S. V. Nesterov

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.

Transverse vibration spectrum of a part of a moving rod under a longitudinal load

We study the natural transverse vibrations of a constant-length part of a rectilinear thin rod moving along a neutral line in unstrained state. The displacement occurs between two fixed coaxial rod guides (clamps) and the distance between them is equal to the length of the vibration part of the rod. Moreover, it is assumed that a constant longitudinal force acts along the neutral line, and two significantly different cases are distinguished: the force extends the rod, and the force compresses the rod.The vibration process is described by a nonself-adjoint boundary value problem. For arbitrary values of the rod displacement velocity and longitudinal forces, the natural frequencies of possible vibration modes are determined and analyzed by a numerical-analytical procedure with a prescribed accuracy. The global properties of the spectrum depending on the velocity, the longitudinal force, and mode number are determined. For higher-order modes, the domains are discovered where the frequencies nonuniquely depend on the rod motion velocity and the value and direction of the longitudinal force and where the lower vibration frequencies can be absent as the rod motion velocity increases. It is shown that the picture of partial vibrations from the standpoint of an immovable observer cardinally differs from the commonly known picture for the immovable rod.The obtained results are interesting for studying vibrations of various elements of moving elastic media, including the case of systems with moving boundaries. These results can also have technical applications in problems of dynamics and strength of devices, machines, and mechanisms in textile industry in the filament and rope manufacturing and in metallurgy, in particular, in rolling of metallic rods and strips, in wire drawing, and in production of plastic articles and paper rolls. The developed technique for calculating the natural frequencies and shape modes can be used to analyze the transverse vibrations of parts of transport pipelines with rapidly flowing liquids.

Natural oscillations of multidimensional systems nonlinear in the spectral parameter

A new method of solving the generalized vector self-conjugated Sturm-Liouville boundary value problems with the boundary conditions of the first kind is proposed and developed. The iterative algorithm is based on a constructive procedure of introduction of a small parameter and an efficient correction of the desired eigenvalue. The matrix coefficients of the equations are assumed to be nonlinearly dependent on the spectral parameter. The criterion of proximity is introduced, and it is shown that this method has an accelerated convergence of the second order with respect to a small parameter for a reasonably close initial approximation. Test examples are considered.

Spectrum of transverse vibrations of a pipeline element under longitudinal load

Transverse eigenvibrations of a pipeline element fixed between two supports and transporting an ideal fluid are investigated. The global properties of the spectrum in dependence on the value of the transported-fluid velocity, the relative linear densities of the pipe material and the fluid, the longitudinal force, and its direction are established.

Natural Vibrations of a Pipeline Segment

Transverse natural vibrations of an extended segment of a pipeline conveying a uniformly moving fluid are studied. The mechanical model under study takes into account the pipe and fluid inertia forces and the moment of the Coriolis and centrifugal forces due to the medium motion. It is assumed that both ends are rigidly fixed and the elastic characteristics are constant along the pipe. A mathematical model is developed on the basis of a generalized procedure of separation of variables, and a boundary value problemfor the eigenvalues and eigenfunctions (natural frequencies and vibration shapes) is posed. Ferrari’s formulas are used to solve the fourth-order complex characteristic equation for the wave parameter, and a closed procedure of numerical-analytical determination of roots of the secular equation for the frequencies is obtained. The frequency curves for the firsts two vibration modes against the dimensionless velocity and inertia parameters are constructed. The forms of the observed motions at different times are obtained. Several effects are revealed indicating that there is a dramatic quantitative and qualitative difference between these vibrations and the standard vibrations corresponding to the case of immovablemedium. We discover the absence of a rectilinear configuration of the axis, the variable number and location of nodes, their inconsistency with the mode number, and some other effects.

The synthesis of an inhomogeneous elastic system with a boundary load

For an adequate description of free and controlled motions of one-dimensional elastic systems with distributed parameters, we consider a suitable model expressed by a linear problem with Robin boundary conditions. It is assumed that the control action is used additively in the equation of motion and in the boundary conditions. The coefficients used in the equation of state and in the boundary conditions may depend on the spectral parameter (frequency), which allows one to take into account the inertial and/or force load at one or both ends of the boundary as well as the elastic properties (the Rayleigh correction) and other imperfections.

Mass defect influence on the longitudinal vibration frequencies and mode shapes of a beam

The problem of the influence of complicated-shape defects on the natural frequencies and mode shapes of elastic systems is considered. The beam is considered as an example to discover the general properties of defects of various physical nature such as of mass, elasticity, and cross-section. Some notions of damage and a criterion that permits defectoscopy by nondestructive inspection techniques are introduced. The problem is solved in more detail for a free beam with a mass defect. The resonance method is used experimentally to determine the mass defect with high accuracy. The efficiency of the developed numerical-analytic and experimental approaches to studying the properties of elastic systems with defects is confirmed.

Natural Vibrations of a Liquid-Transporting Pipeline on an Elastic Base

Flexural free vibrations of an ideal-liquid-transporting pipeline on an elastic base are studied. A numerical-analytical method for finding the pipeline natural frequencies and vibration modes is developed, which permits one to determine the natural frequencies and modes for the case in which the tension or compression (the longitudinal force acting along the pipeline axis), the pipe diameter, and hence the velocity of the incompressible fluid being transported are arbitrary functions of the longitudinal coordinate measured along the pipeline axis. The least natural frequencies are calculated for the case in which the variable elasticity of the base is given by some test functions.

Parametric Oscillations of the Kochin Oscillator with Dissipation

The fundamental properties of parametrically excited oscillatory systems of the Kochin oscillator type with double degeneracy of frequencies are investigated. The qualitative phenomenon of their absence is established for arbitrarily small actions of a dissipative nature. As a result, the resonance motions of the first mode prove to be relatively more substantial; the excitation of subsequent odd modes is difficult in the applied aspect and requires a significant increase in the eigenvalue of the corresponding parameter.

Evolution of Natural Frequencies of Longitudinal Vibrations of a Bar as Its Cross-Section Defect Increases

We study the evolution of characteristics of natural longitudinal vibrations of a circular bar in the case of increasing defect in its cross-section. It is shown that, in the limit case where the defect separates the bar into two independent fragments, the natural frequencies of the initially defect-free bar pass into the natural frequencies of its separate parts. The respective evolution of the natural modes of vibrations is observed. The evolution predicted by the theoretical analysis can be observed experimentally by using the resonance method and constantly increasing the defect till the final separation of the bar into two parts.