The axisymmetric vibrations of a cylindrical incompressible liquid column with a gas bubble

The axisymmetric vibrations of an ideal incompressible liquid column in a rigid circular cylindrical vessel with a spherical gas bubble pulsating near the position of dynamic equilibrium are considered. The boundary-value problem for the liquid velocity potential and the equations for the vibrations of the gas bubble are solved under the conditions on the free surface, sidewall, and the boundary of the gas body. For the case of small amplitudes, the resonance frequencies of the system are determined, and the pressure field in the liquid column is constructed. The results are compared with data known for the gas-accumulation model, data obtained without allowance for the boundedness of the liquid, and experimental data. National Technical University (KPI), Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 74–80, July, 2000.     

Example of motion of a cylindrical solid in a viscous liquid

The problem of motion of a pulsating solid (an infinitely long circular cylinder) in an oscillating viscous liquid in the presence (or absence) of an external stationary force is considered. The perturbation method is applied. It is found that the solution of the time-average motion of a body exists if and only if body pulsations, liquid vibrations, and external forces satisfy a certain relation. The presence of a plane analog of the phenomenon of predominantly unidirectional motion of a compressible solid in an oscillating liquid is established.     

Pulsating motion of an inhomogeneous solid sphere in a vibrating liquid

The problem of rotational vibrations of an inhomogeneous solid body (a sphere) in a uniformly vibrating ideal liquid under gravity is considered. New hydromechanical effects are reported.     

Vibrations of a Cylindrical Shell Filled with a Liquid

The concurrent effect of a harmonic excitation and an airflow on the vibrations and vibroaccelerations of a thin-walled cylindrical shell filled with a liquid is studied     

Forced Nonlinear Vibrations of Cylindrical Shells Filled with a Liquid

Forced nonlinear vibrations of cylindrical shells partially filled with a liquid are studied. The wave motions of the free liquid surface are taken into account. Regular periodic deformation modes of the shell are studied, and frequency ranges where these modes do not occur are determined     

Vibrations of a body in a bounded volume of viscous fluid

The problem of the vibrations of a body in a bounded volume of viscous fluid has been studied on a number of occasions [1–4]. The main attention has been devoted to determining the hydrodynamic characteristics of elements in the form of rods. Analytic solution of the problem is possible only in the simplest cases [2]. In the present paper, in which large Reynolds numbers are considered, the asymptotic method of Vishik and Lyusternik [5] and Chernous' ko [6] is used to consider the general problem of translational vibrations of an axisymmetric body in an axisymmetric volume of fluid. Equations of motion of the body and expressions for the coefficients due to the viscosity of the fluid are obtained. It is shown that in the first approximation these coefficients differ only by a constant factor and are completely determined if the solution to the problem for an ideal fluid is known. Examples are given of the determination of the “viscous” added mass and the damping coefficient for some bodies and cavities. In the case of an ideal fluid, general estimates are obtained for the added mass and also for the influence of nonlinearity. Ritz's method is used to solve the problem of longitudinal vibrations of an ellipsoid of revolution in a circular cylinder. The hydrodynamic coefficients have been determined numerically on a computer. The theoretical results agree well with the results of experimental investigations.     

Viscoelastic vibrations of a triangular plates

The problem of stationary vibrations of a viscoelastic equilateral triangular plate is treated. The vibrations are caused by the action of a uniformly distributed load which varies harmonically and by the motion of the simply supported boundary of the plate as a rigid body which vibrates at the same frequency. The level lines of the vibration amplitude are studied and the graphs of the amplitude distribution along the height of the triangle are given.     

Nonlinear problems of the dynamics of elastic shells partialiy filled with a liquid

Theoretical and experimental investigations of the nonlinear vibrations and dynamic stability of thin shells partially filled with a liquid are reviewed. The paper deals with the basic laws governing the dynamic high-deflection deformation of carrying shell structures and the considerable vibrations of the free liquid surface due to the natural, forced, and parametrically excited vibrations of the combined system and also due to impulse loads acting on the carrying object. The nonlinear dynamic interaction of shells with a liquid filler is analyzed with allowance for the wave motions of the free liquid surface. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 3–34, April, 2000.     

Asymptotic Solution of a Harmonic Contact Problem for a Permeable Stamp on a Liquid Saturated Base

A plane harmonic problem of vertical vibrations of a rigid permeable stamp on a liquid saturated poroelastic base is considered. The equations of two-phase Biot media, which take into account inertial and viscous interactions of phases, are used. The asymptotic properties of the contact stress at low vibration frequencies are studied.     

Stability of high-frequency periodic motions of a heavy rigid body with a horizontally vibrating suspension point

The motion of a heavy rigid body one of whose points (the suspension point) executes horizontal harmonic high-frequency vibrations with small amplitude is considered. The problem of existence of high-frequency periodic motions with period equal to the period of the suspension point vibrations is considered. The stability conditions for the revealed motions are obtained in the linear approximation. The following three special cases of mass distribution in the body are considered; a body whose center of mass lies on the principal axis of inertia, a body whose center of mass lies in the principal plane of inertia, and a dynamically symmetric body.     

