75th birthday of Professor Leonid Isaakovich Belen’kii

Anniversaries and Dates

75th birthday of Professor Leonid Isaakovich Belen’kii     

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.     

Seiches in a channel with a sharp variation in the bottom relief

A mathematical model of seiches is developed for the case of sharp bottom elevation or depression. An effective high-precision numerical and analytical method is applied to determine the natural frequencies and shapes of the lower modes of oscillations. New important hydrodynamic effects of the bottom relief are revealed. The main features of standing waves in a narrow vessel in the presence of sharp bottom elevation are confirmed by laboratory experiments.     

The Dynamics of a Field in a Cavity with a Slowly Oscillating Metal Mirror

We have examined analytically the behavior of an electromagnetic field in a cavity of one of two metal mirrors that is stationary, while the other mirror oscillates in the vicinity of its equilibrium position. The case in which the period of the mirror oscillations substantially exceeds the characteristic roundtrip time of the electromagnetic radiation in the cavity has been considered. To solve the problem, we have applied the method of two time scales. The method makes it possible to solve the problem by using the expansion with respect to a small parameter, for which the ratio of the cavity roundtrip time to the mirror-oscillation period is used. The solution to the problem in the zeroth-order approximation with respect to the small parameter has been obtained and examined. This solution has been shown to be correct up to the mirror-oscillation amplitude being on the order of the average cavity length. We have showed that this scheme can be used to gain and generate electromagnetic radiation. We have compared our results obtained for the oscillating mirror with the results of an exact solution of the problem when the mirror moves uniformly and rectilinearly. We have showed that, in these two cases, a change in the total field energy in the cavity is inversely proportional to the current value of the cavity length.     

On the asymptotics of the solution to a singularly perturbed system of first-order partial differential equations with small nonlinearity in the critical case

A complete asymptotic expansion of the solution to an initial value problem for a singularly perturbed system of hyperbolic equations is constructed and justified. A specific feature of the problem is that its solution has a wavelet zone in a neighborhood of which the asymptotics is described by a parabolic equation.     

Anomalous dependence of the vibration frequencies of a rod in an elastic medium on the rod length

We use numerical-analytic methods to study the influence of the length of a thin inhomogeneous rod on its natural frequencies and the shapes of its plane transverse vibrations. We found that the existence of an external elastic medium described by the Winkler model can lead to an anomalous effect, i.e., to an increase in the natural frequencies of the vibration lower modes as the rod length increases continuously. We discovered rather subtle properties of this phenomenon in the case of variations in the length, the mode number, and the fixation method. We separately studied vibrations for the standard boundary conditions: fixation, hinged fixation, tangential fixation, and free end. We calculated several simple examples illustrating the anomalous dependence of the frequency of the rod natural vibrations in a strongly inhomogeneous elastic medium with different boundary conditions.     

Anomalous Phenomena in Zirconium,Hafnium, and Titanium Salt Solutions,Studied by Capillary Isotachophoresis

Capillary isotachophoresis was used to separate mixtures of solutions containing zirconium, hafnium, titanium, and aluminum salts with various anions. Anomalous phenomena occurring in separation of mixtures in solutions with medium and high concentrations are described and analyzed.     

Novel Stereoselective Synthesis and Molecular and Crystalline Structure of 3-Allyl-4-(4-bromophenyl)-3-cyano-6-oxopiperidine-2-thione

The stereoselective synthesis of 3-allyl-4-(4-bromophenyl)-3-cyano-6-oxopiperidine-2-thione has been carried out by refluxing N-methylmorpholinium 5-[1-(4-bromophenyl)-2-cyano-2-thiocarbamoylethyl]-2,2-dimethyl-1,3-dioxa-6-oxo-4-cyclohexen-4-olate with allyl bromide in ethanol. The structure of the product was confirmed by X-ray investigation.     

Synthesis and X-ray diffraction studies of 3-cyano-2-trifluoromethyl-4H-pyrido[1,2-a]pyrimidine derivatives

Reactions of 2-chloro-1-cyano-2-trifluoromethylethylenes with 2-aminopyridines were studied. According to the data of NMR spectroscopy and X-ray diffraction analysis, the first stage of the reaction involves alkenylation of the exocyclic nitrogen atom of 2-aminopyridines followed by formation of pyrido[1,2-a]pyrimidines. Published inIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 7, pp. 1261–1264, July, 2000.     

Crystal structure of 3-amino-5-acetyl-4,7-dihydro-6-methyl-4-(2-nitrophenyl)-2-cyanothieno[2,3-b]-pyridine and specific features of its molecular packing in crystal

The molecular and crystal structures of 3-amino-5-acetyl-4,7-dihydro-6-methyl-4-(2-nitrophenyl)-2-cyanothieno[2,3-b]-pyridine is determined by X-ray diffraction. The unit-cell parameters are as follows: a = 14.68(1) Å, b = 8.704(5) Å, c = 25.36(1) Å, V = 3241(6) ų, d _calcd = 1.453 g/cm³, Z = 8(1,1), and space group Pna2₁. Theo-nitrophenyl substituent is synperiplanar relative to the hydrogen atom at the C(4) atom of the heterocycle. Two crystallographically independent reference molecules A and B in the structure can be considered a pair of the enantiomers related by an approximate noncrystallographic center of inversion. The degree of approximation of this pseudocenter is discussed. It is shown that the structure can be described as a combination of two systems of translationally equal layers. The pseudosymmetry of the crystal structure and the possibility of forming a similar molecular packing with a higher crystallographic symmetry are considered.