-differential equation. The effect of the viscous components on the regions of dynamic instability is investigated (creep due to the action of the shear stresses is taken into account).Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 714–721, July–August, 1973.     

Parametric resonance of a FG cylindrical thin shell with periodic rotating angular speeds in thermal environment

Parametric resonance of a functionally graded (FG) cylindrical thin shell with periodic rotating angular speeds subjected to thermal environment is studied in this paper. Taking account of the temperature-dependent properties of the shell, the dynamic equations of a rotating FG cylindrical thin shell based upon Love's thin shell theory are built by Hamilton's principle. The multiple scales method is utilized to obtain the instability boundaries of the problem with the consideration of time-varying rotating angular speeds. It is shown that only the combination instability regions exist for a rotating FG cylindrical thin shell. Moreover, some numerical examples are employed to systematically analyze the effects of constant rotating angular speed, material heterogeneity and thermal effects on vibration characteristics, instability regions and critical rotating speeds of the shell. Of great interest in the process is the combined effect of constant rotating angular speed and temperature on instability regions.     

Nonlinear evolution of the travelling waves at the surface of a thin viscoelastic falling film

The problem of hydrodynamic instability of a thin condensate viscoelastic liquid film flowing down on the outer surface of an axially moving vertical cylinder is investigated. In order to improve the accuracy of numerical results, the viscoelastic and heat transfer parameters have been included into the governing equations. Also, the analytical solutions are obtained by utilizing the long-wave perturbation method. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem. It has been revealed that the stability of the film flow is weakened when the radius of cylinder and the temperature difference are reduced. Moreover, it is found that the increment of down-moving motion of the cylinder can enhance the flow stability. Further, the thin film flow can be destabilized by the viscoelastic property. The results show that both supercritical stability and subcritical instability can take place within the film flow system given appropriate conditions. Moreover, the absence of Reynolds number leads to an obvious difference in the behavior of some physical parameters.     

Bending-torsion vibration of a partially submerged cylinder with an arbitrary cross-section

An analytical method is presented to investigate the bending-torsion vibration characteristics of a cylinder with an arbitrary cross-section and partially submerged in water. The compressibility and the free surface waves of the water are considered simultaneously in the analysis. The exact solution of structure–water interaction is obtained mathematically. Firstly, the analytical expression of the velocity potential of the water is derived by using the method of separation of variables. The unknown coefficients in the velocity potential are determined by the longitudinal and circumferential Fourier expansions along the outer surface of the cylinder and are expressed in the form of integral equations including the unknown dynamic bending deflection and torsional angle of the cylinder. Secondly, the force and torque acting on the cylinder per unit length, provided by the water, are obtained by integrating the water dynamic pressure along the circumference of the cylinder. The general solution of bending-torsion vibration of the cylinder under the water dynamic pressure is derived analytically. The integral equations included in the velocity potential of the water can be solved exactly. Finally, the eigenfrequency equation of cylinder–water interaction is obtained by means of the boundary conditions of the cylinder. Some numerical examples for elliptical columns partially submerged in water are provided to show the application of the present method.     

Linear and nonlinear dynamics of a circular cylindrical shell connected to a rigid disk

The dynamics of a circular cylindrical shell carrying a rigid disk on the top and clamped at the base is investigated. The Sanders–Koiter theory is considered to develop a nonlinear analytical model for moderately large shell vibration. A reduced order dynamical system is obtained using Lagrange equations: radial and in-plane displacement fields are expanded by using trial functions that respect the geometric boundary conditions.The theoretical model is compared with experiments and with a finite element model developed with commercial software: comparisons are carried out on linear dynamics.The dynamic stability of the system is studied, when a periodic vertical motion of the base is imposed. Both a perturbation approach and a direct numerical technique are used. The perturbation method allows to obtain instability boundaries by means of elementary formulae; the numerical approach allows to perform a complete analysis of the linear and nonlinear response.     

大气运动基本方程组的稳定性分析

以分层理论提供的基本方法分析大气运动基本方程组的拓扑学特征;证明局地直角坐标系中的大气运动基本方程组在无穷可微函数类中是稳定方程;给出局部解意义下使方程组典型定解问题适定的充要条件;讨论大气动力学中有关“以过去推测未来”以及当涉及应用问题时如何修改定解条件和下垫面的选择等问题;指出在通常假设下,基本方程组中的3个运动方程和连续方程完全决定了这个方程组的性质．     

Dynamic stability of an elastic orthotropic cylindrical shell with allowance for transverse shears

The resolvents for the dynamic stability of an elastic orthotropic cylindrical shell are obtained in accordance with the Ambartsumyan and Timoshenko-type refined theories. The regions of instability given by the classical and refined theories are compared. The dependence of the refinements on the shell parameters, the shear moduli of the material, and the buckling modes are investigated.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 312–320, March–April, 1973.     

Stability of an Elastic Tube Conveying a Non-Newtonian Fluid and Having a Locally Weakened Section

The work is devoted to the stability analysis of the flow of a non-Newtonian powerlaw fluid in an elastic tube. Integrating the equations of motion over the cross section, we obtain a one-dimensional equation that describes long-wave low-frequency motions of the system in which the rheology of the flowing fluid is taken into account. In the first part of the paper, we find a stability criterion for an infinite uniform tube and an absolute instability criterion. We show that instability under which the axial symmetry of motion of the tube is preserved is possible only for a power-law index of n < 0.611, and absolute instability is possible only for n < 1/3; thus, after the loss of stability of a linear viscous medium, the flow cannot preserve the axial symmetry, which agrees with the available results. In the second part of the paper, applying the WKB method, we analyze the stability of a tube whose stiffness varies slowly in space in such a way that there is a “weakened” region of finite length in which the “fluid–tube” system is locally unstable. We prove that the tube is globally unstable if the local instability is absolute; otherwise, the local instability is suppressed by the surrounding locally stable regions. Solving numerically the eigenvalue problem, we demonstrate the high accuracy of the result obtained by the WKB method even for a sufficiently fast variation of stiffness along the tube axis.     

