Basic properties of natural vibrations of an extended segment of a pipeline

Natural transverse vibrations of an extended segment of a pipeline containing a uniformly moving fluid are considered. The mechanical model under study takes into account the inertial forces of the pipe and environment and the moment of Coriolis and centrifugal forces arising because of the medium motion. It is proved that all natural frequencies of the pipeline rigidly clamped at both ends are real (and hence no flutter can arise in this model). For the first three modes, the dependence of the eigenvalues on the fluid flow velocity (varying from zero to the buckling velocity) are constructed, and their properties depending on the inertia parameter are studied. Families of vibration mode shapes of the pipeline are constructed and investigated.     

Natural transverse vibrations of a rotating rod

We study the natural transverse vibration frequencies and modes of a rod rotating about an axis fixed at an end of the rod. The cases of low, moderately high, and asymptotically high angular velocities are considered. The case of a homogeneous rod with clamped left and free right end is considered in detail. A new constructive algorithm based on the notion of “sagittary function” is used to find the dependences of the natural frequencies and mode shapes on the angular velocity for lower vibration modes. We establish evolution to the model corresponding to vibrations of a rapidly rotating thread subjected to the centrifugal inertial forces. It is shown that the natural frequencies grow practically linearly with increasing angular rotation velocity. The results obtained can be of interest in technical applications, e.g., when studying vibrations of sensor elements in high-precision instruments or of rapidly rotating elongated mechanism elements (turbine or propeller blades, etc).     

Natural vibrations in a system of nanotubes

Natural vibrations in a system of parallel micro-and nanotubes attached horizontally to an elastic substrate are analyzed. It is shown that several first eigenfrequencies corresponding to flexural vibrations of a single nanotube can be identified with the use of the linear shell theory within the frequency spectrum of an “integrated system” consisting of a substrate and nanotubes. This allows the flexural rigidity of a single nanotube to be evaluated. The resultant conclusion is supported by finite-element modeling based on the three-dimensional theory of electroelasticity. Results of a modal analysis of gallium arsenide nanotubes are presented. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 160–171, March–April, 2008.     

Steady-state seismic vibrations of a buried pipeline in a viscoelastic soil

Steady-state coupled longitudinal vibrations of a buried pipeline and a viscoelastic soil during an earthquake are studied. Using specific examples of soils described by the Kelvin–Voigt model and by the viscous liquid model, the main qualitative and quantitative effects of viscosity on the behavior and seismic stability of metallic (steel) and concrete pipelines are clarified.     

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.     

Electroelastic vibrations of a hollow spherical segment made of a piezoelectric ceramic

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 10, pp. 41–45, October, 1994.     

Behavior of a floating elastic plate during vibrations of a bottom segment

The Wiener-Hopf technique is used to obtain an analytical solution for the problem of vibrations of a floating semi-infinite elastic plate due to earthquake-induced vibrations of a bottom segment. An explicit solution is obtained ignoring the inertial term. The surface-wave amplitudes and ice-plate deflection are studied numerically as functions of the frequency and position of the vibrating bottom segment, ice thickness, and fluid depth.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 98–108, March–April, 2005.     

Natural vibrations of rectangular plates with a through crack

Summary The paper presents a finite element model of the rectangular plate with a through crack. The crack occurring in the plate is nonpropagating and open. It was assumed that the crack changes only the stiffness of the plate, whereas the mass is unchanged. The method of the formation of the stiffness matrix of a finite element for the rectangular, cracked plate is presented. The effects of the crack location and its length on the changes of the eigenfrequencies of the simply supported and cantilever plate are studied. The results of numerical computations are compared with results of theoretical and experimental data presented in the literature.

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Eigenschwingungen von rechteckigen Platten mit Transverslriß

Übersicht Der Beitrag präsentiert ein Finite-Element-Modell einer rechteckigen Platte mit Transverslriß. Der Riß pflanzt sich nicht fort und bleibt offen. Es wird angenommen, daß der Riß nur die Steifigkeit der Platte verändert, wogegen die Masse konstant bleibt. Eine Methode zur Aufstellung der Steifigkeitsmatrix für ein finites Element der rechteckigen Platte wird vorgestellt. Der Einfluß der Lage des Risses und der Rißlänge auf die Eigenschwingungen der einfach gestützten und freitragenden Platte wird untersucht. Abschließend erfolgt ein Vergleich der numerischen Berechnungen mit den Ergebnissen von theoretischen und experimentellen Literaturdaten.

     

Natural vibrations of a cylindrical shell reinforced with rectangular plates

This paper outlines a technique of determining the natural frequencies and modes of an elastic structure consisting of a cylindrical shell and noncrossing rectangular plates inserted into it. The influence of the position, number, and thickness of the plates on the natural frequencies and modes is analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 89–96, March 2006.     

Natural frequencies and modes of out-of-plane vibrations a fixed ring

The natural frequencies and modes of a flexible elastic ring fixed at one point are determined numerically by solving an eigenvalue problem for the differential operator of the equation of its motion. The ring models a large circular antenna that slowly expands under zero gravity. Bending and torsional vibrations in a plane perpendicular to the antenna plane are studied     

Natural vibrations of a cantilever thin elastic orthotropic cylindrical shell

The natural vibrations of a cantilever thin elastic orthotropic circular cylindrical shell are studied. Dispersion equations for the determination of possible natural frequencies of cantilever closed shells and open shells with Navier hinged boundary conditions at the longitudinal edges are derived from the classical dynamic theory of orthotropic cylindrical shells. It is proved that there are asymptotic relationships between these problems and the problems for a cantilever orthotropic strip plate and for a cantilever rectangular plate and the eigenvalue problem for a semi-infinite closed orthotropic cylindrical shell with free end and for the same but open shell with Navier hinged boundary conditions at the longitudinal edges. A procedure to identify types of vibrations is presented. Orthotropic cylindrical shells with different radii and lengths are used as an example to find approximate values of the dimensionless natural frequency and damping factor for vibration modes __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 68–91, May 2008.     

Equation of small transverse vibrations of an elastic pipeline filled with a transported fluid

The integro-differential equation of small transverse vibrations of a rectilinear elastic pipeline filled with a transported fluid is obtained. The pipeline vibrations are described in the linear setting in the beam approximation. The mutual dynamic influence of motions of the pipeline and the filling fluid is taken into account. A complete trigonometric series method is presented for solving problems with various boundary and initial conditions for the pipeline deflection.