Dynamic stabilization of Rayleigh capillary instability

The effect of high-frequency vibrations on the stability of a cylindrical liquid interface is studied. In the absence of external disturbances the interface will be unstable if the length of the liquid cylinder exceeds the length of the normal boundary section. It is shown that vibrations circularly polarized in the plane of the normal section can suppress the development of instability however great the length of the liquid cylinder. The effect of the density ratio of the liquids and the dimensions of the rigid outer shell on the stability of the system is investigated. It is shown that vibrations can stabilize the cylindrical interface only if the radius of the shell is not too great as compared with the radius of the liquid cylinder. The critical value of the radius ratio is approximately equal to 1.58 and does not depend on the density ratio of the liquids.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–7, November–December, 1991.     

The Problem of Forced Nonlinear Vibrations of Cylindrical Shells Completely Filled with Liquid

The forced nonlinear vibrations of a thin cylindrical shell completely filled with a liquid are studied. A refined mathematical model is used. The model takes into account the nonlinear terms up to the fifth power of the generalized displacement of the shell. The Bogolyubov’Mitropolsky averaging method is used to plot amplitude’frequency response curves for steady-state vibrations. The steady-state vibrations at the frequency of principal harmonic resonance are analyzed for stability__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 52–59, February 2005.     

Plane Problem of Vibrations of an Elastic Floating Plate under Periodic External Loading

The Wiener–Hopf technique is used to construct an analytical solution of the problem of vibrations of a semiinfinite elastic floating plate under periodic external loading. The solution is obtained in explicit form ignoring draft. The dependences of the amplitudes of surface waves and iceplate deflection on the loading distribution and frequency, ice thickness, and liquid depth are studied numerically. It is established that for some types of acting load, no waves propagate in the plate and liquid and the plate vibrations are standing waves localized near the loading region. An example of such vibrations is given and a condition for the occurrence of localized vibrations is found.     

Hydroelastic vibrations of a beam floating on the surface of an incompressible liquid

The article considers the problem of determining the forms and the frequencies of the free vibrations of a floating beam of elliptical cross section, on the basis of simultaneous solution of the equations of mechanical bending vibrations and the equations of hydrodynamics. The difference between the forms of the vibrations of a beam in a liquid and in a void is evaluated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 128–132, January–February, 1973.     

Vibrational dynamics of a light spherical body in a rotating cylinder filled with a fluid

The behavior of a light spherical body in a rotating horizontal cylindrical cavity filled with a low-viscosity fluid is experimentally investigated both in the absence and in the presence of transverse vibrations. The system is in a centrifuged state. Mean rotation of the sphere relative to the cavity is found to exist. In the absence of the vibrations slow lagging rotation of the body is due to the gravity field. With increase in the cavity rotation velocity the intensity of differential rotation reduces. The vibrations lead to the excitation of differential rotation of the body, either anticipating or lagging, due to the resonance excitation of its inertial oscillations. The differential rotation of the body leads to the formation of the cylindrical Taylor-Proudman column. With increase in the column rotation velocity the instability of its boundary manifests itself as an azimuthal two-dimensional wave.     

Vibrations of a complex-shaped panel

The free vibrations of flexible shallow shells with complex planform are studied. To analyze the natural frequencies and modes of linear vibrations, the R-function and Rayleigh–Ritz methods are used. A discrete model is obtained using the Bubnov–Galerkin method. The nonlinear vibrations are studied by combining the nonlinear normal mode method and the multiple-scales method. Skeleton curves of natural vibrations are drawn     

Effect of a Shock Pulse on a Floating Ice Sheet

The vibrations of a viscoelastic plate lying on an elastic liquid base subjected to pulse loading have been studied theoretically and experimentally. The effect of the variable depth of the reservoir, plate thickness, and strain relaxation time on the value of the plate vibration amplitude and the length and curvature of the flexural gravity wave profile are analyzed. Good agreement of theoretical and experimental results is obtained.     

Poynting oscillations of a rigid disk supported by a neo-Hookean shaft

Simultaneous axial and torsional oscillations of a rigid disk attached to an elastomeric shaft are investigated. Five cases are solved exactly. The uncoupled, small amplitude axial and torsional oscillations of the disk are investigated for neo-Hookean and Mooney-Rivlin shafts with static stretch. The finite torsional vibration of the load superimposed on a static stretch of the shaft is studied for the Mooney-Rivlin model. Solutions for both small and finite amplitude, uniaxial vibrations of the body superimposed on a pretwisted neo-Hookean shaft with static stretch are derived. Simple bounds on the period for the finite motion are provided; and various universal frequency relations for neo-Hookean and Mooney-Rivlin materials are identified.Finally, the main problem of finite, uniaxial vibrations accompanied by a small twisting motion is studied for the neo-Hookean model. The exact periodic solution for the axial response is obtained; and the coupled, small torsional motion is then determined by Hill's equation. A stability criterion for the Mathieu-Hill equation is used to obtain stability maps in a physical parameter space. Geometrical conditions sufficient for universal stability of the motion are read from this graph. Instability of the torsional oscillation, the beating phenomenon and exchange of energies, and the relation of the stability diagram to amplitude bounds on the uncoupled, linearized motion sufficient to assure universal stability predicted for small amplitude vibrations, are discussed and described graphically with the aid of a numerical model. It is shown that an unstable configuration may be stabilized by increasing the diameter of the disk.     

Vibrations of Cylindrical Shells Filled with a Liquid

The paper reports on experimental results for a dynamic system of two identical shells filled with a liquid and placed on a common elastic foundation. It is established that vibrations excited in one of the shells synchronize the vibrations of the whole system