The steady motion of an elastic cylinder compressed by a fixed shell

The steady mixed problem of the motion of a transversely isotropic elastic circular cylinder, compressed by a finite elastic shell, is solved by the method of piecewise-homogeneous solutions [1]. One of the relations of generalized orthogonality obtained for homogeneous solutions is used. Two special cases are considered: (1) a semi-infinite shell is placed on a movable cylinder with a specified negative allowance the edge of the shell is stress-free, and there is no preloading, and (2) a concentrated encircling load acts on the shell. The solution of the problem of a semi-infinite shell and the system of piecewise-homogeneous solutions are constructed in quadratures by the Wiener-Hopf method. (A similar problem was investigated in [2] in a static formulation. Steady mixed contact problems were investigated previously in [3–10]).     

Small free vibrations of an infinite cylindrical shell rotating on rollers

Small free vibrations of an infinitely long rotating cylindrical shell being in contact with rigid cylindrical rollers are considered. A system of linear differential equations for the vibrations of such a shell is derived. By using the Fourier transform of the solutions in the circumferential coordinate, a system of algebraic equations for approximately determining the vibration frequencies and mode shapes is obtained. It is shown that, for any number n of uniformly distributed rollers, the approximate values of the first n frequencies and mode shapes can be found explicitly. On the basis of the orthogonal sweep method, an algorithm for numerically solving the boundary value eigenvalue problem describing the vibrations of a rotating shell is developed. Analytical and numerical results are compared. The obtained approximate formulas for frequencies and the numerical algorithm can be used to design centrifugal concentrators for ore enrichment.     

Effect of transverse shear on the stability of an orthotropic cylindrical shell with an elastic core in axial compression

The effect of transverse shear on the stability of a composite shell in axial compression has been investigated. The core is treated as an isotropic elastic cylinder bonded to the inner surface of the shell. The effect of the tangential shearing forces between the shell and the core is taken into account.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 267–273, March–April, 1973.     

The effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk

It is well known that either the asymmetric disk or transverse crack brings parametric inertia (or stiffness) excitation to the rotor-bearing system. When both of them appear in a rotor system, the parametric instability behaviors have not gained sufficient attentions. Thus, the effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk is studied. First, the finite element equations of motion are established for the asymmetric rotor system. Both the open and breathing transverse cracks are taken into account in the model. Then, the discrete state transition matrix (DSTM) method is introduced for numerically acquiring the instability regions. Based upon these, some computations for a practical asymmetric rotor system with open or breathing transverse crack are conducted, respectively. Variations of the primary and combination instability regions induced by the asymmetric disk with the crack depth are observed, and the effect of the orientation angle between the crack and asymmetric disk on various instability regions are discussed in detail. It is shown that for the asymmetric angle around 0, the existence of transverse (either open or breathing) crack has attenuation effect upon the instability regions. Under certain crack depth, the instability regions could be vanished by the transverse crack. When the asymmetric angle is around π/2, increasing the crack depth would enhance the instability regions.     

Dynamic response of fiber-reinforced composite plates

The dynamic stability of orthotropic thick plates subjected to a periodic uniaxial stress and a bending stress is investigated. Both the rotary inertia and the transverse stress are considered in the investigation. The governing equations of motion of Mathieu type are established by applying the Galerkin method with reduced eigenfunction transforms. Based on Bolotin’s method, the dynamic instability regions of graphite- and glass-fiber-reinforced plates are evaluated by solving eigenvalue problems. A dynamic instability index is defined and used as an instability measure to study the influence of various parameters. The effects of material properties and load parameters on the instability region and on the index of dynamic instability of orthotropic plates are discussed.     

Dynamic stability of a porous cylindrical shell

This paper is devoted to a closed cylindrical shell made of a porous-cellular material. The mechanical properties vary continuously on the thickness of a shell. The mechanical model of porosity is as described as presented by Magnucki, Stasiewicz. A shell is simply supported on edges. On the ground of assumed displacement functions the deformation of shell is defined. The displacement field of any cross section and linear geometrical and physical relationships are assumed in cylindrical coordinate system. The components of deformation and stress state were found. Using the Hamilton's principle the system of differential equations of dynamic stability is obtained. The forms of unknown functions are assumed and the system of a differential equations is reduced to a simple ordinary equation of dynamic stability of shell (Mathieu's equation). The derived equation are used for solving a problem of dynamic stability of porous-cellular shell with intensity of load directed in generators of shell. The critical loads are derived for a family of porous shells. The unstable space of family porous shells is found. The influence a coefficient of porosity on the stability regions in Figures is presented. The results obtained for porous shell are compared to a homogeneous isotropic cylindrical shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Method of solving equations of motion of viscoelastic media

A new method is proposed for solving dynamic problems for viscoelastic media based on the introduction of potential functions and transformation of equations of motion. The equations obtained for potential functions are used for constructing the general solution in the case of the effect of moving loads on viscoelastic media with plane-parallel interfaces. The problem of the propagation of Rayleigh surface waves is solved independently of the form of the kernels of the linear operators; a formula is obtained for determining the velocity of the Rayleigh surface wave with an arbitrary form of the viscoelastic operators. A method of experimental determination of the kernels determining the linear viscoelastic operators is proposed.V. V. Kuibyshev Moscow Civil Engineering Institute. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 429–435, May–June, 1973.