Sergei V. Nesterov

The study was aimed to the experimental verification of the authors’ hypothesis that in elderly people an endogenous functional mechanism develops during late ontogeny to protect muscle effectors of the cardiovascular system (CVS) against negative effects of excessive activity of the sympathetic nervous system by partial replacement of its neurotransmitter norepinephrine with humoral catecholamines. Experiments were carried out using our original method of arterial piezopulsometry which allows evaluation of cardiohemodynamic parameters, as well as the spectral power of oscillations of V_maxPP and TNN parameters of arterial blood pressure pulse waves caused by regulatory effects of the autonomic nervous and endocrine systems. Such a replacement is assumed to be crucial for raising the metabolic tolerance of the CVS in elderly people to an age-related increase in oxygen deficiency, as well as for reducing the incidence of tachyarrhythmia, stenocardia and other cardiac pathological conditions under the influence of psychoemotional and physical stressors.

A numerical-analytical iterative method is proposed for solving generalized self-adjoint regular vector Sturm–Liouville problems with Dirichlet boundary conditions. The method is based on eigenvalue (spectral) correction. The matrix coefficients of the equations are assumed to be nonlinear functions of the spectral parameter. For a relatively close initial approximation, the method is shown to have second-order convergence with respect to a small parameter. Test examples are considered, and the model problem of transverse vibrations of a hinged rod with a variable cross section is solved taking into account its rotational inertia.

The results of laboratory experiments on the estimation of the effect of wall and bottom geometry on the frequency, height, and decay rate of standing surface waves in a tank oscillating in the vertical direction are presented. The effect of one or two semi-cylindrical inserts mounted on the face and rear walls of the tank is considered in detail for the cases of a horizontal bottom and a linear shallow on the bottom. The experimental data are interpreted using a mathematical longwave model based on the method of accelerated convergence.

For the analysis of the frequencies and shapes of the natural oscillations of a gas in an elongated rectilinear combustion chamber, this chamber can be treated as a kind of an organ pipe that has the following specific features: 1. the chamber has an inlet and outlet nozzles; 2. a gas mixture burns in the combustion chamber; 3. the combustion materials flow out from the outlet nozzle; 4. the gas flows in such a way that its velocity in the larger part (closer to the outlet nozzle) of the chamber exceeds the speed of sound (Mach number M > 1). There are only separate domains (one or several), where M < 1. The excitation of the natural oscillations of the gas and an increase in the amplitude of such oscillations can lead to instability of the combustion process [1].

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.

The problem of plane wave propagation through a circular hole is studied in the framework of long-wave approximation. The constructive notion of “apparent mass of holes” (Rayleigh; Fok) is used to construct a mathematical model of gas vibrations in an acoustic resonator and determine and analyze the natural frequencies and mode shapes for the velocity potential depending on the relative geometric parameters of the system. The high-precision calculations of the boundary value problem for the natural frequencies and mode shapes in the parametric approximation to the cross-section are based on a numerical-analytical accelerated convergence method. Two models are analyzed and compared, and the basic qualitative properties of gas vibrations are revealed depending on the basic parameters such as the mode number, relative size of the hole, and the dividing wall location.

An original iteration algorithm is used to construct new analytic expressions for computing approximate natural frequencies and shape modes of bending vibrations of a square homogeneous plate clamped along its contour. The errors are estimated by comparing with the results of well-known numerical high-precision computations. The results of analytic computations are also compared with experimental data obtained by the author by the resonance method. The proposed research technique and the obtained high-precision expressions for the natural shape modes can be used in the case of rectangular plates and for other types of boundary conditions. A numerical-analytical method is used to show that the small isoperimetric theorem holds.